A Halpern-Type Iteration Method for Bregman Nonspreading...

Research ArticleA Halpern-Type Iteration Method for Bregman NonspreadingMapping and Monotone Operators in Reflexive Banach Spaces

F U Ogbuisi 12 L O Jolaoso1 and F O Isiogugu 123

1School of Mathematics Statistics and Computer Science University of Kwazulu-Natal Durban South Africa2Department of Mathematics University of Nigeria Nsukka Nigeria3DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) Johannesburg South Africa

Correspondence should be addressed to F U Ogbuisi ferdinardogbuisiunnedung

Received 14 May 2019 Accepted 22 July 2019 Published 2 September 2019

Guest Editor Jianhua Chen

Copyright copy 2019 F U Ogbuisi et al (is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this paper we introduce an iterative method for approximating a common solution of monotone inclusion problem and fixedpoint of Bregman nonspreading mappings in a reflexive Banach space Using the Bregman distance function we study thecomposition of the resolvent of a maximal monotone operator and the antiresolvent of a Bregman inverse strongly monotoneoperator and introduce a Halpern-type iteration for approximating a common zero of a maximal monotone operator and aBregman inverse strongly monotone operator which is also a fixed point of a Bregman nonspreading mapping We further stateand prove a strong convergence result using the iterative algorithm introduced (is result extends many works on finding acommon solution of the monotone inclusion problem and fixed-point problem for nonlinear mappings in a real Hilbert space to areflexive Banach space

1 Introduction

Let E be a real reflexive Banach space with a norm middot andElowast be the dual space of E We denote the value of xlowast isin Elowast atx isin E by langxlowast xrang A mapping A is called a monotonemapping if for any x y isin domA we have

μ isin Ax

] isin Ay⟹ langμ minus ] x minus yrangge 0(1)

A monotone mapping A E⟶ 2Elowast is said to bemaximal monotone if its graph G(A) ≔ (x u) isin

E times Elowast u isin Ax is not properly contained in the graph ofany other monotone operator A basic problem that arises inseveral branches of applied mathematics [1ndash7] is to findx isin E such that

0 isin Ax (2)

One of the methods for solving this problem is the well-known proximal point algorithm (PPA) introduced by

Martinet [8] Let H be a Hilbert space and let I denote theidentity operator on H (e PPA generates for any startingpoint x0 x isin H a sequence xn1113864 1113865 in H by

xn+1 I + λnA( 1113857minus 1

xn n 1 2 (3)

where A is a maximal monotone mapping and λn1113864 1113865 is a givensequence of positive real numbers It has been observed that(3) is equivalent to

0 isin Axn+1 +1λn

xn+1 minus xn( 1113857 n 1 2 (4)

(is algorithm was further developed by Rockafellar [5]who proved that the sequence generated by (3) convergesweakly to an element of Aminus 1(0) when Aminus 1(0) is nonemptyand lim infn⟶infinλn gt 0 Furthermore Rockafellar [5] asked ifthe sequence generated by (3) converges strongly in general(is question was answered in the negative by Guler [9] whopresented an example of a subdifferential for which thesequence generated by (3) converges weakly but notstrongly Also the works of Bruck and Reich [10] and

HindawiJournal of MathematicsVolume 2019 Article ID 8059135 11 pageshttpsdoiorg10115520198059135

Bauschke et al [11] are very important in this direction Formore recent results on PPA see [12ndash14]

(e problem of finding the zeros of the sum of twomonotone mappings A and B is to find a point xlowast isin E suchthat

0 isin (A + B)xlowast (5)

has recently received attention due to its significant im-portance in many physical problems One classical methodfor solving problem (5) is the forward-backward splittingmethod [15] which is as follows for x1 isin E

xn+1 (I + rB)minus 1

xn minus rAxn( 1113857 nge 1 (6)

where rgt 0 (is method combines the proximal point al-gorithm and the gradient projection algorithm In [16]Lions and Mercier introduced the following splitting iter-ative methods in a real Hilbert space H

xn+1 2JAr minus I1113872 1113873 2J

Br minus I1113872 1113873xn nge 1

xn+1 JAr 2J

Br minus I1113872 1113873xn + I minus J

Br1113872 1113873xn nge 1

(7)

where JTr (I + rT)minus 1 (e first one is called Peacemanndash

Rachford algorithm and the second one is calledDouglasndashRachford algorithm [15] It was noted that bothalgorithms converge weakly in general [16 17]

Many authors have studied the approximation of zero ofthe sum of two monotone operators (in Hilbert space) andaccretive operators (in Banach spaces) but the approxi-mation of the sum of two monotone operators in moregeneral Banach spaces other the Hilbert spaces has notenjoyed such popularity

(roughout this paper f E⟶ (minus infin +infin] is a properlower semicontinuous and convex function and the Fenchelconjugate of f is the function flowast Elowast ⟶ (minus infin +infin] de-fined by

flowast

xlowast

( 1113857 sup langxlowast xrang minus f(x) x isin E1113864 1113865 (8)

We denote by domf the domain of f that is the setx isin E f(x)lt +infin1113864 1113865 For any x isin intdomf and y isin E theright-hand derivative of f at x in the direction of t is defined by

fo(x y) ≔ lim

t⟶0+

f(x + ty) minus f(x)

t (9)

(e function f is said to be Gateaux differentiable at x ifthe limit as t⟶ 0+ in (9) exists for any y In this casefo(x y) coincides with nablaf(x) the value of the gradient nablafat x (e function f is said to be Gateaux differentiable if it isGateaux differentiable for any x isin intdomf(e function f isFrechet differentiable at x if the limit is attained with y 1and uniformly Frechet differentiable on a subset C of E if thelimit is attained uniformly for x isin C and y 1

(e function f is said to be Legendre if it satisfies thefollowing two conditions

(L1) intdomfneempty and the subdifferential zf is single-valued in its domain(L2) tdomflowast neempty and zflowast is single-valued on itsdomain

(e class of Legendre functions in infinite dimensionalBanach spaces was first introduced and studied by Bauschkeet al in [18] (eir definition is equivalent to conditions (L1)and (L2) because the space E is assumed to be reflexive (see[18] (eorems 54 and 56 p 634) It is well known that inreflexive Banach spaces nablaf (nablaflowast)minus 1 (see [19] p 83)When this fact is combined with conditions (L1) and (L2)we obtain

rannablaf domnablaflowast int(domf)lowast

rannablaflowast domnablaf int(domf)(10)

It also follows that f is Legendre if and only if flowast isLegendre (see [18] Corollary 55 p 634) and that thefunctions f and flowast are Gateaux differentiable and strictlyconvex in the interior of their respective domains

Several interesting examples of the Legendre functionsare presented in [18 20 21] A very important example ofLegendre function is the function 1s middot s with s isin (1infin)where the Banach space E is smooth and strictly convex andin particular a Hilbert space (roughout this article weassume that the convex function f E⟶ (minus infin +infin] isLegendre

Definition 1 Let f E⟶ (minus infin +infin] be a convex andGateaux differentiable function the function Df domftimes

intdomf⟶ [0infin) which is defined by

Df(y x) ≔ f(y) minus f(x) minus langnablaf(x) y minus xrang (11)

is called the Bregman distance [22ndash24]

(e Bregman distance does not satisfy the well-knownmetric properties but it does have the following importantproperty which is called the three-point identity for anyx isin domf and y z isin intdomf

Df(x y) + Df(y z) minus Df(x z) langnablaf(z) minus nablaf(y) x minus yrang

(12)

Let C be a nonempty subset of a Banach space E andT C⟶ C be amapping then a point x is called fixed pointof T if Tx x (e set of fixed point of T is denoted by F(T)Also a point xlowast isin C is said to be an asymptotic fixed point ofT if C contains a sequence xn1113864 1113865

infinn1 which converges weakly

to xlowast and limn⟶infinxn minus Txn 0 [25] (e set of asymp-totic fixed points of T is denoted by 1113954F(T)

Definition 2 [26 27] Let C be a nonempty closed andconvex subset of E A mapping T C⟶ int(domf) iscalled

(i) Bregman firmly nonexpansive (BFNE for short) if

langnablaf(Tx) minus nablaf(Ty) Tx minus Tyrang

le langnablaf(x) minus nablaf(y) Tx minus Tyrang forallx y isin C(13)

(ii) Bregman strongly nonexpansive (BSNE) with re-spect to a nonempty 1113954F(T) if

2 Journal of Mathematics

Df(p Tx)leDf(p x) (14)

for all p isin 1113954F(T) and x isin C and if whenever xn1113864 1113865infinn1 sub C

is bounded p isin 1113954F(T) and

limn⟶infin

Df p xn( 1113857 minus Df p Txn( 11138571113872 1113873 0 (15)

it follows that

limn⟶infin

Df Txn xn( 1113857 0 (16)

(iii) Bregman quasi-nonexpansive if F(T)neempty and

Df(p Tx)leDf(p x) forallx isin C andp isin F(T) (17)

(iv) Bregman skew quasi-nonexpansive if F(T)neempty and

Df(Tx p)leDf(x p) forallx isin C andp isin F(T) (18)

(v) Bregman nonspreading if

Df(Tx Ty) + Df(Ty Tx)leDf(Tx y)

+ Df(Ty x) forallx y isin C(19)

It is easy to see that every Bregman nonspreadingmapping T with F(T)neempty is Bregman quasi-nonexpansiveAlso Bregman nonspreading mappings include in partic-ular the class of nonspreading functions studied by Taka-hashi et al in [28 29] For more information on Bregmannonspreading mappings see [30]

In a real Hilbert space H the nonlinear mapping T

C⟶ C is said to be

(i) Nonexpansive if

Tx minus Tyle x minus y forallx y isin C (20)

(ii) Quasi-nonexpansive if F(T)neempty and

Tx minus ple x minus p forallx isin C and p isin F(T) (21)

(iii) Nonspreading if

2Tx minus Ty2 le Tx minus y

2+ Ty minus x

2 forallx y isin C

(22)

Clearly every nonspreading mapping Twith F(T)neempty isalso quasi-nonexpansive mapping (e class of non-spreading mappings is very important due to its relationwith maximal monotone operators (see eg [28])

Let B E⟶ 2Elowast be a maximal monotone operator (eresolvent of B Resf

B E⟶ 2E is defined by (see [26])

Resf

B ≔ (nablaf + B)minus 1 ∘ nablaf (23)

It is known that ResfB is a BFNE operator single-valued

and F(ResfB) Bminus 1(0lowast) (see [26]) If f E⟶ R is a

Legendre function which is bounded uniformly Frechetdifferentiable on bounded subsets of E then Resf

