On an open problem of Kyung Soo Kim

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1 Panyanak Fixed Point Theory and Applications ( :186 DOI /s ESEACH Open Access On an open problem of Kyung Soo Kim Bancha Panyanak* * Correspondence: bancha.p@cmu.ac.th Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 5000, Thailand Abstract We prove a convergence theorem of the Mann iteration scheme for a uniformly L-Lipschitzian asymptotically demicontractive mapping in a CAT(κ space with κ > 0. We also obtain a convergence theorem of the Ishikawa iteration scheme for a uniformly L-Lipschitzian asymptotically hemicontractive mapping. Our results provide a complete solution to an open problem raised by Kim (Abstr. Appl. Anal. 013:381715, 013. MSC: 47H09; 49J53 Keywords: Mann iteration; Ishikawa iteration; strong convergence; fixed point; CAT(κ space 1 Introduction oughly speaking, CAT(κ spaces are geodesic spaces of bounded curvature and generalizations of iemannian manifolds of sectional curvature bounded above. The precise definition is given below. The letters C, A, and T stand for Cartan, Alexandrov, and Toponogov, who have made important contributions to the understanding of curvature via inequalities for the distance function, and κ is a real number that we impose it as the curvature bound of the space. Fixed point theory in CAT(κ spaces was first studied by Kirk [, ]. His work was followed by a series of new works by many authors, mainly focusing on CAT( spaces (see e.g., [ ]. Since any CAT(κ space is a CAT(κ space for κ κ, all results for CAT( spaces immediately apply to any CAT(κ space with κ. However, there are only a few articles that contain fixed point results in the setting of CAT(κ spaces with κ >, because in this case the proof seems to be more complicated. The notion of uniformly L-Lipschitzian mappings, which is more general than the notion of asymptotically nonexpansive mappings, was introduced by Goebel and Kirk [ ]. In, Schu [ ] proved the strong convergence of Mann iteration for asymptotically nonexpansive mappings in Hilbert spaces. Qihou [ ] extended Schu s result to the general setting of asymptotically demicontractive mappings and also obtained the strong convergence of Ishikawa iteration for asymptotically hemicontractive mappings. ecently, Kim [ ] proved the analogous results of Qihou in the framework of the so-called CAT( spaces. Precisely, Kim obtained the following theorems. Theorem A Let (X, ρ be a complete CAT( space, C be a nonempty bounded closed convex subset of X, and T : C C be a completely continuous and uniformly L-Lipschitzian 015 Panyanak. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Panyanak Fixed Point Theory and Applications ( :186 Page of 1 asymptotically demicontractive mapping with constant k [0, 1 and sequence {a n } in [1, such that n=1 (a n 1<. Let {α n} be a sequence in [ε,1 k ε] for some ε >0. Given x 1 C, define the iteration scheme { } by +1 =(1 α n α n T n, n 1. Then { } converges strongly to a fixed point of T. Theorem B Let (X, ρ be a complete CAT(0 space, let C be a nonempty bounded closed convex subset of X, and let T : C C be a completely continuous and uniformly L-Lipschitzian asymptotically hemicontractive mapping with sequence {a n } in [1, such that n=1 (a n 1<. Let {α n }, {β n } [0, 1] be such that ε α n β n bforsomeε >0and b (0, 1+L 1. Given x L 1 C, define the iteration scheme { } by +1 =(1 α n α n T n y n, y n =(1 β n β n T n, n 1. Then { } converges strongly to a fixed point of T. In [9], the author raised the following problem. Problem Can we extend Theorems A and B to the general setting of CAT(κspaceswith κ >0? The purpose of the paper is to solve this problem. Our main discoveries are Theorems 3. and 3.6. Preliminaries Let (X, ρ beametricspace.ageodesic path joining x X to y X (or, more briefly, a geodesic from x to y isamapc from a closed interval [0, l] to X such that c(0 = x, c(l =y, andρ(c(t, c(t = t t for all t, t [0, l]. In particular, c is an isometry and ρ(x, y = l. The image c([0,l] of c is called a geodesic segment joining x and y. When it is unique this geodesic segment is denoted by [x, y]. This means that z [x, y] ifandonlyif there exists α [0, 1] such that ρ(x, z =(1 αρ(x, y and ρ(y, z =αρ(x, y. In this case, we write z = αx (1 αy. Thespace(X, ρ issaidtobeageodesic space (D-geodesic space ifeverytwopointsofx (every two points of distance smaller than D are joined by a geodesic, and X is said to be uniquely geodesic (D-uniquely geodesic if there is exactly one geodesic joining x and y for each x, y X (for x, y X with ρ(x, y<d. A subset C of X is said tobe convex if C includes every geodesic segment joining any two of its points. The set C is said to be bounded if diam(c:=sup { ρ(x, y:x, y C } <. Now we introduce the model spaces M n κ, for more details on these spaces the reader is referred to [30, 31]. Let n N. WedenotebyE n the metric space n endowed with the

