# Approximating Fixed Points of Some Maps in Uniformly Convex Metric Spaces

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## Abstract

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.

### Keywords

Convex Subset Fixed Point Theorem Strong Convergence Nonempty Subset Cauchy Sequence## 1. Introduction and Preliminaries

Let Open image in new window be a nonempty subset of a metric space Open image in new window and let Open image in new window be a map. Denote the set of fixed points of Open image in new window Open image in new window by Open image in new window The map Open image in new window is said to be (i) quasi-nonexpansive if Open image in new window and Open image in new window for all Open image in new window and Open image in new window , (ii) Open image in new window -Lipschitz if for some Open image in new window we have Open image in new window for all Open image in new window for Open image in new window it becomes nonexpansive, and (iii) generalized nonexpansive (cf. [1] and the references therein) if

for all Open image in new window where Open image in new window with Open image in new window

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 [2].

Mann and Ishikawa type iterates for nonexpansive and quasi-nonexpansive maps have been extensively studied in uniformly convex Banach spaces [1, 3, 4, 5, 6]. Senter and Dotson [7] established convergence of Mann type iterates of quais-nonexpansive maps under a condition in uniformly convex Banach spaces. In 1973, Goebel et al. [8] proved that generalized nonexpansive self maps have fixed points in uniformly convex Banach spaces. Based on their work, Bose and Mukerjee [1] proved theorems for the convergence of Mann type iterates of generalized nonexpansive maps and obtained a result of Kannan [9] under relaxed conditions. Maiti and Ghosh [6] generalized the results of Bose and Mukerjee [1] for Ishikawa iterates by using modified conditions of Senter and Dotson [7] (see, also [10]). For the sake of completeness, we state the result of Kannan [9] and its generalization by Bose and Mukerjee [1].

Theorem 1.1 (see [9]).

Let Open image in new window be a nonempty, bounded, closed, and convex subset of a uniformly convex Banach space. Let Open image in new window be a map of Open image in new window into itself such that

(i) Open image in new window Open image in new window for all Open image in new window ,

(ii) Open image in new window where Open image in new window is any nonempty convex subset of Open image in new window which is mapped into itself by Open image in new window and Open image in new window is the diameter of Open image in new window

Then the sequence Open image in new window defined by Open image in new window converges to the fixed point of Open image in new window where Open image in new window is any arbitrary point of Open image in new window

Theorem 1.2 (see [1]).

for all Open image in new window where Open image in new window and Open image in new window Define a sequence Open image in new window in Open image in new window for Open image in new window , for all Open image in new window , where Open image in new window Then Open image in new window converges to a fixed point of Open image in new window .

In Theorem 1.2, taking Open image in new window , and Open image in new window for all Open image in new window it becomes Theorem 1.1 without requiring condition (ii).

In 1970, Takahashi [11] introduced a notion of convexity in a metric space Open image in new window as follows: a map Open image in new window Open image in new window is a convex structure in Open image in new window if

for all Open image in new window and Open image in new window A metric space together with a convex structure is said to be convex metric space. A nonempty subset Open image in new window of a convex metric space is convex if Open image in new window for all Open image in new window and Open image in new window In fact, every normed space and its convex subsets are convex metric spaces but the converse is not true, in general (see [11]). Later on, Shimizu and Takahashi [12] obtained some fixed point theorems for nonexpansive maps in convex metric spaces. This notion of convexity has been used in [13, 14, 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, 17, 18, 19].

In the sequel, we assume that Open image in new window is a nonempty convex subset of a convex metric space Open image in new window and Open image in new window is a selfmap on Open image in new window . For an initial value Open image in new window we define the Ishikawa iteration scheme in Open image in new window as follows:

where Open image in new window and Open image in new window are control sequences in Open image in new window

If we choose Open image in new window then (1.3) reduces to the following Mann iteration scheme:

where Open image in new window is a control sequence in Open image in new window

If Open image in new window is a normed space with Open image in new window as its convex subset, then Open image in new window is a convex structure in Open image in new window consequently (1.3) and (1.4), respectively, become

where Open image in new window and Open image in new window are control sequences in Open image in new window

A convex metric space Open image in new window is said to be uniformly convex [11] if for arbitrary positive numbers Open image in new window and Open image in new window , there exists Open image in new window such that

whenever Open image in new window and Open image in new window

In 1989, Maiti and Ghosh [6] generalized the two conditions due to Senter and Dotson [7]. We state all these conditions in convex metric spaces:

*Let* Open image in new window *be a map with nonempty fixed point set* Open image in new window and Open image in new window . *Then* Open image in new window *is said to satisfy the following Condotions*.