B is BSNEand 1113954F(Resf

B) F(Resf

B) (see [31])Assume that the Legendre function f satisfies the fol-

lowing range condition

ran(nablaf minus A)sube rannablaf (24)

An operator A E⟶ 2Elowast is called Bregman inversestrongly monotone (BISM) if (domA) (domf) and for anyx y isin intdomf and each u isin Ax and v isin Ay we have

langu minus vnablaflowast(nablaf(x) minus u) minus nablaflowast(nablaf(y) minus v)rang ge 0 (25)

(e class of BISM mappings is a generalization of theclass of firmly nonexpansive mappings in Hilbert spacesIndeed if f 12 middot 2 then nablaf nablaflowast I where I is theidentity operator and (25) becomes

langu minus v x minus u minus (y minus v)rangge 0 (26)

which meansu minus v

2 le langx minus y u minus vrang (27)

Observe that

domAf

(domA)cap (intdomf)

ranAf sub intdomf

(28)

In other words T is a (single-valued) firmly non-expansive operator

For any operator A E⟶ 2Elowast the antiresolvent op-erator Af E⟶ 2E of A is defined by

Af ≔ nablaflowast ∘ (nablaf minus A) (29)

It is known that the operator A is BISM if and only if theantiresolvent Af is a single-valued BFNE (see [32] Lemma32(c) and (d) p 2109) and F(Af) Aminus 1(0lowast) For examplesand further information on BISM see [32]

Since the monotone inclusion problems have very closeconnections with both the fixed-point problems and theequilibrium problems finding the common solutions ofthese problems has drawn many peoplersquos attention and hasbecome one of the hot topics in the related fields in the pastfew years [33 34] Furthermore interest in finding thecommon solution of these problems has also grown be-cause of the possible application of these problems tomathematical models whose constraints can be present asfixed points of mappings andor monotone inclusionproblems andor equilibrium problems Such a problemoccurs in particular in the practical problems as signalprocessing network resource allocation and image re-covery (see [35 36])

Journal of Mathematics 3

In this paper we introduce an iterative method forapproximating a common solution of monotone inclusionproblem and fixed point of Bregman nonspreading mappingin a reflexive Banach space and prove a strong convergenceof the sequence generated by our iterative algorithm (isresult extends many works on finding common solution ofmonotone inclusion problem and fixed problem of non-linear mapping in a real Hilbert space to a reflexive Banachspace

2 Preliminaries

(e Bregman projection [22] of x isin int(domf) onto thenonempty closed and convex subset C sub int (domf) isdefined as the necessarily unique vector ProjfC(x) isin C

satisfying

Df ProjfC(x) x1113872 1113873 inf Df(y x) y isin C1113966 1113967 (30)

It is known from [37] that z ProjfC(x) if and only if

langnablaf(x) minus nablaf(z) y minus zrangle 0 for ally isin C (31)

We also have

Df yProjfC(x)1113872 1113873 + Df ProjfC(x) x1113872 1113873

le Df(y x) for allx isin E y isin C(32)

Note that if E is a Hilbert space and f(x) 12x2 thenthe Bregman projection of x onto C ieargmin y minus x y isin C1113864 1113865 is the metric projection PC

Lemma 1 [37] Let f be totally convex on int (domf) Let C bea nonempty closed and convex subset of int (domf) andx isin int(domf) if z isin C then the following conditions areequivalent

(i) z Projf

C(x)

(ii) langnablaf(x) minus nablaf(z) z minus yrangge 0 for all y isin C

(iii) Df(y z) + Df(z x)leDf(y x) for all y isin C

Let f E⟶ Rcup +infin be a convex and Gateaux dif-ferentiable function (e function f is said to be totallyconvex at x isin intdomf if its modulus of totally convexity atx that is the function vf int (domf) times [0 +infin) defined by

vf(x t) ≔ inf Df(y x) y isin domf y minus x t1113966 1113967 (33)

is positive for any tgt 0 (e function f is said to be totallyconvex when it is totally convex at every pointx isin int(domf) In addition the function f is said to betotally convex on bounded set if vf(B t) is positive for anynonempty bounded subset B where the modulus of totalconvexity of the function f on the set B is the function vf

int(domf) times [0 +infin) defined by

vf(B t) ≔ inf vf(x t) x isin Bcap domf1113966 1113967 (34)

For further details and examples on totally convexfunctions see [37ndash39]

Let f E⟶ R be a convex Legendre and Gateauxdifferentiable function and let the functionVf E times Elowast ⟶ [0infin) associated with f (see [23 40]) bedefined by

Vf x xlowast

( 1113857 f(x) minus langxlowast xrang + flowast

xlowast

( 1113857 forallx isin E xlowast isin Elowast

(35)

(en Vf is nonnegative and Vf(x xlowast) Df(x

nablaflowast(xlowast)) forallx isin E xlowast isin Elowast Furthermore by the sub-differential inequality we have (see [41])

Vf x xlowast

( 1113857 +langylowastnablaflowast xlowast

( 1113857 minus xrang

leVf x xlowast

+ ylowast

( 1113857 forallx isin E xlowast ylowast isin Elowast

(36)

In addition if f E⟶ (minus infin +infin] is a proper lowersemicontinuous function then flowast Elowast ⟶ (minus infin +infin] is aproper weaklowast lower semicontinuous and convex function(see [42]) Hence Vf is convex in the second variable (usfor all z isin E

Df znablaflowast1113944

N

i1tinablaf xi( 1113857⎛⎝ ⎞⎠le 1113944

N

i1tiDf z xi( 1113857 (37)

where xi1113864 1113865N

i1 sub E and ti1113864 1113865 sub (0 1) with 1113936Ni1ti 1

Lemma 2 (see [43]) Let rgt 0 be a constant and let f

E⟶ R be a continuous uniformly convex function onbounded subsets of E lten

f 1113944infin

k0αkxk

⎛⎝ ⎞⎠le 1113944infin

k0αkf xk( 1113857 minus αiαjρr xi minus xj

1113874 1113875 (38)

for all i j isin Ncup 0 xk isin Br αk isin (0 1) and k isin Ncup 0 with1113936infink0αk 1 where ρr is the gauge of uniform convexity of f

Recall that a function f is said to be sequentially consisted(see [37]) if for any two sequences xn1113864 1113865 and yn1113864 1113865 in E suchthat the first one is bounded

limn⟶infin

Df yn xn( 1113857 0⟹ limn⟶infin

yn minus xn

0 (39)

(e following lemma follows from [44]

Lemma 3 If domf contains at least two points then thefunction f is totally convex on bounded sets if and only if thefunction f is sequentially consistent

Lemma 4 (see [45]) Let f E⟶ (minus infin +infin] be a Leg-endre function and let A E⟶ 2Elowast be a BISM operator suchthat Aminus 1(0lowast)neempty lten the following statements hold

(i) Aminus 1(0lowast) F(Af)

(ii) For any w isin Aminus 1(0lowast) and x isin domAf we have

Df w Af

x1113872 1113873 + Df Af

x x1113872 1113873 le Df(w x) (40)

Remark 1 If the Legendre function f is uniformly Frechetdifferentiable and bounded on bounded subsets of E then


the antiresolvent Af is a single-valued BSNE operator whichsatisfies F(Af) 1113954F(Af) (cf [31])

Lemma 5 (see [46]) If f E⟶ R is uniformly Frechetdifferentiable and bounded on bounded subsets of E then nablafis uniformly continuous on bounded subsets of E from thestrong topology of E to the strong topology of Elowast

Lemma 6 (see [44]) Let f E⟶ R be a Gateaux dif-ferentiable and totally convex function If x1 isin E and thesequence Df(xn x1)1113966 1113967 is bounded then the sequence xn1113864 1113865 isalso bounded

Lemma 7 (see [45]) Assume that f E⟶ R is a Legendrefunction which is uniformly Frechet differentiable andbounded on bounded subset of E Let C be a nonempty closedand convex subset of E Let Ti 1le ileN1113864 1113865 be BSNE operatorswhich satisfy 1113954F(Ti) F(Ti) for each 1le ileN and letT ≔ wnTNminus 1 T1 If

cap F Ti( 1113857 1le ileN1113864 1113865 (41)

and F(T) are nonempty then T is also BSNE withF(T) 1113954F(T)

Lemma 8 (Demiclosedness principle [30]) Let C be anonempty subset of a reflexive Banach space Let g E⟶ R

be a strict convex Gateaux differentiable and locally boundedfunction Let T C⟶ E be a Bregman nonspreadingmapping If xnp in C and lim

n⟶infinTxn minus xn 0 then

p isin F(T)

Lemma 9 (see [47]) Assume an1113864 1113865 is a sequence of non-negative real numbers satisfying

an+1 le 1 minus tn( 1113857an + tnδnforallnge 0 (42)

where tn1113864 1113865 is a sequence in (0 1) and δn1113864 1113865 is a sequence in R

such that

(i) 1113936infinnotn infin

(ii) lim supn⟶infin δn le 0

lten limn⟶infin an 0

Lemma 10 [48] Let an1113864 1113865 be a sequence of real numberssuch that there exists a nondecreasing subsequence ni1113864 1113865 ofn that is ani

le ani+1 for all i isin N lten there exists anondecreasing sequence mk1113864 1113865 sub N such that mk⟶infin andthe following properties are satisfied for all (sufficientlylarge number k isin N) amk

le amk+1 and ak le amk+1mk max jle k aj le aj+11113966 1113967

3 Main Results

Theorem 1 Let C be a nonempty closed and convex subsetof a real reflexive Banach space E and f E⟶ R a Legendrefunction which is bounded uniformly Frechet differentiableand totally convex on bounded subsets of E Let A E⟶ 2Elowast

be a Bregman inverse strongly monotone operator

B E⟶ 2Elowast be a maximal monotone operator andT C⟶ C be a Bregman nonspreading mapping SupposeΓ ≔ F(ResfA ∘Af)capF(T)neempty Let cn1113864 1113865 sub (0 1) and αn1113864 1113865 βnand δn1113864 1113865 be sequences in (0 1) such that αn + βn + δn 1Given u isin E and x1 isin C arbitrarily let xn1113864 1113865 and yn1113864 1113865 besequences in E generated by

yn nablaflowast cnnablaf xn( 1113857 + 1 minus cn( 1113857nablaf Txn( 1113857( 1113857

xn+1 ProjfC nablaflowast αnnablaf(u) + βnnablaf yn( 1113857 + δnnablaf Resf

B ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎨

⎩

(43)