3 Panyanak Fixed Point Theory and Applications ( :186 Page 3 of 1 usual Euclidean distance. We denote by ( the Euclidean scalar product in n,thatis, (x y=x 1 y y n, wherex =(x 1,...,, y =(y 1,...,y n. Let S n denote the n-dimensional sphere defined by S n = { x =(x 1,...,+1 n+1 :(x x=1 }, with metric d S n(x, y=arccos(x y, x, y S n. Let E n,1 denote the vector space n+1 endowed with the symmetric bilinear form which associates to vectors u =(u 1,...,u n+1 andv =(v 1,...,v n+1 therealnumber u v defined by u v = u n+1 v n+1 + n u i v i. i=1 Let H n denote the hyperbolic n-space defined by H n = { u =(u 1,...,u n+1 E n,1 : u u = 1,u n+1 >0 }, with metric d H n such that cosh d H n(x, y= x y, x, y H n. Definition.1 Given κ,wedenotebymκ n the following metric spaces: (i if κ =0then M0 n is the Euclidean space En ; (ii if κ >0then Mκ n is obtained from the spherical space Sn by multiplying the distance function by the constant 1/ κ; (iii if κ <0then Mκ n is obtained from the hyperbolic space Hn by multiplying the distance function by the constant 1/ κ. A geodesic triangle (x, y, z in a geodesic space (X, ρ consists of three points x, y, z in X (the vertices of and three geodesic segments between each pair of vertices (the edges of. A comparison triangle for a geodesic triangle (x, y, zin(x, ρ is a triangle ( x, ȳ, z in Mκ such that ρ(x, y=d M κ ( x, ȳ, ρ(y, z=d M κ (ȳ, z and ρ(z, x=d M κ ( z, x. If κ 0 then such a comparison triangle always exists in Mκ.Ifκ >0thensuchatriangle exists whenever ρ(x, y +ρ(y, z +ρ(z, x <D κ,whered κ = π/ κ.apoint p [ x, ȳ] is called a comparison point for p [x, y]ifρ(x, p=d M κ ( x, p. Ageodesictriangle (x, y, zinx is said to satisfy the CAT(κ inequality if for any p, q (x, y, z and for their comparison points p, q ( x, ȳ, z, one has ρ(p, q d M κ ( p, q. Definition. If κ 0, then X is called a CAT(κ space if X is a geodesic space such that all of its geodesic triangles satisfy the CAT(κinequality.

