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Approximating fixed points for nonself mappings in CAT(0) spaces

Fixed Point Theory and Applications20112011:65

https://doi.org/10.1186/1687-1812-2011-65

Received: 24 May 2011

Accepted: 13 October 2011

Published: 13 October 2011

Abstract

Suppose K is a nonempty closed convex subset of a complete CAT(0) space X with the nearest point projection P from X onto K. Let T : KX be a nonself mapping, satisfying Condition (E) with F(T): = {x K : Tx = x} . Suppose {x n } is generated iteratively by x1 K, x n +1 = P ((1 - α n )x n α n TP [(1 - β n )x n β n Tx n ]),n ≥ 1, where {α n } and {β n } are real sequences in [ε, 1 - ε] for some ε (0, 1). Then, {x n } Δ-converges to some point x in F(T). This extends a result of Laowang and Panyanak [Fixed Point Theory Appl. 367274, 11 (2010)] for nonself mappings satisfying Condition (E).

Keywords

CAT(0) spacesfixed pointcondition (E)nonself mappings

1 Introduction

In 2010, Laowang and Panyanak [1] studied an iterative scheme and proved the following result: let K be a nonempty closed convex subset of a complete CAT(0) space X, (the initials of term "CAT" are in honor of E. Cartan, A.D. Alexanderov and V.A. Toponogov) with the nearest point projection P from X onto K. Let T : KX be a nonexpansive nonself mapping with nonempty fixed point set. If {x n } is generated iteratively by
x 1 K , x n + 1 = P ( ( 1 - α n ) x n α n T P [ ( 1 - β n ) x n β n T x n ] ) ,
(1.1)

where {α n } and {β n } are real sequences in [ε, 1 - ε] for some ε (0, 1), then {x n } is Δ-convergent to a fixed point of T. In this article, this result is extended for nonself mappings satisfying Condition (E).

Let K be a nonempty subset of a CAT(0) space X and T : KX be a mapping. A point x K is called a fixed point of T, if x = Tx. We shall denote the fixed point set of T by F(T). Moreover, T is called nonexpansive if for each x, y K, d(Tx, Ty) ≤ d(x, y).

In 2011, Falset et al. [2] introduced Condition (E) as follows:

Definition 1.1. Let K be a bounded closed convex subset of a complete CAT(0) space X. A mapping T : KX is called to satisfy Condition (E μ ) on C, if there exists μ ≥ 1 such that
d ( x , T y ) μ d ( T x , x ) + d ( x , y )

holds, for all x, y K. It is called, T satisfies Condition (E) on C whenever T satisfies (E μ ) for some μ ≥ 1.

Proposition 1.2 [2]. Every nonexpansive mapping satisfies Condition (E), but the inverse is not true.

Now, we need some fact about CAT(0) spaces as follows:

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 α of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic 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 to 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 Δ(x1, x2, x3) in a geodesic metric space (X, d) consists of three points in X (the vertices of Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for geodesic triangle Δ(x1, x2, x3) in (X, d) is a triangle Δ ̄ ( x 1 , x 2 , x 3 ) : = Δ ( x ̄ 1 , x ̄ 2 , x ̄ 3 ) in the Euclidean plane E2 such that d E 2 ( x ̄ i , x ̄ j ) = d ( x i , x j ) for i, j {1, 2, 3}. A geodesic metric space is said to be a CAT(0) space [3], if all geodesic triangles of appropriate size satisfy the following comparison axiom. Let Δ be a geodesic triangle in X and Δ ̄ be a comparison triangle for Δ. Then Δ is said to satisfy the CAT(0) inequality if for all x, y Δ and all comparison points x ̄ , ȳ Δ ̄ ,
d ( x , y ) d E 2 ( x ̄ , ȳ ) .
(1.2)
If x, y1, y2 are points in a CAT(0) space and y0 is the midpoint of the segment [y1, y2], then the CAT(0) inequality implies
d ( x , y 0 ) 2 1 2 d ( x , y 1 ) 2 + 1 2 d ( x , y 2 ) 2 - 1 4 d ( y 1 , y 2 ) 2 .
(CN)

In fact, a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality (Courbure negative)[[3], p. 163].

Lemma 1.3. Let (X, d) be a CAT(0) space.

1. [[3], Proposition 2.4] Let K 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) K such that d(x, P(x)) = inf{d(x, y): y K}. Moreover, the map xP(x) is a nonexpansive retract from X onto K.

