Approximating Fixed Points of Some Maps in Uniformly Convex Metric Spaces
© Abdul Rahim Khan et al. 2010
Received: 1 October 2009
Accepted: 19 January 2010
Published: 26 January 2010
We study strong convergence of the Ishikawa iterates of qasi-nonexpansive (generalized nonexpansive) maps and some related results in uniformly convex metric spaces. Our work improves and generalizes the corresponding results existing in the literature for uniformly convex Banach spaces.
1. Introduction and Preliminaries
Let be a nonempty subset of a metric space and let be a map. Denote the set of fixed points of by The map is said to be (i) quasi-nonexpansive if and for all and , (ii) -Lipschitz if for some we have for all for it becomes nonexpansive, and (iii) generalized nonexpansive (cf.  and the references therein) if
The concept of quasi-nonexpansiveness is more general than that of nonexpansiveness. A nonexpansive map with at least one fixed point is quasi-nonexpansive but there are quasi-nonexpansive maps which are not nonexpansive .
Mann and Ishikawa type iterates for nonexpansive and quasi-nonexpansive maps have been extensively studied in uniformly convex Banach spaces [1, 3–6]. Senter and Dotson  established convergence of Mann type iterates of quais-nonexpansive maps under a condition in uniformly convex Banach spaces. In 1973, Goebel et al.  proved that generalized nonexpansive self maps have fixed points in uniformly convex Banach spaces. Based on their work, Bose and Mukerjee  proved theorems for the convergence of Mann type iterates of generalized nonexpansive maps and obtained a result of Kannan  under relaxed conditions. Maiti and Ghosh  generalized the results of Bose and Mukerjee  for Ishikawa iterates by using modified conditions of Senter and Dotson  (see, also ). For the sake of completeness, we state the result of Kannan  and its generalization by Bose and Mukerjee .
Theorem 1.1 (see ).
Theorem 1.2 (see ).
In 1970, Takahashi  introduced a notion of convexity in a metric space as follows: a map is a convex structure in if
for all and A metric space together with a convex structure is said to be convex metric space. A nonempty subset of a convex metric space is convex if for all and In fact, every normed space and its convex subsets are convex metric spaces but the converse is not true, in general (see ). Later on, Shimizu and Takahashi  obtained some fixed point theorems for nonexpansive maps in convex metric spaces. This notion of convexity has been used in [13–15] to study Mann and Ishikawa iterations in convex metric spaces. For other fixed point results in the closely related classes of spaces, namely, hyperbolic and hyperconvex metric spaces, we refer to [16–19].
A convex metric space is said to be uniformly convex  if for arbitrary positive numbers and , there exists such that
Note that if satisfies Condition 1 (resp., 3), then it satisfies Condition 2 (resp., 4). We also note that Conditions 1 and 2 become Conditions A and B, respectively, of Senter and Dotson  while Conditions 3 and 4 become Conditions I and II, respectively, of Maiti and Ghosh  in a normed space. Further, Conditions 3 and 4 reduce to Conditions 1 and 2, respectively, when
In this note, we present results under relaxed control conditions which generalize the corresponding results of Kannan , Bose and Mukerjee , and Maiti and Ghosh  from uniformly convex Banach spaces to uniformly convex metric spaces. We present sufficient conditions for the convergence of Ishikawa iterates of Lipschitz maps to their fixed points in convex metric spaces and improve [3, Lemma 2]. A necessary and sufficient condition is obtained for the convergence of a sequence to fixed point of a generalized nonexpansive map in metric spaces.
We need the following fundamental result for the developmant of our results.
Theorem 1.3 (see ).
2. Convergence Analysis
We prove a lemma which plays key role to establish strong convergence of the iterative schemes (1.3) and (1.4).
Now we state and prove Ishikawa type convergence result in uniformly convex metric spaces.
Let be a uniformly convex complete metric space with continuous convex structure and let be its nonempty closed convex subset. Let be a continuous quasi-nonexpansive map of into itself satisfying Condition 3. If is as in (1.3), where and , then converges to a fixed point of .
Let be a uniformly convex complete metric space with continuous convex structure and let be its nonempty closed convex subset. Let be a continuous quasi-nonexpansive map of into itself satisfying Condition 1. If is as in (1.4), where , then converges to a fixed point of .
Next we establish strong convergence of Ishikawa iterates of a generalized nonexpansive map.
In the above theorem, we have assumed that the generalied nonexpansive map has a fixed point. It remains an open questions: what conditions on , and in (*) are sufficient to guarantee the existence of a fixed point of even in the setting of a metric space.
Let be a uniformly convex complete metric space with continuous convex structure and let be its nonempty closed convex subset. Let be a continuous map of into itself with at least one fixed point such that for all . Then the sequence where and converges to a fixed point of
Let be a convex metric space and let be its nonempty convex subset. Let be a -Lipschitz selfmap of Let be the sequence as in (1.3), where and satisfy (i) for all (ii) and (iii) If and then is a fixed point of
Finally, using a generalized nonexpansive map on a metric space , we provide a necessary and sufficient condition for the convergence of an arbitrary sequence in to a fixed point of in terms of the approximating sequence
Since and therefore from the above inequality, it follows that is a Cauchy sequence in In view of closedness of this sequence converges to an element of Also gives that Now using the continuity of we have Hence is a fixed point of
Theorem 2.8 improves Lemma 2 in  from real line to convex metric space setting. Theorem 2.9 is an extension of Theorem 4 in  to metric spaces. If we choose in Theorem 2.9, it is still an improvement of [21, Theorem 4].
The author A. R. Khan is grateful to King Fahd University of Petroleum & Minerals for support during this research.
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