Some iterative algorithms for k-strictly pseudo-contractive mappings in a CAT (0) space

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1 Some iterative algorithms for k-strictly pseudo-cotractive mappigs i a CAT 0) space AYNUR ŞAHİN Sakarya Uiversity Departmet of Mathematics Sakarya, TURKEY ayuce@sakarya.edu.tr METİN BAŞARIR Sakarya Uiversity Departmet of Mathematics Sakarya, TURKEY basarir@sakarya.edu.tr Abstract: I this paper, we prove the -covergece theorems of the cyclic algorithm ad the ew multi-step iteratio for k-strictly pseudo-cotractive mappigs ad give also the strog covergece theorem of the modified Halper s iteratio for these mappigs i a CAT 0) space. Our results exted ad improve the correspodig recet results aouced by may authors i the literature. Key Words: CAT 0) space, fixed poit, strog covergece, -covergece, k-strictly pseudo-cotractive mappig, iterative algorithm. 1 Itroductio Let C be a oempty subset of a real Hilbert space X. Recall that a mappig T : C C is said to be k-strictly pseudo-cotractive if there exists a costat k [0, 1) such that T x T y 2 x y 2 +k I T )x I T )y 2 for all x, y C. A poit x C is called a fixed poit of T if x = T x. We will deote the set of fixed poits of T by F T ). Note that the class of k-strictly pseudocotractios icludes the class of oexpasive mappigs which are mappigs T o C such that T x T y x y, x, y C. That is, T is oexpasive if ad oly if T is 0-strictly pseudo-cotractive. The mappig T is also said to be pseudo-cotractive if k = 1 ad T is said to be strogly pseudo-cotractive if there exists a costat λ 0, 1) such that T λi is pseudo-cotractive. Clearly, the class of k-strictly pseudo-cotractive mappigs is the oe betwee classes of oexpasive mappigs ad pseudo-cotractive mappigs. Also we remark that the class of strogly pseudo-cotractive mappigs is idepedet from the class of k-strictly pseudo-cotractive mappigs see, e.g., [1]-[3]). Recetly, may authors have bee devoted the studies o the problems of fidig fixed poits for k-strictly pseudo-cotractive mappigs see, e.g., [4]-[10]). We defie the cocept of k-strictly pseudocotractive mappig i a CAT 0) space as follows. Let C be a oempty subset of a CAT 0) space X. A mappig T : C C is said to be k-strictly pseudo-cotractive if there exists a costat k [0, 1) such that dt x, T y) 2 dx, y) 2 + k d x, T x) + dy, T y)) 2 1) for all x, y C. Gürsoy, Karakaya ad Rhoades [11] itroduced a ew multi-step iteratio i a Baach space. Recetly, Başarır ad Şahi [12] modified this iteratio i a CAT 0) space as follows. For a arbitrary fixed order k 2, x 0 C, x +1 = )y 1 α T y 1, y 1 = 1 β 1 )y 2 β 1 T y 2, y 2 = 1 β 2 )y 3 β 2 T y 3,. y k 2 y k 1 or, i short, = 1 β k 2 = 1 β k 1 )y k 1 )x β k 1 x 0 C x +1 = )y 1 α T y 1, β k 2 T y k 1, T x, 0, y i = 1 β)y i i+1 βt i y i+1, i = 1, 2,..., k 2, y k 1 = 1 β k 1 )x β k 1 T x, 0. 2) By takig k = 3 ad k = 2 i 2), we obtai the SP-iteratio of Phuegrattaa ad Suatai [13] ad the two-step iteratio of Thiawa [14], respectively. E-ISSN: Volume 13, 2014