4 Panyanak Fixed Point Theory and Applications ( :186 Page 4 of 1 If κ >0,thenX is called a CAT(κ space if X is D κ -geodesic and any geodesic triangle (x, y, zinx with ρ(x, y+ρ(y, z+ρ(z, x<d κ satisfies the CAT(κinequality. Notice that in a CAT(0 space (X, ρ, if x, y, z X then the CAT(0 inequality implies (CN ρ ( x, 1 y 1 z 1 ρ (x, y+ 1 ρ (x, z 1 4 ρ (y, z. This is the (CN inequality of Bruhat and Tits [3]. This inequality is extended by Dhompongsa and Panyanak [9]as (CN ρ ( x,(1 αy αz (1 αρ (x, y+αρ (x, z α(1 αρ (y, z for all α [0, 1] and x, y, z X.Infact,ifX is a geodesic space then the following statements are equivalent: (i X is a CAT(0 space; (ii X satisfies (CN; (iii X satisfies (CN. Let (0, ]. ecall that a geodesic space (X, ρissaidtobe-convex for [33] if for any three points x, y, z X,wehave ρ ( x,(1 αy αz (1 αρ (x, y+αρ (x, z α(1 αρ (y, z. (1 It follows from (CN that a geodesic space (X, ρisacat(0 space if and only if (X, ρ is -convex for =. The following lemma generalizes Proposition 3.1 of Ohta [33]. Lemma.3 Let κ be an arbitrary positive real number and (X, ρ be a CAT(κ space with diam(x π/ η κ for some η (0, π/. Then (X, ρ is -convex for =(π η tan(η. Proof Let x, y, z X. Sincediam(X< π κ, ρ(x, y+ρ(x, z+ρ(y, z<d κ where D κ = π κ. Let (x, y, z be the geodesic triangle constructed from x, y, z and ( x, ȳ, z its comparison triangle. Then ρ(x, y=d M κ ( x, ȳ, ρ(x, z=d M κ ( x, z and ρ(y, z=d M κ (ȳ, z. ( It is sufficient to prove (1 onlythecaseofα = 1/. Let a = d S ( x, ȳ, b = d S ( x, z, c = d S (ȳ, z/, and d = d S ( x, 1 ȳ 1 zanddefine f (a, b, c:= c ( 1 a + 1 b d. By using the same argument in the proof of Proposition 3.1 in [33], we obtain d S ( x, 1 ȳ 1 z 1 d S ( x, ȳ+ 1 d S ( x, z where =(π η tan(η. This implies that d M κ ( ( 1 d 4 S (ȳ, z, ( x, 1ȳ 1 z 1 d Mκ ( x, ȳ+ 1 ( ( 1 d Mκ ( x, z d 4 Mκ (ȳ, z. (3

5 Panyanak Fixed Point Theory and Applications ( :186 Page 5 of 1 By (and(3, we get ρ ( x, 1 y 1 z 1 ρ (x, y+ 1 ρ (x, z ( ( 1 ρ (y, z. 4 This completes the proof. The following lemma is also needed. Lemma.4 Let {s n } and {t n } be sequences of nonnegative real numbers satisfying s n+1 s n + t n for all n N. If n=1 t n < and {s n } has a subsequence converging to 0, then lim n s n =0. Definition.5 Let C be a nonempty subset of a CAT(κspace(X, ρandt : C C be a mapping. We denote by F(T thesetofallfixedpointsoft, i.e., F(T={x C : x = Tx}. Then T is said to (i be completely continuous if T is continuous and for any bounded sequence { } in C, {T } has a convergent subsequence in C; (ii be uniformly L-Lipschitzian if thereexistsa constant L >0such that ρ ( T n x, T n y Lρ(x, y for all x, y C and all n N; (iii be asymptotically demicontractive if F(T andthereexist k [0,1 anda sequence {a n } with lim n a n =1such that ρ ( T n x, p a n ρ (x, p+kρ ( x, T n x for all x C, p F(T and n N; (iv be asymptotically hemicontractive if F(T and there exists a sequence {a n } with lim n a n =1such that ρ ( T n x, p a n ρ (x, p+ρ ( x, T n x for all x C, p F(T and n N. It follows from the definition that every asymptotically demicontractive mapping is asymptotically hemicontractive. For more details as regards these classes of mappings the reader is referred to [7, 8]. Let C be a nonempty convex subset of a CAT(κ space(x, ρ andt : C C be a mapping. Given x 1 C. Algorithm 1 The sequence { } defined by +1 =(1 α n α n T n y n, y n =(1 β n β n T n, n 1, is called an Ishikawa iterative sequence (see [34]. If β n =0foralln N, then Algorithm 1 reduces to the following.

