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Journal Nonlinear Analysis and Application 2015 No.2 (2015) 87-94 Available online at www.ispacs.com/jnaa Volume 2015, Issue 2, Year 2015 Article ID jnaa-00202, 8 Pages doi:10.5899/2015/jnaa-00202 Research Article

Fixed Point Theorems for Nonself Asymptotically Nonexpansive type Mappings in CAT(0) Spaces Balwant Singh Thakur1 , Mohammad Saeed Khan2∗ , Dipti Thakur1 (1) School of Studies in Mathematics, Pt.Ravishankar Shukla University Raipur - 492010 (C.G.), India (2) Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, PCode 123 Al-Khod, Muscat, Sultanate of Oman, Oman c Balwant Singh Thakur, Mohammad Saeed Khan and Dipti Thakur. This is an open access article distributed under the Creative Copyright 2015 ⃝ Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In this paper, we propose a three-step iteration scheme for nonself asymptotically nonexpansive type mappings in a complete CAT(0) metric space and establish necessary and sufficient conditions for convergence of this process to a fixed point of nonself asymptotically nonexpansive type mappings. We also establish a strong convergence result. These results generalize and unify many important results in the literature. Keywords: CAT(0) space, nonself asymptotically nonexpansive type mapping, fixed points, strong convergence, iteration process.

2000 Mathematics Subject Classification. 47H09, 47H10.

1 Introduction Let C be a nonempty subset of metric space (X, d). A mapping T : C → C is said to be (i) Lipschitzian if d(T x, Ty) ≤ kd(x, y) for all x, y ∈ C , k ≥ 0, (ii) nonexpansive if d(T x, Ty) ≤ d(x, y) for all x, y ∈ C (iii) asymptotically nonexpansive if there exists a sequence {kn } in [1, ∞) with lim kn = 1 such that d(T n x, T n y) ≤ kn d(x, y) for all n→∞ x, y in C and n ∈ N, where N denotes the set of positive integers. Class of asymptotically nonexpansive mappings includes a class of nonexpansive mappings as a proper subclass [8], and both the mappings are Lipschitzian. In 1974, Kirk [11] substantially weaken the assumption of asymptotic nonexpansiveness of T by replacing it with an assumption, which may hold even if none of the iterates of T is Lipschitzian. A mapping T : C → C is said to be asymptotically nonexpansive type if for each y ∈ C the following inequality holds: { } n n (1.1) lim sup sup(d (T x, T y) − d (x, y)) ≤ 0 . n→∞

x∈C

Every asymptotically nonexpansive mapping satisfies (1.1), but converse need not be true [11, p.345]. The concept of asymptotically nonexpansive type mappings is more general than that of asymptotically nonexpansive mappings. Iterative approximation of fixed points of nonexpansive, asymptotically nonexpansive and asymptotically nonexpansive type mappings have been studied by various authors in the setting of Hilbert spaces, Banach spaces and convex metric spaces, see [10, 21–26] and reference therein. ∗ Email

address: [email protected]

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A metric space (X, d) is a CAT(0) space if it is geodesically connected, and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. Any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include Pre-Hilbert spaces, R−trees [1], Euclidean buildings [2], the complex Hilbert ball with a hyperbolic metric [9], and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry; see [1]. Fixed point theory in CAT(0) spaces was first studied by Kirk [12, 13]. In recent years, the fixed point theory for single-valued and multivalued mappings in CAT(0) spaces has been studied by many authors; see [4, 5, 14–18] and references therein. Motivated by the works going in this area, in this paper, we study a three-step iterative scheme for nonself asymptotically nonexpansive type mappings and establish necessary and sufficient conditions for strong convergence, we also establish a strong convergence theorem. 2 Preliminaries Let (X, d) be a metric space. A geodesic path joining x ∈ X to y ∈ X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊂ R to X such that c(0) = x, c(l) = y, and d(c(t), c(t ′ )) = |t − t ′ | for all t,t ′ ∈ [0, l]. In particular, c is an isometry and d(x, y) = l. The image of c is called a geodesic (or metric) segment joining x and y. When it is unique this geodesic segment is denoted by [x, y]. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A subset Y ⊆ X is said to be convex if Y includes every geodesic segment joining any two of its points. A geodesic triangle △(x1 , x2 , x3 ) in a geodesic metric space (X, d) consists of three points x1 , x2 , x3 in X (the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A comparison triangle for the geodesic triangle △(x1 , x2 , x3 ) in (X, d) is a triangle △(x1 , x2 , x3 ) := △(x¯1 , x¯2 , x¯3 ) in the Euclidean plane E2 such that dE2 (x¯i , x¯j ) = d(xi , x j ) for i, j ∈ {1, 2, 3}. A geodesic space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom. CAT(0) : Let △ be a geodesic triangle in X and let △ be a comparison triangle for △. Then △ is said to satisfy the CAT(0) inequality if for all x, y ∈ △ and all comparison points x, ¯ y¯ ∈ △, d(x, y) ≤ dE2 (x, ¯ y). ¯ If x, y1 , y2 are points in a CAT(0) space and if y0 is the midpoint of the segment [y1 , y2 ], then the CAT(0) inequality implies 1 1 1 d(x, y0 )2 ≤ d(x, y1 )2 + d(x, y2 )2 − d(y1 , y2 )2 . 2 2 4 This is the (CN) inequality of Bruhat and Tits [3]. In fact, a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality [1, p.163]. CAT(0) spaces may be regarded as a metric version of Hilbert spaces. For example, in any Hilbert space H we have the following extended version of parallelogram law: ∥z − (α x + (1 − α )y∥2 = α ∥z − x∥2 + (1 − α ) ∥z − y∥2 − α (1 − α ) ∥x − y∥2 for any α ∈ [0, 1] and x, y, z ∈ H. Any CAT(0) metric space X has the inequality d (z, α x ⊕ (1 − α )y)2 ≤ α d(x, z)2 + (1 − α )d(z, y)2 − α (1 − α )d(x, y)2

