Iterative Method For Approximating a Common Fixed Point of Infinite Family of Strictly Pseudo Contractive Mappings in Real Hilbert Spaces

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1 Iteratioal Joural of Computatioal ad Applied Mathematics. ISSN Volume 2, Number 2 (207), pp Research Idia Publicatios Iterative Method For Approimatig a Commo Fied Poit of Ifiite Family of Strictly Pseudo Cotractive Mappigs i Real Hilbert Spaces Mollalg Haile Taele * ad B. Krisha Reddy 2 Departmet of Mathematics, College of sciece Osmaia Uiversity, Hyderabad, Idia & Bahir Dar Uiversity, Bahir Dar, Ethiopia *Correspodig Author 2 Departmet of Mathematics, Uiversity College of Sciece Osmaia Uiversity, Hyderabad, Idia Let Abstract T : K H be a coutable ifiite family of -strictly Pseudo cotractive, uiformly wealy closed ad iward mappigs o a o empty, closed ad strictly cove subset K of a real Hilbert space H i to H with F F( T ) is o empty. Let (,) ad for each, h : K be h ( ) if 0 : ( ) T K.The for each, defied by, h ( ) K ma,, we defie the Krasoselsii-Ma type algorithm by ( ) T, where ma, h ( ),,2,... ad we prove the wea ad strog covergece of the sequece to a commo fied poit of the familyt. Also we prove the wea ad strog covergece theorems for the algorithm to the family of oepasive mappigs i uiformly cove Baach space, which is more geeral tha Hilbert space. Keywords ad phrases: Commo fied poit; strictly Pseudo cotractive mappig; oself mappig; Krasoselsii-Ma s iterative method; ifiite family of mappigs subect Classificatio: 47H09; 47H0 ; 47J25;47J05.

2 294 Mollalg Haile Taele ad B. Krisha Reddy. INTRODUCTION Fidig fied poit or commo fied poit (if it eists) is importat i the study of may real world problems, such as; iverse problems; the split feasibility problem ad the cove feasibility problem i sigal processig ad image recostructio ca both be formulated as a problem of fidig fied poits of certai operators(mappigs), respectively. I particular, -strictly pseudo cotractive mappigs are more applicable tha oepasive mappigs i solvig various problems.therefore, it is desirable to develop the algorithms for the class of strictly pseudo cotractive mappigs, which is a itermediate betwee the class of o epasive mappigs ad that of the class of pseudo cotractive mappigs i which the eact solutio of the oliear problem may ot be possible. The class of κ-strictly pseudo cotractive mappigs was first itroduced by Browder ad Petryshy i Hilbert spaces (see, for eample []). Sice the may research efforts have bee made for the study of fied poit ad commo fied poit for family of such mappigs. Here, we study the fied poit iterative method for approimatig a commo fied poit of coutable ifiite family of -strictly pseudo cotractive mappigs i Hilbert space settig ad its etesios. Let K be a oempty, closed ad cove subset of a Hilbert space H ad let T : K H be a mappig. The T is said to be oepasive if T Ty y for arbitrary, y K. Whereas T that : K H is called -strictly pseudo cotractive if there eists 0, such T Ty 2 y 2 ( I T) ( I T) y 2, y K. (.) We see that every oepasive mappig is strictly pseudo cotractive, hece the class of -strictly pseudo cotractive mappigs is more geeral tha the class of oepasive mappigs. If the fied poit set F T) K : T ( is oempty ad T is self ( T : K K ) ad oepasive mappig, Ma i [9] itroduced a iterative method of the form ( ) T for ay K ad 0, (.2) Sice the, a umber of etesive research wors have bee made (See, for eample [3&6] ad their refereces). Reich i [2] studied the wea covergece of the Ma s algorithm i [3].Several attempts have bee made i lowerig the requiremet for the mappig to be self-mappig by assumig T to be o-self at the cost of additioal requiremets o the sequece ad o the domai K (see, for eample, [0,3,5,8,9&2] ad their refereces). However the study was usig the calculatio of metric proectio P : H K which is costly ad i may cases it requires approimatio techique. However, Colao ad Mario i [4] itroduced a ew techique for the coefficiets ad they have proved that the Krasoselsii Ma

