Research ArticleOn Solutions of Variational Inequality Problems viaIterative Methods
Mohammed Ali Alghamdi1 Naseer Shahzad1 and Habtu Zegeye2
1 Department of Mathematics King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia2Department of Mathematics University of Botswana Private Bag 00704 Gaborone Botswana
Correspondence should be addressed to Naseer Shahzad nshahzadkauedusa
Received 12 May 2014 Revised 24 June 2014 Accepted 30 June 2014 Published 4 August 2014
Academic Editor Adrian Petrusel
Copyright copy 2014 Mohammed Ali Alghamdi et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
We investigate an algorithm for a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings andsolutions of a finite family of 120574-inverse strongly accretive mappings Our theorems improve and unify most of the results that havebeen proved in this direction for this important class of nonlinear mappings
1 Introduction
Let119862 be a subset of a real Hilbert space119867 Let119860 119862 rarr 119867 bea nonlinear mapping The variational inequality problem for119860 and 119862 is to
find 119909lowast
isin 119862 such that ⟨119860119909lowast
V minus 119909lowast
⟩ ge 0 forallV isin 119862 (1)
The set of solutions of variational inequality problem isdenoted by VI(119862 119860) that is
VI (119862 119860) = 119909lowast
isin 119862 ⟨119860119909lowast
119909 minus 119909lowast
⟩ ge 0 forall119909 isin 119862 (2)
It is well known that variational inequality theory hasemerged as an important tool in studying a wide classof numerous problems in variational inequalities minimaxproblems optimization physics and the Nash equilibriumproblems in noncooperative games Several numerical meth-ods have been developed for solving variational inequalitiesand related optimization problems see for instance [1ndash5]and the references therein
A mapping 119860 119862 sube 119867 rarr 119867 is said to be 120574-inversestrongly accretive (or 120574-inverse strongly monotone) if thereexists a positive real number 120574 such that
⟨119909 minus 119910 119860119909 minus 119860119910⟩ ge 1205741003817100381710038171003817119860119909 minus 119860119910
10038171003817100381710038172
forall119909 119910 isin 119862 (3)
If 119860 is 120574-inverse strongly accretive then inequality (3)implies that 119860 is Lipschitzian with constant 119871 = 1120574 that
is 119860119909minus119860119910 le (1120574)119909minus119910 for all 119909 119910 isin 119862 If in (3) we havethat 120574 = 0 then 119860 is called accretive (or monotone)
Let119862 be a closed and convex subset of a real Hilbert space119867 A mapping 119879 119862 rarr 119867 is called a contraction mapping ifthere exists 119871 isin [0 1) such that 119879119909 minus 119879119910 le 119871119909 minus 119910 for all119909 119910 isin 119862 If 119871 = 1 then 119879 is called nonexpansive A mapping119879 119862 rarr 119864 is called 120582-strictly pseudocontractive of Browder-Petryshyn type [6] if and only if there exists 120582 isin (0 1) suchthat
1003817100381710038171003817119879119909 minus 11987911991010038171003817100381710038172
le1003817100381710038171003817119909 minus 119910
10038171003817100381710038172
+ 1205821003817100381710038171003817(119868 minus 119879) 119909 minus (119868 minus 119879) 119910
10038171003817100381710038172
forall119909 119910 isin 119862
(4)
119879 is called pseudocontractive if
1003817100381710038171003817119879119909 minus 11987911991010038171003817100381710038172
le1003817100381710038171003817119909 minus 119910
10038171003817100381710038172
+1003817100381710038171003817(119868 minus 119879) 119909 minus (119868 minus 119879) 119910
10038171003817100381710038172
forall119909 119910 isin 119862
(5)
We note that inequalities (4) and (5) can be equivalentlywritten as
⟨119879119909 minus 119879119910 119909 minus 119910⟩ le1003817100381710038171003817119909 minus 119910
10038171003817100381710038172
minus 1198961003817100381710038171003817(119909 minus 119879119909) minus (119910 minus 119879119910)
10038171003817100381710038172
forall119909 119910 isin 119862
(6)
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 424875 10 pageshttpdxdoiorg1011552014424875
2 Abstract and Applied Analysis
for some 119896 gt 0 and
⟨119879119909 minus 119879119910 119909 minus 119910⟩ le1003817100381710038171003817119909 minus 119910
10038171003817100381710038172
forall119909 119910 isin 119862 (7)
respectively We remark that 119879 is pseudocontractive if andonly if 119860 = (119868 minus 119879) is accretive A point 119909 isin 119862 is a fixedpoint of 119879 if 119879119909 = 119909 and we denote by 119865(119879) the set of fixedpoints of 119879 that is 119865(119879) = 119909 isin 119862 119879119909 = 119909
We observe that in a real Hilbert space 119867 a class ofpseudocontractive mappings includes the class of 120582-strictlypseudocontractive mappings and hence the classes of nonex-pansive and contraction mappings
Closely related to the variational inequality problems isthe problem of finding fixed points of nonexpansive map-pings 120582-strict pseudocontraction mappings or pseudocon-tractive mappings which is the current interest in functionalanalysis Several researchers considered a unified approachthat approximates a commonpoint of fixed point of nonlinearproblems and solutions of variational inequality problemsand solutions of variational inequality problems see forexample [7ndash18] and the references therein
In [19] Takahashi and Toyoda studied the problem offinding a common point of fixed points of a nonexpansivemapping and solutions of a variational inequality problem (1)by considering the following iterative algorithm
119879 119862 rarr 119862 is a nonexpansivemapping and119860 119862 rarr 119867 is an120574-inverse strongly accretive mapping They showed that thesequence 119909
119899 generated by (8) converges weakly to some 119911 isin
VI(119862 119860) cap 119865(119878) provided that the control sequences satisfysome restrictions
Iiduka and Takahashi [20] reconsidered the commonelement problem via the following iterative algorithm
where119879 119862 rarr 119862 is a nonexpansivemapping119860 119862 rarr 119867 isa 120574-inverse-strongly accretive mapping 120572
119899 is a sequence in
(0 1) and 120582119899 is a sequence in (0 2120572) They proved that the
sequence 119909119899 strongly converges to some point 119911 isin 119865(119879) cap
VI(119862 119860)Recently Zegeye and Shahzad [21] investigated the prob-
lem of finding a common point of fixed points of a Lipschitzpseudocontractive mapping 119879 and solutions of a variationalinequality problem for 120574-inverse strongly accretive mapping119860 by considering the following iterative algorithm
119899) (120575119899119879119910119899+ 120579119899119909119899+ 120574119899119875119862[119868 minus 120574119860] 119909
119899)]
(10)
where 119875119862
is a metric projection from 119867 onto 119862 and120575119899 120579119899 120574119899 120572119899 120573119899 are in (0 1) satisfying certain
conditions Then they proved that the sequence 119909119899 con-
verges strongly to the minimum-norm point of 119865(119879) cap
VI(119862 119860)A natural question arises whether we can obtain an itera-
tive schemewhich converges strongly to a commonpoint of fixedpoints of a finite family of pseudocontractive mappings andsolutions of a finite family of variational inequality problemsfor 120574-inverse strongly accretive mappings or not
It is our purpose in this paper to introduce an algorithmand prove that the algorithm converges strongly to a commonpoint of fixed points of a finite family of Lipschitz pseudo-contractive mappings and solutions of a finite family of vari-ational inequality problems for 120574-inverse strongly accretivemappings The results obtained in this paper improve andextend the results of Takahashi and Toyoda [19] Iiduka andTakahashi [20] and Zegeye and Shahzad [21]Theorem 32 ofYao et al [22] and some other results in this direction
2 Preliminaries
In what follows we will make use of the following lemmas
Lemma 1 Letting 119867 be a real Hilbert space the followingidentity holds
Lemma 2 (see [23]) Let 119862 be a nonempty closed and convexsubset of a real Hilbert space119867 Let 119860 119862 rarr 119864 be a 120574-inversestrongly accretive mapping Then for 0 lt 120583 lt 2120574 the mapping119860120583119909 = (119909 minus 120583119860119909) is nonexpansive
Lemma 3 (see [24]) Let 119862 be a nonempty closed and convexsubset of a smooth Banach space 119864 Let 119876
119862be a sunny
nonexpansive retraction from 119864 onto 119862 and let 119860 be anaccretive operator of 119862 into 119864 Then for all 120582 gt 0
119881119868 (119862 119860) = 119865 (119876119862(119868 minus 120582119860)) (12)
Lemma 4 (see [25]) Let 119862 be a nonempty closed and convexsubset of a real Hilbert space119867 Let 119879
119894 119862 rarr 119864 119894 = 1 119873
be nonexpansive mappings such that cap119873119894=1
Lemma 5 (see [26]) Let 119862 be a convex subset of a real Hilbertspace119867 Let 119909 isin 119867 Then 119909
0= 119875119862119909 if and only if
⟨119911 minus 1199090 119909 minus 119909
0⟩ le 0 forall119911 isin 119862 (13)
Lemma 6 (see [27]) Let 119862 be a closed convex subset of a realHilbert space 119867 and 119860 119862 rarr 119862 be a continuous pseudo-contractive mapping Then for 0 lt 120583 lt 2120574 the mapping119860120583119909 = (119909 minus 120583119860119909) is nonexpansive
(i) 119865(119879) is a closed convex subset of 119862(ii) (119868minus119879) is demiclosed at zero that is if 119909
119899 is a sequence
in119862 such that 119909119899 119909 and119879119909
119899minus119909119899rarr 0 as 119899 rarr infin
then 119909 = 119879(119909)
Abstract and Applied Analysis 3
Lemma 7 (see [28]) Let119867 be a real Hilbert spaceThen for all119909119894isin 119867 and 120572
119894isin [0 1] for 119894 = 1 2 3 such that 120572
as 119899 rarr infin for all 119895 isin 1 2 119872Now since 119909
119899 is bounded subset of 119867 we can choose
a subsequence 119909119899119898 of 119909
119899 such that 119909
119899119898 119909 and
lim sup119899rarrinfin
⟨119891(119909lowast
) minus 119909lowast
119909119899minus 119909lowast
⟩ = lim119898rarrinfin
⟨119891(119909lowast
) minus
119909lowast
119909119899119898
minus 119909lowast
⟩ Then from (37) and Lemma 6 we have that119909 isin 119865(119879
119895) for each 119895 = 1 2 119872 Hence 119909 isin cap
119872
119895=1119865(119879119895)
In addition since 119866 is nonexpansive from Lemma 6 weget that 119909 isin 119865(119866) and hence by Lemmas 4 and 3 we obtainthat 119909 isin VI(119862 119860
119895) for each 119895 isin 1 2 119873
Therefore by Lemma 5 we immediately obtain that
lim sup119899rarrinfin
⟨119891 (119909lowast
) minus 119909lowast
119909119899minus 119909lowast
⟩
= lim119898rarrinfin
⟨119891 (119909lowast
) minus 119909lowast
119909119899119898
minus 119909lowast
⟩
= ⟨119891 (119909lowast
) minus 119909lowast
119909 minus 119909lowast
⟩ le 0
(38)
Then it follows from (32) (38) and Lemma 9 that 119909119899minus
119909lowast
rarr 0 as 119899 rarr infin Consequently 119909119899rarr 119909lowast
= 119875F(119891(119909lowast
))
Case 2 Suppose that there exists a subsequence 119899119894 of 119899
such that10038171003817100381710038171003817119909119899119894minus 119909lowast10038171003817100381710038171003817lt10038171003817100381710038171003817119909119899119894+1
minus 119909lowast10038171003817100381710038171003817 (39)
for all 119894 isin N Then by Lemma 8 there exists a nondecreasingsequence 119898
1) forall119899 ge 0 for 119871 = max119871119895 1 le 119895 le 119872 Then 119909
119899 converges
strongly to a unique point 119909lowast isin 119862 satisfying 119909lowast = 119875F(119906) whichis the unique solution of the variational inequality ⟨119909lowast minus 119906 119909 minus
119909lowast
⟩ ge 0 for all 119909 isin F
If inTheorem 10 we assume that119873 = 1 and119872 = 1 thenwe get the following corollary which is Theorem 31 of [21]
Corollary 12 Let 119862 be a nonempty closed and convex subsetof a real Hilbert space 119867 Let 119879 119862 rarr 119862 be Lipschitzpseudocontractive mappings with Lipschitz constant 119871 and 119860
119862 rarr 119867 an 120574-inverse strongly accretive mapping Let 119891
119862 rarr 119862 be a contraction with constant 120572 Assume that F =
119865(119879)⋂VI(119862 119860) is nonempty Let a sequence 119909119899 be generated
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119894le119873120574119894
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 119886
119899+ 119887119899+ 119888119899= 1 and 119887
119899+ 119888119899le
120582119899le 120582 lt 1(radic2 + 1) forall119899 ge 0 Then 119909
119899 converges strongly to
point 119909lowast isin F which is the unique solution of the variationalinequality ⟨(119868 minus 119891)(119909
lowast
) 119909 minus 119909lowast
⟩ ge 0 for all 119909 isin F
We note that the method of proof ofTheorem 10 providesthe following theorem which is a convergence theoremfor a minimum norm point of common fixed points of afinite family of Lipschitz pseudocontractive mappings andcommon solutions of a finite family of variational inequalityproblems for accretive mappings
Theorem 15 Let 119862 be a nonempty closed and convex subsetof a real Hilbert space119867 Let 119879
119895 119862 rarr 119862 119895 = 1 2 119872 be
Lipschitz pseudocontractive mappings with Lipschitz constants119871119895 respectively Let 119860
119895 119862 rarr 119867 for 119895 = 1 2 119873
be 120574119895-inverse strongly accretive mappings Assume that F =
[cap119872
119895=1119865(119879119895)]⋂[cap
119873
119895=1VI(119862 119860
119895)] is nonempty Let a sequence
119909119899 be generated from an arbitrary 119909
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119895le119873120574119895
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 and 119887
119899+ 119888119899
le 120582119899
le 120582 lt
1(radic1 + 1198712 + 1) forall119899 ge 0 for 119871 = max119871119895 1 le 119895 le 119872 Then
119909119899 converges strongly to a unique minimum norm point 119909lowast
of F (ie 119909lowast = 119875F(0)) which is the unique solution of thevariational inequality ⟨119909lowast 119909 minus 119909
lowast
⟩ ge 0 for all 119909 isin F
4 Numerical Example
Now we give an example of two Lipschitz pseudocontractivemappings and two 120574-inverse strongly accretive mappingssatisfyingTheorem 10 and some numerical experiment resultto explain the conclusion of the theorem as follows
Example 1 Let 119867 = R with absolute value norm Let 119862 =
[minus2 2] and let 1198791 1198792 119862 rarr 119862 be defined by
1198791119909 =
119909 + 1199092
119909 isin [minus2 0]