2 Acedo ad Xu [15] itroduced a cyclic algorithm i a Hilbert space. We modify this algorithm i a CAT 0) space. Let x 0 C ad α } be a sequece i [a, b] for some a, b 0, 1). The cyclic algorithm geerates a sequece x } i the followig way: x 1 = α 0 x 0 1 α 0 )T 0 x 0, x 2 = α 1 x 1 1 α 1 )T 1 x 1,. x N = α N 1 x N 1 1 α N 1 )T N 1 x N 1, x N+1 = α N x N 1 α N )T 0 x N,. or, shortly, x +1 = α x )T [] x, 0, 3) where T [] = T i, with i = modn), 0 i N 1. By takig T [] = T for all i 3), we obtai the Ma iteratio i [16]. I this paper, motivated by the above results, we prove the demiclosedess priciple for k-strictly pseudo-cotractive mappigs i a CAT 0) space. Also we preset the -covergece theorems of the cyclic algorithm ad the ew multi-step iteratio ad the strog covergece theorem of the modified Halper s iteratio which is itroduced for Hilbert space by Hu [17] for these mappigs i a CAT 0) space. 2 Prelimiaries o CAT 0) space A metric space X is a CAT 0) space if it is geodesically coected ad if every geodesic triagle i X is at least as thi as its compariso triagle i the Euclidea plae. It is well kow that ay complete, simply coected Riemaia maifold havig o-positive sectioal curvature is a CAT 0) space. Other examples iclude Pre-Hilbert spaces see [18]), Euclidea buildigs see [19]), R-trees see [20]), the complex Hilbert ball with a hyperbolic metric see [21]) ad may others. For a thorough discussio of these spaces ad of the fudametal role they play i geometry, we refer the reader to Bridso ad Haefliger [18]. Fixed poit theory i a CAT 0) space has bee first studied by Kirk see [22], [23]). He showed that every oexpasive mappig defied o a bouded closed covex subset of a complete CAT 0) space always has a fixed poit. Sice the the fixed poit theory i a CAT 0) space has bee rapidly developed ad may papers have appeared see, e.g., [24]-[32]). It is worth metioig that fixed poit theorems i a CAT 0) space specially i R-trees) ca be applied to graph theory, biology ad computer sciece see, e.g., [20], [33]-[36]). Let X, d) be a metric space. A geodesic path joiig x X to y X or more briefly, a geodesic from x to y) is a map c from a closed iterval [0, l] R to X such that c0) = x, cl) = y ad dct), ct )) = t t for all t, t [0, l]. I particular, c is a isometry ad dx, y) = l. The image of c is called a geodesic or metric) segmet joiig x ad y. Whe it is uique, this geodesic is deoted by [x, y]. The space X, d) is said to be a geodesic space if every two poits of X are joied by a geodesic ad X is said to be a uiquely geodesic if there is exactly oe geodesic joiig x to y for each x, y X. A geodesic triagle x 1, x 2, x 3 ) i a geodesic metric space X, d) cosist of three poits i X the vertices of ) ad a geodesic segmet betwee each pair of vertices the edges of ). A compariso triagle for geodesic triagle x 1, x 2, x 3 ) i X, d) is a triagle x 1, x 2, x 3 ) = x 1, x 2, x 3 ) i the Euclidea plae R 2 such that d R 2x i, x j ) = dx i, x j ) for i, j 1, 2, 3}. Such a triagle always exists see [18]). A geodesic metric space is said to be a CAT 0) space [18] if all geodesic triagles of appropriate size satisfy the followig compariso axiom: Let be a geodesic triagle i X ad be a compariso triagle for. The, is said to satisfy the CAT 0) iequality if for all x, y ad all compariso poits x, y, dx, y) d R 2x, y). If x, y 1, y 2 are poits i a CAT 0) space ad if y 0 is the midpoit of the segmet [y 1, y 2 ], the the CAT 0) iequality implies that d x, y 0 ) dx, y 1) dx, y 2) dy 1, y 2 ) 2. This is the CN) iequality of Bruhat ad Tits [37]. I fact see [18, p.163]), a geodesic metric space is a CAT 0) space if ad oly if it satisfies the CN) iequality. It is worth metioig that the results i a CAT 0) space ca be applied to ay CAT k) space with k 0 sice ay CAT k) space is a CAT k ) space for every k k see [18, p.165]). Let x, y X ad by Lemma 2.1iv) of [27] for each t [0, 1], there exists a uique poit z [x, y] such that dx, z) = tdx, y), dy, z) = 1 t)dx, y). 4) E-ISSN: Volume 13, 2014