2. [[4], Lemma 2.1] For x, y X and t [0, 1], there exists a unique point z [x, y] such that
d ( x , z ) = t d ( x , y ) , d ( y , z ) = ( 1 - t ) d ( x , y )

one uses the notation (1 - t)x ty for the unique point z.

3. [[4], Lemma 2.4] For x, y, z X and t [0, 1], one has
d ( ( 1 - t ) x t y , z ) ( 1 - t ) d ( x , z ) + t d ( y , z ) .
[[4], Lemma 2.5] For x, y, z X and t [0, 1], one has
d ( ( 1 - t ) x t y , z ) 2 ( 1 - t ) d ( x , z ) 2 + t d ( y , z ) 2 - t ( 1 - t ) d ( x , y ) 2 .
Let {x n } be a bounded sequence in a CAT(0) space X. For x X, we set
r ( x , { x n } ) = lim sup n  d ( x , x n ) .
The asymptotic radius
r ( { x n } ) = inf { r ( x , { x n } ) : x X } ,
and the asymptotic center A({x n }) of {x n } is the set
A ( { x n } ) = { x X : r ( x , { x n } ) = r ( { x n } ) } .

It is known [[5], Proposition 7], in a CAT(0) space X, A({x n }) consists of exactly one point.

Definition 1.4. [[6], Definition 3.1] A sequence {x n } in a CAT(0) space X is said Δ-converges to x X, if x is the unique asymptotic center of {u n } for every subsequence {u n } of {x n }. In this case, one can write Δ - lim n x n = x and call x the Δ - lim of {x n }.

Lemma 1.5. Let (X, d) be a CAT(0) space.

1. [[6], p. 3690] Every bounded sequence in X has a Δ-convergent subsequence.

2. [[7], Proposition 2.1] If K is a closed convex subset of X and if {x n } is a bounded sequence in K, then the asymptotic center of {x n } is in K.

3. [[4], Lemma 2.8] If {x n } is a bounded sequence in X with A({x n }) = {x} and {u n } is a subsequence of {x n } with A({u n }) = {u} and the sequence {d(x n , u)} converges, then x = u.

2 Main results

The following lemma was proved by Dhompongsa and Panyanak in the case of nonexpansive [[4], Lemma 2.10].

Lemma 2.1. Let K be a nonempty closed convex subset of a complete CAT(0) space X, and T : KX be a nonself mapping, satisfying Condition (E). Suppose {x n } is a bounded sequence in K such that lim n d(x n , Tx n ) = 0 and {d(x n , v)} converges for all v F (T). Then
ω w ( x n ) F ( T ) ,

where ω w ( x n ) : = A ( { u n } ) and the union is taken over all subsequences {u n } of {x n }. Moreover, ω w (x n ) consists of exactly one point.

Proof. Let u ω w (x n ), then there exists a subsequence {u n } of {x n } such that A({u n }) = {u}. By part (1) and (2) of Lemma 1.5, there exists a subsequence {v n } of {u n } such that Δ - lim n v n = v K. We show v F (T). In order to prove this, by Condition (E), one can write
d ( x n , T v ) μ d ( T x n , x n ) + d ( x n , v )
for some μ ≥ 1. Therefore
lim sup n d ( x n , T v ) lim sup n ( μ d ( T x n , x n ) + d ( x n , v ) ) = lim sup n d ( x n , v ) .

The uniqueness of asymptotic center, implies v K and T(v) = v. By part (3) Lemma 1.5, u = v. Therefore ω w (x n ) F(T). Let {u n } be a subsequence of {x n } with A({u n }) = {u} and A({x n }) = {x}. Since u ω w (x n ) F(T), {d(x n , v)} converges. By part (3) Lemma 1.5, x = u. This shows that ω w (x n ) consists of exactly one point. □

Theorem 2.2. Let K be a nonempty closed convex subset of a complete CAT(0) space X, and T : KX be a nonself mapping, satisfying Condition (E) with x F(T) = {x K : Tx = x}. Let {α n } and {β n } be sequences in [ε, 1 - ε] for some ε (0, 1). Starting from arbitrary x1 K, define the sequence {x n } by x n +1 = P((1 - α n )x n α n TP[(1 - β n )x n β n Tx n ]), n ≥ 1. Then lim n →∞ d(x n , x) exists.