119909 119909 isin (0 2]
1198792119909 =
119909 119909 isin [minus21
2]
119909 minus (16
9) (119909 minus
1
2)
2
119909 isin (1
2 2]
(49)
Clearly for 119909 119910 isin 119862 we have that
⟨(119868 minus 1198791) 119909 minus (119868 minus 119879
1) 119910 119909 minus 119910⟩ ge 0
⟨(119868 minus 1198792) 119909 minus (119868 minus 119879
2) 119910 119909 minus 119910⟩ ge 0
(50)
which show that bothmappings are pseudocontractive Nextwe show that 119879
1is Lipschitz with 119871 = 5 If 119909 119910 isin [minus2 0] then
10038161003816100381610038161198791119909 minus 11987911199101003816100381610038161003816 =
with 120574 = 15 andVI(119862 1198601) = [12 2] Similarly we can show
that 1198602is 120574-inverse strongly accretive mapping with 120574 = 12
and VI(119862 1198602) = [minus2 23]
Note that we have119865(1198791)cap119865(119879
2)capVI(119862 119860
1)capVI(119862 119860
2) =
12Thus taking 120572
119899= 1(10119899 + 100) 120582
119899= 2(119899 + 100) +
0065 119887119899
= 119888119899
= 1(119899 + 100) + 001 119886119899
= 1 minus 2(119899 +
100) minus 002 and 119891(119909) = 119906 isin 119862 we observe that conditionsof Theorem 10 are satisfied and Scheme (17) provides thefollowing Table 1 and Figures 1(a) and 1(b) for 119906 = 06 and119906 = 08 respectively
We observe that the data provides strong convergenceof the sequence to the common point of fixed points ofboth pseudocontractive mappings and solutions of both vari-ational inequality problems for 120574-inverse strongly accretivemappings
Remark 2 Theorem 10 provides an iteration scheme whichconverges strongly to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problemsin Hilbert spaces
Remark 3 Theorem 10 improves Theorem 31 of Takahashiand Toyoda [19] Iiduka and Takahashi [20] and Zegeye andShahzad [21] and Theorem 32 of Yao et al [22] in the sensethat our convergence is to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problems
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1671301434 The authors therefore acknowledge
Abstract and Applied Analysis 9
0 05 1 15 20
0102030405060708091
x0= 1
Iterations n
x = 05
times104
Iteratesxn
(a)
0 1 2 3 4 5minus15
minus1
minus05
0
05
1
Iterations n
x0= 1
x = 05
times104
Iteratesxn
(b)
Figure 1
with thanksDSR technical and financial supportThe authorsalso thank the referees for their valuable comments andsuggestions which improved the presentation of this paper
References
[1] D Kinderlehrer and G Stampaccia An Iteration to VariationalInequalities and Their Applications Academic Press New YorkNY USA 1990
[2] H Zegeye and N Shahzad ldquoA hybrid scheme for finite familiesof equilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no1 pp 263ndash272 2011
[3] H Zegeye and N Shahzad ldquoApproximating common solutionof variational inequality problems for two monotone mappingsin Banach spacesrdquo Optimization Letters vol 5 no 4 pp 691ndash704 2011
[4] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor variational inequality problems and quasi-120601-asymptoticallynonexpansive mappingsrdquo Journal of Global Optimization vol54 no 1 pp 101ndash116 2012
[5] H Zegeye and N Shahzad ldquoAn iteration to a common pointof solution of variational inequality and fixed point-problemsin Banach spacesrdquo Journal of Applied Mathematics vol 2012Article ID 504503 19 pages 2012
[6] F E Browder and W V Petryshyn ldquoConstruction of fixedpoints of nonlinear mappings in Hilbert spacerdquo Journal ofMathematical Analysis and Applications vol 20 pp 197ndash2281967
[7] L C Ceng A Petrusel and J C Yao ldquoComposite viscosityapproximation methods for equilibrium problem variationalinequality and common fixed pointsrdquo Journal of Nonlinear andConvex Analysis vol 15 no 2 pp 219ndash240 2014
[8] L C Ceng A Petrusel M M Wong and J C Yao ldquoHybridalgorithms for solving variational inequalities variational inclu-sions mixed equilibria and fixed point problemsrdquoAbstract andApplied Analysis vol 2014 Article ID 208717 22 pages 2014
[9] L Ceng A Petrusel and M Wong ldquoStrong convergencetheorem for a generalized equilibrium problem and a pseudo-contractive mapping in a Hilbert spacerdquo Taiwanese Journal ofMathematics vol 14 no 5 pp 1881ndash1901 2010
[10] D R Sahu V Colao and G Marino ldquoStrong convergencetheorems for approximating common fixed points of familiesof nonexpansive mappings and applicationsrdquo Journal of GlobalOptimization vol 56 no 4 pp 1631ndash1651 2013
[11] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[12] Y Yao Y Liou and N Shahzad ldquoConstruction of iterativemethods for variational inequality and fixed point problemsrdquoNumerical Functional Analysis and Optimization vol 33 no 10pp 1250ndash1267 2012
[13] Y Yao G Marino and L Muglia ldquoA modified Korpelevichrsquosmethod convergent to the minimum-norm solution of a varia-tional inequalityrdquoOptimization vol 63 no 4 pp 559ndash569 2014
[14] Y Yao and M Postolache ldquoIterative methods for pseudomono-tone variational inequalities and fixed-point problemsrdquo Journalof OptimizationTheory and Applications vol 155 no 1 pp 273ndash287 2012
[15] H Zegeye E U Ofoedu and N Shahzad ldquoConvergencetheorems for equilibrium problem variational inequality prob-lem and countably infinite relatively quasi-nonexpansive map-pingsrdquo Applied Mathematics and Computation vol 216 no 12pp 3439ndash3449 2010
[16] H Zegeye andN Shahzad ldquoAhybrid approximationmethod forequilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 619ndash6302010
[17] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor a solution of finite families of equilibrium and variationalinequality problemsrdquo Optimization vol 63 no 2 pp 207ndash2232014
[18] H Zegeye andN Shahzad ldquoExtragradientmethod for solutionsof variational inequality problems in Banach spacesrdquo Abstractand Applied Analysis vol 2013 Article ID 832548 8 pages 2013
10 Abstract and Applied Analysis
[19] W Takahashi andM Toyoda ldquoWeak convergence theorems fornonexpansive mappings and monotone mappingsrdquo Journal ofOptimization Theory and Applications vol 118 no 2 pp 417ndash428 2003
[20] H Iiduka and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and inverse-strongly monotonemappingsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 61 no 3 pp 341ndash350 2005
[21] H Zegeye and N Shahzad ldquoSolutions of variational inequalityproblems in the set of fixed points of pseudocontractive map-pingsrdquo Carpathian Journal of Mathematics vol 30 no 2 pp257ndash265 2014
[22] Y Yao Y C Liou and S M Kang ldquoAlgorithms construction forvariational inequalitiesrdquo Fixed Point Theory and Applicationsvol 2011 Article ID 794203 12 pages 2011
[23] Y Cai Y Tang and L Liu ldquoIterative algorithms for minimum-norm fixed point of non-expansive mapping in Hilbert spacerdquoFixed Point Theory and Applications vol 2012 article 49 2012
[24] K Aoyama H Iiduka andW Takahashi ldquoWeak convergence ofan iterative sequence for accretive operators in Banach spacesrdquoFixed PointTheory andApplications vol 2006 Article ID 3539013 pages 2006
[25] C E Chidume H Zegeye and E Prempeh ldquoStrong conver-gence theorems for a finite family of nonexpansive mappings inBanach spacesrdquoCommunications onAppliedNonlinearAnalysisvol 11 no 2 pp 25ndash32 2004
[26] Y I Alber ldquoMetric and generalized projection operators inBanach spaces properties and applicationsrdquo in Theory andApplications of Nonlinear Operators of Accretive and MonotoneType Lecture Notes in Pure and Applied Mathematics pp 15ndash50 Marcel Dekker New York NY USA 1996
[27] Q B Zhang and C Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008
[28] H Zegeye and N Shahzad ldquoConvergence of Mannrsquos typeiteration method for generalized asymptotically nonexpansivemappingsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4007ndash4014 2011
[29] P E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[30] H K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
⟨119879119909 minus 119879119910 119909 minus 119910⟩ le1003817100381710038171003817119909 minus 119910
10038171003817100381710038172
forall119909 119910 isin 119862 (7)
respectively We remark that 119879 is pseudocontractive if andonly if 119860 = (119868 minus 119879) is accretive A point 119909 isin 119862 is a fixedpoint of 119879 if 119879119909 = 119909 and we denote by 119865(119879) the set of fixedpoints of 119879 that is 119865(119879) = 119909 isin 119862 119879119909 = 119909
We observe that in a real Hilbert space 119867 a class ofpseudocontractive mappings includes the class of 120582-strictlypseudocontractive mappings and hence the classes of nonex-pansive and contraction mappings
Closely related to the variational inequality problems isthe problem of finding fixed points of nonexpansive map-pings 120582-strict pseudocontraction mappings or pseudocon-tractive mappings which is the current interest in functionalanalysis Several researchers considered a unified approachthat approximates a commonpoint of fixed point of nonlinearproblems and solutions of variational inequality problemsand solutions of variational inequality problems see forexample [7ndash18] and the references therein
In [19] Takahashi and Toyoda studied the problem offinding a common point of fixed points of a nonexpansivemapping and solutions of a variational inequality problem (1)by considering the following iterative algorithm
119879 119862 rarr 119862 is a nonexpansivemapping and119860 119862 rarr 119867 is an120574-inverse strongly accretive mapping They showed that thesequence 119909
119899 generated by (8) converges weakly to some 119911 isin
VI(119862 119860) cap 119865(119878) provided that the control sequences satisfysome restrictions
Iiduka and Takahashi [20] reconsidered the commonelement problem via the following iterative algorithm
where119879 119862 rarr 119862 is a nonexpansivemapping119860 119862 rarr 119867 isa 120574-inverse-strongly accretive mapping 120572
119899 is a sequence in
(0 1) and 120582119899 is a sequence in (0 2120572) They proved that the
sequence 119909119899 strongly converges to some point 119911 isin 119865(119879) cap
VI(119862 119860)Recently Zegeye and Shahzad [21] investigated the prob-
lem of finding a common point of fixed points of a Lipschitzpseudocontractive mapping 119879 and solutions of a variationalinequality problem for 120574-inverse strongly accretive mapping119860 by considering the following iterative algorithm
119899) (120575119899119879119910119899+ 120579119899119909119899+ 120574119899119875119862[119868 minus 120574119860] 119909
119899)]
(10)
where 119875119862
is a metric projection from 119867 onto 119862 and120575119899 120579119899 120574119899 120572119899 120573119899 are in (0 1) satisfying certain
conditions Then they proved that the sequence 119909119899 con-
verges strongly to the minimum-norm point of 119865(119879) cap
VI(119862 119860)A natural question arises whether we can obtain an itera-
tive schemewhich converges strongly to a commonpoint of fixedpoints of a finite family of pseudocontractive mappings andsolutions of a finite family of variational inequality problemsfor 120574-inverse strongly accretive mappings or not
It is our purpose in this paper to introduce an algorithmand prove that the algorithm converges strongly to a commonpoint of fixed points of a finite family of Lipschitz pseudo-contractive mappings and solutions of a finite family of vari-ational inequality problems for 120574-inverse strongly accretivemappings The results obtained in this paper improve andextend the results of Takahashi and Toyoda [19] Iiduka andTakahashi [20] and Zegeye and Shahzad [21]Theorem 32 ofYao et al [22] and some other results in this direction
2 Preliminaries
In what follows we will make use of the following lemmas
Lemma 1 Letting 119867 be a real Hilbert space the followingidentity holds
Lemma 2 (see [23]) Let 119862 be a nonempty closed and convexsubset of a real Hilbert space119867 Let 119860 119862 rarr 119864 be a 120574-inversestrongly accretive mapping Then for 0 lt 120583 lt 2120574 the mapping119860120583119909 = (119909 minus 120583119860119909) is nonexpansive
Lemma 3 (see [24]) Let 119862 be a nonempty closed and convexsubset of a smooth Banach space 119864 Let 119876
119862be a sunny
nonexpansive retraction from 119864 onto 119862 and let 119860 be anaccretive operator of 119862 into 119864 Then for all 120582 gt 0
119881119868 (119862 119860) = 119865 (119876119862(119868 minus 120582119860)) (12)
Lemma 4 (see [25]) Let 119862 be a nonempty closed and convexsubset of a real Hilbert space119867 Let 119879
119894 119862 rarr 119864 119894 = 1 119873
be nonexpansive mappings such that cap119873119894=1
Lemma 5 (see [26]) Let 119862 be a convex subset of a real Hilbertspace119867 Let 119909 isin 119867 Then 119909
0= 119875119862119909 if and only if
⟨119911 minus 1199090 119909 minus 119909
0⟩ le 0 forall119911 isin 119862 (13)
Lemma 6 (see [27]) Let 119862 be a closed convex subset of a realHilbert space 119867 and 119860 119862 rarr 119862 be a continuous pseudo-contractive mapping Then for 0 lt 120583 lt 2120574 the mapping119860120583119909 = (119909 minus 120583119860119909) is nonexpansive
(i) 119865(119879) is a closed convex subset of 119862(ii) (119868minus119879) is demiclosed at zero that is if 119909
119899 is a sequence
in119862 such that 119909119899 119909 and119879119909
119899minus119909119899rarr 0 as 119899 rarr infin
then 119909 = 119879(119909)
Abstract and Applied Analysis 3
Lemma 7 (see [28]) Let119867 be a real Hilbert spaceThen for all119909119894isin 119867 and 120572
119894isin [0 1] for 119894 = 1 2 3 such that 120572
as 119899 rarr infin for all 119895 isin 1 2 119872Now since 119909
119899 is bounded subset of 119867 we can choose
a subsequence 119909119899119898 of 119909
119899 such that 119909
119899119898 119909 and
lim sup119899rarrinfin
⟨119891(119909lowast
) minus 119909lowast
119909119899minus 119909lowast
⟩ = lim119898rarrinfin
⟨119891(119909lowast
) minus
119909lowast
119909119899119898
minus 119909lowast
⟩ Then from (37) and Lemma 6 we have that119909 isin 119865(119879
119895) for each 119895 = 1 2 119872 Hence 119909 isin cap
119872
119895=1119865(119879119895)
In addition since 119866 is nonexpansive from Lemma 6 weget that 119909 isin 119865(119866) and hence by Lemmas 4 and 3 we obtainthat 119909 isin VI(119862 119860
119895) for each 119895 isin 1 2 119873
Therefore by Lemma 5 we immediately obtain that
lim sup119899rarrinfin
⟨119891 (119909lowast
) minus 119909lowast
119909119899minus 119909lowast
⟩
= lim119898rarrinfin
⟨119891 (119909lowast
) minus 119909lowast
119909119899119898
minus 119909lowast
⟩
= ⟨119891 (119909lowast
) minus 119909lowast
119909 minus 119909lowast
⟩ le 0
(38)