(2.2)

for any α ∈ [0, 1], x, y ∈ X. 1 If α = , then the inequality (2.2) becomes the (CN) inequality. 2 Lemma 2.1. [6] Let (X, d) be a CAT(0) space, then :

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(i) For x, y ∈ X and t ∈ [0, 1], there exists a unique point z ∈ [x, y] such that d(x, z) = td(x, y) and d(y, z) = (1 − t)d(x, y).

(2.3)

We use the notation (1 − t)x ⊕ ty for the unique point z satisfying (2.3). (ii) For x, y, z ∈ X and t ∈ [0, 1], we have d((1 − t)x ⊕ ty, z) ≤ (1 − t)d(x, z) + td(y, z).

(2.4)

We shall denote with Fix(T ) the set of fixed points of a mapping T . A mapping P : X → C is said to be retraction if C ⊂ X and P restricted to C is the identity, i.e. Px = x for any x ∈ C. Clearly P2 = P, the set C is called a retract of X. Lemma 2.2. [1, Chapter II.2, Proposition 2.4] Let C be a convex subset of X which is complete in the induced metric. Then, for every x ∈ X, there exists a unique point P(x) ∈ C such that d(x, P(x)) = d(x,C) := inf{d(x, y) : y ∈ C}. Moreover, the map x → P(x) is a nonexpansive retract from X onto C. We now define nonself asymptotically nonexpansive type mapping in a CAT(0) space. Definition 2.1. Let C be a nonempty subset of a CAT(0) space X. A map T : C → X is said to be asymptotically nonexpansive type, if for each y ∈ C, [ ] lim sup sup{d(T (PT )n−1 x, T (PT )n−1 y) − d(x, y)} ≤ 0 , (2.5) n→∞

x∈C

where P is the nonexpansive retraction of X onto C. Glowinski and Le Tallec [7] used three-step iterative schemes to find the approximate solutions of the elasto-viscoplasticity problem, liquid crystal theory and eigenvalue computation, and shown that the three-step iterative schemes gives better numerical results than the two-step and one-step iterative schemes. Noor [19] introduced a three-step iterative scheme and studied the approximate solution of variational inclusion in Hilbert spaces. Xu and Noor [27] studied three-step iterative scheme to approximate fixed points of asymptotically nonexpansive self mappings in Banach spaces. Motivated by the above facts, we now consider following three-step iteration for nonself asymptotically nonexpansive type mapping as follows :  x0 ∈ C,    ) (   xn+1 = P (1 − αn )xn ⊕ αn T (PT )n−1 yn (2.6) ( )  yn = P (1 − βn )xn ⊕ βn T (PT )n−1 zn    ( )  zn = P (1 − γn )xn ⊕ γn T (PT )n−1 xn where {αn }, {βn } and {γn } are sequences in [ε , 1 − ε ] for some ε ∈ (0, 1). In the next section, we establish necessary and sufficient conditions for strong convergence of the sequence {xn } given by (2.6) to a fixed point of nonself asymptotically nonexpansive type mapping. 3

Main results

Lemma 3.1. Let C be a nonempty convex subset of a CAT(0) space X and let T : C → X be a mapping of asymptotically nonexpansive type with Fix(T ) ̸= 0/ and {xn } be sequence defined by (2.6). If x∗ ∈ Fix(T ), then d(xn+1 , x∗ ) ≤ d(xn , x∗ ) + 3 sup{d(T (PT )n−1 x, x∗ ) − d(x, x∗ )};

n ∈ N.