3 From ow o, we will use the otatio 1 t) x ty for the uique poit z satisfyig 4). We ow collect some elemetary facts about CAT 0) spaces which will be used i sequel the proofs of our mai results. Lemma 1 Let X be a CAT 0) space. The i) see [27, Lemma 2.4]) for each x, y, z X ad t [0, 1], oe has d 1 t)x ty, z) 1 t)dx, z) + tdy, z), ii) see [27, Lemma 2.5]) for each x, y, z X ad t [0, 1], oe has d 1 t)x ty, z) 2 1 t)dx, z) 2 + tdy, z) 2 t1 t)dx, y) 2. 3 Demiclosedess priciple for k- strictly pseudo-cotractive mappigs I 1976 Lim [38] itroduced a cocept of covergece i a geeral metric space settig which is called - covergece. Later, Kirk ad Payaak [39] used the cocept of -covergece itroduced by Lim [38] to prove o the CAT 0) space aalogs of some Baach space results which ivolve weak covergece. Also, Dhompogsa ad Payaak [27] obtaied the -covergece theorems for the Picard, Ma ad Ishikawa iteratios i a CAT 0) space for oexpasive mappigs uder some appropriate coditios. We ow give the defiitio ad collect some basic properties of the -covergece. Let X be a CAT 0) space ad x } be a bouded sequece i X. For x X, we set r x, x }) = lim sup dx, x ). The asymptotic radius r x }) of x } is give by r x }) = if r x, x }) : x X}. The asymptotic ceter A x }) of x } is the set A x }) = x X : r x, x }) = r x })}. It is kow that i a complete CAT 0) space, A x }) cosists of exactly oe poit see [40, Propositio 7]). Defiitio 2 [38], [39]) A sequece x } i a CAT 0) space X is said to be -coverget to x X if x is the uique asymptotic ceter of u } for every subsequece u } of x }. I this case, we write -lim x = x ad x is called the -limit of x }. Lemma 3 i) Every bouded sequece i a complete CAT 0) space always has a -coverget subsequece. see [39, p.3690]) ii) Let C be a oempty closed covex subset of a complete CAT 0) space ad let x } be a bouded sequece i C. The the asymptotic ceter of x } is i C. see [41, Propositio 2.1]) Lemma 4 [27, Lemma 2.8]) If x } is a bouded sequece i a complete CAT 0) space with Ax }) = x}, u } is a subsequece of x } with Au }) = u} ad the sequece dx, u)} is coverget the x = u. Let C be a closed covex subset of a CAT 0) space X ad x } be a bouded sequece i C. We deote the otatio x } w Φw) = if Φx) 5) x C where Φx) = lim sup dx, x). Najaras ad Payaak [42] gave a coectio betwee the covergece ad -covergece. Propositio 5 [42, Propositio 3.12]) Let C be a closed covex subset of a CAT 0) space X ad x } be a bouded sequece i C. The -lim x = p implies that x } p. The purpose of this sectio is to prove demiclosedess priciple for k -strictly pseudo-cotractive mappigs i a CAT 0) space by usig the covergece defied i 5). Theorem 6 Let C be a oempty closed covex subset of a complete CAT 0) space X ad T : C C be a [ k-strictly ) pseudo-cotractive mappig such that k 0, 1 2 ad F T ). Let x } be a bouded sequece i C such that -lim x = w ad lim dx, T x ) = 0. The T w = w. Proof: By the hypothesis, -lim x = w. From Propositio 5, we get x } w. The we obtai Ax }) = w} by Lemma 3 ii) see [42]). Sice lim dx, T x ) = 0, the we get Φx) = lim sup dx, x) = lim sup dt x, x) 6) for all x C. I 6) by takig x = T w, we have ΦT w) 2 = lim sup dt x, T w) 2 lim supdx, w) 2 +kdx, T x ) + dw, T w)) 2 } lim sup dx, w) 2 +k lim sup dx, T x ) + dw, T w)) 2 = Φw) 2 + kdw, T w) 2 7) E-ISSN: Volume 13, 2014