Proof. By part (1) of Lemma 1.3, the nearest point projection P from X onto K is nonexpansive. Then,
d ( x n + 1 , x ) = d ( P ( ( 1 - α n ) x n α n T P [ ( 1 - β n ) x n β n T x n ] ) , P x ) d ( ( 1 - α n ) x n α n T P [ ( 1 - β n ) x n β n T x n ] , x ) = ( 1 - α n ) d ( x n , x ) + α n d ( T P [ ( 1 - β n ) x n β n T x n ] , x ) .
But by Condition (E), for some μ ≥ 1, we have
( 1 - α n ) d ( x n , x ) + α n d ( T P [ ( 1 - β n ) x n β n T x n ] , x ) ( 1 - α n ) d ( x n , x ) + α n ( μ d ( T x , x ) + d ( P [ ( 1 - β n ) x n β n T x n ] , x ) ) ( 1 - α n ) d ( x n , x ) + α n [ ( 1 - β n ) d ( x n , x ) + β n d ( x n , x ) ] = d ( x n , x ) .

Consequently, d(x n +1, x) ≤ d(x n , x). Then d(x n , x) ≤ d(x1, x) for all n ≥ 1. This implies { d ( x n , x ) } n = 1 is bounded and decreasing. Hence, lim n →∞ d(x n , x) exists. □

Theorem 2.3. Let K be a nonempty closed convex subset of a complete CAT(0) space X, and T : KX be a nonself mapping, satisfying Condition (E) with F(T) ≠ . Let {α n } and {β n } be sequences in [ε, 1 - ε] for some ε (0, 1). Starting from arbitrary x1 K, define the sequence {x n } by x n +1 = P((1 - α n )x n α n TP [(1 - β n )x n β n Tx n ]), n ≥ 1. Then lim n →∞ d(x n , Tx n ) = 0.

Proof. Let x F(T). By Theorem 2.2, lim n →∞ d(x n , x) exists. Set
lim n d ( x n , x ) = r .
If r = 0, by the Condition (E), for some μ ≥ 1,
d ( x n , T x n ) d ( x , x n ) + d ( x , T x n ) d ( x , x n ) + μ d ( x , T x ) + d ( x , x n ) .

Therefore lim n →∞ d(x n , Tx n ) = 0.

If r > 0, set y n = P [(1 - β n )x n β n Tx n ]. By part (4) of Lemma 1.3,
d ( y n , x ) 2 = d ( P [ ( 1 - β n ) x n β n T x n ] , P x ) 2 d ( [ ( 1 - β n ) x n β n T x n ] , x ) 2 ( 1 - β n ) d ( x n , x ) 2 + β n d ( T x n , x ) 2 - β n ( 1 - β n ) d ( x n , T x n ) 2 ( 1 - β n ) d ( x n , x ) 2 + β n d ( T x n , x ) 2 .
(2.3)
Using Condition (E), for some μ ≥ 1,
( 1 - β n ) d ( x n , x ) 2 + β n d ( T x n , x ) 2 ( 1 - β n ) d ( x n , x ) 2 + β n ( μ d ( T x , x ) + d ( x n , x ) ) 2 = d ( x n , x ) 2 .
(2.4)
Therefore by inequities (2.3) and (2.4), one can get
d ( y n , x ) d ( x n , x ) .
(2.5)
Part (4) of Lemma 1.3, shows
d ( x n + 1 , x ) 2 = d ( P [ ( 1 - α n ) x n α n T y n ] , P x ) 2 d ( ( 1 - α n ) x n α n T y n , x ) 2 ( 1 - α n ) d ( x n , x ) 2 + α n d ( T y n , x ) 2 - α n ( 1 - α n ) d ( x n , T y n ) 2 ( 1 - α n ) d ( x n , x ) 2 + α n ( μ d ( T x , x ) + d ( y n , x ) ) 2 - α n ( 1 - α n ) d ( x n , T y n ) 2 = ( 1 - α n ) d ( x n , x ) 2 + α n d ( y n , x ) 2 - α n ( 1 - α n ) d ( x n , T y n ) 2 ( 1 - α n ) d ( x n , x ) 2 + α n d ( x n , x ) 2 - α n ( 1 - α n ) d ( x n , T y n ) 2 = d ( x n , x ) 2 - α n ( 1 - α n ) d ( x n , T y n ) 2 .
Therefore
d ( x n + 1 , x ) 2 d ( x n , x ) 2 - W ( α n ) d ( x n , T y n ) 2 ,

where W(α) = α(1 - α). Since α [ε, 1 - ε], W(α n ) ≥ ε2.