x∈C

Proof. Let x∗ ∈ Fix(T ). Using (2.6) and (2.4), we have

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) ) ( ( d(xn+1 , x∗ ) = d P (1 − αn )xn ⊕ αn T (PT )n−1 yn , Px∗ ( ( ) ) ≤ d (1 − αn )xn ⊕ αn T (PT )n−1 yn , x∗ [ ( ) ] ≤ (1 − αn )d(xn , x∗ ) + αn d T (PT )n−1 yn , x∗ − d(yn , x∗ ) + αn d(yn , x∗ ) [ ( ) ] ≤ (1 − αn )d(xn , x∗ ) + αn sup d T (PT )n−1 x, x∗ − d(x, x∗ ) x∈C

+ αn d(yn , x∗ ) .

(3.7)

Now, ( ( ) ) d(yn , x∗ ) = d P (1 − βn )xn ⊕ βn T (PT )n−1 zn , Px∗ ( ( ) ) ≤ d (1 − βn )xn ⊕ βn T (PT )n−1 zn , x∗ [ ( ) ] ≤ (1 − βn )d(xn , x∗ ) + βn d T (PT )n−1 zn , x∗ − d(zn , x∗ ) + βn d(zn , x∗ ) [ ( ) ] ≤ (1 − βn )d(xn , x∗ ) + βn sup d T (PT )n−1 x, x∗ − d(x, x∗ ) x∈C ∗

+ βn d(zn , x ) .

(3.8)

Similarly, we get [ ( ) ] d(zn , x∗ ) ≤ d(xn , x∗ ) + γn sup d T (PT )n−1 x, x∗ − d(x, x∗ ) .

(3.9)

x∈C

Substituting (3.9) into (3.8), we get d(yn , x∗ ) ≤ d(xn , x∗ ) + 2 sup[d(T (PT )n−1 x, x∗ ) − d(x, x∗ )] .

(3.10)

x∈C

Substituting (3.10) into (3.7), we get d(xn+1 , x∗ ) ≤ d(xn , x∗ ) + 3 sup[d(T (PT )n−1 x, x∗ ) − d(x, x∗ )] . x∈C

This completes the proof. We need following lemma to prove next result. Lemma 3.2. [24] Suppose that {an } and {bn } are two sequenes of nonnegative numbers such that an+1 ≤ an + bn , for all n ≥ 1. If ∑∞ n=1 bn converges, then limn→∞ an exists. Lemma 3.3. Let C be a nonempty convex subset of a complete CAT(0) space X and let T : C → X be a mapping of asymptotically nonexpansive type with Fix(T ) ̸= 0/ and sequence {xn } be defined by (2.6). Then (i) limn→∞ d(xn , x∗ ) exists for all x∗ ∈ Fix(T ); (ii) limn→∞ d(xn , Fix(T )) exists. Proof. Since T is asymptotically nonexpansive type mapping, for each x∗ ∈ Fix(T ), we have [ ] ( ) lim sup sup{d T (PT )n−1 x, T (PT )n−1 x∗ − d(x, x∗ )}) ≤ 0 , n→∞

x∈C

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we can choose no such that n ≥ no imply [ { ( ) }] 1 k−1 ∗ ∗ sup sup d T (PT ) x, x − d(x, x ) ≤ 2 . 3n k≥n x∈C By Lemma 3.1, we have d(xn+1 , x∗ ) ≤ d(xn , x∗ ) +

1 . n2

Therefore for n, m ≥ n0 , we have d(xm+n , x∗ ) ≤ d(xn , x∗ ) +

n+m−1

∑

i=n

1 . i2

(3.11)

Taking infimum over all x∗ ∈ Fix(T ), we have d(xm+n , Fix(T )) ≤ d(xn , Fix(T )) +

n+m−1

∑

i=n

1 . i2

(3.12)

If follows from Lemma 3.2, (3.11) and (3.12) that lim d(xn , x∗ ) and

n→∞

lim d(xn , Fix(T ))

n→∞

exists. Lemma 3.4. Let C be a nonempty closed convex subset of a complete CAT(0) space X, let T : C → X be asymptotically nonexpansive type mapping with Fix(T ) ̸= 0, / and the sequence {xn } is defined by (2.6). If limn→∞ d(xn , Fix(T )) = 0, then {xn } is a Cauchy sequence. Proof. Since limn→∞ d(xn , Fix(T )) = 0, then for all ε > 0 there exists k(ε ) ∈ N such that for all n ≥ k(ε ) d(xn , Fix(T ))
κ [1, page 165], therefore the results in this paper can be applied to any CAT(κ ) space with κ < 0.

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