4 The CN) iequality implies that d x, w T w 2 ) dx, w) dx, T w) dw, T w)2. Lettig ad takig superior limit o the both sides of the above iequality, we get ) w T w 2 Φ Φw) ΦT w)2 1 4 dw, T w)2. Sice Ax }) = w}, we have w T w Φw) 2 Φ 2 which implies that ) Φw) ΦT w)2 1 4 dw, T w)2. dw, T w) 2 2ΦT w) 2 2Φw) 2. 8) By 7) [ ad ) 8), we get 1 2k)dw, T w) 2 0. Sice k 0, 1 2, the we have T w = w as desired. Now, we prove the -covergece of the ew multi-step iteratio for k-strictly pseudo-cotractive mappigs i a CAT 0) space. Theorem 7 Let C be a oempty closed covex subset of a complete CAT 0) space X ad T : C C be a k-strictly [ ) pseudo-cotractive mappig such that k 0, 1 2 ad F T ). Let α } ad β} i, i = 1, 2,..., k 2 be sequeces i [a, b] for some a, b 0, 1) ad k < 1 b. Let x } be a sequece defied by 2). The the sequece x } is coverget to a fixed poit of T. Proof: Let p F T ). From 1), 2) ad Lemma 1, we have dx +1, p) 2 = d )y 1 α T y, 1 p) 2 )dy, 1 p) 2 + α dt y, 1 p) 2 α )dy, 1 T y) 1 2 )dy, 1 p) 2 +α dy, 1 p) 2 + kdy, 1 T y) 1 2} α )dy 1, T y 1 ) 2 = dy 1, p) 2 α ) k)dy 1, T y 1 ) 2 dy 1, p) 2. Also, we obtai dy, 1 p) 2 = d1 β)y 1 2 βt 1 y, 2 p) 2 1 β)dy 1, 2 p) 2 + βdt 1 y, 2 p) 2 1 β)dy 1, 2 p) 2 dy, 2 p) 2 + kdy, 2 T y) 2 2} +β 1 β 1 1 β 1 )dy 2, T y 2 ) 2 = dy 2, p) 2 β 1 1 β 1 ) k)dy 2, T y 2 ) 2 dy 2, p) 2. Cotiuig the above process we have dx +1, p) dy, 2 p)... dy k 1, p) dx, p). 9) This iequality guaratees that the sequece x } is bouded ad lim dx, p) exists for all p F T ). Let lim dx, p) = r. By usig 9), we get lim dyk 1, p) = r. By Lemma 1, we also have dy k 1 1 β k 1, p) 2 = d1 β k 1 )x β k 1 T x, p) 2 )dx, p) 2 + β k 1 dt x, p) 2 β k 1 1 β k 1 )dx, T x ) 2 1 β k 1 )dx, p) 2 dx, p) 2 + kdx, T x ) 2} +β k 1 β k 1 1 β k 1 )dx, T x ) 2 = dx, p) 2 β k 1 1 β k 1 ) k)dx, T x ) 2, which implies that dx, T x ) 2 1 [ dx, p) 2 dy k 1, p) 2]. a1 b) k) Thus lim dx, T x ) = 0. To show that the sequece x } is -coverget to a fixed poit of T, we prove that W x ) = A u }) F T ) u} x} ad W x ) cosists of exactly oe poit. Let u W x ). The, there exists a subsequece u } of x } such that Au }) = u}. By Lemma 3, there exists a subsequece v } of u } such that -lim v = v K. By Theorem 6, we have v F T ) ad by Lemma 4, we have u = v F T ). This shows that W x ) F T ). Now, we prove that W x ) cosists of exactly oe poit. Let u } E-ISSN: Volume 13, 2014

5 be a subsequece of x } with Au }) = u} ad let Ax }) = x}. We have already see that u = v ad v F T ). Fially, sice dx, v)} is coverget, we have x = v F T ) by Lemma 4. This shows W x ) = x}. This completes the proof. Also, we prove the -covergece of the cyclic algorithm for k-strictly pseudo-cotractive mappigs i a CAT 0) space. Theorem 8 Let C be a oempty closed covex subset of a complete CAT 0) space X ad N 1 be a iteger. Let, for each 0 i N 1, T i : C C be k i -strictly pseudo-cotractive mappigs for some 0 k i < 1 2. Let k = max k i; 0 i N 1}, α } be a sequece i [a, b] for some a, b 0, 1) ad k < a. Let F = N 1 i=0 F T i). For x 0 C, let x } be a sequece defied by 3). The the sequece x } is -coverget to a commo fixed poit of the family T i } N 1 i=0. Proof: Let p F. Usig 1), 3) ad Lemma 1, we have dx +1, p) 2 = dα x )T [] x, p) 2 α dx, p) 2 + )dt [] x, p) 2 α )dx, T [] x ) 2 α dx, p) 2 + ) dx, p) 2 + kdx, T [] x ) 2} α )dx, T [] x ) 2 = dx, p) 2 )α k)dx, T [] x ) 2 10) dx, p) 2. This iequality guaratees that the sequece x } is bouded ad lim dx, p) exists for all p F. By 10), we also have dx, T [] x ) 2 1 [dx, p) 2 dx +1, p) 2] )α k) 1 [ dx, p) 2 dx +1, p) 2]. 