6 Panyanak Fixed Point Theory and Applications ( :186 Page 6 of 1 Algorithm The sequence { } defined by +1 =(1 α n α n T n, n 1, is called a Mann iterative sequence (see [35]. 3 Main results We first discuss the strong convergence of Mann iteration for uniformly L-Lipschitzian asymptotically demicontractive mappings. The following lemma follows immediately from Lemma 6 of [9]and[30], p.176. Lemma 3.1 Let κ >0and (X, ρ be a CAT(κ space with diam(x π/ η κ for some η (0, π/. Let C be a nonempty convex subset of X, T : C C be a uniformly L-Lipschitzian mapping, and {α n }, {β n } be sequences in [0, 1]. Given x 1 C, define the iteration scheme { } by +1 =(1 α n α n T n y n, y n =(1 β n β n T n, n 1. Then ρ(, T ρ (, T n + L ( 1+L + L ρ ( 1, T n 1 1 for all n 1. The following theorem is one of our main results. Theorem 3. Let κ >0and (X, ρ be a CAT(κ space with diam(x π/ η κ for some η (0, π/. Let C be a nonempty closed convex subset of X, and T : C Cbeacompletely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with constant k [0, 1 and sequence {a n } in [1, such that n=1 (a n 1<. Let {α n} be a sequence in [ε, / k ε] for some ε >0where =(π η tan(η. Given x 1 C, define the iteration scheme { } by +1 =(1 α n α n T n, n 1. Then { } converges strongly to a fixed point of T. Proof Let p F(T. By (1, we have ρ (+1, p (1 α n ρ (, p+α n ρ ( T n, p α n(1 α n ρ (, T n. It follows from the asymptotically demicontractiveness of T that ρ (+1, p (1 α n ρ (, p+α n [ a n ρ (, p+kρ (, T n ] α n(1 α n ρ (, T n

7 Panyanak Fixed Point Theory and Applications ( :186 Page 7 of 1 = ρ ( (, p+α n a n 1 ( ρ (, p α n α n k ρ (, T n ρ ( (, p+α n a n 1 ( ρ (, p α n α n k ρ (, T n. (4 Since ε α n / k ε,wehaveε / α n k.thus, ε α n (/ α n k. (5 By (4and(5, we have ρ (+1, p ρ (, p+α n ( a n 1 ρ (, p ε ρ (, T n Therefore, ρ (, p+ π (a n 1 4κ ε ρ (, T n ρ (, p ρ (+1, p+ π (a n 1. 4κ ε ρ (, T n. (6 Since n=1 (a n 1<, n=1 ρ (, T n <, which implies that lim n ρ(, T n = 0. By Lemma 3.1,wehave lim n ρ(, T =0. (7 Since T is completely continuous, {T } has a convergent subsequence in C. By(7, { } has a convergent subsequence, say k q C.Moreover, ρ(q, Tq ρ(q, k +ρ(k, Tk +ρ(tk, Tq 0 ask. That is q F(T. It follows from (6that ρ (+1, p ρ (, p+ π (a n 1. 4κ Since n=1 (a n 1<, by Lemma.4 we have q. This completes the proof. Corollary 3.3 (Theorem 7 of [9] Let (X, ρ be a CAT(0 space, C be a nonempty bounded closed convex subset of X, and T : C C be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with constant k [0, 1 and sequence {a n } in [1, such that n=1 (a n 1<. Let {α n} be a sequence in [ε,1 k ε] for some ε >0.Given x 1 C, define the iteration scheme { } by +1 =(1 α n α n T n, n 1. Then { } converges strongly to a fixed point of T. Proof It is well known that every convex subset of a CAT(0 space, equipped with the induced metric, is a CAT(0 space. Then (C, ρ isacat(0 space and hence it is a CAT(κ