3 Iterative Method For Approimatig a Commo Fied Poit of Ifiite Family 295 algorithm (.2) is well-defied for choice of the sequece. They also have proved both wea ad strog covergece results for the algorithm (.2) whe K is a strictly cove subset of H ad T is iward. To be precise, we put their result as follow. They defie iward mappig as; Defiitio. A mappig coditio) if for ay K T : K H is said to be iward (or to satisfy the iward T IK( ) c( u ) : c & u K ad T is said to, IK( satisfy wealy iward coditio if T )( the closure of IK( ). Theorem. CM [3] ) Let K be a cove, closed ad oempty subset of a Hilbert space H ad T : K H be a mappig ad let for ay give K, h : K be h( ) if 0 : ( ) T K.The the algorithm defied by defied by 0 K 0 ma, h( 0), 2 ( ) T ma, h is well-defied ad assume that; K is strictly cove set,t is oepasive, oself ad Iward mappig with F(T) is o empty, the coverges wealy to p F F(T). Moreover, if ( ), the the covergece is strog. Meawhile, they proposed a ope questio to approimatio for a commo fied poit of a coutable family of oself ad oepasive mappigs. Motivated by the wor of Colao ad Mario ad their ope questio, several attempts have bee made to geeralize the theorem of CM to approimatio for fiite or ifiite family of oself ad oepasive mappigs (see for eample,[8 ad 6 ] respectively ad their refereces ). To metio a few, Haile ad Reddy i [8] costructed Krasoselsii Ma type algorithm ad prove wea ad strog covergece theorems for approimatig a commo fied poit of the fiite family of oself, oepasive ad iward mappigs. Moreover, it was earlier Gao et al i [6] costructed Krasoselsii Ma type algorithm for approimatig a commo fied poit of coutable ifiite family of oself, oepasive ad iward mappigs. They also proved wea ad strog covergece theorem by imposig additioal coditios such as; (i) T : K H to be uiformly wealy closed (ii) The pair (F, K) satisfies S- coditio. where the coditios (i) ad (ii) ca be redefied as;

4 296 Mollalg Haile Taele ad B. Krisha Reddy Defiitio.2 Let F, K be two closed ad cove oempty sets i a Hilbert space H ad F K. For ay sequece K coverges strogly to a elemet, if \ F, implies that K, is ot Feer-mootoe with respect to the set F K, we called that, the pair (F, K) satisfies S-coditio. T : Defiitio.3 Let K H be sequece of mappigs with oempty commo fied poit set F F( T ). The the familyt is said to be uiformly wealy K such that lim T 0 the wea closed if for ay coverget sequece cluster Poits of K belog to F. To be precise we put the result of Guo et al i [6] i the followig theorem. Theorem.2 GLY Let K be a cove, closed ad oempty subset of a Hilbert space H ad let T : K H be a uiformly wealy closed coutable family of o self oepasive Mappigs. For ay K, h : K be defied by 0 : ( T K. The, the sequece h ( ) if ) algorithm, K ma, h ( ), 2 ( ) T ; ma, h. defie by the is well-defied. Let that K be strictly cove ad each T satisfies the iward coditio ad such that F F( T ) is oempty. The if there eist a, b (0, ) such that [ a, b for all wealy coverges to a commo fied poit p F. ] Moreover, if, the ( ) ad (F, K) satisfies S-coditio, the covergece is strog. We raise ope questio that, is it possible to eted the theorem of GLY for the class of -strictly pseudo cotractive mappigs which is more geeral class tha that of the class of oepasive mappigs which has ot bee studied?. Approimatig a commo fied poit for the class of -strictly strictly pseudo cotractive mappigs has bee etesively studied for fiite ad ifiite family as well. (See, for eample [2,7,