Then it follows from (32) (38) and Lemma 9 that 119909119899minus
119909lowast
rarr 0 as 119899 rarr infin Consequently 119909119899rarr 119909lowast
= 119875F(119891(119909lowast
))
Case 2 Suppose that there exists a subsequence 119899119894 of 119899
such that10038171003817100381710038171003817119909119899119894minus 119909lowast10038171003817100381710038171003817lt10038171003817100381710038171003817119909119899119894+1
minus 119909lowast10038171003817100381710038171003817 (39)
for all 119894 isin N Then by Lemma 8 there exists a nondecreasingsequence 119898
1) forall119899 ge 0 for 119871 = max119871119895 1 le 119895 le 119872 Then 119909
119899 converges
strongly to a unique point 119909lowast isin 119862 satisfying 119909lowast = 119875F(119906) whichis the unique solution of the variational inequality ⟨119909lowast minus 119906 119909 minus
119909lowast
⟩ ge 0 for all 119909 isin F
If inTheorem 10 we assume that119873 = 1 and119872 = 1 thenwe get the following corollary which is Theorem 31 of [21]
Corollary 12 Let 119862 be a nonempty closed and convex subsetof a real Hilbert space 119867 Let 119879 119862 rarr 119862 be Lipschitzpseudocontractive mappings with Lipschitz constant 119871 and 119860
119862 rarr 119867 an 120574-inverse strongly accretive mapping Let 119891
119862 rarr 119862 be a contraction with constant 120572 Assume that F =
119865(119879)⋂VI(119862 119860) is nonempty Let a sequence 119909119899 be generated
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119894le119873120574119894
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 119886
119899+ 119887119899+ 119888119899= 1 and 119887
119899+ 119888119899le
120582119899le 120582 lt 1(radic2 + 1) forall119899 ge 0 Then 119909
119899 converges strongly to
point 119909lowast isin F which is the unique solution of the variationalinequality ⟨(119868 minus 119891)(119909
lowast
) 119909 minus 119909lowast
⟩ ge 0 for all 119909 isin F
We note that the method of proof ofTheorem 10 providesthe following theorem which is a convergence theoremfor a minimum norm point of common fixed points of afinite family of Lipschitz pseudocontractive mappings andcommon solutions of a finite family of variational inequalityproblems for accretive mappings
Theorem 15 Let 119862 be a nonempty closed and convex subsetof a real Hilbert space119867 Let 119879
119895 119862 rarr 119862 119895 = 1 2 119872 be
Lipschitz pseudocontractive mappings with Lipschitz constants119871119895 respectively Let 119860
119895 119862 rarr 119867 for 119895 = 1 2 119873
be 120574119895-inverse strongly accretive mappings Assume that F =
[cap119872
119895=1119865(119879119895)]⋂[cap
119873
119895=1VI(119862 119860
119895)] is nonempty Let a sequence
119909119899 be generated from an arbitrary 119909
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119895le119873120574119895
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 and 119887
119899+ 119888119899
le 120582119899
le 120582 lt
1(radic1 + 1198712 + 1) forall119899 ge 0 for 119871 = max119871119895 1 le 119895 le 119872 Then
119909119899 converges strongly to a unique minimum norm point 119909lowast
of F (ie 119909lowast = 119875F(0)) which is the unique solution of thevariational inequality ⟨119909lowast 119909 minus 119909
lowast
⟩ ge 0 for all 119909 isin F
4 Numerical Example
Now we give an example of two Lipschitz pseudocontractivemappings and two 120574-inverse strongly accretive mappingssatisfyingTheorem 10 and some numerical experiment resultto explain the conclusion of the theorem as follows
Example 1 Let 119867 = R with absolute value norm Let 119862 =
[minus2 2] and let 1198791 1198792 119862 rarr 119862 be defined by
1198791119909 =
119909 + 1199092
119909 isin [minus2 0]
119909 119909 isin (0 2]
1198792119909 =
119909 119909 isin [minus21
2]
119909 minus (16
9) (119909 minus
1
2)
2
119909 isin (1
2 2]
(49)
Clearly for 119909 119910 isin 119862 we have that
⟨(119868 minus 1198791) 119909 minus (119868 minus 119879
1) 119910 119909 minus 119910⟩ ge 0
⟨(119868 minus 1198792) 119909 minus (119868 minus 119879
2) 119910 119909 minus 119910⟩ ge 0
(50)
which show that bothmappings are pseudocontractive Nextwe show that 119879
1is Lipschitz with 119871 = 5 If 119909 119910 isin [minus2 0] then
10038161003816100381610038161198791119909 minus 11987911199101003816100381610038161003816 =
with 120574 = 15 andVI(119862 1198601) = [12 2] Similarly we can show
that 1198602is 120574-inverse strongly accretive mapping with 120574 = 12
and VI(119862 1198602) = [minus2 23]
Note that we have119865(1198791)cap119865(119879
2)capVI(119862 119860
1)capVI(119862 119860
2) =
12Thus taking 120572
119899= 1(10119899 + 100) 120582
119899= 2(119899 + 100) +
0065 119887119899
= 119888119899
= 1(119899 + 100) + 001 119886119899
= 1 minus 2(119899 +
100) minus 002 and 119891(119909) = 119906 isin 119862 we observe that conditionsof Theorem 10 are satisfied and Scheme (17) provides thefollowing Table 1 and Figures 1(a) and 1(b) for 119906 = 06 and119906 = 08 respectively
We observe that the data provides strong convergenceof the sequence to the common point of fixed points ofboth pseudocontractive mappings and solutions of both vari-ational inequality problems for 120574-inverse strongly accretivemappings
Remark 2 Theorem 10 provides an iteration scheme whichconverges strongly to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problemsin Hilbert spaces
Remark 3 Theorem 10 improves Theorem 31 of Takahashiand Toyoda [19] Iiduka and Takahashi [20] and Zegeye andShahzad [21] and Theorem 32 of Yao et al [22] in the sensethat our convergence is to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problems
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1671301434 The authors therefore acknowledge
Abstract and Applied Analysis 9
0 05 1 15 20
0102030405060708091
x0= 1
Iterations n
x = 05
times104
Iteratesxn
(a)
0 1 2 3 4 5minus15
minus1
minus05
0
05
1
Iterations n
x0= 1
x = 05
times104
Iteratesxn
(b)
Figure 1
with thanksDSR technical and financial supportThe authorsalso thank the referees for their valuable comments andsuggestions which improved the presentation of this paper
References
[1] D Kinderlehrer and G Stampaccia An Iteration to VariationalInequalities and Their Applications Academic Press New YorkNY USA 1990
[2] H Zegeye and N Shahzad ldquoA hybrid scheme for finite familiesof equilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no1 pp 263ndash272 2011
[3] H Zegeye and N Shahzad ldquoApproximating common solutionof variational inequality problems for two monotone mappingsin Banach spacesrdquo Optimization Letters vol 5 no 4 pp 691ndash704 2011
[4] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor variational inequality problems and quasi-120601-asymptoticallynonexpansive mappingsrdquo Journal of Global Optimization vol54 no 1 pp 101ndash116 2012
[5] H Zegeye and N Shahzad ldquoAn iteration to a common pointof solution of variational inequality and fixed point-problemsin Banach spacesrdquo Journal of Applied Mathematics vol 2012Article ID 504503 19 pages 2012
[6] F E Browder and W V Petryshyn ldquoConstruction of fixedpoints of nonlinear mappings in Hilbert spacerdquo Journal ofMathematical Analysis and Applications vol 20 pp 197ndash2281967
[7] L C Ceng A Petrusel and J C Yao ldquoComposite viscosityapproximation methods for equilibrium problem variationalinequality and common fixed pointsrdquo Journal of Nonlinear andConvex Analysis vol 15 no 2 pp 219ndash240 2014
[8] L C Ceng A Petrusel M M Wong and J C Yao ldquoHybridalgorithms for solving variational inequalities variational inclu-sions mixed equilibria and fixed point problemsrdquoAbstract andApplied Analysis vol 2014 Article ID 208717 22 pages 2014
[9] L Ceng A Petrusel and M Wong ldquoStrong convergencetheorem for a generalized equilibrium problem and a pseudo-contractive mapping in a Hilbert spacerdquo Taiwanese Journal ofMathematics vol 14 no 5 pp 1881ndash1901 2010
[10] D R Sahu V Colao and G Marino ldquoStrong convergencetheorems for approximating common fixed points of familiesof nonexpansive mappings and applicationsrdquo Journal of GlobalOptimization vol 56 no 4 pp 1631ndash1651 2013
[11] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[12] Y Yao Y Liou and N Shahzad ldquoConstruction of iterativemethods for variational inequality and fixed point problemsrdquoNumerical Functional Analysis and Optimization vol 33 no 10pp 1250ndash1267 2012
[13] Y Yao G Marino and L Muglia ldquoA modified Korpelevichrsquosmethod convergent to the minimum-norm solution of a varia-tional inequalityrdquoOptimization vol 63 no 4 pp 559ndash569 2014
[14] Y Yao and M Postolache ldquoIterative methods for pseudomono-tone variational inequalities and fixed-point problemsrdquo Journalof OptimizationTheory and Applications vol 155 no 1 pp 273ndash287 2012
[15] H Zegeye E U Ofoedu and N Shahzad ldquoConvergencetheorems for equilibrium problem variational inequality prob-lem and countably infinite relatively quasi-nonexpansive map-pingsrdquo Applied Mathematics and Computation vol 216 no 12pp 3439ndash3449 2010
[16] H Zegeye andN Shahzad ldquoAhybrid approximationmethod forequilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 619ndash6302010
[17] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor a solution of finite families of equilibrium and variationalinequality problemsrdquo Optimization vol 63 no 2 pp 207ndash2232014
[18] H Zegeye andN Shahzad ldquoExtragradientmethod for solutionsof variational inequality problems in Banach spacesrdquo Abstractand Applied Analysis vol 2013 Article ID 832548 8 pages 2013
10 Abstract and Applied Analysis
[19] W Takahashi andM Toyoda ldquoWeak convergence theorems fornonexpansive mappings and monotone mappingsrdquo Journal ofOptimization Theory and Applications vol 118 no 2 pp 417ndash428 2003
[20] H Iiduka and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and inverse-strongly monotonemappingsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 61 no 3 pp 341ndash350 2005
[21] H Zegeye and N Shahzad ldquoSolutions of variational inequalityproblems in the set of fixed points of pseudocontractive map-pingsrdquo Carpathian Journal of Mathematics vol 30 no 2 pp257ndash265 2014
[22] Y Yao Y C Liou and S M Kang ldquoAlgorithms construction forvariational inequalitiesrdquo Fixed Point Theory and Applicationsvol 2011 Article ID 794203 12 pages 2011
[23] Y Cai Y Tang and L Liu ldquoIterative algorithms for minimum-norm fixed point of non-expansive mapping in Hilbert spacerdquoFixed Point Theory and Applications vol 2012 article 49 2012
[24] K Aoyama H Iiduka andW Takahashi ldquoWeak convergence ofan iterative sequence for accretive operators in Banach spacesrdquoFixed PointTheory andApplications vol 2006 Article ID 3539013 pages 2006
[25] C E Chidume H Zegeye and E Prempeh ldquoStrong conver-gence theorems for a finite family of nonexpansive mappings inBanach spacesrdquoCommunications onAppliedNonlinearAnalysisvol 11 no 2 pp 25ndash32 2004
[26] Y I Alber ldquoMetric and generalized projection operators inBanach spaces properties and applicationsrdquo in Theory andApplications of Nonlinear Operators of Accretive and MonotoneType Lecture Notes in Pure and Applied Mathematics pp 15ndash50 Marcel Dekker New York NY USA 1996
[27] Q B Zhang and C Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008
[28] H Zegeye and N Shahzad ldquoConvergence of Mannrsquos typeiteration method for generalized asymptotically nonexpansivemappingsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4007ndash4014 2011
[29] P E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[30] H K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
as 119899 rarr infin for all 119895 isin 1 2 119872Now since 119909
119899 is bounded subset of 119867 we can choose
a subsequence 119909119899119898 of 119909
119899 such that 119909
119899119898 119909 and
lim sup119899rarrinfin
⟨119891(119909lowast
) minus 119909lowast
119909119899minus 119909lowast
⟩ = lim119898rarrinfin
⟨119891(119909lowast
) minus
119909lowast
119909119899119898
minus 119909lowast
⟩ Then from (37) and Lemma 6 we have that119909 isin 119865(119879
119895) for each 119895 = 1 2 119872 Hence 119909 isin cap
119872
119895=1119865(119879119895)
In addition since 119866 is nonexpansive from Lemma 6 weget that 119909 isin 119865(119866) and hence by Lemmas 4 and 3 we obtainthat 119909 isin VI(119862 119860
119895) for each 119895 isin 1 2 119873
Therefore by Lemma 5 we immediately obtain that
lim sup119899rarrinfin
⟨119891 (119909lowast
) minus 119909lowast
119909119899minus 119909lowast
⟩
= lim119898rarrinfin
⟨119891 (119909lowast
) minus 119909lowast
119909119899119898
minus 119909lowast
⟩
= ⟨119891 (119909lowast
) minus 119909lowast
119909 minus 119909lowast
⟩ le 0
(38)
Then it follows from (32) (38) and Lemma 9 that 119909119899minus
119909lowast
rarr 0 as 119899 rarr infin Consequently 119909119899rarr 119909lowast
= 119875F(119891(119909lowast
))
Case 2 Suppose that there exists a subsequence 119899119894 of 119899
such that10038171003817100381710038171003817119909119899119894minus 119909lowast10038171003817100381710038171003817lt10038171003817100381710038171003817119909119899119894+1
minus 119909lowast10038171003817100381710038171003817 (39)
for all 119894 isin N Then by Lemma 8 there exists a nondecreasingsequence 119898
1) forall119899 ge 0 for 119871 = max119871119895 1 le 119895 le 119872 Then 119909
119899 converges
strongly to a unique point 119909lowast isin 119862 satisfying 119909lowast = 119875F(119906) whichis the unique solution of the variational inequality ⟨119909lowast minus 119906 119909 minus
119909lowast
⟩ ge 0 for all 119909 isin F
If inTheorem 10 we assume that119873 = 1 and119872 = 1 thenwe get the following corollary which is Theorem 31 of [21]
Corollary 12 Let 119862 be a nonempty closed and convex subsetof a real Hilbert space 119867 Let 119879 119862 rarr 119862 be Lipschitzpseudocontractive mappings with Lipschitz constant 119871 and 119860
119862 rarr 119867 an 120574-inverse strongly accretive mapping Let 119891
119862 rarr 119862 be a contraction with constant 120572 Assume that F =
119865(119879)⋂VI(119862 119860) is nonempty Let a sequence 119909119899 be generated
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119894le119873120574119894
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 119886
119899+ 119887119899+ 119888119899= 1 and 119887
119899+ 119888119899le
120582119899le 120582 lt 1(radic2 + 1) forall119899 ge 0 Then 119909
119899 converges strongly to
point 119909lowast isin F which is the unique solution of the variationalinequality ⟨(119868 minus 119891)(119909
lowast
) 119909 minus 119909lowast
⟩ ge 0 for all 119909 isin F
