Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 406727 5 pageshttpdxdoiorg1011552013406727
Research ArticlePositive Fixed Points for Semipositone Operators inOrdered Banach Spaces and Applications
Zengqin Zhao and Xinsheng Du
School of Mathematical Sciences Qufu Normal University Qufu Shandong 273165 China
Correspondence should be addressed to Zengqin Zhao zqzhaomailqfnueducn
Received 23 January 2013 Accepted 7 April 2013
Academic Editor Kunquan Lan
Copyright copy 2013 Z Zhao and X Du This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The theory of semipositone integral equations and semipositone ordinary differential equations has been emerging as an importantarea of investigation in recent years but the research on semipositone operators in abstract spaces is yet rare By employing a well-known fixed point index theorem and combining it with a translation substitution we study the existence of positive fixed pointsfor a semipositone operator in ordered Banach space Lastly we apply the results to Hammerstein integral equations of polynomialtype
1 Introduction
Existence of fixed points for positive operators have beenstudied by many authors see [1ndash9] and their references Thetheory of semipositone integral equations and semipositoneordinary differential equations has been emerging as animportant area of investigation in recent years (see [10ndash17])But the research on semipositone operators in abstract spacesare yet rare up to now
Inspired by a number of semipositone problems forintegral equations and ordinary differential equations westudy the existence of positive fixed points for semipositoneoperators in ordered Banach spaces Then the results areapplied to Hammerstein integral equations of polynomialtype
Let 119864 be a real Banach space with the norm sdot 119875 a coneof119864 and ldquolerdquo the partial ordering defined by119875 120579 denoting thezero element of 119864 119875+ = 119875 120579 [119886 119887] = 119909 isin 119864 | 119886 le 119909 le 119887
Recall that cone 119875 is said to be normal if there exists apositive constant 119873 such that 120579 le 119909 le 119910 implies 119909 le
119873 119910 the smallest 119873 is called the normal constant of119875 An element 119911 isin 119864 is called the least upper bound (iesupremum) of set 119863 sub 119864 if it satisfies two conditions (i)119909 le 119911 for any 119909 isin 119863 (ii) 119909 le 119910 119909 isin 119863 implies 119911 le 119910We denote the least upper bound of 119863 by sup119863 that is119911 = sup119863
Definition 1 Cone 119875 sub 119864 is said to be minihedral if sup119909 119910exists for each pair of elements 119909 119910 isin 119864 For any 119909 in 119864 wedefine 119909+ = sup119909 120579
Definition 2 (see [1 3]) Let 119864119894be real Banach spaces 119875
119894cones
of 119864119894 119894 = 1 2 119879 119875
1rarr 1198752 and 120572 isin 119877 Then we say 119879 is 120572-
convex if and only if 119879(119905119906) le 119905120572119879119906 for all (119906 119905) isin 119875
1times (0 1)
Definition 3 Let 119864119894be real Banach spaces 119875
119894cones of 119864
119894 and
119894 = 1 2 1198751sub 119863 sub 119864
1 119879 119863 rarr 119864
2 119879 is said to be
nondecreasing if 1199091le 1199092(1199091 1199092isin 119863) implies 119879119909
1le 1198791199092
119879 is said to be positive if 119879119909 isin 1198752for any 119909 isin 119875
1 119879 is said
to be semipositone if (i) there exists an element 1199090isin 1198751such
that 119865(1199090) notin 1198752and (ii) there exists an element 119902 isin 119864
2such
that 119879119909 + 119902 isin 1198752for any 119909 isin 119875
1
In order to prove the main results we need the followinglemma which is obtained in [18]