8 Panyanak Fixed Point Theory and Applications ( :186 Page 8 of 1 space for all κ >0.NoticealsothatC is -convex for =.SinceC is bounded, we can choose η (0, π/ and κ >0sothatdiam(C π/ η κ. The conclusion follows from Theorem 3.. Next, we prove the strong convergence of Ishikawa iteration for uniformly L-Lipschitzian asymptotically hemicontractive mappings. The following lemmas are also needed. Lemma 3.4 Let κ >0and (X, ρ be a CAT(κ space with diam(x π/ η κ for some η (0, π/. Let =(π η tan(η, C be a nonempty convex subset of X, and T : C Cbea uniformly L-Lipschitzian and asymptotically hemicontractive mapping with sequence {a n } in [1,. Given x 1 C, define the iteration scheme { } by +1 =(1 α n α n T n y n, y n =(1 β n β n T n, n 1, where {α n } and {β n } are sequences in [0, 1]. Then the following inequality holds: ρ (+1, p [ 1+α n (a n 1(1 + a n β n ] ρ (, p [ α n β n (1 β n(1 + a n ( a n + L βn ] ρ (, T n [ ] α n (1 α n (1 β n ρ (, T n y n for all p F(T. Proof Let p F(T. By (1, we have ρ (+1, p (1 α n ρ (, p+α n ρ ( T n y n, p α n(1 α n ρ (, T n y n (8 and ρ (y n, p (1 β n ρ (, p+β n ρ ( T n, p β n(1 β n ρ (, T n. (9 Since T is asymptotically hemicontractive, ρ ( T n y n, p a n ρ (y n, p+ρ ( y n, T n y n (10 and ρ ( T n, p a n ρ (, p+ρ (, T n. (11 It follows from (9and(11that ρ (y n, p (1 β n ρ (, p+β n [ an ρ (, p+ρ (, T n ] β n(1 β n ρ (, T n = ( 1+β n (a n 1 ρ (, p+β n (1 (1 β n ρ (, T n. (1

9 Panyanak Fixed Point Theory and Applications ( :186 Page 9 of 1 Substituting (1into(10andusing(1, weget ρ ( T n y n, p ( a n 1+βn (a n 1 ρ (, p + a n β n (1 (1 β n ρ (, T n + ρ ( y n, T n y n ( a n 1+βn (a n 1 ρ (, p+a n β n (1 (1 β n ρ (, T n +(1 β n ρ (, T n y n + βn ρ ( T n, T n y n β n(1 β n ρ (, T n ( a n 1+βn (a n 1 ρ (, p [ ] + a n β n a n β n (1 β n β n (1 β n ρ (, T n +(1 β n ρ (, T n y n + βn L ρ (, y n ( a n 1+βn (a n 1 ρ (, p [ ] + a n β n a n β n (1 β n β n (1 β n+β 3 L ρ (, T n +(1 β n ρ (, T n y n. (13 Substituting (13 into(8, we obtain ρ (+1, p (1 α n ρ ( (, p+α n a n 1+βn (a n 1 ρ (, p [ ] + α n a n β n a n β n (1 β n β n (1 β n+β 3 L ρ (, T n + α n (1 β n ρ (, T n y n α n(1 α n ρ (, T n y n = [ 1+α n (a n 1(1 + a n β n ] ρ (, p [ α n β n (1 β n(1 + a n ( a n + L βn ] ρ (, T n [ ] α n (1 α n (1 β n ρ (, T n y n. This completes the proof. Lemma 3.5 Let κ >0and (X, ρ be a CAT(κ space with diam(x π/ η κ for some η (0, π/. Let C be a nonempty convex subset of X, and T : C C be a uniformly L-Lipschitzian and asymptotically hemicontractive mapping with sequence {a n } in [1, such that n=1 (a n 1<. Let {α n }, {β n } [0, 1] be such that 1 β n 1 α n where =(π η tan(η and α n, β n [ε, b] for some ε >0and b (0, +4L 4L. Given x L 1 C, define the iteration scheme { } by +1 =(1 α n α n T n y n, y n =(1 β n β n T n, n 1.