We note that the method of proof ofTheorem 10 providesthe following theorem which is a convergence theoremfor a minimum norm point of common fixed points of afinite family of Lipschitz pseudocontractive mappings andcommon solutions of a finite family of variational inequalityproblems for accretive mappings
Theorem 15 Let 119862 be a nonempty closed and convex subsetof a real Hilbert space119867 Let 119879
119895 119862 rarr 119862 119895 = 1 2 119872 be
Lipschitz pseudocontractive mappings with Lipschitz constants119871119895 respectively Let 119860
119895 119862 rarr 119867 for 119895 = 1 2 119873
be 120574119895-inverse strongly accretive mappings Assume that F =
[cap119872
119895=1119865(119879119895)]⋂[cap
119873
119895=1VI(119862 119860
119895)] is nonempty Let a sequence
119909119899 be generated from an arbitrary 119909
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119895le119873120574119895
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 and 119887
119899+ 119888119899
le 120582119899
le 120582 lt
1(radic1 + 1198712 + 1) forall119899 ge 0 for 119871 = max119871119895 1 le 119895 le 119872 Then
119909119899 converges strongly to a unique minimum norm point 119909lowast
of F (ie 119909lowast = 119875F(0)) which is the unique solution of thevariational inequality ⟨119909lowast 119909 minus 119909
lowast
⟩ ge 0 for all 119909 isin F
4 Numerical Example
Now we give an example of two Lipschitz pseudocontractivemappings and two 120574-inverse strongly accretive mappingssatisfyingTheorem 10 and some numerical experiment resultto explain the conclusion of the theorem as follows
Example 1 Let 119867 = R with absolute value norm Let 119862 =
[minus2 2] and let 1198791 1198792 119862 rarr 119862 be defined by
1198791119909 =
119909 + 1199092
119909 isin [minus2 0]
119909 119909 isin (0 2]
1198792119909 =
119909 119909 isin [minus21
2]
119909 minus (16
9) (119909 minus
1
2)
2
119909 isin (1
2 2]
(49)
Clearly for 119909 119910 isin 119862 we have that
⟨(119868 minus 1198791) 119909 minus (119868 minus 119879
1) 119910 119909 minus 119910⟩ ge 0
⟨(119868 minus 1198792) 119909 minus (119868 minus 119879
2) 119910 119909 minus 119910⟩ ge 0
(50)
which show that bothmappings are pseudocontractive Nextwe show that 119879
1is Lipschitz with 119871 = 5 If 119909 119910 isin [minus2 0] then
10038161003816100381610038161198791119909 minus 11987911199101003816100381610038161003816 =
with 120574 = 15 andVI(119862 1198601) = [12 2] Similarly we can show
that 1198602is 120574-inverse strongly accretive mapping with 120574 = 12
and VI(119862 1198602) = [minus2 23]
Note that we have119865(1198791)cap119865(119879
2)capVI(119862 119860
1)capVI(119862 119860
2) =
12Thus taking 120572
119899= 1(10119899 + 100) 120582
119899= 2(119899 + 100) +
0065 119887119899
= 119888119899
= 1(119899 + 100) + 001 119886119899
= 1 minus 2(119899 +
100) minus 002 and 119891(119909) = 119906 isin 119862 we observe that conditionsof Theorem 10 are satisfied and Scheme (17) provides thefollowing Table 1 and Figures 1(a) and 1(b) for 119906 = 06 and119906 = 08 respectively
We observe that the data provides strong convergenceof the sequence to the common point of fixed points ofboth pseudocontractive mappings and solutions of both vari-ational inequality problems for 120574-inverse strongly accretivemappings
Remark 2 Theorem 10 provides an iteration scheme whichconverges strongly to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problemsin Hilbert spaces
Remark 3 Theorem 10 improves Theorem 31 of Takahashiand Toyoda [19] Iiduka and Takahashi [20] and Zegeye andShahzad [21] and Theorem 32 of Yao et al [22] in the sensethat our convergence is to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problems
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1671301434 The authors therefore acknowledge
Abstract and Applied Analysis 9
0 05 1 15 20
0102030405060708091
x0= 1
Iterations n
x = 05
times104
Iteratesxn
(a)
0 1 2 3 4 5minus15
minus1
minus05
0
05
1
Iterations n
x0= 1
x = 05
times104
Iteratesxn
(b)
Figure 1
with thanksDSR technical and financial supportThe authorsalso thank the referees for their valuable comments andsuggestions which improved the presentation of this paper
References
[1] D Kinderlehrer and G Stampaccia An Iteration to VariationalInequalities and Their Applications Academic Press New YorkNY USA 1990
[2] H Zegeye and N Shahzad ldquoA hybrid scheme for finite familiesof equilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no1 pp 263ndash272 2011
[3] H Zegeye and N Shahzad ldquoApproximating common solutionof variational inequality problems for two monotone mappingsin Banach spacesrdquo Optimization Letters vol 5 no 4 pp 691ndash704 2011
[4] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor variational inequality problems and quasi-120601-asymptoticallynonexpansive mappingsrdquo Journal of Global Optimization vol54 no 1 pp 101ndash116 2012
[5] H Zegeye and N Shahzad ldquoAn iteration to a common pointof solution of variational inequality and fixed point-problemsin Banach spacesrdquo Journal of Applied Mathematics vol 2012Article ID 504503 19 pages 2012
[6] F E Browder and W V Petryshyn ldquoConstruction of fixedpoints of nonlinear mappings in Hilbert spacerdquo Journal ofMathematical Analysis and Applications vol 20 pp 197ndash2281967
[7] L C Ceng A Petrusel and J C Yao ldquoComposite viscosityapproximation methods for equilibrium problem variationalinequality and common fixed pointsrdquo Journal of Nonlinear andConvex Analysis vol 15 no 2 pp 219ndash240 2014
[8] L C Ceng A Petrusel M M Wong and J C Yao ldquoHybridalgorithms for solving variational inequalities variational inclu-sions mixed equilibria and fixed point problemsrdquoAbstract andApplied Analysis vol 2014 Article ID 208717 22 pages 2014
[9] L Ceng A Petrusel and M Wong ldquoStrong convergencetheorem for a generalized equilibrium problem and a pseudo-contractive mapping in a Hilbert spacerdquo Taiwanese Journal ofMathematics vol 14 no 5 pp 1881ndash1901 2010
[10] D R Sahu V Colao and G Marino ldquoStrong convergencetheorems for approximating common fixed points of familiesof nonexpansive mappings and applicationsrdquo Journal of GlobalOptimization vol 56 no 4 pp 1631ndash1651 2013
[11] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[12] Y Yao Y Liou and N Shahzad ldquoConstruction of iterativemethods for variational inequality and fixed point problemsrdquoNumerical Functional Analysis and Optimization vol 33 no 10pp 1250ndash1267 2012
[13] Y Yao G Marino and L Muglia ldquoA modified Korpelevichrsquosmethod convergent to the minimum-norm solution of a varia-tional inequalityrdquoOptimization vol 63 no 4 pp 559ndash569 2014
[14] Y Yao and M Postolache ldquoIterative methods for pseudomono-tone variational inequalities and fixed-point problemsrdquo Journalof OptimizationTheory and Applications vol 155 no 1 pp 273ndash287 2012
[15] H Zegeye E U Ofoedu and N Shahzad ldquoConvergencetheorems for equilibrium problem variational inequality prob-lem and countably infinite relatively quasi-nonexpansive map-pingsrdquo Applied Mathematics and Computation vol 216 no 12pp 3439ndash3449 2010
[16] H Zegeye andN Shahzad ldquoAhybrid approximationmethod forequilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 619ndash6302010
[17] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor a solution of finite families of equilibrium and variationalinequality problemsrdquo Optimization vol 63 no 2 pp 207ndash2232014
[18] H Zegeye andN Shahzad ldquoExtragradientmethod for solutionsof variational inequality problems in Banach spacesrdquo Abstractand Applied Analysis vol 2013 Article ID 832548 8 pages 2013
10 Abstract and Applied Analysis
[19] W Takahashi andM Toyoda ldquoWeak convergence theorems fornonexpansive mappings and monotone mappingsrdquo Journal ofOptimization Theory and Applications vol 118 no 2 pp 417ndash428 2003
[20] H Iiduka and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and inverse-strongly monotonemappingsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 61 no 3 pp 341ndash350 2005
[21] H Zegeye and N Shahzad ldquoSolutions of variational inequalityproblems in the set of fixed points of pseudocontractive map-pingsrdquo Carpathian Journal of Mathematics vol 30 no 2 pp257ndash265 2014
[22] Y Yao Y C Liou and S M Kang ldquoAlgorithms construction forvariational inequalitiesrdquo Fixed Point Theory and Applicationsvol 2011 Article ID 794203 12 pages 2011
[23] Y Cai Y Tang and L Liu ldquoIterative algorithms for minimum-norm fixed point of non-expansive mapping in Hilbert spacerdquoFixed Point Theory and Applications vol 2012 article 49 2012
[24] K Aoyama H Iiduka andW Takahashi ldquoWeak convergence ofan iterative sequence for accretive operators in Banach spacesrdquoFixed PointTheory andApplications vol 2006 Article ID 3539013 pages 2006
[25] C E Chidume H Zegeye and E Prempeh ldquoStrong conver-gence theorems for a finite family of nonexpansive mappings inBanach spacesrdquoCommunications onAppliedNonlinearAnalysisvol 11 no 2 pp 25ndash32 2004
[26] Y I Alber ldquoMetric and generalized projection operators inBanach spaces properties and applicationsrdquo in Theory andApplications of Nonlinear Operators of Accretive and MonotoneType Lecture Notes in Pure and Applied Mathematics pp 15ndash50 Marcel Dekker New York NY USA 1996
[27] Q B Zhang and C Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008
[28] H Zegeye and N Shahzad ldquoConvergence of Mannrsquos typeiteration method for generalized asymptotically nonexpansivemappingsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4007ndash4014 2011
[29] P E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[30] H K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
as 119899 rarr infin for all 119895 isin 1 2 119872Now since 119909
119899 is bounded subset of 119867 we can choose
a subsequence 119909119899119898 of 119909
119899 such that 119909
119899119898 119909 and
lim sup119899rarrinfin
⟨119891(119909lowast
) minus 119909lowast
119909119899minus 119909lowast
⟩ = lim119898rarrinfin
⟨119891(119909lowast
) minus
119909lowast
119909119899119898
minus 119909lowast
⟩ Then from (37) and Lemma 6 we have that119909 isin 119865(119879
119895) for each 119895 = 1 2 119872 Hence 119909 isin cap
119872
119895=1119865(119879119895)
In addition since 119866 is nonexpansive from Lemma 6 weget that 119909 isin 119865(119866) and hence by Lemmas 4 and 3 we obtainthat 119909 isin VI(119862 119860
119895) for each 119895 isin 1 2 119873
Therefore by Lemma 5 we immediately obtain that
lim sup119899rarrinfin
⟨119891 (119909lowast
) minus 119909lowast
119909119899minus 119909lowast
⟩
= lim119898rarrinfin
⟨119891 (119909lowast
) minus 119909lowast
119909119899119898
minus 119909lowast
⟩
= ⟨119891 (119909lowast
) minus 119909lowast
119909 minus 119909lowast
⟩ le 0
(38)
Then it follows from (32) (38) and Lemma 9 that 119909119899minus
119909lowast
rarr 0 as 119899 rarr infin Consequently 119909119899rarr 119909lowast
= 119875F(119891(119909lowast
))
Case 2 Suppose that there exists a subsequence 119899119894 of 119899
such that10038171003817100381710038171003817119909119899119894minus 119909lowast10038171003817100381710038171003817lt10038171003817100381710038171003817119909119899119894+1
minus 119909lowast10038171003817100381710038171003817 (39)
for all 119894 isin N Then by Lemma 8 there exists a nondecreasingsequence 119898
1) forall119899 ge 0 for 119871 = max119871119895 1 le 119895 le 119872 Then 119909
119899 converges
strongly to a unique point 119909lowast isin 119862 satisfying 119909lowast = 119875F(119906) whichis the unique solution of the variational inequality ⟨119909lowast minus 119906 119909 minus
119909lowast
⟩ ge 0 for all 119909 isin F
If inTheorem 10 we assume that119873 = 1 and119872 = 1 thenwe get the following corollary which is Theorem 31 of [21]
Corollary 12 Let 119862 be a nonempty closed and convex subsetof a real Hilbert space 119867 Let 119879 119862 rarr 119862 be Lipschitzpseudocontractive mappings with Lipschitz constant 119871 and 119860
119862 rarr 119867 an 120574-inverse strongly accretive mapping Let 119891
119862 rarr 119862 be a contraction with constant 120572 Assume that F =
119865(119879)⋂VI(119862 119860) is nonempty Let a sequence 119909119899 be generated
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119894le119873120574119894
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 119886
119899+ 119887119899+ 119888119899= 1 and 119887
119899+ 119888119899le
120582119899le 120582 lt 1(radic2 + 1) forall119899 ge 0 Then 119909
119899 converges strongly to
point 119909lowast isin F which is the unique solution of the variationalinequality ⟨(119868 minus 119891)(119909
lowast
) 119909 minus 119909lowast
⟩ ge 0 for all 119909 isin F
We note that the method of proof ofTheorem 10 providesthe following theorem which is a convergence theoremfor a minimum norm point of common fixed points of afinite family of Lipschitz pseudocontractive mappings andcommon solutions of a finite family of variational inequalityproblems for accretive mappings
Theorem 15 Let 119862 be a nonempty closed and convex subsetof a real Hilbert space119867 Let 119879
119895 119862 rarr 119862 119895 = 1 2 119872 be
Lipschitz pseudocontractive mappings with Lipschitz constants119871119895 respectively Let 119860
119895 119862 rarr 119867 for 119895 = 1 2 119873
be 120574119895-inverse strongly accretive mappings Assume that F =
[cap119872
119895=1119865(119879119895)]⋂[cap
119873
119895=1VI(119862 119860
119895)] is nonempty Let a sequence
119909119899 be generated from an arbitrary 119909
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119895le119873120574119895
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 and 119887
119899+ 119888119899
le 120582119899
le 120582 lt
1(radic1 + 1198712 + 1) forall119899 ge 0 for 119871 = max119871119895 1 le 119895 le 119872 Then
119909119899 converges strongly to a unique minimum norm point 119909lowast
of F (ie 119909lowast = 119875F(0)) which is the unique solution of thevariational inequality ⟨119909lowast 119909 minus 119909
lowast
⟩ ge 0 for all 119909 isin F
4 Numerical Example
Now we give an example of two Lipschitz pseudocontractivemappings and two 120574-inverse strongly accretive mappingssatisfyingTheorem 10 and some numerical experiment resultto explain the conclusion of the theorem as follows
Example 1 Let 119867 = R with absolute value norm Let 119862 =
[minus2 2] and let 1198791 1198792 119862 rarr 119862 be defined by
1198791119909 =
119909 + 1199092
119909 isin [minus2 0]
119909 119909 isin (0 2]
1198792119909 =
119909 119909 isin [minus21
2]
119909 minus (16
9) (119909 minus
1
2)
2
119909 isin (1
2 2]
(49)
Clearly for 119909 119910 isin 119862 we have that
⟨(119868 minus 1198791) 119909 minus (119868 minus 119879