5 Iterative Method For Approimatig a Commo Fied Poit of Ifiite Family 297 8,,4&20] ad their refereces). However, all the studies was for self mappigs. Cosequetly, Haile ad Reddy i [7] costructed Krasoselsii Ma type algorithm for approimatig a commo fied poit for fiite family of -strictly pseudo cotractive mappigs. They also proved wea ad strog covergece theorem ad proposed a ope questio for the possibility of approimatig commo fied poit for the coutable ifiite family. Thus, it is the purpose of this paper to approimate a commo fied poit for the coutable ifiite family of -strictly pseudo cotractive, oself, iward mappigs i Hilbert spaces which is a positive aswer to our questio. 2. PRELIMINARY CONCEPTS T : Let K be a o empty subset of a real Hilbert space ad mappigs, the we shall eed the followig assumptios; K H be family of Lemma 2. (See, for eample [4] lemma 3.& [6] lemma 3.) Let for each,2,..., : K H be o self mappigs. If we defie h : C by T K h ( ) if 0, : ( ) T K, the a) for ay K, h ( ) 0, b) for ay K ad (), ad h ( ) 0 if ad oly if ( ) K ; h, ( ) T ( ) K ; c) If T is iward mappig,the h ( ) for ay K ; d) If T K, the h ( ) ( h ( )) T K Lemma 2.2 (see, for eample, Reich [2] ) Let, y i a uiformly cove Baach space E be two sequeces, if there eists a costat r 0 such that lim sup r, lim sup r ad lim ( ) y r, where [, ] (0, ) for some y (0,), the y 0. Lemma 2.3 (Etesio of lemma 2. for oself mappig [7]). Let K be a closed ad cove subset of a Hilbert space H. Let T : K H be a mappig o K. The if T is a κ-strict-pseudo cotractive, the T satisfies the Lipschitz coditio T Ty y, for all, y K. Lemma 2.4 (See, for eample [7]) Let T : K H be -strictly pseudo cotractive for some (0, ) ad (,), the T : K H defied by T ( ) T is o epasive ad F( T ) F( T). T

6 298 Mollalg Haile Taele ad B. Krisha Reddy Defiitio 2. (see, for eample [3] pp 6) A uiformly cove space E is a ormed space E for which for every 0 2, there is a 0, such that for every y, y S E :, if y ( y), the 2.For each, y E the modulus of coveity of E is defied by y E ( t) if, y & y t,0 t 2 ad 2 E is said to be uiformly cove if ( t) 0 for all 0 t 2. E Hilbert spaces, the Lebesgue L p, the sequece l p, for p (, ) are eamples of uiformly cove Baach spaces. For p 2 L p ad l p are ot Hilbert spaces. 3. MAIN RESULT Let T : K H be -strictly pseudo cotractive, oself, iward mappigs, the our obective is to costruct iterative method for approimatig a commo fied poit of the family. We will have the followig mai theorem. T : K H be uiformly wealy closed. If for each,2,... ad (0, ), T is defied by T ( ), the T is uiformly wealy closed. Lemma 3. Let T : Proof: Suppose K H is uiformly wealy closed. Thus, for ay sequece i K such that wealy ad T 0 strogly, the F F( T ). Suppose T 0strogly, the, sice T ( ) ( ( ) T ) T ad (0, ) we have T 0 strogly, hece F F( T ). Sice, F( T ) F( T ), we have F F( T ). This completes the proof of the lemma.

