Approximating fixed points for generalized nonexpansive mapping in spaces
© Uddin et al.; licensee Springer. 2014
Received: 23 January 2014
Accepted: 16 April 2014
Published: 2 May 2014
Takahashi and Kim (Math. Jpn. 48:1-9, 1998) used the Ishikawa iteration process to prove some convergence theorems for nonexpansive mappings in Banach spaces. The aim of this paper is to prove similar results in spaces for generalized nonexpansive mappings, which, in turn, generalize the corresponding results of Takahashi and Kim (Math. Jpn. 48:1-9, 1998), Laokul and Panyanak (Int. J. Math. Anal. 3(25-28):1305-1315, 2009), Razani and Salahifard (Bull. Iran. Math. Soc. 37(1):235-246, 2011) and some others.
For a comparison of the preceding two iterative schemes in one-dimensional case, we refer the reader to Rhoades  wherein it is shown that under suitable conditions (see part (a) of Theorem 3) the rate of convergence of the Ishikawa iteration is faster than that of the Mann iteration procedure. Iterative techniques for approximating fixed points of nonexpansive single-valued mappings have been investigated by various authors using the Mann as well as Ishikawa iteration schemes. By now, there exists an extensive literature on the iterative fixed points for various classes of mappings. For an up-to-date account of the literature on this theme, we refer the readers to Berinde .
and for some a, b with ,
and for some a, b with .
Utilizing the forgoing iterative scheme, Takahashi and Kim  proved weak as well as strong convergence theorems for a nonexpansive mapping in Banach spaces.
Recently, García-Falset et al.  introduced two generalizations of nonexpansive mappings which in turn include Suzuki generalized nonexpansive mappings contained in  and also utilized the same to prove some fixed point theorems.
The following definitions are relevant to our subsequent discussions.
Definition 1.1 ()
We say that T satisfies the condition whenever T satisfies the condition for some .
Definition 1.2 ()
The following theorem is essentially due to García-Falset et al. .
Theorem 1.3 ()
C is weakly compact and X satisfies the Opial condition.
C is compact.
C is weakly compact and X is (UCED).
Then there exists such that .
A metric space is a space if it is geodesically connected and every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a space. Other examples are pre-Hilbert spaces, R-trees, the Hilbert ball (see ) and many others. For more details on spaces, one can consult [9–11]. Fixed point theory in spaces was initiated by Kirk  who proved some theorems for nonexpansive mappings. Since then the fixed point theory for single-valued as well as multi-valued mappings has intensively been developed in spaces (e.g. [13–16]). Further relevant background material on spaces is included in the next section.
The purpose of this paper is to prove some weak and strong convergence theorems of the iterative scheme (1) in spaces for generalized nonexpansive mappings enabling us to enlarge the class of underlying mappings as well as the class of spaces in the corresponding results of Takahashi and Kim .
In this section, to make our presentation self-contained, we collect relevant definitions and results. In a metric space , a geodesic path joining and is a map c from a closed interval to X such that , and for all . In particular, the mapping c is an isometry and . The image of c is called a geodesic segment joining x and y, which is denoted by , whenever such a segment exists uniquely. For any , we denote the point by , where if and . The space is called a geodesic space if any 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 . A subset C of X is called convex if C contains every geodesic segment joining any two points in C.
A geodesic triangle in a geodesic metric space consists of three points of X (as the vertices of △) and a geodesic segment between each pair of points (as the edges of △). A comparison triangle for in is a triangle in the Euclidean plane such that for . A point is said to be comparison point for if . Comparison points on and are defined in the same way.
A geodesic metric space X is called a space if all geodesic triangles satisfy the following comparison axiom ( inequality):
Any convex subset of a Euclidean space , when endowed with the induced metric is a space.
Every pre-Hilbert space is a space.
If a normed real vector space X is space, then it is a pre-Hilbert space.
The Hilbert ball with the hyperbolic metric is a space.
If and are spaces, then is also a space.
It is well known that in a space, consists of exactly one point (see Proposition 5 of ).
Definition 2.1 A sequence in X is said to Δ-converge to if x is the unique asymptotic center of for every subsequence of . In this case we write and x is the Δ-limit of .