1) 119910 119909 minus 119910⟩ ge 0
⟨(119868 minus 1198792) 119909 minus (119868 minus 119879
2) 119910 119909 minus 119910⟩ ge 0
(50)
which show that bothmappings are pseudocontractive Nextwe show that 119879
1is Lipschitz with 119871 = 5 If 119909 119910 isin [minus2 0] then
10038161003816100381610038161198791119909 minus 11987911199101003816100381610038161003816 =
with 120574 = 15 andVI(119862 1198601) = [12 2] Similarly we can show
that 1198602is 120574-inverse strongly accretive mapping with 120574 = 12
and VI(119862 1198602) = [minus2 23]
Note that we have119865(1198791)cap119865(119879
2)capVI(119862 119860
1)capVI(119862 119860
2) =
12Thus taking 120572
119899= 1(10119899 + 100) 120582
119899= 2(119899 + 100) +
0065 119887119899
= 119888119899
= 1(119899 + 100) + 001 119886119899
= 1 minus 2(119899 +
100) minus 002 and 119891(119909) = 119906 isin 119862 we observe that conditionsof Theorem 10 are satisfied and Scheme (17) provides thefollowing Table 1 and Figures 1(a) and 1(b) for 119906 = 06 and119906 = 08 respectively
We observe that the data provides strong convergenceof the sequence to the common point of fixed points ofboth pseudocontractive mappings and solutions of both vari-ational inequality problems for 120574-inverse strongly accretivemappings
Remark 2 Theorem 10 provides an iteration scheme whichconverges strongly to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problemsin Hilbert spaces
Remark 3 Theorem 10 improves Theorem 31 of Takahashiand Toyoda [19] Iiduka and Takahashi [20] and Zegeye andShahzad [21] and Theorem 32 of Yao et al [22] in the sensethat our convergence is to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problems
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1671301434 The authors therefore acknowledge
Abstract and Applied Analysis 9
0 05 1 15 20
0102030405060708091
x0= 1
Iterations n
x = 05
times104
Iteratesxn
(a)
0 1 2 3 4 5minus15
minus1
minus05
0
05
1
Iterations n
x0= 1
x = 05
times104
Iteratesxn
(b)
Figure 1
with thanksDSR technical and financial supportThe authorsalso thank the referees for their valuable comments andsuggestions which improved the presentation of this paper
References
[1] D Kinderlehrer and G Stampaccia An Iteration to VariationalInequalities and Their Applications Academic Press New YorkNY USA 1990
[2] H Zegeye and N Shahzad ldquoA hybrid scheme for finite familiesof equilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no1 pp 263ndash272 2011
[3] H Zegeye and N Shahzad ldquoApproximating common solutionof variational inequality problems for two monotone mappingsin Banach spacesrdquo Optimization Letters vol 5 no 4 pp 691ndash704 2011
[4] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor variational inequality problems and quasi-120601-asymptoticallynonexpansive mappingsrdquo Journal of Global Optimization vol54 no 1 pp 101ndash116 2012
[5] H Zegeye and N Shahzad ldquoAn iteration to a common pointof solution of variational inequality and fixed point-problemsin Banach spacesrdquo Journal of Applied Mathematics vol 2012Article ID 504503 19 pages 2012
[6] F E Browder and W V Petryshyn ldquoConstruction of fixedpoints of nonlinear mappings in Hilbert spacerdquo Journal ofMathematical Analysis and Applications vol 20 pp 197ndash2281967
[7] L C Ceng A Petrusel and J C Yao ldquoComposite viscosityapproximation methods for equilibrium problem variationalinequality and common fixed pointsrdquo Journal of Nonlinear andConvex Analysis vol 15 no 2 pp 219ndash240 2014
[8] L C Ceng A Petrusel M M Wong and J C Yao ldquoHybridalgorithms for solving variational inequalities variational inclu-sions mixed equilibria and fixed point problemsrdquoAbstract andApplied Analysis vol 2014 Article ID 208717 22 pages 2014
[9] L Ceng A Petrusel and M Wong ldquoStrong convergencetheorem for a generalized equilibrium problem and a pseudo-contractive mapping in a Hilbert spacerdquo Taiwanese Journal ofMathematics vol 14 no 5 pp 1881ndash1901 2010
[10] D R Sahu V Colao and G Marino ldquoStrong convergencetheorems for approximating common fixed points of familiesof nonexpansive mappings and applicationsrdquo Journal of GlobalOptimization vol 56 no 4 pp 1631ndash1651 2013
[11] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[12] Y Yao Y Liou and N Shahzad ldquoConstruction of iterativemethods for variational inequality and fixed point problemsrdquoNumerical Functional Analysis and Optimization vol 33 no 10pp 1250ndash1267 2012
[13] Y Yao G Marino and L Muglia ldquoA modified Korpelevichrsquosmethod convergent to the minimum-norm solution of a varia-tional inequalityrdquoOptimization vol 63 no 4 pp 559ndash569 2014
[14] Y Yao and M Postolache ldquoIterative methods for pseudomono-tone variational inequalities and fixed-point problemsrdquo Journalof OptimizationTheory and Applications vol 155 no 1 pp 273ndash287 2012
[15] H Zegeye E U Ofoedu and N Shahzad ldquoConvergencetheorems for equilibrium problem variational inequality prob-lem and countably infinite relatively quasi-nonexpansive map-pingsrdquo Applied Mathematics and Computation vol 216 no 12pp 3439ndash3449 2010
[16] H Zegeye andN Shahzad ldquoAhybrid approximationmethod forequilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 619ndash6302010
[17] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor a solution of finite families of equilibrium and variationalinequality problemsrdquo Optimization vol 63 no 2 pp 207ndash2232014
[18] H Zegeye andN Shahzad ldquoExtragradientmethod for solutionsof variational inequality problems in Banach spacesrdquo Abstractand Applied Analysis vol 2013 Article ID 832548 8 pages 2013
10 Abstract and Applied Analysis
[19] W Takahashi andM Toyoda ldquoWeak convergence theorems fornonexpansive mappings and monotone mappingsrdquo Journal ofOptimization Theory and Applications vol 118 no 2 pp 417ndash428 2003
[20] H Iiduka and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and inverse-strongly monotonemappingsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 61 no 3 pp 341ndash350 2005
[21] H Zegeye and N Shahzad ldquoSolutions of variational inequalityproblems in the set of fixed points of pseudocontractive map-pingsrdquo Carpathian Journal of Mathematics vol 30 no 2 pp257ndash265 2014
[22] Y Yao Y C Liou and S M Kang ldquoAlgorithms construction forvariational inequalitiesrdquo Fixed Point Theory and Applicationsvol 2011 Article ID 794203 12 pages 2011
[23] Y Cai Y Tang and L Liu ldquoIterative algorithms for minimum-norm fixed point of non-expansive mapping in Hilbert spacerdquoFixed Point Theory and Applications vol 2012 article 49 2012
[24] K Aoyama H Iiduka andW Takahashi ldquoWeak convergence ofan iterative sequence for accretive operators in Banach spacesrdquoFixed PointTheory andApplications vol 2006 Article ID 3539013 pages 2006
[25] C E Chidume H Zegeye and E Prempeh ldquoStrong conver-gence theorems for a finite family of nonexpansive mappings inBanach spacesrdquoCommunications onAppliedNonlinearAnalysisvol 11 no 2 pp 25ndash32 2004
[26] Y I Alber ldquoMetric and generalized projection operators inBanach spaces properties and applicationsrdquo in Theory andApplications of Nonlinear Operators of Accretive and MonotoneType Lecture Notes in Pure and Applied Mathematics pp 15ndash50 Marcel Dekker New York NY USA 1996
[27] Q B Zhang and C Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008
[28] H Zegeye and N Shahzad ldquoConvergence of Mannrsquos typeiteration method for generalized asymptotically nonexpansivemappingsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4007ndash4014 2011
[29] P E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[30] H K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
as 119899 rarr infin for all 119895 isin 1 2 119872Now since 119909
119899 is bounded subset of 119867 we can choose
a subsequence 119909119899119898 of 119909
119899 such that 119909
119899119898 119909 and
lim sup119899rarrinfin
⟨119891(119909lowast
) minus 119909lowast
119909119899minus 119909lowast
⟩ = lim119898rarrinfin
⟨119891(119909lowast
) minus
119909lowast
119909119899119898
minus 119909lowast
⟩ Then from (37) and Lemma 6 we have that119909 isin 119865(119879
119895) for each 119895 = 1 2 119872 Hence 119909 isin cap
119872
119895=1119865(119879119895)
In addition since 119866 is nonexpansive from Lemma 6 weget that 119909 isin 119865(119866) and hence by Lemmas 4 and 3 we obtainthat 119909 isin VI(119862 119860
119895) for each 119895 isin 1 2 119873
Therefore by Lemma 5 we immediately obtain that
lim sup119899rarrinfin
⟨119891 (119909lowast
) minus 119909lowast
119909119899minus 119909lowast
⟩
= lim119898rarrinfin
⟨119891 (119909lowast
) minus 119909lowast
119909119899119898
minus 119909lowast
⟩
= ⟨119891 (119909lowast
) minus 119909lowast
119909 minus 119909lowast
⟩ le 0
(38)
Then it follows from (32) (38) and Lemma 9 that 119909119899minus
119909lowast
rarr 0 as 119899 rarr infin Consequently 119909119899rarr 119909lowast
= 119875F(119891(119909lowast
))
Case 2 Suppose that there exists a subsequence 119899119894 of 119899
such that10038171003817100381710038171003817119909119899119894minus 119909lowast10038171003817100381710038171003817lt10038171003817100381710038171003817119909119899119894+1
minus 119909lowast10038171003817100381710038171003817 (39)
for all 119894 isin N Then by Lemma 8 there exists a nondecreasingsequence 119898
1) forall119899 ge 0 for 119871 = max119871119895 1 le 119895 le 119872 Then 119909
119899 converges
strongly to a unique point 119909lowast isin 119862 satisfying 119909lowast = 119875F(119906) whichis the unique solution of the variational inequality ⟨119909lowast minus 119906 119909 minus
119909lowast
⟩ ge 0 for all 119909 isin F
If inTheorem 10 we assume that119873 = 1 and119872 = 1 thenwe get the following corollary which is Theorem 31 of [21]
Corollary 12 Let 119862 be a nonempty closed and convex subsetof a real Hilbert space 119867 Let 119879 119862 rarr 119862 be Lipschitzpseudocontractive mappings with Lipschitz constant 119871 and 119860
119862 rarr 119867 an 120574-inverse strongly accretive mapping Let 119891
119862 rarr 119862 be a contraction with constant 120572 Assume that F =
119865(119879)⋂VI(119862 119860) is nonempty Let a sequence 119909119899 be generated
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119894le119873120574119894
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 119886
119899+ 119887119899+ 119888119899= 1 and 119887
119899+ 119888119899le
120582119899le 120582 lt 1(radic2 + 1) forall119899 ge 0 Then 119909
119899 converges strongly to
point 119909lowast isin F which is the unique solution of the variationalinequality ⟨(119868 minus 119891)(119909
lowast
) 119909 minus 119909lowast
⟩ ge 0 for all 119909 isin F
We note that the method of proof ofTheorem 10 providesthe following theorem which is a convergence theoremfor a minimum norm point of common fixed points of afinite family of Lipschitz pseudocontractive mappings andcommon solutions of a finite family of variational inequalityproblems for accretive mappings
Theorem 15 Let 119862 be a nonempty closed and convex subsetof a real Hilbert space119867 Let 119879
119895 119862 rarr 119862 119895 = 1 2 119872 be
Lipschitz pseudocontractive mappings with Lipschitz constants119871119895 respectively Let 119860
119895 119862 rarr 119867 for 119895 = 1 2 119873
be 120574119895-inverse strongly accretive mappings Assume that F =
[cap119872
119895=1119865(119879119895)]⋂[cap
119873
119895=1VI(119862 119860
119895)] is nonempty Let a sequence
119909119899 be generated from an arbitrary 119909
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119895le119873120574119895
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 and 119887
119899+ 119888119899
le 120582119899
le 120582 lt
1(radic1 + 1198712 + 1) forall119899 ge 0 for 119871 = max119871119895 1 le 119895 le 119872 Then
119909119899 converges strongly to a unique minimum norm point 119909lowast
of F (ie 119909lowast = 119875F(0)) which is the unique solution of thevariational inequality ⟨119909lowast 119909 minus 119909
lowast
⟩ ge 0 for all 119909 isin F
4 Numerical Example
Now we give an example of two Lipschitz pseudocontractivemappings and two 120574-inverse strongly accretive mappingssatisfyingTheorem 10 and some numerical experiment resultto explain the conclusion of the theorem as follows
Example 1 Let 119867 = R with absolute value norm Let 119862 =
[minus2 2] and let 1198791 1198792 119862 rarr 119862 be defined by
1198791119909 =
119909 + 1199092
119909 isin [minus2 0]
119909 119909 isin (0 2]
1198792119909 =
119909 119909 isin [minus21
2]
119909 minus (16
9) (119909 minus
1
2)
2
119909 isin (1
2 2]
(49)
Clearly for 119909 119910 isin 119862 we have that
⟨(119868 minus 1198791) 119909 minus (119868 minus 119879
1) 119910 119909 minus 119910⟩ ge 0
⟨(119868 minus 1198792) 119909 minus (119868 minus 119879
2) 119910 119909 minus 119910⟩ ge 0
(50)
which show that bothmappings are pseudocontractive Nextwe show that 119879
1is Lipschitz with 119871 = 5 If 119909 119910 isin [minus2 0] then
10038161003816100381610038161198791119909 minus 11987911199101003816100381610038161003816 =
with 120574 = 15 andVI(119862 1198601) = [12 2] Similarly we can show
that 1198602is 120574-inverse strongly accretive mapping with 120574 = 12
and VI(119862 1198602) = [minus2 23]
Note that we have119865(1198791)cap119865(119879
2)capVI(119862 119860
1)capVI(119862 119860
2) =
12Thus taking 120572
119899= 1(10119899 + 100) 120582
119899= 2(119899 + 100) +
0065 119887119899
= 119888119899
= 1(119899 + 100) + 001 119886119899
= 1 minus 2(119899 +
100) minus 002 and 119891(119909) = 119906 isin 119862 we observe that conditionsof Theorem 10 are satisfied and Scheme (17) provides thefollowing Table 1 and Figures 1(a) and 1(b) for 119906 = 06 and119906 = 08 respectively
We observe that the data provides strong convergenceof the sequence to the common point of fixed points ofboth pseudocontractive mappings and solutions of both vari-ational inequality problems for 120574-inverse strongly accretivemappings
Remark 2 Theorem 10 provides an iteration scheme whichconverges strongly to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problemsin Hilbert spaces
Remark 3 Theorem 10 improves Theorem 31 of Takahashiand Toyoda [19] Iiduka and Takahashi [20] and Zegeye andShahzad [21] and Theorem 32 of Yao et al [22] in the sensethat our convergence is to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problems
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1671301434 The authors therefore acknowledge
Abstract and Applied Analysis 9
0 05 1 15 20
0102030405060708091
x0= 1
Iterations n
x = 05
times104
Iteratesxn
(a)
0 1 2 3 4 5minus15
minus1
minus05
0
05
1
Iterations n