Therefore
ε 2 n = 1 d ( x n , T y n ) 2 d ( x 1 , x ) 2 < .

This implies lim n →∞ d(x n , Ty n ) = 0.

By Condition (E), for some μ ≥ 1, we have
d ( x n , x ) d ( x n , T y n ) + d ( T y n , x ) d ( x n , T y n ) + μ d ( T x , x ) + d ( y n , x ) = d ( x n , T y n ) + d ( y n , x ) .
Hence
r lim inf n  d ( y n , x ) .
On the other hand, from (2.5),
lim sup n  d ( y n , x ) r .
This implies
lim n d ( y n , x ) = r .
Thus (2.5) shows
lim n d ( ( 1 β n ) x n β n T x n ] , x ) = r .
Since T satisfies Condition (E), we have
d ( T x n , x ) μ d ( T x , x ) + d ( x n , x ) = d ( x n , x )
Thus
lim sup n  d ( T x n , x ) r .

Now, by [[1], Lemma 2.9], lim n →∞ d(x n , Tx n ) = 0. □

Theorem 2.4. Let K be a nonempty closed convex subset of a complete CAT(0) space X, and T : KX be a nonself mapping, satisfying Condition (E) with F(T) ≠ . Assume {α n } and {β n } are sequences in [ε, 1 - ε] for some ε (0, 1). Starting from arbitrary x1 K, define the sequence {x n } by x n +1 = P((1 - α n )x n α n TP[(1 - β n )x n β n Tx n ]), n ≥ 1. Then {x n } is Δ-convergent to some point x in F(T).

Proof. By Theorem 2.3, lim n →∞ d(x n , Tx n ) = 0. The proof of Theorem 2.2 shows {d(x n , v)} is bounded and decreasing for each v F (T), and so it is convergent. By Lemma 2.1, ω w (x n ) consists exactly one point which is a fixed point of T. Consequently, the sequence {x n } is Δ-convergent to some point x in F(T). □

The following definition is recalled from [8].

Definition 2.5. A mapping T : KX is said to satisfy Condition I, if there exists a nondecreasing function f : [0, ∞) → [0, ∞) with f(0) = 0 and f(r) > 0 for all r > 0 such that
d ( x , T x ) f ( d ( x , F ( T ) ) ) ,

where x K.

With respect to the above definition, we have the following theorem [[1], Theorem 3.4].

Theorem 2.6. Let K be a nonempty closed convex subset of a complete CAT(0) space X, and T : KX be a nonself mapping, satisfying condition (E) with F(T) ≠ . Assume {α n } and {β n } are sequences in [ε, 1 - ε] for some ε (0, 1). Starting from arbitrary x1 K, define the sequence {x n } by x n +1 = P((1 - α n )x n α n TP [(1 - β n )x n β n Tx n ]), n ≥ 1. If T satisfies condition I, then {x n } converges strongly to a fixed point of T.

We state another strong convergence theorem [[1], Theorem 3.5] as follows:

Theorem 2.7. Let K be a nonempty compact convex subset of a complete CAT(0) space X, and T : KX be a nonself mapping, satisfying condition (E) with F(T) ≠ . Assume {α n } and {β n } are sequences in [ε, 1 - ε] for some ε (0, 1). Starting from arbitrary x1 K, define the sequence {x n } by x n +1 = P((1 - α n )x n α n TP[(1 - β n )x n β n Tx n ]),n ≥ 1. Then, {x n } converges strongly to a fixed point of T.

Another result in [1] is to obtain the Δ-convergence of a defined sequence, to a common fixed point of two nonexpansive self-mappings. According to the present setting, we can state the following result.

Theorem 2.8. Let K be a nonempty closed convex subset of a complete CAT(0) space X, and S, T : KX be two nonself mappings, satisfying Condition (E) with F(S) ∩ F(T) ≠ . Assume {α n } and {β n } are sequences in [ε, 1 - ε] for some ε (0, 1). Starting from arbitrary x1 K, define the sequence {x n } by x n +1 = (1 - α n )x n α n S[(1 - β n )x n β n Tx n ], n ≥ 1. Then {x n } is Δ-convergent to a common fixed point of S and T.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

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Copyright

© Razani and Shabani; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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