Suppose the following conditions are satisfied

(i) limn⟶infin

αn 0 and 1113936infinn1αn infin

(ii) (1 minus αn)alt δn αn le blt 1 a isin (0 12)

(iii) 0le clt lim infn⟶infin cn le lim supn⟶infin cn lt 1

Then xn1113864 1113865 converges strongly to ProjfΓu where ProjfΓ isthe Bregman projection of E onto Γ

Proof First we observe that F(ResfB ∘Af) (A + B)minus 10 and

F(ResfB ∘Af) F(Resf

B)capF(Af) (us since ResfB and Af

are BSNE operators and F(Resf

B)capF(Af) (A + B)minus 10neempty

it then follows from Lemma 7 that Resf

B ∘Af is BSNE andF(Resf

B ∘Af) 1113954F(Resf

B ∘Af)

We next show that xn1113864 1113865 and yn1113864 1113865 are boundedLet p isin Γ then from (43) we have

Df p yn( 1113857 Df pnablaflowast cnnablaf xn( 1113857( 1113857 + 1 minus cn( 1113857nablaf Txn( 1113857( 1113857

le cnDf p xn( 1113857 + 1 minus cn( 1113857Df p Txn( 1113857

le cnDf p xn( 1113857 + 1 minus cn( 1113857Df p xn( 1113857

Df p xn( 1113857

(44)

Also

Df p xn+1( 1113857leDf1113874pnablaflowast1113876αnnablaf(u) + βnnablaf yn( 1113857

+ δnnablaf Resf

B ∘Af

yn( 11138571113872 111387311138771113875

le αnDf(p u) + βnDf p yn( 1113857

+ δnDf pResfB ∘A

fyn1113872 1113873

le αnDf(p u) + βnDf p yn( 1113857 + δnDf p yn( 1113857

αnDf(p u) + 1 minus αn( 1113857Df p yn( 1113857

αnDf(p u) + 1 minus αn( 1113857Df p xn( 1113857

lemax Df(p u) Df p xn( 11138571113966 1113967

⋮

lemax Df(p u) Df p x1( 11138571113966 1113967

(45)

Hence Df(p xn)1113966 1113967 is bounded (erefore by Lemma 6xn1113864 1113865 is also bounded and consequently yn1113864 1113865 is also bounded


We now show that xn converges strongly tox ProjfΓ (u) To do this we first show that if there exists asubsequence xni

1113966 1113967 of xn1113864 1113865 such that xni q isin C then q isin Γ

Let s sup nablaf(xn) nablaf(Txn)1113864 1113865 and ρlowasts Elowast ⟶ R

be the gauge of uniform convexity of the conjugate functionflowast From Lemma 2 and (9) we have

Df p yn( 1113857leDf pnablaflowast cnnablaf xn( 1113857( 1113857 + 1 minus cn( 1113857nablaf Txn( 1113857( 1113857

Vf p cnnablaf xn( 1113857 + 1 minus cn( 1113857nablaf Txn( 1113857( 1113857

f(p) minus langp cnnablaf xn( 1113857 + 1 minus cn( 1113857nablaf Txn( 1113857rang+ flowast

cnnablaf xn( 1113857 + 1 minus cn( 1113857nablaf Txn( 1113857( 1113857

le cnf(p) minus cnlangpnablaf xn( 1113857rang + cnflowast nablaf xn( 1113857( 1113857

+ 1 minus cn( 1113857f(p) minus 1 minus cn( 1113857langpnablaf Txn( 1113857rang+ 1 minus cn( 1113857f

lowast nablaf Txn( 1113857( 1113857 minus cn 1 minus cn( 1113857ρlowastsmiddot nablaf xn( 1113857 minus nablaf Txn( 1113857

1113872 1113873

cnDf p xn( 1113857 + 1 minus cn( 1113857Df p Txn( 1113857

minus cn 1 minus cn( 1113857ρlowasts nablaf xn( 1113857 minus nablaf Txn( 1113857

1113872 1113873

leDf p xn( 1113857 minus cn 1 minus cn( 1113857ρlowastsmiddot nablaf xn( 1113857 minus nablaf Txn( 1113857

1113872 1113873

(46)(us from (45) we have

Df p xn+1( 1113857le αnDf(p u) + 1 minus αn( 1113857

middot Df p xn( 1113857 minus cn 1 minus cn( 1113857ρlowasts nablaf xn( 1113857 minus nablaf Txn( 1113857

1113872 11138731113960 1113961

(47)

We consider the following two cases for the rest of theproof

Case A Suppose Df(p xn)1113966 1113967 is monotonicallynonincreasing (en Df(p xn)1113966 1113967 converges andDf(p xn) minus Df(p xn+1)⟶ 0 as n⟶infin (us from(47) we have

1 minus αn( 1113857 1 minus cn( 1113857cnρlowasts nablaf xn( 1113857 minus nablaf Txn( 1113857

1113872 1113873

le αn Df(p u) minus Df p xn( 11138571113872 1113873 + Df p xn( 1113857 minus Df p xn+1( 1113857

(48)

Since αn⟶ 0 n⟶infin then we have

limn⟶infin

cn 1 minus cn( 1113857ρlowasts nablaf xn( 1113857 minus nablaf Txn( 1113857

1113872 1113873 0 (49)

and hence by condition (iii) and the property of ρlowasts we have

limn⟶infinnablaf xn( 1113857 minus nablaf Txn( 1113857

0 (50)

Since nablaflowast is uniformly norm-to-norm continuous onbounded subset of Elowast we have

limn⟶infin

xn minus Txn

0 (51)

Again

nablaf xn( 1113857 minus nablaf yn( 1113857

nablaf xn( 1113857 minus cnnablaf xn( 1113857 + 1 minus cn( 1113857nablaf Txn( 1113857( 1113857

1 minus cn( 1113857 nablaf xn( 1113857 minus nablaf Txn( 1113857

⟶ 0 n⟶infin

(52)

Since nablaflowast is uniformly norm-to-norm continuous onbounded subsets of Elowast we have that

limn⟶infin

xn minus yn

0 (53)

Now let wn nablaflowast(βn1 minus αnnablaf(yn) + δn1 minus αnnablaf(Resf

B ∘Afyn)) then

Df p wn( 1113857 Df pnablaflowastβn

1 minus αn

nablaf yn( 1113857 +δn

1 minus αn

nablaf ResfB ∘A

fyn1113872 11138731113890 11138911113888 1113889

leβn

1 minus αn

Df p yn( 1113857 +δn

1 minus αn

Df pResf

B ∘Af

yn1113872 1113873

leβn + δn

1 minus αn

Df p yn( 1113857

Df p yn( 1113857

(54)

(erefore we have

0leDf p xn( 1113857 minus Df p wn( 1113857

Df p xn( 1113857 minus Df p xn+1( 1113857 + Df p xn+1( 1113857 minus Df p wn( 1113857

leDf p xn( 1113857 minus Df p xn+1( 1113857 + αnDf(p u)

+ 1 minus αn( 1113857Df p wn( 1113857 minus Df p wn( 1113857

Df p xn( 1113857 minus Df p xn+1( 1113857

+ αn Df(p u) minus Df p wn( 11138571113960 1113961⟶ 0 as n⟶infin

(55)

More so

Df p wn( 1113857leβn

1 minus αn

Df p yn( 1113857 +δn

1 minus αn

Df pResfB ∘A

fyn1113872 1113873

Df p yn( 1113857 minus 1 minusβn

1 minus αn

1113888 1113889Df p yn( 1113857

+δn

1 minus αn

Df pResfB ∘A

fyn1113872 1113873

leDf p xn( 1113857 +δn

1 minus αn

1113890Df pResfB ∘A

fyn1113872 1113873

minus Df p yn( 11138571113891

(56)

Since (1 minus αn)alt δn and αn le blt 1 we have


a Df p yn( 1113857 minus Df pResfB ∘A

fyn1113872 11138731113872 1113873 lt

δn

1 minus αn

1113890Df p yn( 1113857

minus Df pResf

B ∘Af

yn1113872 11138731113891

leDf p xn( 1113857

minus Df p wn( 1113857⟶ 0

as n⟶infin

(57)

(us

Df p yn( 1113857 minus Df pResfB ∘A

fyn1113872 1113873⟶ 0 as n⟶infin

(58)

(erefore since Resf

B ∘Af is BSNE we have thatlimn⟶infinDf(ynResf

B ∘Afyn) 0 which implies that

limn⟶infin

yn minus ResfB ∘A

fyn

0 (59)

Setting un nablaflowast[αnnablaf(u) + βnnablaf(yn) + δnnablaf(ResfB ∘

Afyn)] for each nge 1 we have

Df yn un( 1113857 Df1113874ynnablaflowast1113876αnnablaf(u) + βnnablaf yn( 1113857

+ δnnablaf ResfB ∘A

fyn1113872 111387311138771113875

le αnDf yn u( 1113857 + βnDf yn yn( 1113857 + δnDf

middot ynResfB ∘A

fyn1113872 1113873⟶ 0

(60)

(us

limn⟶infin

yn minus un

0 (61)

(erefore from (47) we have

un minus xn

le un minus yn

+ yn minus xn

⟶ 0 n⟶infin

(62)

Moreover since xn+1 ProjfCun then

Df p xn+1( 1113857 + Df xn+1 un( 1113857leDf p un( 1113857 (63)

and therefore we have that

Df xn+1 un( 1113857leDf p un( 1113857 minus Df p xn+1( 1113857

le αnDf(p u) + βnDf p yn( 1113857 + δnDf pResfB ∘A

fyn1113872 1113873

minus Df p xn+1( 1113857

αnDf(p u) + 1 minus αn( 1113857Df p yn( 1113857 minus Df p xn+1( 1113857

le αn Df(p u) minus Df p xn( 11138571113872 1113873 + Df p xn( 1113857

minus Df p xn+1( 1113857⟶ 0 n⟶infin

(64)

which implies

xn+1 minus un

⟶ 0 n⟶infin (65)

Hence

xn+1 minus xn

le xn+1 minus un

+ un minus xn

⟶ 0 n⟶infin

(66)

Since xn1113864 1113865 is bounded there exists a subsequence xni1113966 1113967 of

xn1113864 1113865 such that xni1113966 1113967 converges weakly to q isin C as n⟶infin

Since limn⟶infin

xniminus Txni

0 it follows from Lemma 8

that q isin F(T) Also since xniminus yni

⟶ 0 it implies thatyni

also converges weakly to q isin E (ereforefrom (59) we have that q isin F(Resf

B ∘Af) and henceq isin Γ F(T)capF(Resf

B ∘Af)