x0= 1
x = 05
times104
Iteratesxn
(b)
Figure 1
with thanksDSR technical and financial supportThe authorsalso thank the referees for their valuable comments andsuggestions which improved the presentation of this paper
References
[1] D Kinderlehrer and G Stampaccia An Iteration to VariationalInequalities and Their Applications Academic Press New YorkNY USA 1990
[2] H Zegeye and N Shahzad ldquoA hybrid scheme for finite familiesof equilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no1 pp 263ndash272 2011
[3] H Zegeye and N Shahzad ldquoApproximating common solutionof variational inequality problems for two monotone mappingsin Banach spacesrdquo Optimization Letters vol 5 no 4 pp 691ndash704 2011
[4] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor variational inequality problems and quasi-120601-asymptoticallynonexpansive mappingsrdquo Journal of Global Optimization vol54 no 1 pp 101ndash116 2012
[5] H Zegeye and N Shahzad ldquoAn iteration to a common pointof solution of variational inequality and fixed point-problemsin Banach spacesrdquo Journal of Applied Mathematics vol 2012Article ID 504503 19 pages 2012
[6] F E Browder and W V Petryshyn ldquoConstruction of fixedpoints of nonlinear mappings in Hilbert spacerdquo Journal ofMathematical Analysis and Applications vol 20 pp 197ndash2281967
[7] L C Ceng A Petrusel and J C Yao ldquoComposite viscosityapproximation methods for equilibrium problem variationalinequality and common fixed pointsrdquo Journal of Nonlinear andConvex Analysis vol 15 no 2 pp 219ndash240 2014
[8] L C Ceng A Petrusel M M Wong and J C Yao ldquoHybridalgorithms for solving variational inequalities variational inclu-sions mixed equilibria and fixed point problemsrdquoAbstract andApplied Analysis vol 2014 Article ID 208717 22 pages 2014
[9] L Ceng A Petrusel and M Wong ldquoStrong convergencetheorem for a generalized equilibrium problem and a pseudo-contractive mapping in a Hilbert spacerdquo Taiwanese Journal ofMathematics vol 14 no 5 pp 1881ndash1901 2010
[10] D R Sahu V Colao and G Marino ldquoStrong convergencetheorems for approximating common fixed points of familiesof nonexpansive mappings and applicationsrdquo Journal of GlobalOptimization vol 56 no 4 pp 1631ndash1651 2013
[11] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[12] Y Yao Y Liou and N Shahzad ldquoConstruction of iterativemethods for variational inequality and fixed point problemsrdquoNumerical Functional Analysis and Optimization vol 33 no 10pp 1250ndash1267 2012
[13] Y Yao G Marino and L Muglia ldquoA modified Korpelevichrsquosmethod convergent to the minimum-norm solution of a varia-tional inequalityrdquoOptimization vol 63 no 4 pp 559ndash569 2014
[14] Y Yao and M Postolache ldquoIterative methods for pseudomono-tone variational inequalities and fixed-point problemsrdquo Journalof OptimizationTheory and Applications vol 155 no 1 pp 273ndash287 2012
[15] H Zegeye E U Ofoedu and N Shahzad ldquoConvergencetheorems for equilibrium problem variational inequality prob-lem and countably infinite relatively quasi-nonexpansive map-pingsrdquo Applied Mathematics and Computation vol 216 no 12pp 3439ndash3449 2010
[16] H Zegeye andN Shahzad ldquoAhybrid approximationmethod forequilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 619ndash6302010
[17] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor a solution of finite families of equilibrium and variationalinequality problemsrdquo Optimization vol 63 no 2 pp 207ndash2232014
[18] H Zegeye andN Shahzad ldquoExtragradientmethod for solutionsof variational inequality problems in Banach spacesrdquo Abstractand Applied Analysis vol 2013 Article ID 832548 8 pages 2013
10 Abstract and Applied Analysis
[19] W Takahashi andM Toyoda ldquoWeak convergence theorems fornonexpansive mappings and monotone mappingsrdquo Journal ofOptimization Theory and Applications vol 118 no 2 pp 417ndash428 2003
[20] H Iiduka and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and inverse-strongly monotonemappingsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 61 no 3 pp 341ndash350 2005
[21] H Zegeye and N Shahzad ldquoSolutions of variational inequalityproblems in the set of fixed points of pseudocontractive map-pingsrdquo Carpathian Journal of Mathematics vol 30 no 2 pp257ndash265 2014
[22] Y Yao Y C Liou and S M Kang ldquoAlgorithms construction forvariational inequalitiesrdquo Fixed Point Theory and Applicationsvol 2011 Article ID 794203 12 pages 2011
[23] Y Cai Y Tang and L Liu ldquoIterative algorithms for minimum-norm fixed point of non-expansive mapping in Hilbert spacerdquoFixed Point Theory and Applications vol 2012 article 49 2012
[24] K Aoyama H Iiduka andW Takahashi ldquoWeak convergence ofan iterative sequence for accretive operators in Banach spacesrdquoFixed PointTheory andApplications vol 2006 Article ID 3539013 pages 2006
[25] C E Chidume H Zegeye and E Prempeh ldquoStrong conver-gence theorems for a finite family of nonexpansive mappings inBanach spacesrdquoCommunications onAppliedNonlinearAnalysisvol 11 no 2 pp 25ndash32 2004
[26] Y I Alber ldquoMetric and generalized projection operators inBanach spaces properties and applicationsrdquo in Theory andApplications of Nonlinear Operators of Accretive and MonotoneType Lecture Notes in Pure and Applied Mathematics pp 15ndash50 Marcel Dekker New York NY USA 1996
[27] Q B Zhang and C Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008
[28] H Zegeye and N Shahzad ldquoConvergence of Mannrsquos typeiteration method for generalized asymptotically nonexpansivemappingsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4007ndash4014 2011
[29] P E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[30] H K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
as 119899 rarr infin for all 119895 isin 1 2 119872Now since 119909
119899 is bounded subset of 119867 we can choose
a subsequence 119909119899119898 of 119909
119899 such that 119909
119899119898 119909 and
lim sup119899rarrinfin
⟨119891(119909lowast
) minus 119909lowast
119909119899minus 119909lowast
⟩ = lim119898rarrinfin
⟨119891(119909lowast
) minus
119909lowast
119909119899119898
minus 119909lowast
⟩ Then from (37) and Lemma 6 we have that119909 isin 119865(119879
119895) for each 119895 = 1 2 119872 Hence 119909 isin cap
119872
119895=1119865(119879119895)
In addition since 119866 is nonexpansive from Lemma 6 weget that 119909 isin 119865(119866) and hence by Lemmas 4 and 3 we obtainthat 119909 isin VI(119862 119860
119895) for each 119895 isin 1 2 119873
Therefore by Lemma 5 we immediately obtain that
lim sup119899rarrinfin
⟨119891 (119909lowast
) minus 119909lowast
119909119899minus 119909lowast
⟩
= lim119898rarrinfin
⟨119891 (119909lowast
) minus 119909lowast
119909119899119898
minus 119909lowast
⟩
= ⟨119891 (119909lowast
) minus 119909lowast
119909 minus 119909lowast
⟩ le 0
(38)
Then it follows from (32) (38) and Lemma 9 that 119909119899minus
119909lowast
rarr 0 as 119899 rarr infin Consequently 119909119899rarr 119909lowast
= 119875F(119891(119909lowast
))
Case 2 Suppose that there exists a subsequence 119899119894 of 119899
such that10038171003817100381710038171003817119909119899119894minus 119909lowast10038171003817100381710038171003817lt10038171003817100381710038171003817119909119899119894+1
minus 119909lowast10038171003817100381710038171003817 (39)
for all 119894 isin N Then by Lemma 8 there exists a nondecreasingsequence 119898
1) forall119899 ge 0 for 119871 = max119871119895 1 le 119895 le 119872 Then 119909
119899 converges
strongly to a unique point 119909lowast isin 119862 satisfying 119909lowast = 119875F(119906) whichis the unique solution of the variational inequality ⟨119909lowast minus 119906 119909 minus
119909lowast
⟩ ge 0 for all 119909 isin F
If inTheorem 10 we assume that119873 = 1 and119872 = 1 thenwe get the following corollary which is Theorem 31 of [21]
Corollary 12 Let 119862 be a nonempty closed and convex subsetof a real Hilbert space 119867 Let 119879 119862 rarr 119862 be Lipschitzpseudocontractive mappings with Lipschitz constant 119871 and 119860
119862 rarr 119867 an 120574-inverse strongly accretive mapping Let 119891
119862 rarr 119862 be a contraction with constant 120572 Assume that F =
119865(119879)⋂VI(119862 119860) is nonempty Let a sequence 119909119899 be generated
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119894le119873120574119894
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 119886
119899+ 119887119899+ 119888119899= 1 and 119887
119899+ 119888119899le
120582119899le 120582 lt 1(radic2 + 1) forall119899 ge 0 Then 119909
119899 converges strongly to
point 119909lowast isin F which is the unique solution of the variationalinequality ⟨(119868 minus 119891)(119909
lowast
) 119909 minus 119909lowast
⟩ ge 0 for all 119909 isin F
We note that the method of proof ofTheorem 10 providesthe following theorem which is a convergence theoremfor a minimum norm point of common fixed points of afinite family of Lipschitz pseudocontractive mappings andcommon solutions of a finite family of variational inequalityproblems for accretive mappings
Theorem 15 Let 119862 be a nonempty closed and convex subsetof a real Hilbert space119867 Let 119879
119895 119862 rarr 119862 119895 = 1 2 119872 be
Lipschitz pseudocontractive mappings with Lipschitz constants119871119895 respectively Let 119860
119895 119862 rarr 119867 for 119895 = 1 2 119873
be 120574119895-inverse strongly accretive mappings Assume that F =
[cap119872
119895=1119865(119879119895)]⋂[cap
119873
119895=1VI(119862 119860
119895)] is nonempty Let a sequence
119909119899 be generated from an arbitrary 119909
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119895le119873120574119895
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 and 119887
119899+ 119888119899
le 120582119899
le 120582 lt
1(radic1 + 1198712 + 1) forall119899 ge 0 for 119871 = max119871119895 1 le 119895 le 119872 Then
119909119899 converges strongly to a unique minimum norm point 119909lowast
of F (ie 119909lowast = 119875F(0)) which is the unique solution of thevariational inequality ⟨119909lowast 119909 minus 119909
lowast
⟩ ge 0 for all 119909 isin F
4 Numerical Example
Now we give an example of two Lipschitz pseudocontractivemappings and two 120574-inverse strongly accretive mappingssatisfyingTheorem 10 and some numerical experiment resultto explain the conclusion of the theorem as follows
Example 1 Let 119867 = R with absolute value norm Let 119862 =
[minus2 2] and let 1198791 1198792 119862 rarr 119862 be defined by
1198791119909 =
119909 + 1199092
119909 isin [minus2 0]
119909 119909 isin (0 2]
1198792119909 =
119909 119909 isin [minus21
2]
119909 minus (16
9) (119909 minus
1
2)
2
119909 isin (1
2 2]
(49)
Clearly for 119909 119910 isin 119862 we have that
⟨(119868 minus 1198791) 119909 minus (119868 minus 119879
1) 119910 119909 minus 119910⟩ ge 0
⟨(119868 minus 1198792) 119909 minus (119868 minus 119879
2) 119910 119909 minus 119910⟩ ge 0
(50)
which show that bothmappings are pseudocontractive Nextwe show that 119879
1is Lipschitz with 119871 = 5 If 119909 119910 isin [minus2 0] then
10038161003816100381610038161198791119909 minus 11987911199101003816100381610038161003816 =
with 120574 = 15 andVI(119862 1198601) = [12 2] Similarly we can show
that 1198602is 120574-inverse strongly accretive mapping with 120574 = 12
and VI(119862 1198602) = [minus2 23]
Note that we have119865(1198791)cap119865(119879
2)capVI(119862 119860
1)capVI(119862 119860
2) =
12Thus taking 120572
119899= 1(10119899 + 100) 120582
119899= 2(119899 + 100) +
0065 119887119899
= 119888119899
= 1(119899 + 100) + 001 119886119899
= 1 minus 2(119899 +
100) minus 002 and 119891(119909) = 119906 isin 119862 we observe that conditionsof Theorem 10 are satisfied and Scheme (17) provides thefollowing Table 1 and Figures 1(a) and 1(b) for 119906 = 06 and119906 = 08 respectively
We observe that the data provides strong convergenceof the sequence to the common point of fixed points ofboth pseudocontractive mappings and solutions of both vari-ational inequality problems for 120574-inverse strongly accretivemappings
Remark 2 Theorem 10 provides an iteration scheme whichconverges strongly to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problemsin Hilbert spaces
Remark 3 Theorem 10 improves Theorem 31 of Takahashiand Toyoda [19] Iiduka and Takahashi [20] and Zegeye andShahzad [21] and Theorem 32 of Yao et al [22] in the sensethat our convergence is to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problems
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1671301434 The authors therefore acknowledge
Abstract and Applied Analysis 9
0 05 1 15 20
0102030405060708091
x0= 1
Iterations n
x = 05
times104
Iteratesxn
(a)
0 1 2 3 4 5minus15
minus1
minus05
0
05
1
Iterations n
x0= 1
x = 05
times104
Iteratesxn
(b)
Figure 1
with thanksDSR technical and financial supportThe authorsalso thank the referees for their valuable comments andsuggestions which improved the presentation of this paper
References
[1] D Kinderlehrer and G Stampaccia An Iteration to VariationalInequalities and Their Applications Academic Press New YorkNY USA 1990
[2] H Zegeye and N Shahzad ldquoA hybrid scheme for finite familiesof equilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no1 pp 263ndash272 2011
[3] H Zegeye and N Shahzad ldquoApproximating common solutionof variational inequality problems for two monotone mappingsin Banach spacesrdquo Optimization Letters vol 5 no 4 pp 691ndash704 2011
[4] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor variational inequality problems and quasi-120601-asymptoticallynonexpansive mappingsrdquo Journal of Global Optimization vol54 no 1 pp 101ndash116 2012
[5] H Zegeye and N Shahzad ldquoAn iteration to a common pointof solution of variational inequality and fixed point-problemsin Banach spacesrdquo Journal of Applied Mathematics vol 2012Article ID 504503 19 pages 2012
[6] F E Browder and W V Petryshyn ldquoConstruction of fixedpoints of nonlinear mappings in Hilbert spacerdquo Journal ofMathematical Analysis and Applications vol 20 pp 197ndash2281967
[7] L C Ceng A Petrusel and J C Yao ldquoComposite viscosityapproximation methods for equilibrium problem variationalinequality and common fixed pointsrdquo Journal of Nonlinear andConvex Analysis vol 15 no 2 pp 219ndash240 2014
[8] L C Ceng A Petrusel M M Wong and J C Yao ldquoHybridalgorithms for solving variational inequalities variational inclu-sions mixed equilibria and fixed point problemsrdquoAbstract andApplied Analysis vol 2014 Article ID 208717 22 pages 2014
[9] L Ceng A Petrusel and M Wong ldquoStrong convergencetheorem for a generalized equilibrium problem and a pseudo-contractive mapping in a Hilbert spacerdquo Taiwanese Journal ofMathematics vol 14 no 5 pp 1881ndash1901 2010