10 Panyanak Fixed Point Theory and Applications ( :186 Page 10 of 1 Then lim n ρ(, T =0. (14 Proof First, we prove that lim n ρ(, T n =0.Since 1 β n 1 α n, by Lemma 3.4 we have ρ (+1, p ρ (, p α n (a n 1(1 + a n β n ρ (, p α n β n [ (1 β n(1 + a n ( a n + L β n ] ρ (, T n. Since {α n (1 + a n β n ρ (, p} n=1 is a bounded sequence, there exists M >0suchthat ρ (+1, p ρ (, p (a n 1M [ α n β n (1 β n(1 + a n ( a n + L βn ] ρ (, T n. (15 Let D = (1 b (1+L b >0.Sincelim n a n =1,thereexistsanaturalnumberN such that (1 β n(1 + a n ( a n + L βn (1 b(1 + a n ( a n + L b D >0 (16 for all n N. Supposethatlim n ρ(, T n 0. Then there exist ε 0 >0andasubsequence {i } of { } such that ρ ( i, T n i i ε0. (17 Without loss of generality, we let n 1 N.From(15, we have α n β n [ (1 β n(1+a n ( a n +L β n ] ρ (, T n (an 1M +ρ (, p ρ (+1, p. Then i [ α nl β nl (1 β n l (1 + a nl ( a nl + L βn ] l ρ ( l, T n l l l=1 = n i m=n 1 α m β m n i [ (1 β m(1 + a m ( a m + L βm ] ρ ( x m, T m x m m=n 1 (a m 1M + ρ (1, p ρ (i +1, p. From this, together with (16, (17andthefactthatε α n β n,weobtain i ε D ε 0 n i m=n 1 (a m 1M + ρ (1, p ρ (i +1, p. (18 If we take i, the right side of (18 is bounded while the left side is unbounded. This is a contradiction. Therefore lim n ρ(, T n =0,andhencelim n ρ(, T =0by Lemma 3.1.

11 Panyanak Fixed Point Theory and Applications ( :186 Page 11 of 1 Theorem 3.6 Let κ >0and (X, ρ be a CAT(κ space with diam(x π/ η κ for some η (0, π/. Let C be a nonempty closed convex subset of X, and T : C Cbeacompletely continuous and uniformly L-Lipschitzian asymptotically hemicontractive mapping with sequence {a n } in [1, such that n=1 (a n 1<. Let {α n }, {β n } [0, 1] be such that 1 β n 1 α n where =(π η tan(η and α n, β n [ε, b] for some ε >0and b (0, +4L 4L. L Given x 1 C, define the iteration scheme { } by +1 =(1 α n α n T n y n, y n =(1 β n β n T n, n 1. Then { } converges strongly to a fixed point of T. Proof Since T is completely continuous, {T } has a convergent subsequence in C. By using Lemma 3.5, we can show that { } has a convergent subsequence, say k q C. Hence q F(T by(14 and the continuity of T. It followsfrom (15 and(16 that ρ (+1, p ρ (, p+(a n 1M. Since n=1 (a n 1<, by Lemma.4 we have q. This completes the proof. As consequences of Theorem 3.6, we obtain the following. Corollary 3.7 Let κ >0and (X, ρ be a CAT(κ space with diam(x π/ η κ for some η (0, π/. Let C be a nonempty closed convex subset of X, and T : C Cbeacompletely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with sequence {a n } in [1, such that n=1 (a n 1<. Let {α n}, {β n } [0, 1] be such that 1 β n 1 α n where =(π η tan(η and α n, β n [ε, b] for some ε >0and b (0, +4L 4L. L Given x 1 C, define the iteration scheme { } by +1 =(1 α n α n T n y n, y n =(1 β n β n T n, n 1. Then { } converges strongly to a fixed point of T. Corollary 3.8 (Theorem 11 of [9] Let (X, ρ be a CAT(0 space, let C be a nonempty bounded closed convex subset of X, and let T : C C be a completely continuous and uniformly L-Lipschitzian asymptotically hemicontractive mapping with sequence {a n } in [1, such that n=1 (a n 1<. Let {α n }, {β n } [0, 1] be such that ε α n β n bfor some ε >0and b (0,. Given x 1 C, define the iteration scheme { } by 1+L 1 L +1 =(1 α n α n T n y n, y n =(1 β n β n T n, n 1. Then { } converges strongly to a fixed point of T.

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