1 b)a k) Sice lim dx, p) exists, we obtai lim dx, T [] x ) = 0. The rest of the proof closely follows the proof of Theorem 7 ad is therefore omitted. 4 The strog covergece theorem for the modified Halper s iteratio I [17], Hu itroduced a modified Halper s iteratio. We modify this iteratio i a CAT 0) space as follows. For a arbitrary iitial value x 0 C ad a fixed achor u C, the sequece x } is defied by x+1 = α u )y, y = β 1 α x γ 1 α T x, 0, 11) where α }, β }, γ } are three real sequeces i 0, 1) satisfyig α + β + γ = 1. Clearly, the iterative sequece 11) is a atural geeralizatio of the well kow iteratios. i) If we take β = 0 for all i 11), the the sequece 11) reduces to the Halper s iteratio i [43]. ii) If we take α = 0 for all i 11), the the sequece 11) reduces to the Ma iteratio i [16]. I this sectio, we prove the strog covergece of the modified Halper s iteratio i a CAT 0) space. Recall that a cotiuous liear fuctioal µ o l, the Baach space of bouded real sequeces, is called a Baach limit if µ = µ1, 1,...) = 1 ad µa ) = µa +1 ) for all a } =1 l. Lemma 9 see [44, Propositio 2]) Let a } l be such that µa ) 0 for all Baach limits µ ad lim sup a +1 a ) 0. The, lim sup a 0. Lemma 10 Let C be a oempty closed covex subset of a complete CAT 0) space X, T : C C be a k-strictly pseudo-cotractive mappig with k [0, 1) ad S : C C be a mappig defied by Sz = kz 1 k) T z, for z C. Let u C be fixed. For each t [0, 1], the mappig S t : C C defied by S t z = tu 1 t) Sz = tu 1 t) kz 1 k) T z) for z C, has a uique fixed poit z t C, that is, z t = S t z t ) = tu 1 t) Sz t ). 12) Proof: As it has bee prove i [45], if T is a k- strictly pseudo-cotractive mappig with k [0, 1), S is a oexpasive mappig such that F S) = F T ). The, from Lemma 2.1 i [29], the mappig S t has a uique fixed poit z t C. E-ISSN: Volume 13, 2014

6 Lemma 11 Let X, C, T ad S be as i Lemma 10. The, F T ) if ad oly if z t } give by 12) remais bouded as t 0. I this case, the followig statemets hold: 1) z t } coverges to the uique fixed poit z of T which is earest to u; 2) du, z) 2 µdu, x ) 2 for all Baach limits µ ad all bouded sequeces x } with lim dx, T x ) = 0. Proof: If F T ), the we have F S) = F T ). Also, if lim dx, T x ) = 0, we obtai that dx, Sx ) = dx, kx 1 k) T x ) 1 k)dx, T x ) 0 as. Thus, from Lemma 2.2 i [29], the rest of the proof of this lemma ca be see. The followig lemma ca be foud i [46]. Lemma 12 see [46, Lemma 2.1]) Let a } be a sequece of o-egative real umbers satisfyig the coditio a +1 1 γ )a + γ σ, 0, where γ } ad σ } are sequeces of real umbers such that 1) γ } [0, 1] ad =1 γ =, 2) either lim sup σ 0 or =1 γ σ <. The, lim a = 0. We are ow ready to prove our mai result. Theorem 13 Let C be a oempty closed covex subset of a complete CAT 0) space X ad T : C C be a k-strictly pseudo-cotractive mappig such that 0 k < β 1 α < 1 ad F T ). Let x } be a sequece defied by 11). Suppose that α }, β } ad γ } satisfy the followig coditios: C1) lim α = 0, C2) =1 α =, C3) lim β k ad lim γ 0. The the sequece x } coverges strogly