[10] D R Sahu V Colao and G Marino ldquoStrong convergencetheorems for approximating common fixed points of familiesof nonexpansive mappings and applicationsrdquo Journal of GlobalOptimization vol 56 no 4 pp 1631ndash1651 2013
[11] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[12] Y Yao Y Liou and N Shahzad ldquoConstruction of iterativemethods for variational inequality and fixed point problemsrdquoNumerical Functional Analysis and Optimization vol 33 no 10pp 1250ndash1267 2012
[13] Y Yao G Marino and L Muglia ldquoA modified Korpelevichrsquosmethod convergent to the minimum-norm solution of a varia-tional inequalityrdquoOptimization vol 63 no 4 pp 559ndash569 2014
[14] Y Yao and M Postolache ldquoIterative methods for pseudomono-tone variational inequalities and fixed-point problemsrdquo Journalof OptimizationTheory and Applications vol 155 no 1 pp 273ndash287 2012
[15] H Zegeye E U Ofoedu and N Shahzad ldquoConvergencetheorems for equilibrium problem variational inequality prob-lem and countably infinite relatively quasi-nonexpansive map-pingsrdquo Applied Mathematics and Computation vol 216 no 12pp 3439ndash3449 2010
[16] H Zegeye andN Shahzad ldquoAhybrid approximationmethod forequilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 619ndash6302010
[17] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor a solution of finite families of equilibrium and variationalinequality problemsrdquo Optimization vol 63 no 2 pp 207ndash2232014
[18] H Zegeye andN Shahzad ldquoExtragradientmethod for solutionsof variational inequality problems in Banach spacesrdquo Abstractand Applied Analysis vol 2013 Article ID 832548 8 pages 2013
10 Abstract and Applied Analysis
[19] W Takahashi andM Toyoda ldquoWeak convergence theorems fornonexpansive mappings and monotone mappingsrdquo Journal ofOptimization Theory and Applications vol 118 no 2 pp 417ndash428 2003
[20] H Iiduka and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and inverse-strongly monotonemappingsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 61 no 3 pp 341ndash350 2005
[21] H Zegeye and N Shahzad ldquoSolutions of variational inequalityproblems in the set of fixed points of pseudocontractive map-pingsrdquo Carpathian Journal of Mathematics vol 30 no 2 pp257ndash265 2014
[22] Y Yao Y C Liou and S M Kang ldquoAlgorithms construction forvariational inequalitiesrdquo Fixed Point Theory and Applicationsvol 2011 Article ID 794203 12 pages 2011
[23] Y Cai Y Tang and L Liu ldquoIterative algorithms for minimum-norm fixed point of non-expansive mapping in Hilbert spacerdquoFixed Point Theory and Applications vol 2012 article 49 2012
[24] K Aoyama H Iiduka andW Takahashi ldquoWeak convergence ofan iterative sequence for accretive operators in Banach spacesrdquoFixed PointTheory andApplications vol 2006 Article ID 3539013 pages 2006
[25] C E Chidume H Zegeye and E Prempeh ldquoStrong conver-gence theorems for a finite family of nonexpansive mappings inBanach spacesrdquoCommunications onAppliedNonlinearAnalysisvol 11 no 2 pp 25ndash32 2004
[26] Y I Alber ldquoMetric and generalized projection operators inBanach spaces properties and applicationsrdquo in Theory andApplications of Nonlinear Operators of Accretive and MonotoneType Lecture Notes in Pure and Applied Mathematics pp 15ndash50 Marcel Dekker New York NY USA 1996
[27] Q B Zhang and C Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008
[28] H Zegeye and N Shahzad ldquoConvergence of Mannrsquos typeiteration method for generalized asymptotically nonexpansivemappingsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4007ndash4014 2011
[29] P E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[30] H K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119894le119873120574119894
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 119886
119899+ 119887119899+ 119888119899= 1 and 119887
119899+ 119888119899le
120582119899le 120582 lt 1(radic2 + 1) forall119899 ge 0 Then 119909
119899 converges strongly to
point 119909lowast isin F which is the unique solution of the variationalinequality ⟨(119868 minus 119891)(119909
lowast
) 119909 minus 119909lowast
⟩ ge 0 for all 119909 isin F
We note that the method of proof ofTheorem 10 providesthe following theorem which is a convergence theoremfor a minimum norm point of common fixed points of afinite family of Lipschitz pseudocontractive mappings andcommon solutions of a finite family of variational inequalityproblems for accretive mappings
Theorem 15 Let 119862 be a nonempty closed and convex subsetof a real Hilbert space119867 Let 119879
119895 119862 rarr 119862 119895 = 1 2 119872 be
Lipschitz pseudocontractive mappings with Lipschitz constants119871119895 respectively Let 119860
119895 119862 rarr 119867 for 119895 = 1 2 119873
be 120574119895-inverse strongly accretive mappings Assume that F =
[cap119872
119895=1119865(119879119895)]⋂[cap
119873
119895=1VI(119862 119860
119895)] is nonempty Let a sequence
119909119899 be generated from an arbitrary 119909
sdot sdot sdot + 119890119873119875119862[119868 minus 120574119860
119903] for 120574 isin (0 2120574
0) for 120574
0= min
1le119895le119873120574119895
with 1198900+ 1198901+ sdot sdot sdot + 119890
119903= 1 and 119887
119899+ 119888119899
le 120582119899
le 120582 lt
1(radic1 + 1198712 + 1) forall119899 ge 0 for 119871 = max119871119895 1 le 119895 le 119872 Then
119909119899 converges strongly to a unique minimum norm point 119909lowast
of F (ie 119909lowast = 119875F(0)) which is the unique solution of thevariational inequality ⟨119909lowast 119909 minus 119909
lowast
⟩ ge 0 for all 119909 isin F
4 Numerical Example
Now we give an example of two Lipschitz pseudocontractivemappings and two 120574-inverse strongly accretive mappingssatisfyingTheorem 10 and some numerical experiment resultto explain the conclusion of the theorem as follows
Example 1 Let 119867 = R with absolute value norm Let 119862 =
[minus2 2] and let 1198791 1198792 119862 rarr 119862 be defined by
1198791119909 =
119909 + 1199092
119909 isin [minus2 0]
119909 119909 isin (0 2]
1198792119909 =
119909 119909 isin [minus21
2]
119909 minus (16
9) (119909 minus
1
2)
2
119909 isin (1
2 2]
(49)
Clearly for 119909 119910 isin 119862 we have that
⟨(119868 minus 1198791) 119909 minus (119868 minus 119879
1) 119910 119909 minus 119910⟩ ge 0
⟨(119868 minus 1198792) 119909 minus (119868 minus 119879
2) 119910 119909 minus 119910⟩ ge 0
(50)
which show that bothmappings are pseudocontractive Nextwe show that 119879
1is Lipschitz with 119871 = 5 If 119909 119910 isin [minus2 0] then
10038161003816100381610038161198791119909 minus 11987911199101003816100381610038161003816 =
with 120574 = 15 andVI(119862 1198601) = [12 2] Similarly we can show
that 1198602is 120574-inverse strongly accretive mapping with 120574 = 12
and VI(119862 1198602) = [minus2 23]
Note that we have119865(1198791)cap119865(119879
2)capVI(119862 119860
1)capVI(119862 119860
2) =
12Thus taking 120572
119899= 1(10119899 + 100) 120582
119899= 2(119899 + 100) +
0065 119887119899
= 119888119899
= 1(119899 + 100) + 001 119886119899
= 1 minus 2(119899 +
100) minus 002 and 119891(119909) = 119906 isin 119862 we observe that conditionsof Theorem 10 are satisfied and Scheme (17) provides thefollowing Table 1 and Figures 1(a) and 1(b) for 119906 = 06 and119906 = 08 respectively
We observe that the data provides strong convergenceof the sequence to the common point of fixed points ofboth pseudocontractive mappings and solutions of both vari-ational inequality problems for 120574-inverse strongly accretivemappings
Remark 2 Theorem 10 provides an iteration scheme whichconverges strongly to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problemsin Hilbert spaces
Remark 3 Theorem 10 improves Theorem 31 of Takahashiand Toyoda [19] Iiduka and Takahashi [20] and Zegeye andShahzad [21] and Theorem 32 of Yao et al [22] in the sensethat our convergence is to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problems
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1671301434 The authors therefore acknowledge
Abstract and Applied Analysis 9
0 05 1 15 20
0102030405060708091
x0= 1
Iterations n
x = 05
times104
Iteratesxn
(a)
0 1 2 3 4 5minus15
minus1
minus05
0
05
1
Iterations n
x0= 1
x = 05
times104
Iteratesxn
(b)
Figure 1
with thanksDSR technical and financial supportThe authorsalso thank the referees for their valuable comments andsuggestions which improved the presentation of this paper
References
[1] D Kinderlehrer and G Stampaccia An Iteration to VariationalInequalities and Their Applications Academic Press New YorkNY USA 1990
[2] H Zegeye and N Shahzad ldquoA hybrid scheme for finite familiesof equilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no1 pp 263ndash272 2011
[3] H Zegeye and N Shahzad ldquoApproximating common solutionof variational inequality problems for two monotone mappingsin Banach spacesrdquo Optimization Letters vol 5 no 4 pp 691ndash704 2011
[4] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor variational inequality problems and quasi-120601-asymptoticallynonexpansive mappingsrdquo Journal of Global Optimization vol54 no 1 pp 101ndash116 2012
[5] H Zegeye and N Shahzad ldquoAn iteration to a common pointof solution of variational inequality and fixed point-problemsin Banach spacesrdquo Journal of Applied Mathematics vol 2012Article ID 504503 19 pages 2012
[6] F E Browder and W V Petryshyn ldquoConstruction of fixedpoints of nonlinear mappings in Hilbert spacerdquo Journal ofMathematical Analysis and Applications vol 20 pp 197ndash2281967
[7] L C Ceng A Petrusel and J C Yao ldquoComposite viscosityapproximation methods for equilibrium problem variationalinequality and common fixed pointsrdquo Journal of Nonlinear andConvex Analysis vol 15 no 2 pp 219ndash240 2014
[8] L C Ceng A Petrusel M M Wong and J C Yao ldquoHybridalgorithms for solving variational inequalities variational inclu-sions mixed equilibria and fixed point problemsrdquoAbstract andApplied Analysis vol 2014 Article ID 208717 22 pages 2014
[9] L Ceng A Petrusel and M Wong ldquoStrong convergencetheorem for a generalized equilibrium problem and a pseudo-contractive mapping in a Hilbert spacerdquo Taiwanese Journal ofMathematics vol 14 no 5 pp 1881ndash1901 2010
[10] D R Sahu V Colao and G Marino ldquoStrong convergencetheorems for approximating common fixed points of familiesof nonexpansive mappings and applicationsrdquo Journal of GlobalOptimization vol 56 no 4 pp 1631ndash1651 2013
[11] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[12] Y Yao Y Liou and N Shahzad ldquoConstruction of iterativemethods for variational inequality and fixed point problemsrdquoNumerical Functional Analysis and Optimization vol 33 no 10pp 1250ndash1267 2012
[13] Y Yao G Marino and L Muglia ldquoA modified Korpelevichrsquosmethod convergent to the minimum-norm solution of a varia-tional inequalityrdquoOptimization vol 63 no 4 pp 559ndash569 2014
[14] Y Yao and M Postolache ldquoIterative methods for pseudomono-tone variational inequalities and fixed-point problemsrdquo Journalof OptimizationTheory and Applications vol 155 no 1 pp 273ndash287 2012
[15] H Zegeye E U Ofoedu and N Shahzad ldquoConvergencetheorems for equilibrium problem variational inequality prob-lem and countably infinite relatively quasi-nonexpansive map-pingsrdquo Applied Mathematics and Computation vol 216 no 12pp 3439ndash3449 2010
[16] H Zegeye andN Shahzad ldquoAhybrid approximationmethod forequilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 619ndash6302010
[17] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor a solution of finite families of equilibrium and variationalinequality problemsrdquo Optimization vol 63 no 2 pp 207ndash2232014
[18] H Zegeye andN Shahzad ldquoExtragradientmethod for solutionsof variational inequality problems in Banach spacesrdquo Abstractand Applied Analysis vol 2013 Article ID 832548 8 pages 2013
10 Abstract and Applied Analysis
[19] W Takahashi andM Toyoda ldquoWeak convergence theorems fornonexpansive mappings and monotone mappingsrdquo Journal ofOptimization Theory and Applications vol 118 no 2 pp 417ndash428 2003
[20] H Iiduka and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and inverse-strongly monotonemappingsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 61 no 3 pp 341ndash350 2005
[21] H Zegeye and N Shahzad ldquoSolutions of variational inequalityproblems in the set of fixed points of pseudocontractive map-pingsrdquo Carpathian Journal of Mathematics vol 30 no 2 pp257ndash265 2014
[22] Y Yao Y C Liou and S M Kang ldquoAlgorithms construction forvariational inequalitiesrdquo Fixed Point Theory and Applicationsvol 2011 Article ID 794203 12 pages 2011
[23] Y Cai Y Tang and L Liu ldquoIterative algorithms for minimum-norm fixed point of non-expansive mapping in Hilbert spacerdquoFixed Point Theory and Applications vol 2012 article 49 2012
[24] K Aoyama H Iiduka andW Takahashi ldquoWeak convergence ofan iterative sequence for accretive operators in Banach spacesrdquoFixed PointTheory andApplications vol 2006 Article ID 3539013 pages 2006
[25] C E Chidume H Zegeye and E Prempeh ldquoStrong conver-gence theorems for a finite family of nonexpansive mappings inBanach spacesrdquoCommunications onAppliedNonlinearAnalysisvol 11 no 2 pp 25ndash32 2004
[26] Y I Alber ldquoMetric and generalized projection operators inBanach spaces properties and applicationsrdquo in Theory andApplications of Nonlinear Operators of Accretive and MonotoneType Lecture Notes in Pure and Applied Mathematics pp 15ndash50 Marcel Dekker New York NY USA 1996
[27] Q B Zhang and C Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008
[28] H Zegeye and N Shahzad ldquoConvergence of Mannrsquos typeiteration method for generalized asymptotically nonexpansivemappingsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4007ndash4014 2011
[29] P E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[30] H K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
with 120574 = 15 andVI(119862 1198601) = [12 2] Similarly we can show
that 1198602is 120574-inverse strongly accretive mapping with 120574 = 12
and VI(119862 1198602) = [minus2 23]
Note that we have119865(1198791)cap119865(119879
2)capVI(119862 119860
1)capVI(119862 119860
2) =
12Thus taking 120572
119899= 1(10119899 + 100) 120582
119899= 2(119899 + 100) +
0065 119887119899
= 119888119899
= 1(119899 + 100) + 001 119886119899
= 1 minus 2(119899 +
100) minus 002 and 119891(119909) = 119906 isin 119862 we observe that conditionsof Theorem 10 are satisfied and Scheme (17) provides thefollowing Table 1 and Figures 1(a) and 1(b) for 119906 = 06 and119906 = 08 respectively