Lemma4 Let119864 be a real Banach space andΩ a bounded opensubset of 119864 with 120579 isin Ω and 119860 Ω cap 119876 rarr 119876 is a completelycontinuous operator where 119876 is a cone in 119864
(i) Suppose that 119860119906 = 120583119906 for all 119906 isin 120597Ω cap 119876 120583 ge 1 thenthe fixed point index 119894(119860Ω cap 119876119876) = 1
(ii) Suppose that119860119906 ≰ 119906 for all 119906 isin 120597Ωcap119876 then 119894(119860Ωcap
119876119876) = 0
2 Abstract and Applied Analysis
The research on ordered Banach spaces cones fixed pointindex and the above lemma can be seen in [18 19]
2 Main Results and Their Proofs
Theorem 5 Let 119864119894be Banach space 119875
119894sub 119864119894cones and 119894 =
1 2 Suppose that operator 119860 1198641rarr 1198642can be expressed as
119860 = 119861119865 where the cone 119875119894and the operator 119865 and 119861 satisfy the
following conditions
(H1) when 1198751is normal and minihedral 119875
2is normal
(H2) when 119865 1198641rarr 1198642is continuous there exist 119892 isin 119875
+
2
119902 isin 1198642 a nondecreasing 120572-convex operator 119866 119875
1rarr
1198752 (120572 gt 1) and a bounded functional 119867 119875
1rarr
[0 +infin) such that
119866119906 le 119865119906 + 119902 le 119867 (119906) 119892 forall119906 isin 1198751 (1)
(H3) when 119861 1198642rarr 1198641is linear completely continuous
Then for 120582119860 = 119861(120582119865) the conditions in Theorem 5 aresatisfied Thus 120582119860 has a positive fixed point that is 119906 = 120582119860
has a positive solution and the proof is complete
We consider the integral equation
119906 (119909) = int119866
119896 (119909 119910)(
119898
sum
119894=1
119886119894(119910) 119906(119910)
120572119894
+ 119902 (119910)
times (119906(119910)120574
minus 119906(119910)120575
minus 1199080))119889119910
(31)
where 119866 is a bounded closed domain in 119877119899 and 120572119894ge 0 119886
119894(119909)
119902(119909) isin 119871(119866 [0infin)) 119894 = 1 2 119898 119896(119909 119910) is nonnegativecontinuous on 119866 times 119866
Theorem 7 Suppose that among 120572119894(119894 = 1 2 119898) there
exists 1205721198940
gt 1 such that inf119909isin119866
1198861198940
(119909) gt 0 and there existnontrivial nonnegative functions 119886(119909) 119887(119909) isin 119862(119866) and apositive number 119888 120574 120575 119908
0such that
119888119886 (119909) 119887 (119910) le 119896 (119909 119910) le 119886 (119909)
119896 (119909 119910) le 119887 (119910) forall119909 119910 isin 119866
From (35) and (36) we know that (H1) is satisfied By (47)and (48) we obtain that (H2) and (H3) are satisfied Equations(49) and (50) imply that (H4) is satisfied Therefore usingTheorem 5 the integral equation (31) has a positive solutionin 119862(119866)
Acknowledgments
The authors are very grateful to the referee for his or hervaluable suggestions This research was supported by theNational Natural Science Foundation of China (10871116)the Doctoral Program Foundation of Education Ministry ofChina (20103705120002) Shandong Provincial Natural Sci-ence Foundation China (ZR2012AM006) and the Programfor Scientific Research Innovation Team in Colleges andUniversities of Shandong Province
References
[1] A J B Potter ldquoApplications of Hilbertrsquos projective metric tocertain classes of non-homogeneous operatorsrdquo The QuarterlyJournal of Mathematics Oxford Second Series vol 28 no 109pp 93ndash99 1977
[2] C Zhai and C Guo ldquoOn 120572-convex operatorsrdquo Journal ofMathematical Analysis and Applications vol 316 no 2 pp 556ndash565 2006
[3] Z Zhao and X Du ldquoFixed points of generalized 119890-concave(generalized 119890-convex) operators and their applicationsrdquo Jour-nal of Mathematical Analysis and Applications vol 334 no 2pp 1426ndash1438 2007