Next we show that xn1113864 1113865 converges strongly tox ProjfΓ (u)

Now from (43) we have

Df x xn+1( 1113857leDf1113874xnablaflowast1113876αnnablaf(u) + βnnablafyn

+ δnnablafResfB ∘A

fyn11138771113875

Vf x αnnablaf(u) + βnnablaf yn( 1113857 + δnnablaf ResfB ∘A

f1113872 1113873yn1113872 1113873

leVf1113874x αnnablaf(u) + βnnablaf yn( 1113857 + δnnablaf ResfB ∘A

f1113872 1113873yn

minus αn(nablaf(u) minus nablaf(x))1113875

minus 1113866 minus αn(nablaf(u) minus nablaf(x))nablaflowast

middot αnnablaf(u) + βnnablaf yn( 1113857 + δnnablaf ResfB ∘A

f1113872 1113873yn1113960 1113961 minus x1113867

Vf x αnnablaf(x) + βnnablaf yn( 1113857 + δnnablaf ResfB ∘A

f1113872 1113873yn1113872 1113873

+ αnlangnablaf(u) minus nablaf(w) un minus xrang

Df1113874xnablaflowast1113876αnnablaf(x) + βnnablaf yn( 1113857


f1113872 1113873yn11138771113875 + αn1113866nablaf(u)

minus nablaf(x) un minus x1113867

αnDf(x x) + βnDf x yn( 1113857 + δnDf xResf

B ∘Af

yn1113872 1113873

+ αnlangnablaf(u) minus nablaf(x) un minus xrang

le βnDf x yn( 1113857 + δnDf x yn( 1113857


1 minus αn( 1113857Df x yn( 1113857 + αnlangnablaf(u) minus nablaf(x) un minus xrang

le 1 minus αn( 1113857Df x xn( 1113857 + αnlangnablaf(u) minus nablaf(x) un minus xrang

(67)Choose a subsequence xnj

1113882 1113883 of xn1113864 1113865 such that

lim supn⟶infinlangnablaf(u) minus nablaf(x) xn minus xrang lim

j⟶infin1113866nablaf(u)

minus nablaf(x) xnjminus x1113867

(68)

Since xnj q it follows from Lemma 1(ii) that





1113866nablaf(u) minus nablaf(x)

q minus x1113867le 0

(69)

Since un minus xn⟶ 0 n⟶infin thenlim sup

n⟶infinlangnablaf(u) minus nablaf(x) un minus xrang le 0 (70)

Hence by Lemma 9 and (67) we conclude thatDf(x xn)⟶ 0 n⟶infin (erefore xn1113864 1113865 convergesstrongly to x ProjfΓ(u)

Case B Suppose that there exists a subsequence nj1113966 1113967 of n

such that

Df xnj w1113874 1113875ltDf xnj+1 w1113874 1113875 (71)

for all j isin N (en by Lemma 10 there exists a non-decreasing sequence mk1113864 1113865 sub N with mk⟶infin as n⟶infinsuch that

Df p xmk1113872 1113873leDf p xmk+11113872 1113873

Df p xk( 1113857leDf p xmk+11113872 1113873(72)

for all k isin N Following the same line of arguments as in CaseI we have that

limk⟶infin

Txmkminus xmk

0

limk⟶infin

ResfBA

fymk

minus ymk

0

limk⟶infin

wmkminus xmk

0

lim supk⟶infinlangnablaf(u) minus nablaf(p) wmk

minus prang le 0

(73)

From (67) we have

Df p xmk+11113872 1113873le 1 minus αmk1113872 1113873Df p xmk

1113872 1113873

+ αmklangnablaf(u) minus nablaf(p) wmk

minus prang(74)

Since Df(p xmk)leDf(p xmk+1) it follows from (74)

that

αmkDf p xmk

1113872 1113873leDf p xmk1113872 1113873 minus Df p xmk+11113872 1113873

+ αmklangnablaf(u) minus nablaf x

lowast( 1113857 wmk

minus prang

le αmklangnablaf(u) minus nablaf(p) wmk

minus prang

(75)

Since αmkgt 0 we obtain

Df p xmk1113872 1113873le langnablaf(u) minus nablaf(p) wmk

minus prang (76)

(en from (73) it follows that Df(p xmk)⟶ 0 as

k⟶infin Combining Df(p xmk)⟶ 0 with (74) we

obtain Df(p xmk+1)⟶ 0 as k⟶infin SinceDf(p xk)leDf (p xmk+1) for all k isin N we have xk⟶ p ask⟶infin which implies that xn⟶ p as n⟶infin

(erefore from the above two cases we conclude thatxn1113864 1113865 converges strongly to x ProjfΓu

(is completes the proof

Corollary 1 Let C be a nonempty closed and convex subsetof a real reflexive Banach space E and f E⟶ R a Legendrefunction which is bounded uniformly Frechet differentiableand totally convex on bounded subsets of E Let A E⟶ 2Elowast

be a Bregman inverse strongly monotone operatorB E⟶ 2Elowast be a maximal monotone operator andT C⟶ C be a Bregman firmly nonexpansive mappingSuppose Γ ≔ F(ResfA ∘Af)capF(T)neempty Let cn1113864 1113865 sub (0 1) andαn1113864 1113865 βn and δn1113864 1113865 be sequences in (0 1) such thatαn + βn + δn 1 Given u isin E and x1 isin C arbitrarily let xn1113864 1113865

and yn1113864 1113865 be sequences in E generated by



B ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎨

⎩

(77)


(i) limn⟶infin





Corollary 2 Let C be a nonempty closed and convex subset of areal Hilbert space H Let A H⟶ H be a single-valued 1-inverse stronglymonotone operatorB E⟶ 2Elowast be amaximalmonotone operator and T C⟶ C be a firmly nonexpansivemapping Suppose Γ ≔ F((I + B)minus 1(I minus A))capF(T)neempty Letcn1113864 1113865 sub (0 1) and αn1113864 1113865 βn and δn1113864 1113865 be sequences in (0 1) suchthat αn + βn + δn 1 Given u isin E and x1 isin C arbitrarily letxn1113864 1113865 and yn1113864 1113865 be sequences in E generated by

yn cnxn + 1 minus cn( 1113857Txn

xn+1 PC αnu + βnyn + δn(I + B)minus 1(I minus A)yn1113872 1113873 nge 1

⎧⎨

⎩

(78)


(i) limn⟶infin




Then xn1113864 1113865 converges strongly to PΓu where PΓ is themetric projection of H onto Γ


4 Application

In this section we apply our result to obtain a commonsolution of variational inequality problem (VIP) and equi-librium problem (EP) in real reflexive Banach spaces

Let C be a nonempty closed and convex subset of a realreflexive Banach space E Suppose g C times C⟶ R is abifunction that satisfies the following conditions

A1 g(x x) 0 forallx isin C

A2 g(x y) + g(y x)le 0 forallx y isin C

A3 lim suptdarr0g(tz + (1 minus t)x y)leg(x y) forallx y

z isin C

A4 g(x ) is convex and lower semicontinuous foreach x isin C

(e equilibrium problem with respect to g is to findx isin C such that

g(x y)ge 0 forally isin C (79)

We denote the set of solutions of (79) by EP(g) (eresolvent of a bifunction g C times C⟶ R that satisfies A1 minus

A4 (see [49]) is the operator Tfg E⟶ 2C defined by

Tfg(x) ≔ z isin C g(z y) +langnablaf(z) minus nablaf(x) y minus zrangge 0 forally isin C1113864 1113865

(80)

Lemma 11 ([27] Lemma 1 2) Let f E⟶ (minus infininfin) bea coercive Legendre function and let C be a nonempty closedand convex subset of E Suppose the bifunctiong C times C⟶ R satisfies A1 minus A4 then

(1) dom(Tfg) E

(2) Tfg is single valued

(3) Tfg is Bregman firmly nonexpansive

(4) F(Tfg) EP(g)

(5) EP(g) is a closed and convex subset of C(6) Df(u Tf

g(x)) + Df(Tfg(x) x)leDf(u x) for all

x isin E and for all u isin F(Tfg)

Let A E⟶ Elowast be a Bregman inverse stronglymonotone mapping and let C be a nonempty closed andconvex subset of domA (e variational inequality problemcorresponding to A is to find x isin C such that

langAxlowast y minus x

lowastrang ge 0 forally isin C (81)

The set of solutions of (81) is denoted by VI(C A)

Lemma 12 (see [25 46]) Let A E⟶ Elowast be a Bregmaninverse strongly monotone mapping and f E⟶ (minus infininfin]

be a Legendre and totally convex function that satisfies therange condition If C is a nonempty closed and convex subsetof domAcap int(domf) then

(1) Pf

C ∘Af is Bregman relatively nonexpansive mapping

(2) F(Pf

C ∘Af) VI(C A)

Now let iC be the indicator function of a closed convexsubset C of E defined by

iC(x) 0 x isin C

+infin otherwise1113896 (82)

The subdifferential of the indicator functionziC(x) NC(x) where C is a closed subset of a Banachspace E and NC sub Elowast is the normal cone defined by

NC(x) v isin Elowast langv x minus xrangle 0 for allx isin C x isin C

empty x notin C1113896

(83)

The normal cone NC is maximal monotone and theresolvent of the normal cone corresponds to the Bregmanprojection (see [50] Example 44) that is Resf

NC ProjfC

Therefore if we let B NC and T Tfg then the iter-

ative algorithm (77) becomesyn nablaflowast cnnablaf xn( 1113857 + 1 minus cn( 1113857nablaf Tf

gxn1113872 11138731113872 1113873

xn+1 ProjfC nablaflowast αnnablaf(u) + βnnablaf yn( 1113857 + δnnablaf ProjfC ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎪⎨

⎪⎩

(84)

Thus from Corollary 1 we obtain a strong convergenceresult for approximating a point x isin VI(C A)capEP(g)

Data Availability

No data were used to support this study

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e work of the first author is based on the researchsupported wholly by the National Research Foundation(NRF) of South Africa (Grant no 111992) (e third authoracknowledges the financial support from the Departmentof Science and Technology and National Research Foun-dation Republic of South Africa Center of Excellence inMathematical and Statistical Sciences (DST-NRF CoE-MaSS) (postdoctoral fellowship) (Grant no BA 2018012)Opinions expressed and conclusions arrived are those ofthe authors and are not necessarily to be attributed to theNRF and CoE-MaSS

References

[1] P L Combettes ldquoFejer monotonicity in convex optimiza-tionrdquo in Encyclopedia of Optimisation C A Floudes andP M Pardolis Eds vol 2 pp 106ndash114 Kluwer Boston MAUSA 2009