We observe that the data provides strong convergenceof the sequence to the common point of fixed points ofboth pseudocontractive mappings and solutions of both vari-ational inequality problems for 120574-inverse strongly accretivemappings
Remark 2 Theorem 10 provides an iteration scheme whichconverges strongly to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problemsin Hilbert spaces
Remark 3 Theorem 10 improves Theorem 31 of Takahashiand Toyoda [19] Iiduka and Takahashi [20] and Zegeye andShahzad [21] and Theorem 32 of Yao et al [22] in the sensethat our convergence is to a common point of fixed points of afinite family of Lipschitzian pseudocontractivemappings andsolutions of a finite family of variational inequality problems
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah undergrant no 1671301434 The authors therefore acknowledge
Abstract and Applied Analysis 9
0 05 1 15 20
0102030405060708091
x0= 1
Iterations n
x = 05
times104
Iteratesxn
(a)
0 1 2 3 4 5minus15
minus1
minus05
0
05
1
Iterations n
x0= 1
x = 05
times104
Iteratesxn
(b)
Figure 1
with thanksDSR technical and financial supportThe authorsalso thank the referees for their valuable comments andsuggestions which improved the presentation of this paper
References
[1] D Kinderlehrer and G Stampaccia An Iteration to VariationalInequalities and Their Applications Academic Press New YorkNY USA 1990
[2] H Zegeye and N Shahzad ldquoA hybrid scheme for finite familiesof equilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no1 pp 263ndash272 2011
[3] H Zegeye and N Shahzad ldquoApproximating common solutionof variational inequality problems for two monotone mappingsin Banach spacesrdquo Optimization Letters vol 5 no 4 pp 691ndash704 2011
[4] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor variational inequality problems and quasi-120601-asymptoticallynonexpansive mappingsrdquo Journal of Global Optimization vol54 no 1 pp 101ndash116 2012
[5] H Zegeye and N Shahzad ldquoAn iteration to a common pointof solution of variational inequality and fixed point-problemsin Banach spacesrdquo Journal of Applied Mathematics vol 2012Article ID 504503 19 pages 2012
[6] F E Browder and W V Petryshyn ldquoConstruction of fixedpoints of nonlinear mappings in Hilbert spacerdquo Journal ofMathematical Analysis and Applications vol 20 pp 197ndash2281967
[7] L C Ceng A Petrusel and J C Yao ldquoComposite viscosityapproximation methods for equilibrium problem variationalinequality and common fixed pointsrdquo Journal of Nonlinear andConvex Analysis vol 15 no 2 pp 219ndash240 2014
[8] L C Ceng A Petrusel M M Wong and J C Yao ldquoHybridalgorithms for solving variational inequalities variational inclu-sions mixed equilibria and fixed point problemsrdquoAbstract andApplied Analysis vol 2014 Article ID 208717 22 pages 2014
[9] L Ceng A Petrusel and M Wong ldquoStrong convergencetheorem for a generalized equilibrium problem and a pseudo-contractive mapping in a Hilbert spacerdquo Taiwanese Journal ofMathematics vol 14 no 5 pp 1881ndash1901 2010
[10] D R Sahu V Colao and G Marino ldquoStrong convergencetheorems for approximating common fixed points of familiesof nonexpansive mappings and applicationsrdquo Journal of GlobalOptimization vol 56 no 4 pp 1631ndash1651 2013
[11] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[12] Y Yao Y Liou and N Shahzad ldquoConstruction of iterativemethods for variational inequality and fixed point problemsrdquoNumerical Functional Analysis and Optimization vol 33 no 10pp 1250ndash1267 2012
[13] Y Yao G Marino and L Muglia ldquoA modified Korpelevichrsquosmethod convergent to the minimum-norm solution of a varia-tional inequalityrdquoOptimization vol 63 no 4 pp 559ndash569 2014
[14] Y Yao and M Postolache ldquoIterative methods for pseudomono-tone variational inequalities and fixed-point problemsrdquo Journalof OptimizationTheory and Applications vol 155 no 1 pp 273ndash287 2012
[15] H Zegeye E U Ofoedu and N Shahzad ldquoConvergencetheorems for equilibrium problem variational inequality prob-lem and countably infinite relatively quasi-nonexpansive map-pingsrdquo Applied Mathematics and Computation vol 216 no 12pp 3439ndash3449 2010
[16] H Zegeye andN Shahzad ldquoAhybrid approximationmethod forequilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 619ndash6302010
[17] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor a solution of finite families of equilibrium and variationalinequality problemsrdquo Optimization vol 63 no 2 pp 207ndash2232014
[18] H Zegeye andN Shahzad ldquoExtragradientmethod for solutionsof variational inequality problems in Banach spacesrdquo Abstractand Applied Analysis vol 2013 Article ID 832548 8 pages 2013
10 Abstract and Applied Analysis
[19] W Takahashi andM Toyoda ldquoWeak convergence theorems fornonexpansive mappings and monotone mappingsrdquo Journal ofOptimization Theory and Applications vol 118 no 2 pp 417ndash428 2003
[20] H Iiduka and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and inverse-strongly monotonemappingsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 61 no 3 pp 341ndash350 2005
[21] H Zegeye and N Shahzad ldquoSolutions of variational inequalityproblems in the set of fixed points of pseudocontractive map-pingsrdquo Carpathian Journal of Mathematics vol 30 no 2 pp257ndash265 2014
[22] Y Yao Y C Liou and S M Kang ldquoAlgorithms construction forvariational inequalitiesrdquo Fixed Point Theory and Applicationsvol 2011 Article ID 794203 12 pages 2011
[23] Y Cai Y Tang and L Liu ldquoIterative algorithms for minimum-norm fixed point of non-expansive mapping in Hilbert spacerdquoFixed Point Theory and Applications vol 2012 article 49 2012
[24] K Aoyama H Iiduka andW Takahashi ldquoWeak convergence ofan iterative sequence for accretive operators in Banach spacesrdquoFixed PointTheory andApplications vol 2006 Article ID 3539013 pages 2006
[25] C E Chidume H Zegeye and E Prempeh ldquoStrong conver-gence theorems for a finite family of nonexpansive mappings inBanach spacesrdquoCommunications onAppliedNonlinearAnalysisvol 11 no 2 pp 25ndash32 2004
[26] Y I Alber ldquoMetric and generalized projection operators inBanach spaces properties and applicationsrdquo in Theory andApplications of Nonlinear Operators of Accretive and MonotoneType Lecture Notes in Pure and Applied Mathematics pp 15ndash50 Marcel Dekker New York NY USA 1996
[27] Q B Zhang and C Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008
[28] H Zegeye and N Shahzad ldquoConvergence of Mannrsquos typeiteration method for generalized asymptotically nonexpansivemappingsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4007ndash4014 2011
[29] P E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[30] H K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
with thanksDSR technical and financial supportThe authorsalso thank the referees for their valuable comments andsuggestions which improved the presentation of this paper
References
[1] D Kinderlehrer and G Stampaccia An Iteration to VariationalInequalities and Their Applications Academic Press New YorkNY USA 1990
[2] H Zegeye and N Shahzad ldquoA hybrid scheme for finite familiesof equilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no1 pp 263ndash272 2011
[3] H Zegeye and N Shahzad ldquoApproximating common solutionof variational inequality problems for two monotone mappingsin Banach spacesrdquo Optimization Letters vol 5 no 4 pp 691ndash704 2011
[4] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor variational inequality problems and quasi-120601-asymptoticallynonexpansive mappingsrdquo Journal of Global Optimization vol54 no 1 pp 101ndash116 2012
[5] H Zegeye and N Shahzad ldquoAn iteration to a common pointof solution of variational inequality and fixed point-problemsin Banach spacesrdquo Journal of Applied Mathematics vol 2012Article ID 504503 19 pages 2012
[6] F E Browder and W V Petryshyn ldquoConstruction of fixedpoints of nonlinear mappings in Hilbert spacerdquo Journal ofMathematical Analysis and Applications vol 20 pp 197ndash2281967
[7] L C Ceng A Petrusel and J C Yao ldquoComposite viscosityapproximation methods for equilibrium problem variationalinequality and common fixed pointsrdquo Journal of Nonlinear andConvex Analysis vol 15 no 2 pp 219ndash240 2014
[8] L C Ceng A Petrusel M M Wong and J C Yao ldquoHybridalgorithms for solving variational inequalities variational inclu-sions mixed equilibria and fixed point problemsrdquoAbstract andApplied Analysis vol 2014 Article ID 208717 22 pages 2014
[9] L Ceng A Petrusel and M Wong ldquoStrong convergencetheorem for a generalized equilibrium problem and a pseudo-contractive mapping in a Hilbert spacerdquo Taiwanese Journal ofMathematics vol 14 no 5 pp 1881ndash1901 2010
[10] D R Sahu V Colao and G Marino ldquoStrong convergencetheorems for approximating common fixed points of familiesof nonexpansive mappings and applicationsrdquo Journal of GlobalOptimization vol 56 no 4 pp 1631ndash1651 2013
[11] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[12] Y Yao Y Liou and N Shahzad ldquoConstruction of iterativemethods for variational inequality and fixed point problemsrdquoNumerical Functional Analysis and Optimization vol 33 no 10pp 1250ndash1267 2012
[13] Y Yao G Marino and L Muglia ldquoA modified Korpelevichrsquosmethod convergent to the minimum-norm solution of a varia-tional inequalityrdquoOptimization vol 63 no 4 pp 559ndash569 2014
[14] Y Yao and M Postolache ldquoIterative methods for pseudomono-tone variational inequalities and fixed-point problemsrdquo Journalof OptimizationTheory and Applications vol 155 no 1 pp 273ndash287 2012
[15] H Zegeye E U Ofoedu and N Shahzad ldquoConvergencetheorems for equilibrium problem variational inequality prob-lem and countably infinite relatively quasi-nonexpansive map-pingsrdquo Applied Mathematics and Computation vol 216 no 12pp 3439ndash3449 2010
[16] H Zegeye andN Shahzad ldquoAhybrid approximationmethod forequilibrium variational inequality and fixed point problemsrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 619ndash6302010
[17] H Zegeye and N Shahzad ldquoStrong convergence theoremsfor a solution of finite families of equilibrium and variationalinequality problemsrdquo Optimization vol 63 no 2 pp 207ndash2232014
[18] H Zegeye andN Shahzad ldquoExtragradientmethod for solutionsof variational inequality problems in Banach spacesrdquo Abstractand Applied Analysis vol 2013 Article ID 832548 8 pages 2013
10 Abstract and Applied Analysis
[19] W Takahashi andM Toyoda ldquoWeak convergence theorems fornonexpansive mappings and monotone mappingsrdquo Journal ofOptimization Theory and Applications vol 118 no 2 pp 417ndash428 2003
[20] H Iiduka and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and inverse-strongly monotonemappingsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 61 no 3 pp 341ndash350 2005
[21] H Zegeye and N Shahzad ldquoSolutions of variational inequalityproblems in the set of fixed points of pseudocontractive map-pingsrdquo Carpathian Journal of Mathematics vol 30 no 2 pp257ndash265 2014
[22] Y Yao Y C Liou and S M Kang ldquoAlgorithms construction forvariational inequalitiesrdquo Fixed Point Theory and Applicationsvol 2011 Article ID 794203 12 pages 2011
[23] Y Cai Y Tang and L Liu ldquoIterative algorithms for minimum-norm fixed point of non-expansive mapping in Hilbert spacerdquoFixed Point Theory and Applications vol 2012 article 49 2012
[24] K Aoyama H Iiduka andW Takahashi ldquoWeak convergence ofan iterative sequence for accretive operators in Banach spacesrdquoFixed PointTheory andApplications vol 2006 Article ID 3539013 pages 2006
[25] C E Chidume H Zegeye and E Prempeh ldquoStrong conver-gence theorems for a finite family of nonexpansive mappings inBanach spacesrdquoCommunications onAppliedNonlinearAnalysisvol 11 no 2 pp 25ndash32 2004
[26] Y I Alber ldquoMetric and generalized projection operators inBanach spaces properties and applicationsrdquo in Theory andApplications of Nonlinear Operators of Accretive and MonotoneType Lecture Notes in Pure and Applied Mathematics pp 15ndash50 Marcel Dekker New York NY USA 1996
[27] Q B Zhang and C Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008
[28] H Zegeye and N Shahzad ldquoConvergence of Mannrsquos typeiteration method for generalized asymptotically nonexpansivemappingsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4007ndash4014 2011
[29] P E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[30] H K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
[19] W Takahashi andM Toyoda ldquoWeak convergence theorems fornonexpansive mappings and monotone mappingsrdquo Journal ofOptimization Theory and Applications vol 118 no 2 pp 417ndash428 2003
[20] H Iiduka and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and inverse-strongly monotonemappingsrdquoNonlinear AnalysisTheory Methods amp Applicationsvol 61 no 3 pp 341ndash350 2005
[21] H Zegeye and N Shahzad ldquoSolutions of variational inequalityproblems in the set of fixed points of pseudocontractive map-pingsrdquo Carpathian Journal of Mathematics vol 30 no 2 pp257ndash265 2014
[22] Y Yao Y C Liou and S M Kang ldquoAlgorithms construction forvariational inequalitiesrdquo Fixed Point Theory and Applicationsvol 2011 Article ID 794203 12 pages 2011
[23] Y Cai Y Tang and L Liu ldquoIterative algorithms for minimum-norm fixed point of non-expansive mapping in Hilbert spacerdquoFixed Point Theory and Applications vol 2012 article 49 2012
[24] K Aoyama H Iiduka andW Takahashi ldquoWeak convergence ofan iterative sequence for accretive operators in Banach spacesrdquoFixed PointTheory andApplications vol 2006 Article ID 3539013 pages 2006
[25] C E Chidume H Zegeye and E Prempeh ldquoStrong conver-gence theorems for a finite family of nonexpansive mappings inBanach spacesrdquoCommunications onAppliedNonlinearAnalysisvol 11 no 2 pp 25ndash32 2004
[26] Y I Alber ldquoMetric and generalized projection operators inBanach spaces properties and applicationsrdquo in Theory andApplications of Nonlinear Operators of Accretive and MonotoneType Lecture Notes in Pure and Applied Mathematics pp 15ndash50 Marcel Dekker New York NY USA 1996
[27] Q B Zhang and C Z Cheng ldquoStrong convergence theorem fora family of Lipschitz pseudocontractive mappings in a HilbertspacerdquoMathematical and Computer Modelling vol 48 no 3-4pp 480ndash485 2008
[28] H Zegeye and N Shahzad ldquoConvergence of Mannrsquos typeiteration method for generalized asymptotically nonexpansivemappingsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4007ndash4014 2011
[29] P E Mainge ldquoStrong convergence of projected subgradientmethods for nonsmooth and nonstrictly convexminimizationrdquoSet-Valued Analysis vol 16 no 7-8 pp 899ndash912 2008
[30] H K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