[4] J R L Webb ldquoRemarks on 1199060-positive operatorsrdquo Journal of
Fixed Point Theory and Applications vol 5 no 1 pp 37ndash452009
[5] Z Zhao ldquoMultiple fixed points of a sum operator and applica-tionsrdquo Journal of Mathematical Analysis and Applications vol360 no 1 pp 1ndash6 2009
[6] Z Zhao and X Chen ldquoFixed points of decreasing operators inordered Banach spaces and applications to nonlinear secondorder elliptic equationsrdquo Computers ampMathematics with Appli-cations vol 58 no 6 pp 1223ndash1229 2009
[7] C-B Zhai and X-M Cao ldquoFixed point theorems for 120591 minus 120593-concave operators and applicationsrdquo Computers ampMathematicswith Applications vol 59 no 1 pp 532ndash538 2010
[8] Z Zhao ldquoExistence and uniqueness of fixed points for somemixed monotone operatorsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 73 no 6 pp 1481ndash1490 2010
[9] Z Zhao ldquoFixed points of 120591 minus 120593-convex operators and applica-tionsrdquo Applied Mathematics Letters vol 23 no 5 pp 561ndash5662010
[10] R P Agarwal S R Grace and D OrsquoRegan ldquoExistence of pos-itive solutions to semipositone Fredholm integral equationsrdquoFunkcialaj Ekvacioj vol 45 no 2 pp 223ndash235 2002
Abstract and Applied Analysis 5
[11] K Q Lan ldquoPositive solutions of semi-positone Hammersteinintegral equations and applicationsrdquo Communications on Pureand Applied Analysis vol 6 no 2 pp 441ndash451 2007
[12] K Q Lan ldquoEigenvalues of semi-positoneHammerstein integralequations and applications to boundary value problemsrdquo Non-linear Analysis Theory Methods amp Applications vol 71 no 12pp 5979ndash5993 2009
[13] J R Graef and L Kong ldquoPositive solutions for third ordersemipositone boundary value problemsrdquo Applied MathematicsLetters vol 22 no 8 pp 1154ndash1160 2009
[14] J R LWebb andG Infante ldquoSemi-positone nonlocal boundaryvalue problems of arbitrary orderrdquoCommunications onPure andApplied Analysis vol 9 no 2 pp 563ndash581 2010
[15] A Dogan J R Graef and L Kong ldquoHigher order semipositonemulti-point boundary value problems on time scalesrdquo Comput-ers amp Mathematics with Applications vol 60 no 1 pp 23ndash352010
[16] D R Anderson and C Zhai ldquoPositive solutions to semi-positone second-order three-point problems on time scalesrdquoApplied Mathematics and Computation vol 215 no 10 pp3713ndash3720 2010
[17] X Xian D OrsquoRegan and C Yanfang ldquoStructure of positivesolution sets of semi-positone singular boundary value prob-lemsrdquo Nonlinear Analysis Theory Methods amp Applications vol72 no 7-8 pp 3535ndash3550 2010
[18] D J Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones vol 5 of Notes and Reports in Mathematics inScience and Engineering Academic Press Boston Mass USA1988
[19] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985
Then for 120582119860 = 119861(120582119865) the conditions in Theorem 5 aresatisfied Thus 120582119860 has a positive fixed point that is 119906 = 120582119860
has a positive solution and the proof is complete
We consider the integral equation
119906 (119909) = int119866
119896 (119909 119910)(
119898
sum
119894=1
119886119894(119910) 119906(119910)
120572119894
+ 119902 (119910)
times (119906(119910)120574
minus 119906(119910)120575
minus 1199080))119889119910
(31)
where 119866 is a bounded closed domain in 119877119899 and 120572119894ge 0 119886
119894(119909)
119902(119909) isin 119871(119866 [0infin)) 119894 = 1 2 119898 119896(119909 119910) is nonnegativecontinuous on 119866 times 119866
Theorem 7 Suppose that among 120572119894(119894 = 1 2 119898) there
exists 1205721198940
gt 1 such that inf119909isin119866
1198861198940
(119909) gt 0 and there existnontrivial nonnegative functions 119886(119909) 119887(119909) isin 119862(119866) and apositive number 119888 120574 120575 119908