7 Iterative Method For Approimatig a Commo Fied Poit of Ifiite Family 299 Theorem. 3.2 Let K be a o empty, closed ad cove subset of a real Hilbert space H ad let : i K H be a uiformly wealy closed coutable family of oself,- T i strictly pseudo cotractive ad iward mappigs with i F F( ) is o empty. Let (,) ad for i,2,..., let T i ( ) Ti ad we defie h ( ) if 0 : ( ) T K. Let N F F( ) is oempty. The the i sequece give by K, ma, h ( ), ( ) T ma, h i i is well-defied ad if [, ] (0, ) (0,) coverges wealy to some elemet p of N F F( T i T i for some T ). Moreover, if ( ) ad (F, K) satisfies S-coditio, the the covergece is strog. Proof: For each (0, ), ad for each, T is - strictly pseudo cotractive, iward mappig, the by lemma 3.6 ad theorem 3.7 i [7] T ( ) T is iward ad oepasive mappigs. Thus T is uiformly wealy closed, oself, o epasive ad iward mappigs. Hece by theorem.2 i [20] ad F F( T ) F( T ), we complete the proof. The result ca be eteded i to more geeral spaces such as real uiformly cove Baach spaces with the assumptios of opial s coditio ; Let K be a o empty subset of a real Baach space E. The we shall have the followig defiitio; Defiitio 3. A mappig coditio) if for ay K T : K E is said to be iward (or to satisfy the iward T IK( ) c( u ) : c & u K ad T is said to, IK( satisfy wealy iward coditio if T )( the closure of IK( ). Let T : K E h ) if [0,]: be family of mappigs ad ( ( ) T K. The, we will have the followig theorem. Theorem 3.3 Let K be a cove, closed ad oempty subset of a real uiformly cove Baach space E ad let : K E be a uiformly wealy closed coutable T

8 300 Mollalg Haile Taele ad B. Krisha Reddy family of o-self ad oepasive mappigs. The the algorithm defied i theorem.2 is well-defied. Let that K be strictly cove ad each T satisfies the iward coditio ad such that F F( T ) is oempty. The if there eist a, b (0, ) such that [ a, b] for all wealy coverges to a commo fied poit, the p F provided that E satisfies opial s coditio. Moreover, if ( ) ad (F, K) satisfies S-coditio, the covergece is strog. Proof. Let p F p F( T p ( ) T ) Ø. The p Thus p is decreasig ad bouded below, ad hece coverges to some r 0. Thus, lim p r lim p ( p) ( )( T p). Sice Thus, T p T T p p. lim sup T p r, hece by the lemma 2.2 we have lim T 0. Moreover, the sequece is bouded, hece has a wealy coverget subsequece which coverges wealy to K wealy closed, F., sice K is closed. Sice It remais to show wealy. Suppose ot, there is of such that q, similarly q F. Suppose p q. Sice E satisfies opial s coditio i T is uiformly lim p lim p lim q lim q lim p lim p which is a cotradictio.

9 Iterative Method For Approimatig a Commo Fied Poit of Ifiite Family 30 Thus, F wealy. Moreover, if bouded ad Thus, ( ), the lim.sice the sequeces ( ) T, we have lim 0 ad. is Cauchy sequece i E, sice K is closed subset of E ad Coverges i orm to some K. It suffices to show that F. T are is i K, For each, T is iward implies that h ( ), thus for [ h ( ),) we have Sice lim ( ) T K. ad ma, h such that lim h ( )., there is a subsequece ( ) Sice h ( ) ( ) h, h h T K ( ) ( ( )) lim ( h ( h T ) ( ( )) ), thus by lemma 2. K. F, the sequece of Sice K i K is feer mootoe with respect to F ad (F,K) satisfies S-coditio, F. Therefore, F strogly, which completes the proof. ad 4. CONCLUSION Our theorems geeralize may results such as our theorem 3.2 geeralize theorem.2 to the class of -strictly pseudo cotractive mappigs,which is more geeral class tha the class of o epasive mappigs. Theorem 3.3 geeralizes theorem.2 to uiformly cove Baach space, which is more geeral tha Hilbert space. Meawhile, we raise ope questios; Questio Is it possible to eted theorem 3.2 ad 3.3 to uiformly smooth Baach spaces, refleive Baach spaces ad geeral Baach spaces? If so uder what coditios? Questio 2 Is it possible to eted Theorem 3.2 ad 3.3 to the class of Pseudo cotractive mappigs? If so uder what coditios?