[2] A Kaplon and R Tichatschke ldquoA general view on proximalpoint methods to variational inequalities in Hilbert spaces-


iterative regularization and approximationrdquo Journal ofNonlinear and Convex Analysis vol 2 pp 305ndash332 2001

[3] B Lemaire ldquo(e proximal point algorithmrdquo in New Methodsin Optimization and lteir Industrial Users InternationalSeries of Numerical Mathematics J P Penst Ed vol 87pp 73ndash87 Birkhauser Boston MA USA 1989

[4] R T Rockafellar ldquoArgumented Lagrangians and applicationof the proximal point algorithm in convex programmingrdquoMathematics of Operations Research vol 1 no 2 pp 97ndash1161976

[5] R T Rockafellar ldquoMonotone operators and the proximalpoint algorithmrdquo SIAM Journal on Control and Optimizationvol 14 pp 877ndash898 1976

[6] P Tossings ldquo(e perturbed proximal point algorithm andsome of its applicationsrdquo Applied Mathematics amp Optimi-zation vol 29 no 2 pp 125ndash159 1994

[7] E Zeidler Nonlinear Functional Analysis and Its ApplicationsIIBmdashNonlinear Monotone Operators Spinger-Verlag NewYork NY USA 1995

[8] B Martinet ldquoBreve communication Regularisationdrsquoinequations variationnelles par approximations succes-sivesrdquo Revue Franccedilaise Drsquoinformatique et de RechercheOperationnelle Serie Rouge vol 4 no R3 pp 154ndash158 1970

[9] O Guler ldquoOn the convergence of the proximal point algo-rithm for convex minimizationrdquo SIAM Journal on Controland Optimization vol 29 no 2 pp 403ndash419 1991

[10] R E Bruck and S Reich ldquoNonexpansive projections andresolvents of accretive operators in Banach spacesrdquo HoustonJournal of Mathematics vol 3 pp 459ndash470 1977

[11] H H Bauschke E Matouskova and S Reich ldquoProjection andproximal point methods convergence results and counter-examplesrdquo Nonlinear Analysis lteory Methods amp Applica-tions vol 56 no 5 pp 715ndash738 2004

[12] M Eslamian and J Vahidi ldquoGeneral proximal-point algo-rithm for monotone operatorsrdquo Ukrainian MathematicalJournal vol 68 no 11 pp 1715ndash1726 2017

[13] G Morosanu ldquoA proximal point algorithm revisted andextendedrdquo Journal of Optimization lteory and Applicationsvol 161 no 2 pp 478ndash489 2014

[14] Y Shehu ldquoConvergence theorems for maximal monotoneoperators and fixed point problems in Banach spacesrdquo AppliedMathematics and Computation vol 239 pp 285ndash298 2014

[15] G Lopez V Martın-Marquez F Wang and H-K XuldquoForward-Backward splitting methods for accretive operatorsin Banach spacerdquo Abstract and Applied Analysis vol 2012Article ID 109236 25 pages 2012

[16] P L Lions and BMercier ldquoSplitting algorithms for the sum oftwo nonlinear operatorsrdquo SIAM Journal on NumericalAnalysis vol 16 no 6 pp 964ndash979 1978

[17] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator lteory in Hilbert Spaces CMS Books inMathematics Spinger New York NY USA 2011

[18] H H Bauschke J M Borwein and P L Combettes ldquoEs-sential smoothness essential strict convexity and Legendrefunctions in Banach spacesrdquo Communications in Contem-porary Mathematics vol 3 no 4 pp 615ndash647 2001

[19] J F Bonnans and A Shapiro Pertubation Analysis of Opti-mization Problems Spinger-Verlag New York NY USA2000

[20] H H Bauschke and J M Borwein ldquoLegendre functions andthe method of random Bregman projectionsrdquo Journal ofConvex Analysis vol 4 pp 27ndash67 1997

[21] D Reem and S Reich ldquoSolutions to inexact resolvent in-clusion problems with applications to nonlinear analysis and

optimizationrdquo Rendiconti del Circolo Matematico di Palermovol 67 no 2 pp 337ndash371 2018

[22] L M Bregman ldquo(e relaxation method of finding thecommon point of convex sets and its application to the so-lution of problems in convex programmingrdquo USSR Com-putational Mathematics and Mathematical Physics vol 7no 3 pp 200ndash217 1967

[23] Y Censor and A Lent ldquoAn iterative row-action method forinterval convex programmingrdquo Journal of Optimizationlteory and Applications vol 34 no 3 pp 321ndash353 1981

[24] D Reem S Reich and A De Pierro ldquoRe-examination ofBregman functions and new properties of their divergencesrdquoOptimization vol 68 no 1 pp 279ndash348 2019

[25] S Reich A Weak Convergence lteorem for the AlternatingMethod with Bregman Distances CRC Press Boca Raton FLUSA 1996

[26] H H Bauschke J M Borwein and P L CombettesldquoBregmanmonotone optimization algorithmsrdquo SIAM Journalon Control and Optimization vol 42 no 2 pp 596ndash636 2003

[27] S Reich and S Sabach ldquoTwo strong convergence theorems forBregman strongly nonexpansive operator in reflexive Banachspacerdquo Nonlinear Analysis lteory Methods amp Applicationsvol 13 no 1 pp 122ndash135 2010

[28] F Kohsaka and W Takahashi ldquoFixed point theorems for aclass of nonlinear mappings related to maximal monotoneoperators in Banach spacesrdquo Archiv der Mathematik vol 91no 2 pp 166ndash177 2008

[29] W Takahashi N-C Wang and J-C Yao ldquoFixed pointtheorems and Convergence theorems for generalized non-spreading mappings in Banach spacesrdquo Journal of Fixed Pointlteory and Applications vol 11 no 1 pp 159ndash183 2012

[30] E Naraghirad N-C Ching and J-C Yao ldquoApplications ofbregman-opial property to bregman nonspreading mappingsin Banach spacesrdquo Abstract and Applied Analysis vol 2014Article ID 272867 14 pages 2014

[31] S ReichS Sabach et al ldquoExistence and approximation of fixedpoints of Bregman firmly nonexpansive mappings in reflexiveBanach spacesrdquo in Fixed Point Algorithm for InverseProblems in Science and Engineering vol 49 pp 301ndash316 HH Bauschke etal Eds Spinger New York NY USA 2011

[32] D Butnariu and G Kassay ldquoA proximal-projection methodfor finding zeros of set-valued operatorsrdquo SIAM Journal onControl and Optimization vol 47 no 4 pp 2096ndash2136 2008

[33] S Plubtieng and R Punpaeng ldquoA new iterative method forequilibrium problems and fixed point problems of non-expansive mappings and monotone mappingsrdquo AppliedMathematics and Computation vol 197 no 2 pp 548ndash5582008

[34] X Qin Y J Cho and S M Kang ldquoConvergence theorems ofcommon elements for equilibrium problems and fixed pointproblems in Banach spacesrdquo Journal of Computational andApplied Mathematics vol 225 pp 20ndash30 2009

[35] H Iiduka ldquoA new iterative algorithm for the variationalinequality problem over the fixed point set of a firmly non-expansive mappingrdquo Optimization vol 59 no 6 pp 873ndash885 2010

[36] P E Mainge ldquoA hybrid extragradient-viscosity method formonotone operators and fixed point problemsrdquo SIAM Journalon Control and Optimization vol 47 no 3 pp 1499ndash15152008

[37] D Butnariu and E Resmerita ldquoBregman distances totallyconvex function and a method for solving operator equationsin Banach spacesrdquo Abstract and Applied Analysis vol 2006Article ID 84919 39 pages 2006


[38] J M Borwein S Reich and S Sabach ldquoA characterization ofBregman firmly nonexpansive operators using a newmonotonicity conceptrdquo Journal of Nonlinear and ConvexAnalysis vol 12 pp 161ndash184 2011

[39] D Butnariu and A N Iusem Totally Convex Functions forFixed Points Computation and Infinite Dimensional Optimi-zation Kluwer Academic Dordrecht Netherlands 2000

[40] Y I Alber ldquoMetric and generalized projection operators inBanach spaces properties and applicationsrdquo in lteory andApplications of Nonlinear Operator of Accretive andMonotoneType A G Kartsatos Ed pp 15ndash50 Marcel Dekker NewYork NY USA 1996

[41] F Kohsaka and W Takahashi ldquoProximal point algorithmswith Bregman functions in Banach spacesrdquo Journal of Non-linear and Convex Analysis vol 6 pp 505ndash523 2005

[42] R P Phelps ldquoConvex functions monotone operators anddifferentiabilityrdquo in Lecture Notes in Mathematics vol 1364Springer-Verlag Berlin Germany 2nd Edition 1993

[43] E Naraghirad and J-C Yao ldquoBregman weak relativelynonexpansive mappings in Banach spacesrdquo Fixed Pointlteory and Applications vol 2013 no 1 p 43 2013

[44] S Reich and S Sabach ldquoTwo strong convergence theorems fora proximal method in reflexive Banach spacerdquo NumericalFunctional Analysis and Optimization vol 31 no 1 pp 22ndash44 2010

[45] G Kassay S Reich and S Sabach ldquoIterative methods forsolving systems of variational inequalities in reflexive Banachspacesrdquo SIAM Journal on Optimization vol 21 no 4pp 1319ndash1344 2011

[46] S Reich and S Sabach ldquoA strong convergence theorem for aproximal-type algorithm in reflexive Banach spacerdquo Journal ofNonlinear and Convex Analysis vol 10 pp 491ndash485 2009

[47] H K Xu ldquoAn iterative approach to quadratic optimizationrdquoJournal of Optimization lteory and Applications vol 116no 3 pp 659ndash678 2003

[48] P E Mainge ldquo(e viscosity approximation process for quasi-nonexpansive mappings in Hilbert spacesrdquo Computers ampMathematics with Applications vol 59 no 1 pp 74ndash79 2010

[49] S Reich and S Sabach ldquoA projection method for solvingnonlinear problems in reflexive Banach spacesrdquo Journal ofFixed Pointlteory and Applications vol 9 no 1 pp 101ndash1162011

[50] H H Bauschke X Wang and L Yao ldquoGeneral resolvents formonotone operators characterization and extensionrdquo inBiomedical Mathematics Promising Directions in Imaginglterapy Planning and Inverse Problems pp 57ndash74 MedicalPhysics Publishing Madison WI USA 2010


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Bauschke et al [11] are very important in this direction Formore recent results on PPA see [12ndash14]

(e problem of finding the zeros of the sum of twomonotone mappings A and B is to find a point xlowast isin E suchthat