0such that
119888119886 (119909) 119887 (119910) le 119896 (119909 119910) le 119886 (119909)
119896 (119909 119910) le 119887 (119910) forall119909 119910 isin 119866
From (35) and (36) we know that (H1) is satisfied By (47)and (48) we obtain that (H2) and (H3) are satisfied Equations(49) and (50) imply that (H4) is satisfied Therefore usingTheorem 5 the integral equation (31) has a positive solutionin 119862(119866)
Acknowledgments
The authors are very grateful to the referee for his or hervaluable suggestions This research was supported by theNational Natural Science Foundation of China (10871116)the Doctoral Program Foundation of Education Ministry ofChina (20103705120002) Shandong Provincial Natural Sci-ence Foundation China (ZR2012AM006) and the Programfor Scientific Research Innovation Team in Colleges andUniversities of Shandong Province
References
[1] A J B Potter ldquoApplications of Hilbertrsquos projective metric tocertain classes of non-homogeneous operatorsrdquo The QuarterlyJournal of Mathematics Oxford Second Series vol 28 no 109pp 93ndash99 1977
[2] C Zhai and C Guo ldquoOn 120572-convex operatorsrdquo Journal ofMathematical Analysis and Applications vol 316 no 2 pp 556ndash565 2006
[3] Z Zhao and X Du ldquoFixed points of generalized 119890-concave(generalized 119890-convex) operators and their applicationsrdquo Jour-nal of Mathematical Analysis and Applications vol 334 no 2pp 1426ndash1438 2007
[4] J R L Webb ldquoRemarks on 1199060-positive operatorsrdquo Journal of
Fixed Point Theory and Applications vol 5 no 1 pp 37ndash452009
[5] Z Zhao ldquoMultiple fixed points of a sum operator and applica-tionsrdquo Journal of Mathematical Analysis and Applications vol360 no 1 pp 1ndash6 2009
[6] Z Zhao and X Chen ldquoFixed points of decreasing operators inordered Banach spaces and applications to nonlinear secondorder elliptic equationsrdquo Computers ampMathematics with Appli-cations vol 58 no 6 pp 1223ndash1229 2009
[7] C-B Zhai and X-M Cao ldquoFixed point theorems for 120591 minus 120593-concave operators and applicationsrdquo Computers ampMathematicswith Applications vol 59 no 1 pp 532ndash538 2010
[8] Z Zhao ldquoExistence and uniqueness of fixed points for somemixed monotone operatorsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 73 no 6 pp 1481ndash1490 2010
[9] Z Zhao ldquoFixed points of 120591 minus 120593-convex operators and applica-tionsrdquo Applied Mathematics Letters vol 23 no 5 pp 561ndash5662010
[10] R P Agarwal S R Grace and D OrsquoRegan ldquoExistence of pos-itive solutions to semipositone Fredholm integral equationsrdquoFunkcialaj Ekvacioj vol 45 no 2 pp 223ndash235 2002
Abstract and Applied Analysis 5
[11] K Q Lan ldquoPositive solutions of semi-positone Hammersteinintegral equations and applicationsrdquo Communications on Pureand Applied Analysis vol 6 no 2 pp 441ndash451 2007
[12] K Q Lan ldquoEigenvalues of semi-positoneHammerstein integralequations and applications to boundary value problemsrdquo Non-linear Analysis Theory Methods amp Applications vol 71 no 12pp 5979ndash5993 2009
[13] J R Graef and L Kong ldquoPositive solutions for third ordersemipositone boundary value problemsrdquo Applied MathematicsLetters vol 22 no 8 pp 1154ndash1160 2009
[14] J R LWebb andG Infante ldquoSemi-positone nonlocal boundaryvalue problems of arbitrary orderrdquoCommunications onPure andApplied Analysis vol 9 no 2 pp 563ndash581 2010
[15] A Dogan J R Graef and L Kong ldquoHigher order semipositonemulti-point boundary value problems on time scalesrdquo Comput-ers amp Mathematics with Applications vol 60 no 1 pp 23ndash352010
[16] D R Anderson and C Zhai ldquoPositive solutions to semi-positone second-order three-point problems on time scalesrdquoApplied Mathematics and Computation vol 215 no 10 pp3713ndash3720 2010