to a fixed poit of T. Proof: We divide the proof ito three steps. I the first step we show that x }, y } ad T x } are bouded sequeces. I the secod step we show that lim dx, T x ) = 0. Fially, we show that x } coverges to a fixed poit z F T ) which is earest to u. First step: Take ay p F T ), the, from Lemma 1 ad 11), we have dy, p) 2 β dx, p) 2 + γ dt x, p) 2 β γ ) 2 dx, T x ) 2 β dx, p) 2 + γ dx, p) 2 + kdx, T x ) 2) β γ ) 2 dx, T x ) 2 = dx, p) 2 γ ) β k dx, T x ) 2 dx, p) 2. Also, we obtai dx +1, p) 2 α du, p) 2 + ) dy, p) 2 α ) du, y ) 2 α du, p) 2 + ) dx, p) 2 γ ) } β k dx, T x ) 2 α ) du, y ) 2 = α du, p) 2 + ) dx, p) 2 γ k dx, T x ) 2 α ) du, y ) 2 13) α du, p) 2 + ) dx, p) 2 max du, p) 2, dx, p) 2}. By iductio, dx +1, p) 2 max du, p) 2, dx 0, p) 2}. This proves the boudedess of the sequece x }, which leads to the boudedess of T x } ad y }. Secod step: I fact, we have from 13) for some appropriate costat M > 0) that dx +1, p) 2 α du, p) 2 + ) dx, p) 2 γ k dx, T x ) 2 = α du, p) 2 dx, p) 2 ) + dx, p) 2 γ k dx, T x ) 2 E-ISSN: Volume 13, 2014

7 α M + dx, p) 2 γ k dx, T x ) 2, which implies that γ k dx, T x ) 2 α M dx, p) 2 dx +1, p) 2. 14) ) If γ β 1 α k dx, T x ) 2 α M 0, the dx, T x ) 2 α )M, γ β 1 α k ad hece the desired result is obtaied by the coditios C1) ad C3). ) If γ β 1 α k dx, T x ) 2 α M > 0, the followig 14), we have m =0 [ γ k dx 0, p) 2 dx m+1, p) 2 dx 0, p) 2. That is =0 Thus lim [ γ [ γ The we get Third step: obtai k k ] dx, T x ) 2 α M ] dx, T x ) 2 α M <. ] dx, T x ) 2 α M = 0. lim dx, T x ) = 0. 15) Usig the coditio C1) ad 15), we dx +1, x ) dx +1, T x ) + dt x, x ) α du, T x ) + )dy, T x ) +dt x, x ) ) β α du, T x ) + ) dx, T x ) +dt x, x ) = α du, T x ) + β + 1) dx, T x ) 0, as. Also, from 15), we have dx, y ) γ dx, T x ) 0, as. 16) Let z = lim t 0 z t, where z t is give by 12) i Lemma 10. The, z is the poit of F T ) which is earest to u. By Lemma 11 2), we have µ du, z) 2 du, x ) 2) 0 for all Baach limits µ. Let a = du, z) 2 du, x ) 2. Moreover, sice lim dx +1, x ) = 0, we get lim sup a +1 a ) = 0. By Lemma 9, we obtai lim sup du, z) 2 du, x ) 2) 0. 17) It follows from the coditio C1) ad 16) that lim sup du, z) 2 ) du, y ) 2) = lim sup By 17) ad 18), we have lim sup du, z) 2 du, x ) 2). 18) du, z) 2 ) du, y ) 2) 0. 19) We observe that dx +1, z) 2 α du, z) 2 + ) dy, z) 2 α ) du, y ) 2 α du, z) 2 + ) dx, z) 2 α ) du, y ) 2 = ) dx, z) 2 +α [du, z) 2 ) du, y ) 2]. It follows from the coditio C2) ad 19), usig Lemma 12, that lim dx, z) = 0. This completes the proof of Theorem 13. We obtai the followig corollary as a direct cosequece of Theorem 13. Corollary 14 Let X, C ad T be as Theorem 13. Let α } be a real sequece i 0, 1) satisfyig the coditios C1) ad C2). For a costat δ k, 1), a arbitrary iitial value x 0 C ad a fixed achor u C, let the sequece x } be defied by x +1 = α u ) δx 1 δ)t x ), 20) for all 0. The the sequece x } is strogly coverget to a fixed poit of T E-ISSN: Volume 13, 2014