0 isin (A + B)xlowast (5)

has recently received attention due to its significant im-portance in many physical problems One classical methodfor solving problem (5) is the forward-backward splittingmethod [15] which is as follows for x1 isin E

xn+1 (I + rB)minus 1

xn minus rAxn( 1113857 nge 1 (6)

where rgt 0 (is method combines the proximal point al-gorithm and the gradient projection algorithm In [16]Lions and Mercier introduced the following splitting iter-ative methods in a real Hilbert space H

xn+1 2JAr minus I1113872 1113873 2J

Br minus I1113872 1113873xn nge 1

xn+1 JAr 2J

Br minus I1113872 1113873xn + I minus J

Br1113872 1113873xn nge 1

(7)

where JTr (I + rT)minus 1 (e first one is called Peacemanndash

Rachford algorithm and the second one is calledDouglasndashRachford algorithm [15] It was noted that bothalgorithms converge weakly in general [16 17]

Many authors have studied the approximation of zero ofthe sum of two monotone operators (in Hilbert space) andaccretive operators (in Banach spaces) but the approxi-mation of the sum of two monotone operators in moregeneral Banach spaces other the Hilbert spaces has notenjoyed such popularity

(roughout this paper f E⟶ (minus infin +infin] is a properlower semicontinuous and convex function and the Fenchelconjugate of f is the function flowast Elowast ⟶ (minus infin +infin] de-fined by

flowast

xlowast

( 1113857 sup langxlowast xrang minus f(x) x isin E1113864 1113865 (8)

We denote by domf the domain of f that is the setx isin E f(x)lt +infin1113864 1113865 For any x isin intdomf and y isin E theright-hand derivative of f at x in the direction of t is defined by

fo(x y) ≔ lim

t⟶0+

f(x + ty) minus f(x)

t (9)

(e function f is said to be Gateaux differentiable at x ifthe limit as t⟶ 0+ in (9) exists for any y In this casefo(x y) coincides with nablaf(x) the value of the gradient nablafat x (e function f is said to be Gateaux differentiable if it isGateaux differentiable for any x isin intdomf(e function f isFrechet differentiable at x if the limit is attained with y 1and uniformly Frechet differentiable on a subset C of E if thelimit is attained uniformly for x isin C and y 1

(e function f is said to be Legendre if it satisfies thefollowing two conditions

(L1) intdomfneempty and the subdifferential zf is single-valued in its domain(L2) tdomflowast neempty and zflowast is single-valued on itsdomain

(e class of Legendre functions in infinite dimensionalBanach spaces was first introduced and studied by Bauschkeet al in [18] (eir definition is equivalent to conditions (L1)and (L2) because the space E is assumed to be reflexive (see[18] (eorems 54 and 56 p 634) It is well known that inreflexive Banach spaces nablaf (nablaflowast)minus 1 (see [19] p 83)When this fact is combined with conditions (L1) and (L2)we obtain

rannablaf domnablaflowast int(domf)lowast

rannablaflowast domnablaf int(domf)(10)

It also follows that f is Legendre if and only if flowast isLegendre (see [18] Corollary 55 p 634) and that thefunctions f and flowast are Gateaux differentiable and strictlyconvex in the interior of their respective domains

Several interesting examples of the Legendre functionsare presented in [18 20 21] A very important example ofLegendre function is the function 1s middot s with s isin (1infin)where the Banach space E is smooth and strictly convex andin particular a Hilbert space (roughout this article weassume that the convex function f E⟶ (minus infin +infin] isLegendre

Definition 1 Let f E⟶ (minus infin +infin] be a convex andGateaux differentiable function the function Df domftimes

intdomf⟶ [0infin) which is defined by

Df(y x) ≔ f(y) minus f(x) minus langnablaf(x) y minus xrang (11)

is called the Bregman distance [22ndash24]

(e Bregman distance does not satisfy the well-knownmetric properties but it does have the following importantproperty which is called the three-point identity for anyx isin domf and y z isin intdomf

Df(x y) + Df(y z) minus Df(x z) langnablaf(z) minus nablaf(y) x minus yrang

(12)

Let C be a nonempty subset of a Banach space E andT C⟶ C be amapping then a point x is called fixed pointof T if Tx x (e set of fixed point of T is denoted by F(T)Also a point xlowast isin C is said to be an asymptotic fixed point ofT if C contains a sequence xn1113864 1113865

infinn1 which converges weakly

to xlowast and limn⟶infinxn minus Txn 0 [25] (e set of asymp-totic fixed points of T is denoted by 1113954F(T)

Definition 2 [26 27] Let C be a nonempty closed andconvex subset of E A mapping T C⟶ int(domf) iscalled

(i) Bregman firmly nonexpansive (BFNE for short) if

langnablaf(Tx) minus nablaf(Ty) Tx minus Tyrang

le langnablaf(x) minus nablaf(y) Tx minus Tyrang forallx y isin C(13)

(ii) Bregman strongly nonexpansive (BSNE) with re-spect to a nonempty 1113954F(T) if





limn⟶infin


it follows that

limn⟶infin

Df Txn xn( 1113857 0 (16)











(i) Nonexpansive if






2+ Ty minus x

2 forallx y isin C

(22)




Resf






B) F(Resf










Observe that

domAf

(domA)cap (intdomf)

ranAf sub intdomf

(28)








2 Preliminaries


satisfying




We also have





(i) z Projf

C(x)










Vf x xlowast


xlowast


(35)



Vf x xlowast



leVf x xlowast

+ ylowast


(36)



N

i1tinablaf xi( 1113857⎛⎝ ⎞⎠le 1113944

N

i1tiDf z xi( 1113857 (37)

where xi1113864 1113865N




f 1113944infin

k0αkxk

⎛⎝ ⎞⎠le 1113944infin


1113874 1113875 (38)



limn⟶infin


yn minus xn

0 (39)






Df w Af

x1113872 1113873 + Df Af

x x1113872 1113873 le Df(w x) (40)







cap F Ti( 1113857 1le ileN1113864 1113865 (41)





p isin F(T)




such that







3 Main Results






B ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎨

⎩

(43)


(i) limn⟶infin













B ∘Af)





Df p xn( 1113857

(44)

Also


+ δnnablaf Resf

B ∘Af

yn( 11138571113872 111387311138771113875


+ δnDf pResfB ∘A

fyn1113872 1113873




lemax Df(p u) Df p xn( 11138571113966 1113967

⋮

lemax Df(p u) Df p x1( 11138571113966 1113967

(45)














1113872 1113873



1113872 1113873


1113872 1113873




1113872 11138731113960 1113961

(47)




1113872 1113873


(48)


limn⟶infin


1113872 1113873 0 (49)



0 (50)


limn⟶infin

xn minus Txn

0 (51)

Again




⟶ 0 n⟶infin

(52)


limn⟶infin

xn minus yn

0 (53)


B ∘Afyn)) then


1 minus αn


1 minus αn

nablaf ResfB ∘A

fyn1113872 11138731113890 11138911113888 1113889

leβn

1 minus αn

Df p yn( 1113857 +δn

1 minus αn

Df pResf

B ∘Af

yn1113872 1113873

leβn + δn

1 minus αn

Df p yn( 1113857

Df p yn( 1113857

(54)

(erefore we have







(55)

More so


1 minus αn

Df p yn( 1113857 +δn

1 minus αn

Df pResfB ∘A

fyn1113872 1113873


1 minus αn

1113888 1113889Df p yn( 1113857

+δn

1 minus αn

Df pResfB ∘A

fyn1113872 1113873


1 minus αn


fyn1113872 1113873

minus Df p yn( 11138571113891

(56)




fyn1113872 11138731113872 1113873 lt

δn

1 minus αn

1113890Df p yn( 1113857

minus Df pResf

B ∘Af

yn1113872 11138731113891

leDf p xn( 1113857


as n⟶infin

(57)

(us


fyn1113872 1113873⟶ 0 as n⟶infin

(58)

(erefore since Resf



limn⟶infin

yn minus ResfB ∘A

fyn

0 (59)





fyn1113872 111387311138771113875


middot ynResfB ∘A

fyn1113872 1113873⟶ 0

(60)

(us

limn⟶infin

yn minus un

0 (61)


un minus xn

le un minus yn

+ yn minus xn

⟶ 0 n⟶infin

(62)






fyn1113872 1113873





(64)

which implies

xn+1 minus un

⟶ 0 n⟶infin (65)

Hence

xn+1 minus xn

le xn+1 minus un

+ un minus xn

⟶ 0 n⟶infin

(66)



Since limn⟶infin

xniminus Txni






B ∘Af)





fyn11138771113875


f1113872 1113873yn1113872 1113873


f1113872 1113873yn




f1113872 1113873yn1113960 1113961 minus x1113867


f1113872 1113873yn1113872 1113873




f1113872 1113873yn11138771113875 + αn1113866nablaf(u)



B ∘Af

yn1113872 1113873







1113882 1113883 of xn1113864 1113865 such that




(68)








(69)





such that

Df xnj w1113874 1113875ltDf xnj+1 w1113874 1113875 (71)


Df p xmk1113872 1113873leDf p xmk+11113872 1113873

Df p xk( 1113857leDf p xmk+11113872 1113873(72)


limk⟶infin

Txmkminus xmk

0

limk⟶infin

ResfBA

fymk

minus ymk

0

limk⟶infin

wmkminus xmk

0


minus prang le 0

(73)

From (67) we have


1113872 1113873


minus prang(74)


that

αmkDf p xmk



lowast( 1113857 wmk

minus prang


minus prang

(75)



minus prang (76)











B ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎨

⎩

(77)


(i) limn⟶infin








⎧⎨

⎩

(78)


(i) limn⟶infin






4 Application






z isin C







(80)


(1) dom(Tfg) E



(4) F(Tfg) EP(g)










(1) Pf


(2) F(Pf

C ∘Af) VI(C A)


iC(x) 0 x isin C





(83)


NC ProjfC



gxn1113872 11138731113872 1113873


⎧⎪⎨

⎪⎩

(84)


Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018











limn⟶infin


it follows that

limn⟶infin

Df Txn xn( 1113857 0 (16)











(i) Nonexpansive if






2+ Ty minus x

2 forallx y isin C

(22)




Resf






B) F(Resf










Observe that

domAf

(domA)cap (intdomf)

ranAf sub intdomf

(28)








2 Preliminaries


satisfying




We also have





(i) z Projf

C(x)










Vf x xlowast


xlowast


(35)



Vf x xlowast



leVf x xlowast

+ ylowast


(36)