[17] X Xian D OrsquoRegan and C Yanfang ldquoStructure of positivesolution sets of semi-positone singular boundary value prob-lemsrdquo Nonlinear Analysis Theory Methods amp Applications vol72 no 7-8 pp 3535ndash3550 2010
[18] D J Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones vol 5 of Notes and Reports in Mathematics inScience and Engineering Academic Press Boston Mass USA1988
[19] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985
Then for 120582119860 = 119861(120582119865) the conditions in Theorem 5 aresatisfied Thus 120582119860 has a positive fixed point that is 119906 = 120582119860
has a positive solution and the proof is complete
We consider the integral equation
119906 (119909) = int119866
119896 (119909 119910)(
119898
sum
119894=1
119886119894(119910) 119906(119910)
120572119894
+ 119902 (119910)
times (119906(119910)120574
minus 119906(119910)120575
minus 1199080))119889119910
(31)
where 119866 is a bounded closed domain in 119877119899 and 120572119894ge 0 119886
119894(119909)
119902(119909) isin 119871(119866 [0infin)) 119894 = 1 2 119898 119896(119909 119910) is nonnegativecontinuous on 119866 times 119866
Theorem 7 Suppose that among 120572119894(119894 = 1 2 119898) there
exists 1205721198940
gt 1 such that inf119909isin119866
1198861198940
(119909) gt 0 and there existnontrivial nonnegative functions 119886(119909) 119887(119909) isin 119862(119866) and apositive number 119888 120574 120575 119908
0such that
119888119886 (119909) 119887 (119910) le 119896 (119909 119910) le 119886 (119909)
119896 (119909 119910) le 119887 (119910) forall119909 119910 isin 119866
From (35) and (36) we know that (H1) is satisfied By (47)and (48) we obtain that (H2) and (H3) are satisfied Equations(49) and (50) imply that (H4) is satisfied Therefore usingTheorem 5 the integral equation (31) has a positive solutionin 119862(119866)
Acknowledgments
The authors are very grateful to the referee for his or hervaluable suggestions This research was supported by theNational Natural Science Foundation of China (10871116)the Doctoral Program Foundation of Education Ministry ofChina (20103705120002) Shandong Provincial Natural Sci-ence Foundation China (ZR2012AM006) and the Programfor Scientific Research Innovation Team in Colleges andUniversities of Shandong Province
References
[1] A J B Potter ldquoApplications of Hilbertrsquos projective metric tocertain classes of non-homogeneous operatorsrdquo The QuarterlyJournal of Mathematics Oxford Second Series vol 28 no 109pp 93ndash99 1977
[2] C Zhai and C Guo ldquoOn 120572-convex operatorsrdquo Journal ofMathematical Analysis and Applications vol 316 no 2 pp 556ndash565 2006
[3] Z Zhao and X Du ldquoFixed points of generalized 119890-concave(generalized 119890-convex) operators and their applicationsrdquo Jour-nal of Mathematical Analysis and Applications vol 334 no 2pp 1426ndash1438 2007
[4] J R L Webb ldquoRemarks on 1199060-positive operatorsrdquo Journal of
Fixed Point Theory and Applications vol 5 no 1 pp 37ndash452009
[5] Z Zhao ldquoMultiple fixed points of a sum operator and applica-tionsrdquo Journal of Mathematical Analysis and Applications vol360 no 1 pp 1ndash6 2009
[6] Z Zhao and X Chen ldquoFixed points of decreasing operators inordered Banach spaces and applications to nonlinear secondorder elliptic equationsrdquo Computers ampMathematics with Appli-cations vol 58 no 6 pp 1223ndash1229 2009
[7] C-B Zhai and X-M Cao ldquoFixed point theorems for 120591 minus 120593-concave operators and applicationsrdquo Computers ampMathematicswith Applications vol 59 no 1 pp 532ndash538 2010
[8] Z Zhao ldquoExistence and uniqueness of fixed points for somemixed monotone operatorsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 73 no 6 pp 1481ndash1490 2010