Thus every space satisfies the Opial property. Now we collect some basic facts about spaces which will be used throughout the text frequently.
Lemma 2.2 ()
Every bounded sequence in a complete space admits a Δ-convergent subsequence.
Lemma 2.3 ()
If C is closed convex subset of a complete space and if is a bounded sequence in C, then the asymptotic center of is in C.
Lemma 2.4 ()
We use the notation for the unique point z of the above lemma.
Recently García-Falset et al.  introduced two generalizations of the condition in Banach spaces. Now, we state their condition in the framework of spaces.
We say that T satisfies the condition whenever T satisfies the condition for some .
In the case , then the condition implies the condition . The following example shows that the class of mappings satisfying the conditions and (for some ) is larger than the class of mappings satisfying the condition .
Example 2.8 ()
Then the mapping T satisfies the condition but it fails the condition whenever . Moreover, T satisfies the condition for .
The following theorem is an analog of Theorem 4 in  to spaces.
Theorem 2.9 Let C be a bounded convex subset of a complete space X. If satisfies the condition on C for some , then there exists an approximating fixed point sequence for T.
where the sequences and are sequences in .
The following lemma is a consequence of Lemma 2.9 of  which will be used to prove our main results.
This result is an analog of a result of the weak convergence theorem of Takahashi and Kim  for a generalized nonexpansive mapping in a Banach space. A strong convergence theorem is also proved. In the process, the corresponding results of Takahashi and Kim , Laokul and Panyanak , Razani and Salahifard , and others are generalized and improved.
3 Main results
Before proving our main results, firstly we re-write Theorem 1.3 in the setting of spaces which is essentially Theorem 3.2 of .
Theorem 3.1 Let C be a bounded, closed, and convex subset of a complete space X. If satisfies the conditions and for some , then T has a fixed point.
Now, to accomplish our main results, we prove the following lemma.
Lemma 3.2 Let C be a nonempty closed convex subset of a complete space X and be a mapping which satisfies the condition for some . If is a sequence defined by (2) and the sequences and satisfy the condition described in (3), then exists for all .
which shows that the sequence is decreasing and bounded below so that exists. □
Lemma 3.3 Let C be a nonempty closed convex subset of a complete space X and let satisfy the conditions and on C. If is a sequence defined by (2) and the sequences and satisfy the condition described in (3), then is nonempty if and only if is bounded and .
Owing to the condition , we have and .
Case I: If and , then by the foregoing discussion and by Lemma 2.10, we have .
As , in view of the condition we have .
Making use of the observation , we have .
Case 2: and .
Again in view of Lemma 2.10, we have .
Owing to the uniqueness of asymptotic center, , so that x is fixed point of T. □
Theorem 3.4 Let C be a nonempty closed convex subset of a complete space X and be a mapping which satisfies conditions for some and on C with . If the sequences , and are described as in (2) and (3), then the sequence Δ-converges to a fixed point of T.
which is a contradiction so that the conclusion follows. □
In view of Theorem 3.1, we have the following corollary of the preceding theorem.
Corollary 3.5 Let C be a nonempty bounded, closed and convex subset of a complete space X and be a mapping which satisfies conditions for some and on C. If sequences , , and are described by (2) and (3), then the sequence Δ-converges to a fixed point of T.
Theorem 3.6 Let C be a nonempty closed convex subset of a complete space X and be a mapping which satisfies conditions for some and on C. Moreover, T satisfies the condition with . If sequences , , and are defined as in (2) and (3), respectively, then converges strongly to some fixed point of T.
so that . Since exists, we have . □
Corollary 3.7 Let C be a nonempty bounded, closed, and convex subset of a complete space X and let be a mapping which satisfies the conditions for some and on C. Moreover, T satisfies the condition . If the sequences , and are described by (2) and (3), then converges strongly to some fixed point of T.
The authors thank the anonymous referees and handling editor of the manuscript for their valuable suggestions and fruitful comments. The first author is grateful to University Grants commission, India for the financial assistance in the form of Maulana Azad National Fellowship. The research of second author is supported by Jazan University, Saudi Arabia.
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