N

i1tinablaf xi( 1113857⎛⎝ ⎞⎠le 1113944

N

i1tiDf z xi( 1113857 (37)

where xi1113864 1113865N




f 1113944infin

k0αkxk

⎛⎝ ⎞⎠le 1113944infin


1113874 1113875 (38)



limn⟶infin


yn minus xn

0 (39)






Df w Af

x1113872 1113873 + Df Af

x x1113872 1113873 le Df(w x) (40)







cap F Ti( 1113857 1le ileN1113864 1113865 (41)





p isin F(T)




such that







3 Main Results






B ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎨

⎩

(43)


(i) limn⟶infin













B ∘Af)





Df p xn( 1113857

(44)

Also


+ δnnablaf Resf

B ∘Af

yn( 11138571113872 111387311138771113875


+ δnDf pResfB ∘A

fyn1113872 1113873




lemax Df(p u) Df p xn( 11138571113966 1113967

⋮

lemax Df(p u) Df p x1( 11138571113966 1113967

(45)














1113872 1113873



1113872 1113873


1113872 1113873




1113872 11138731113960 1113961

(47)




1113872 1113873


(48)


limn⟶infin


1113872 1113873 0 (49)



0 (50)


limn⟶infin

xn minus Txn

0 (51)

Again




⟶ 0 n⟶infin

(52)


limn⟶infin

xn minus yn

0 (53)


B ∘Afyn)) then


1 minus αn


1 minus αn

nablaf ResfB ∘A

fyn1113872 11138731113890 11138911113888 1113889

leβn

1 minus αn

Df p yn( 1113857 +δn

1 minus αn

Df pResf

B ∘Af

yn1113872 1113873

leβn + δn

1 minus αn

Df p yn( 1113857

Df p yn( 1113857

(54)

(erefore we have







(55)

More so


1 minus αn

Df p yn( 1113857 +δn

1 minus αn

Df pResfB ∘A

fyn1113872 1113873


1 minus αn

1113888 1113889Df p yn( 1113857

+δn

1 minus αn

Df pResfB ∘A

fyn1113872 1113873


1 minus αn


fyn1113872 1113873

minus Df p yn( 11138571113891

(56)




fyn1113872 11138731113872 1113873 lt

δn

1 minus αn

1113890Df p yn( 1113857

minus Df pResf

B ∘Af

yn1113872 11138731113891

leDf p xn( 1113857


as n⟶infin

(57)

(us


fyn1113872 1113873⟶ 0 as n⟶infin

(58)

(erefore since Resf



limn⟶infin

yn minus ResfB ∘A

fyn

0 (59)





fyn1113872 111387311138771113875


middot ynResfB ∘A

fyn1113872 1113873⟶ 0

(60)

(us

limn⟶infin

yn minus un

0 (61)


un minus xn

le un minus yn

+ yn minus xn

⟶ 0 n⟶infin

(62)






fyn1113872 1113873





(64)

which implies

xn+1 minus un

⟶ 0 n⟶infin (65)

Hence

xn+1 minus xn

le xn+1 minus un

+ un minus xn

⟶ 0 n⟶infin

(66)



Since limn⟶infin

xniminus Txni






B ∘Af)





fyn11138771113875


f1113872 1113873yn1113872 1113873


f1113872 1113873yn




f1113872 1113873yn1113960 1113961 minus x1113867


f1113872 1113873yn1113872 1113873




f1113872 1113873yn11138771113875 + αn1113866nablaf(u)



B ∘Af

yn1113872 1113873







1113882 1113883 of xn1113864 1113865 such that




(68)








(69)





such that

Df xnj w1113874 1113875ltDf xnj+1 w1113874 1113875 (71)


Df p xmk1113872 1113873leDf p xmk+11113872 1113873

Df p xk( 1113857leDf p xmk+11113872 1113873(72)


limk⟶infin

Txmkminus xmk

0

limk⟶infin

ResfBA

fymk

minus ymk

0

limk⟶infin

wmkminus xmk

0


minus prang le 0

(73)

From (67) we have


1113872 1113873


minus prang(74)


that

αmkDf p xmk



lowast( 1113857 wmk

minus prang


minus prang

(75)



minus prang (76)











B ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎨

⎩

(77)


(i) limn⟶infin








⎧⎨

⎩

(78)


(i) limn⟶infin






4 Application






z isin C







(80)


(1) dom(Tfg) E



(4) F(Tfg) EP(g)










(1) Pf


(2) F(Pf

C ∘Af) VI(C A)


iC(x) 0 x isin C





(83)


NC ProjfC



gxn1113872 11138731113872 1113873


⎧⎪⎨

⎪⎩

(84)


Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018









2 Preliminaries


satisfying




We also have





(i) z Projf

C(x)










Vf x xlowast


xlowast


(35)



Vf x xlowast



leVf x xlowast

+ ylowast


(36)



N

i1tinablaf xi( 1113857⎛⎝ ⎞⎠le 1113944

N

i1tiDf z xi( 1113857 (37)

where xi1113864 1113865N




f 1113944infin

k0αkxk

⎛⎝ ⎞⎠le 1113944infin


1113874 1113875 (38)



limn⟶infin


yn minus xn

0 (39)






Df w Af

x1113872 1113873 + Df Af

x x1113872 1113873 le Df(w x) (40)







cap F Ti( 1113857 1le ileN1113864 1113865 (41)





p isin F(T)




such that







3 Main Results






B ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎨

⎩

(43)


(i) limn⟶infin













B ∘Af)





Df p xn( 1113857

(44)

Also


+ δnnablaf Resf

B ∘Af

yn( 11138571113872 111387311138771113875


+ δnDf pResfB ∘A

fyn1113872 1113873




lemax Df(p u) Df p xn( 11138571113966 1113967

⋮

lemax Df(p u) Df p x1( 11138571113966 1113967

(45)














1113872 1113873



1113872 1113873


1113872 1113873




1113872 11138731113960 1113961

(47)




1113872 1113873


(48)


limn⟶infin


1113872 1113873 0 (49)



0 (50)


limn⟶infin

xn minus Txn

0 (51)

Again




⟶ 0 n⟶infin

(52)


limn⟶infin

xn minus yn

0 (53)


B ∘Afyn)) then


1 minus αn


1 minus αn

nablaf ResfB ∘A

fyn1113872 11138731113890 11138911113888 1113889

leβn

1 minus αn

Df p yn( 1113857 +δn

1 minus αn

Df pResf

B ∘Af

yn1113872 1113873

leβn + δn

1 minus αn

Df p yn( 1113857

Df p yn( 1113857

(54)

(erefore we have







(55)

More so


1 minus αn

Df p yn( 1113857 +δn

1 minus αn

Df pResfB ∘A

fyn1113872 1113873


1 minus αn

1113888 1113889Df p yn( 1113857

+δn

1 minus αn

Df pResfB ∘A

fyn1113872 1113873


1 minus αn


fyn1113872 1113873

minus Df p yn( 11138571113891

(56)




fyn1113872 11138731113872 1113873 lt

δn

1 minus αn

1113890Df p yn( 1113857

minus Df pResf

B ∘Af

yn1113872 11138731113891

leDf p xn( 1113857


as n⟶infin

(57)

(us


fyn1113872 1113873⟶ 0 as n⟶infin

(58)

(erefore since Resf



limn⟶infin

yn minus ResfB ∘A

fyn

0 (59)





fyn1113872 111387311138771113875


middot ynResfB ∘A

fyn1113872 1113873⟶ 0

(60)

(us

limn⟶infin

yn minus un

0 (61)


un minus xn

le un minus yn

+ yn minus xn

⟶ 0 n⟶infin

(62)






fyn1113872 1113873





(64)

which implies

xn+1 minus un

⟶ 0 n⟶infin (65)

Hence

xn+1 minus xn

le xn+1 minus un

+ un minus xn

⟶ 0 n⟶infin

(66)



Since limn⟶infin

xniminus Txni






B ∘Af)





fyn11138771113875


f1113872 1113873yn1113872 1113873


f1113872 1113873yn




f1113872 1113873yn1113960 1113961 minus x1113867


f1113872 1113873yn1113872 1113873




f1113872 1113873yn11138771113875 + αn1113866nablaf(u)



B ∘Af

yn1113872 1113873







1113882 1113883 of xn1113864 1113865 such that




(68)








(69)





such that

Df xnj w1113874 1113875ltDf xnj+1 w1113874 1113875 (71)


Df p xmk1113872 1113873leDf p xmk+11113872 1113873

Df p xk( 1113857leDf p xmk+11113872 1113873(72)


limk⟶infin

Txmkminus xmk

0

limk⟶infin

ResfBA

fymk

minus ymk

0

limk⟶infin

wmkminus xmk

0


minus prang le 0

(73)

From (67) we have


1113872 1113873


minus prang(74)


that

αmkDf p xmk



lowast( 1113857 wmk

minus prang


minus prang

(75)



minus prang (76)











B ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎨

⎩

(77)


(i) limn⟶infin








⎧⎨

⎩

(78)


(i) limn⟶infin






4 Application






z isin C







(80)


(1) dom(Tfg) E



(4) F(Tfg) EP(g)










(1) Pf


(2) F(Pf

C ∘Af) VI(C A)


iC(x) 0 x isin C





(83)


NC ProjfC



gxn1113872 11138731113872 1113873


⎧⎪⎨

⎪⎩

(84)


Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018












cap F Ti( 1113857 1le ileN1113864 1113865 (41)





p isin F(T)




such that







3 Main Results






B ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎨

⎩

(43)


(i) limn⟶infin













B ∘Af)





Df p xn( 1113857

(44)

Also


+ δnnablaf Resf

B ∘Af

yn( 11138571113872 111387311138771113875


+ δnDf pResfB ∘A

fyn1113872 1113873




lemax Df(p u) Df p xn( 11138571113966 1113967

⋮

lemax Df(p u) Df p x1( 11138571113966 1113967

(45)














1113872 1113873



1113872 1113873


1113872 1113873




1113872 11138731113960 1113961

(47)




1113872 1113873


(48)


limn⟶infin


1113872 1113873 0 (49)



0 (50)


limn⟶infin

xn minus Txn

0 (51)

Again




⟶ 0 n⟶infin

(52)


limn⟶infin

xn minus yn

0 (53)


B ∘Afyn)) then


1 minus αn


1 minus αn

nablaf ResfB ∘A

fyn1113872 11138731113890 11138911113888 1113889

leβn

1 minus αn

Df p yn( 1113857 +δn

1 minus αn

Df pResf

B ∘Af

yn1113872 1113873

leβn + δn

1 minus αn

Df p yn( 1113857

Df p yn( 1113857

(54)