[9] Z Zhao ldquoFixed points of 120591 minus 120593-convex operators and applica-tionsrdquo Applied Mathematics Letters vol 23 no 5 pp 561ndash5662010
[10] R P Agarwal S R Grace and D OrsquoRegan ldquoExistence of pos-itive solutions to semipositone Fredholm integral equationsrdquoFunkcialaj Ekvacioj vol 45 no 2 pp 223ndash235 2002
Abstract and Applied Analysis 5
[11] K Q Lan ldquoPositive solutions of semi-positone Hammersteinintegral equations and applicationsrdquo Communications on Pureand Applied Analysis vol 6 no 2 pp 441ndash451 2007
[12] K Q Lan ldquoEigenvalues of semi-positoneHammerstein integralequations and applications to boundary value problemsrdquo Non-linear Analysis Theory Methods amp Applications vol 71 no 12pp 5979ndash5993 2009
[13] J R Graef and L Kong ldquoPositive solutions for third ordersemipositone boundary value problemsrdquo Applied MathematicsLetters vol 22 no 8 pp 1154ndash1160 2009
[14] J R LWebb andG Infante ldquoSemi-positone nonlocal boundaryvalue problems of arbitrary orderrdquoCommunications onPure andApplied Analysis vol 9 no 2 pp 563ndash581 2010
[15] A Dogan J R Graef and L Kong ldquoHigher order semipositonemulti-point boundary value problems on time scalesrdquo Comput-ers amp Mathematics with Applications vol 60 no 1 pp 23ndash352010
[16] D R Anderson and C Zhai ldquoPositive solutions to semi-positone second-order three-point problems on time scalesrdquoApplied Mathematics and Computation vol 215 no 10 pp3713ndash3720 2010
[17] X Xian D OrsquoRegan and C Yanfang ldquoStructure of positivesolution sets of semi-positone singular boundary value prob-lemsrdquo Nonlinear Analysis Theory Methods amp Applications vol72 no 7-8 pp 3535ndash3550 2010
[18] D J Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones vol 5 of Notes and Reports in Mathematics inScience and Engineering Academic Press Boston Mass USA1988
[19] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985
From (35) and (36) we know that (H1) is satisfied By (47)and (48) we obtain that (H2) and (H3) are satisfied Equations(49) and (50) imply that (H4) is satisfied Therefore usingTheorem 5 the integral equation (31) has a positive solutionin 119862(119866)
Acknowledgments
The authors are very grateful to the referee for his or hervaluable suggestions This research was supported by theNational Natural Science Foundation of China (10871116)the Doctoral Program Foundation of Education Ministry ofChina (20103705120002) Shandong Provincial Natural Sci-ence Foundation China (ZR2012AM006) and the Programfor Scientific Research Innovation Team in Colleges andUniversities of Shandong Province
References
[1] A J B Potter ldquoApplications of Hilbertrsquos projective metric tocertain classes of non-homogeneous operatorsrdquo The QuarterlyJournal of Mathematics Oxford Second Series vol 28 no 109pp 93ndash99 1977
[2] C Zhai and C Guo ldquoOn 120572-convex operatorsrdquo Journal ofMathematical Analysis and Applications vol 316 no 2 pp 556ndash565 2006
[3] Z Zhao and X Du ldquoFixed points of generalized 119890-concave(generalized 119890-convex) operators and their applicationsrdquo Jour-nal of Mathematical Analysis and Applications vol 334 no 2pp 1426ndash1438 2007
[4] J R L Webb ldquoRemarks on 1199060-positive operatorsrdquo Journal of
Fixed Point Theory and Applications vol 5 no 1 pp 37ndash452009
[5] Z Zhao ldquoMultiple fixed points of a sum operator and applica-tionsrdquo Journal of Mathematical Analysis and Applications vol360 no 1 pp 1ndash6 2009
[6] Z Zhao and X Chen ldquoFixed points of decreasing operators inordered Banach spaces and applications to nonlinear secondorder elliptic equationsrdquo Computers ampMathematics with Appli-cations vol 58 no 6 pp 1223ndash1229 2009