8 Proof: If, i the proof of Theorem 13, we take β = )δ ad γ = )1 δ), the we get the desired coclusio. Remark 15 The results i this sectio cotai the strog covergece theorems of the iterative sequeces 11) ad 20) for oexpasive mappigs i a CAT 0) space. Also, our results cotai the correspodig theorems proved for these iterative sequeces i a Hilbert space. Ackowledgemets: This paper has bee preseted i The EUROPMENT coferece Pure Mathematics - Applied Mathematics 2014 i Veice, Italy, March 15-17, This paper was supported by Sakarya Uiversity Scietific Research Foudatio Project umber: ). Refereces: [1] F. E. Browder, Fixed poit theorems for ocompact mappigs i Hilbert spaces, Proc. Natl. Acad. Sci., USA, Vol. 53, 1965, pp [2] F. E. Browder, Covergece of approximats to fixed poits of oexpasive oliear mappigs i Baach spaces, Arch. Ratio. Mech. Aal., Vol. 24, 1967, pp [3] F. E. Browder, W. V. Petryshy, Costructio of fixed poits of oliear mappigs i Hilbert space, J. Math. Aal. Appl., Vol. 20, 1967, pp [4] I. Icha, Strog covergece theorems for a ew iterative method of k-strictly pseudocotractive mappigs i Hilbert spaces, Computers ad Mathematics with Applicatios, Vol. 58, 2009, pp [5] J. S. Jug, Some results o a geeral iterative method for k-strictly pseudo-cotractive mappigs, Fixed Poit Theory ad Applicatios, 2011:24, doi: / , 11 pages, [6] M. Tia ad L. Liu, A geeral iterative algorithm for equilibrium problems ad strict pseudocotractios i Hilbert spaces, WSEAS Trasactios o Mathematics, Vol. 10, 2011, pp [7] P. Dua ad J. Zhao, A geeral iterative algorithm for system of equilibrium problems ad ifiitely may strict pseudo-cotractios, WSEAS Trasactios o Mathematics, Vol. 11, 2012, pp [8] J. Zhao, C. Yag ad G. Liu, A ew iterative method for equilibrium problems, fixed poit problems of ifiitely oexpasive mappigs ad a geeral system of variatioal iequalities, WSEAS Trasactios o Mathematics, Vol. 11, 2012, pp [9] S. Li, L. Li, L. Zhag ad X. He, Strog covergece of modified Halper s iteratio for a k-strictly pseudocotractive mappig, Joural of Iequalities ad Applicatios, 2013:98, doi: / x , [10] Y. Qig, S. Y. Cho ad M. Shag, Strog covergece of a iterative process for a family of strictly pseudocotractive mappigs, Fixed Poit Theory ad Applicatios, 2013:117, doi: / , 10 pages, [11] F. Gürsoy, V. Karakaya ad B. E. Rhoades, Data depedece results of ew multi-step ad S- iterative schemes for cotractive-like operators, Fixed Poit Theory ad Applicatios, 2013:76, doi: / , [12] M. Başarır ad A. Şahi, O the strog ad δ- covergece of ew multi-step ad S-iteratio processes i a CAT0) space. J. Iequal. Appl., 2013:482, doi: / X , 2013, 13 pages. [13] W. Phuegrattaa ad S. Suatai, O the rate of covergece of Ma, Ishikawa, Noor ad SPiteratios for cotiuous fuctios o a arbitrary iterval, J. Comput. Appl. Math., Vol. 235, 2011, pp [14] S. Thiawa, Commo fixed poits of ew iteratios for two asymptotically oexpasive oself-mappigs i a Baach space, J. Comput. Appl. Math., Vol. 224, 2009, pp [15] G. L. Acedo ad H. K. Xu, Iterative methods for strict pseudo-cotractios i Hilbert spaces, Noliear Aal., Vol. 67, No. 7, 2007, pp [16] W. R. Ma, Mea value methods i iteratio, Proceedigs of the America Mathematical Society, Vol. 4, 1953, pp [17] L. G. Hu, Strog covergece of a modified Halper s iteratio for oexpasive mappigs, Fixed Poit Theory ad Applicatios, Vol. 2008, Article ID , 2008, 9 pages. [18] M. R. Bridso ad A. Haefliger, Metric Spaces of No-Positive Curvature, Vol. 319, Grudlehre der Mathematische Wisseschafte, Spriger, Berli, Germay, [19] K.S. Brow, Buildigs, Spriger, NewYork, NY, USA, [20] W. A. Kirk, Fixed poit theorems i CAT 0) spaces ad R-trees, Fixed Poit Theory ad Applicatios, Vol. 2004, No. 4, 2004, pp E-ISSN: Volume 13, 2014

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