(erefore we have







(55)

More so


1 minus αn

Df p yn( 1113857 +δn

1 minus αn

Df pResfB ∘A

fyn1113872 1113873


1 minus αn

1113888 1113889Df p yn( 1113857

+δn

1 minus αn

Df pResfB ∘A

fyn1113872 1113873


1 minus αn


fyn1113872 1113873

minus Df p yn( 11138571113891

(56)




fyn1113872 11138731113872 1113873 lt

δn

1 minus αn

1113890Df p yn( 1113857

minus Df pResf

B ∘Af

yn1113872 11138731113891

leDf p xn( 1113857


as n⟶infin

(57)

(us


fyn1113872 1113873⟶ 0 as n⟶infin

(58)

(erefore since Resf



limn⟶infin

yn minus ResfB ∘A

fyn

0 (59)





fyn1113872 111387311138771113875


middot ynResfB ∘A

fyn1113872 1113873⟶ 0

(60)

(us

limn⟶infin

yn minus un

0 (61)


un minus xn

le un minus yn

+ yn minus xn

⟶ 0 n⟶infin

(62)






fyn1113872 1113873





(64)

which implies

xn+1 minus un

⟶ 0 n⟶infin (65)

Hence

xn+1 minus xn

le xn+1 minus un

+ un minus xn

⟶ 0 n⟶infin

(66)



Since limn⟶infin

xniminus Txni






B ∘Af)





fyn11138771113875


f1113872 1113873yn1113872 1113873


f1113872 1113873yn




f1113872 1113873yn1113960 1113961 minus x1113867


f1113872 1113873yn1113872 1113873




f1113872 1113873yn11138771113875 + αn1113866nablaf(u)



B ∘Af

yn1113872 1113873







1113882 1113883 of xn1113864 1113865 such that




(68)








(69)





such that

Df xnj w1113874 1113875ltDf xnj+1 w1113874 1113875 (71)


Df p xmk1113872 1113873leDf p xmk+11113872 1113873

Df p xk( 1113857leDf p xmk+11113872 1113873(72)


limk⟶infin

Txmkminus xmk

0

limk⟶infin

ResfBA

fymk

minus ymk

0

limk⟶infin

wmkminus xmk

0


minus prang le 0

(73)

From (67) we have


1113872 1113873


minus prang(74)


that

αmkDf p xmk



lowast( 1113857 wmk

minus prang


minus prang

(75)



minus prang (76)











B ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎨

⎩

(77)


(i) limn⟶infin








⎧⎨

⎩

(78)


(i) limn⟶infin






4 Application






z isin C







(80)


(1) dom(Tfg) E



(4) F(Tfg) EP(g)










(1) Pf


(2) F(Pf

C ∘Af) VI(C A)


iC(x) 0 x isin C





(83)


NC ProjfC



gxn1113872 11138731113872 1113873


⎧⎪⎨

⎪⎩

(84)


Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018



















1113872 1113873



1113872 1113873


1113872 1113873




1113872 11138731113960 1113961

(47)




1113872 1113873


(48)


limn⟶infin


1113872 1113873 0 (49)



0 (50)


limn⟶infin

xn minus Txn

0 (51)

Again




⟶ 0 n⟶infin

(52)


limn⟶infin

xn minus yn

0 (53)


B ∘Afyn)) then


1 minus αn


1 minus αn

nablaf ResfB ∘A

fyn1113872 11138731113890 11138911113888 1113889

leβn

1 minus αn

Df p yn( 1113857 +δn

1 minus αn

Df pResf

B ∘Af

yn1113872 1113873

leβn + δn

1 minus αn

Df p yn( 1113857

Df p yn( 1113857

(54)

(erefore we have







(55)

More so


1 minus αn

Df p yn( 1113857 +δn

1 minus αn

Df pResfB ∘A

fyn1113872 1113873


1 minus αn

1113888 1113889Df p yn( 1113857

+δn

1 minus αn

Df pResfB ∘A

fyn1113872 1113873


1 minus αn


fyn1113872 1113873

minus Df p yn( 11138571113891

(56)




fyn1113872 11138731113872 1113873 lt

δn

1 minus αn

1113890Df p yn( 1113857

minus Df pResf

B ∘Af

yn1113872 11138731113891

leDf p xn( 1113857


as n⟶infin

(57)

(us


fyn1113872 1113873⟶ 0 as n⟶infin

(58)

(erefore since Resf



limn⟶infin

yn minus ResfB ∘A

fyn

0 (59)





fyn1113872 111387311138771113875


middot ynResfB ∘A

fyn1113872 1113873⟶ 0

(60)

(us

limn⟶infin

yn minus un

0 (61)


un minus xn

le un minus yn

+ yn minus xn

⟶ 0 n⟶infin

(62)






fyn1113872 1113873





(64)

which implies

xn+1 minus un

⟶ 0 n⟶infin (65)

Hence

xn+1 minus xn

le xn+1 minus un

+ un minus xn

⟶ 0 n⟶infin

(66)



Since limn⟶infin

xniminus Txni






B ∘Af)





fyn11138771113875


f1113872 1113873yn1113872 1113873


f1113872 1113873yn




f1113872 1113873yn1113960 1113961 minus x1113867


f1113872 1113873yn1113872 1113873




f1113872 1113873yn11138771113875 + αn1113866nablaf(u)



B ∘Af

yn1113872 1113873







1113882 1113883 of xn1113864 1113865 such that




(68)








(69)





such that

Df xnj w1113874 1113875ltDf xnj+1 w1113874 1113875 (71)


Df p xmk1113872 1113873leDf p xmk+11113872 1113873

Df p xk( 1113857leDf p xmk+11113872 1113873(72)


limk⟶infin

Txmkminus xmk

0

limk⟶infin

ResfBA

fymk

minus ymk

0

limk⟶infin

wmkminus xmk

0


minus prang le 0

(73)

From (67) we have


1113872 1113873


minus prang(74)


that

αmkDf p xmk



lowast( 1113857 wmk

minus prang


minus prang

(75)



minus prang (76)











B ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎨

⎩

(77)


(i) limn⟶infin








⎧⎨

⎩

(78)


(i) limn⟶infin






4 Application






z isin C







(80)


(1) dom(Tfg) E



(4) F(Tfg) EP(g)










(1) Pf


(2) F(Pf

C ∘Af) VI(C A)


iC(x) 0 x isin C





(83)


NC ProjfC



gxn1113872 11138731113872 1113873


⎧⎪⎨

⎪⎩

(84)


Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018









fyn1113872 11138731113872 1113873 lt

δn

1 minus αn

1113890Df p yn( 1113857

minus Df pResf

B ∘Af

yn1113872 11138731113891

leDf p xn( 1113857


as n⟶infin

(57)

(us


fyn1113872 1113873⟶ 0 as n⟶infin

(58)

(erefore since Resf



limn⟶infin

yn minus ResfB ∘A

fyn

0 (59)





fyn1113872 111387311138771113875


middot ynResfB ∘A

fyn1113872 1113873⟶ 0

(60)

(us

limn⟶infin

yn minus un

0 (61)


un minus xn

le un minus yn

+ yn minus xn

⟶ 0 n⟶infin

(62)






fyn1113872 1113873





(64)

which implies

xn+1 minus un

⟶ 0 n⟶infin (65)

Hence

xn+1 minus xn

le xn+1 minus un

+ un minus xn

⟶ 0 n⟶infin

(66)



Since limn⟶infin

xniminus Txni






B ∘Af)





fyn11138771113875


f1113872 1113873yn1113872 1113873


f1113872 1113873yn




f1113872 1113873yn1113960 1113961 minus x1113867


f1113872 1113873yn1113872 1113873




f1113872 1113873yn11138771113875 + αn1113866nablaf(u)



B ∘Af

yn1113872 1113873







1113882 1113883 of xn1113864 1113865 such that




(68)








(69)





such that

Df xnj w1113874 1113875ltDf xnj+1 w1113874 1113875 (71)


Df p xmk1113872 1113873leDf p xmk+11113872 1113873

Df p xk( 1113857leDf p xmk+11113872 1113873(72)


limk⟶infin

Txmkminus xmk

0

limk⟶infin

ResfBA

fymk

minus ymk

0

limk⟶infin

wmkminus xmk

0


minus prang le 0

(73)

From (67) we have


1113872 1113873


minus prang(74)


that

αmkDf p xmk



lowast( 1113857 wmk

minus prang


minus prang

(75)



minus prang (76)











B ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎨

⎩

(77)


(i) limn⟶infin








⎧⎨

⎩

(78)


(i) limn⟶infin






4 Application






z isin C







(80)


(1) dom(Tfg) E



(4) F(Tfg) EP(g)










(1) Pf


(2) F(Pf

C ∘Af) VI(C A)


iC(x) 0 x isin C





(83)


NC ProjfC



gxn1113872 11138731113872 1113873


⎧⎪⎨

⎪⎩

(84)


Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018













(69)





such that

Df xnj w1113874 1113875ltDf xnj+1 w1113874 1113875 (71)


Df p xmk1113872 1113873leDf p xmk+11113872 1113873

Df p xk( 1113857leDf p xmk+11113872 1113873(72)


limk⟶infin

Txmkminus xmk

0

limk⟶infin

ResfBA

fymk

minus ymk

0

limk⟶infin

wmkminus xmk

0


minus prang le 0

(73)

From (67) we have


1113872 1113873


minus prang(74)


that

αmkDf p xmk



lowast( 1113857 wmk

minus prang


minus prang

(75)



minus prang (76)











B ∘Af yn( 11138571113872 11138731113872 11138731113872 1113873 nge 1

⎧⎨

⎩

(77)


(i) limn⟶infin








⎧⎨

⎩

(78)


(i) limn⟶infin






4 Application






z isin C







(80)


(1) dom(Tfg) E



(4) F(Tfg) EP(g)










(1) Pf


(2) F(Pf

C ∘Af) VI(C A)


iC(x) 0 x isin C





(83)


NC ProjfC



gxn1113872 11138731113872 1113873


⎧⎪⎨

⎪⎩

(84)


Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018








4 Application






z isin C







(80)


(1) dom(Tfg) E



(4) F(Tfg) EP(g)










(1) Pf


(2) F(Pf

C ∘Af) VI(C A)


iC(x) 0 x isin C





(83)


NC ProjfC



gxn1113872 11138731113872 1113873


⎧⎪⎨

⎪⎩

(84)


Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018



































































Journal of












Journal of







Volume 2018






Volume 2018





























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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A Halpern-Type Iteration Method for Bregman Nonspreading...

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