[7] C-B Zhai and X-M Cao ldquoFixed point theorems for 120591 minus 120593-concave operators and applicationsrdquo Computers ampMathematicswith Applications vol 59 no 1 pp 532ndash538 2010
[8] Z Zhao ldquoExistence and uniqueness of fixed points for somemixed monotone operatorsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 73 no 6 pp 1481ndash1490 2010
[9] Z Zhao ldquoFixed points of 120591 minus 120593-convex operators and applica-tionsrdquo Applied Mathematics Letters vol 23 no 5 pp 561ndash5662010
[10] R P Agarwal S R Grace and D OrsquoRegan ldquoExistence of pos-itive solutions to semipositone Fredholm integral equationsrdquoFunkcialaj Ekvacioj vol 45 no 2 pp 223ndash235 2002
Abstract and Applied Analysis 5
[11] K Q Lan ldquoPositive solutions of semi-positone Hammersteinintegral equations and applicationsrdquo Communications on Pureand Applied Analysis vol 6 no 2 pp 441ndash451 2007
[12] K Q Lan ldquoEigenvalues of semi-positoneHammerstein integralequations and applications to boundary value problemsrdquo Non-linear Analysis Theory Methods amp Applications vol 71 no 12pp 5979ndash5993 2009
[13] J R Graef and L Kong ldquoPositive solutions for third ordersemipositone boundary value problemsrdquo Applied MathematicsLetters vol 22 no 8 pp 1154ndash1160 2009
[14] J R LWebb andG Infante ldquoSemi-positone nonlocal boundaryvalue problems of arbitrary orderrdquoCommunications onPure andApplied Analysis vol 9 no 2 pp 563ndash581 2010
[15] A Dogan J R Graef and L Kong ldquoHigher order semipositonemulti-point boundary value problems on time scalesrdquo Comput-ers amp Mathematics with Applications vol 60 no 1 pp 23ndash352010
[16] D R Anderson and C Zhai ldquoPositive solutions to semi-positone second-order three-point problems on time scalesrdquoApplied Mathematics and Computation vol 215 no 10 pp3713ndash3720 2010
[17] X Xian D OrsquoRegan and C Yanfang ldquoStructure of positivesolution sets of semi-positone singular boundary value prob-lemsrdquo Nonlinear Analysis Theory Methods amp Applications vol72 no 7-8 pp 3535ndash3550 2010
[18] D J Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones vol 5 of Notes and Reports in Mathematics inScience and Engineering Academic Press Boston Mass USA1988
[19] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985
[11] K Q Lan ldquoPositive solutions of semi-positone Hammersteinintegral equations and applicationsrdquo Communications on Pureand Applied Analysis vol 6 no 2 pp 441ndash451 2007
[12] K Q Lan ldquoEigenvalues of semi-positoneHammerstein integralequations and applications to boundary value problemsrdquo Non-linear Analysis Theory Methods amp Applications vol 71 no 12pp 5979ndash5993 2009
[13] J R Graef and L Kong ldquoPositive solutions for third ordersemipositone boundary value problemsrdquo Applied MathematicsLetters vol 22 no 8 pp 1154ndash1160 2009
[14] J R LWebb andG Infante ldquoSemi-positone nonlocal boundaryvalue problems of arbitrary orderrdquoCommunications onPure andApplied Analysis vol 9 no 2 pp 563ndash581 2010
[15] A Dogan J R Graef and L Kong ldquoHigher order semipositonemulti-point boundary value problems on time scalesrdquo Comput-ers amp Mathematics with Applications vol 60 no 1 pp 23ndash352010
[16] D R Anderson and C Zhai ldquoPositive solutions to semi-positone second-order three-point problems on time scalesrdquoApplied Mathematics and Computation vol 215 no 10 pp3713ndash3720 2010
[17] X Xian D OrsquoRegan and C Yanfang ldquoStructure of positivesolution sets of semi-positone singular boundary value prob-lemsrdquo Nonlinear Analysis Theory Methods amp Applications vol72 no 7-8 pp 3535ndash3550 2010
[18] D J Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones vol 5 of Notes and Reports in Mathematics inScience and Engineering Academic Press Boston Mass USA1988
[19] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985