10 302 Mollalg Haile Taele ad B. Krisha Reddy AUTHORS CONTRIBUTIONS Both authors cotributed equally ad sigificatly i writig this article. Both authors read ad approved the fial mauscript. COMPETING INTERESTS The authors declare that they have o competig iterests. REFERENCES [] BROWDER.F.E, PETRYSHYN.W.V. Costructio of Fied Poits of Noliear Mappigs i Hilbert Space. Joural of Mathematical aalysis ad applicatios. 20, (967). [2] CHE.C, LI.M, PAN.X Covergece Theorems for Equilibrium Problems ad Fied-Poit Problems of a Ifiite Family of i-strictly Pseudo cotractive Mappig i Hilbert Spaces, Joural of Applied Mathematics,202, Article ID 46476, doi:0.55/202/46476 (202). [3] CHIDUME. C.E. Geometric Properties of Baach Spaces ad Noliear Iteratios. Lecture Notes i Mathematics, vol965. Spriger, Berli (2009). [4] COLAO V, MARINO G. Krasoselsii Ma method for o-self mappigs. Fied Poit Theory Appl. 205:3(205). [5] GUO.M, LI. X, SU.Y. O a ope questio of V. Colao ad G. Mario preseted i the paper Krasoselsii Ma method for o self mappigs SprigerPlus,5:328.doi:0.86/s (206). [6] HICKS TL, KUBICEK JD. O the Ma iteratio process i a Hilbert space. J Math Aal Appl. 59(3): (977). [7] KATCHANG.P, KUMAM.P. A iterative algorithm for commo fied poits for oepasive semigroups ad strictly pseudo-cotractive mappigs with optimizatio problems, J Glob Optim (203). [8] KIM.J.K, BUONG.N. A ew iterative method for equilibrium problems ad fied poit problems for ifiite family of oself strictly pseudo cotractive mappigs, Fied Poit Theory ad Applicatios, DOI: 0.86/ (203). [9 ] MANN.WR. Mea value methods i iteratio. Proc Am Math Soc 4(3):506 5(953). [0] MARINO G, TROMBETTA G. O approimatig fied poits for oepasive mappigs. Idia J Math. 34:9 98(992). [] QIN.X, SHANG.M, QING.Y. Commo fied poits of a family of strictly pseudo cotractive mappigs, Fied Poit Theory ad Applicatios, DOI: 0.86/ (203).

11 Iterative Method For Approimatig a Commo Fied Poit of Ifiite Family 303 [2] REICH.S. Wea covergece theorems for oepasive mappigs i Baach spaces. J Math Aal Appl. 67(2): (979). [3] SONG Y, CHEN R. Viscosity approimatio methods for oepasive oself-mappigs. J Math Aal Appl. 32():36 326(2006). [4] SU.Y,LI.S. Composite implicit iteratio process for commo fied poits of a fiite family of strictly pseudo cotractive maps, J. Math. Aal. Appl.320, (2006). [5] TAKAHASHI W, KIM G-E. Strog covergece of approimats to fied poits of oepasive oself-mappigs i Baach spaces. Noliear Aal. 32(3): (998). [6] TAKELE.M.H, REDDY.B.K. Approimatio of commo fied poit of fiite family of o self ad oepasive mappigs i Hilbert space; Idia our. Mathematics ad Mathematical scieces, Accepted 3() (207). [7] TAKELE.M.H. AND REDDY. B.K. Fied poit theorems for approimatig a commo fied poit for a family of oself, strictly pseudo cotractive ad iward mappigs i real Hilbert spaces, commuicated (207). [8] XU H-K, YIN X-M. Strog covergece theorems for oepasive oselfmappigs. Noliear Aal 24(2): (995). [9] XU.H-K. Approimatig curves of oepasive oself-mappigs i Baach spaces. Comptes Redus Acad Sci Paris Ser I Math. 325(2):5 56(997). [20] YANG.L.P. Modified Noor iteratios for ifiite family of strict pseudocotractio mappigs, Bulleti of the Iraia Mathematical Society, 37 (), 43-6 (20). [2] ZHOU H, WANG P. Viscosity approimatio methods for oepasive oself-mappigs without boudary coditios. Fied Poit Theory Appl. 204:6(204).

12 304 Mollalg Haile Taele ad B. Krisha Reddy