Condition 1.

If there is a nondecreasing function Open image in new window with Open image in new window and Open image in new window for all Open image in new window such that Open image in new window for Open image in new window .

Condition 2.

If there exists a real number Open image in new window such that Open image in new window for Open image in new window .

Condition 3.

If there is a nondecreasing function Open image in new window with Open image in new window and Open image in new window for all Open image in new window such that Open image in new window for Open image in new window and all corresponding Open image in new window where Open image in new window .

Condition 4.

If there exists a real number Open image in new window such that Open image in new window for Open image in new window and all corresponding Open image in new window where Open image in new window

Note that if Open image in new window 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 [7] while Conditions 3 and 4 become Conditions I and II, respectively, of Maiti and Ghosh [6] in a normed space. Further, Conditions 3 and 4 reduce to Conditions 1 and 2, respectively, when Open image in new window

In this note, we present results under relaxed control conditions which generalize the corresponding results of Kannan [9], Bose and Mukerjee [1], and Maiti and Ghosh [6] from uniformly convex Banach spaces to uniformly convex metric spaces. We present sufficient conditions for the convergence of Ishikawa iterates of Open image in new window 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 [20]).

for all Open image in new window and Open image in new window

## 2. Convergence Analysis

We prove a lemma which plays key role to establish strong convergence of the iterative schemes (1.3) and (1.4).

Lemma 2.1.

Let Open image in new window be a uniformly convex metric space. Let Open image in new window be a nonempty closed convex subset of Open image in new window a quasi-nonexpansive map and Open image in new window as in (1.3). If Open image in new window and Open image in new window then Open image in new window

Proof.

This implies that the sequence Open image in new window is nonincreasing and bounded below. Thus Open image in new window exists. We may assume that Open image in new window

For any Open image in new window , we have that

Since Open image in new window exists, so Open image in new window is bounded and hence Open image in new window exists. We show that Open image in new window . Assume that Open image in new window

Then

Let Open image in new window . Then, we have

This is contradiction and hence Open image in new window

In the light of above result, we can construct subsequences Open image in new window and Open image in new window of Open image in new window and Open image in new window , respectively, such that Open image in new window and hence Open image in new window

Now we state and prove Ishikawa type convergence result in uniformly convex metric spaces.

Theorem 2.2.

Let Open image in new window be a uniformly convex complete metric space with continuous convex structure and let Open image in new window be its nonempty closed convex subset. Let Open image in new window be a continuous quasi-nonexpansive map of Open image in new window into itself satisfying Condition 3. If Open image in new window is as in (1.3), where Open image in new window and Open image in new window , then Open image in new window converges to a fixed point of Open image in new window .

Proof.

Using the properties of Open image in new window we have Open image in new window . As Open image in new window exists, therefore Open image in new window

Now, we show that Open image in new window is a Cauchy sequence. For Open image in new window there exists a constant Open image in new window such that for all Open image in new window we have Open image in new window In particular, Open image in new window That is, Open image in new window There must exist Open image in new window Open image in new window such that Open image in new window Now, for Open image in new window , we have that

This proves that Open image in new window is a Cauchy sequence in Open image in new window . Since Open image in new window is a closed subset of a complete metric space Open image in new window therefore it must converge to a point Open image in new window in Open image in new window .

Finally, we prove that Open image in new window is a fixed point of Open image in new window

Since

therefore Open image in new window As Open image in new window is closed, so Open image in new window

Choose Open image in new window for all Open image in new window in the above theorem; it reduces to the following Mann type convergence result.

Theorem 2.3.

Let Open image in new window be a uniformly convex complete metric space with continuous convex structure and let Open image in new window be its nonempty closed convex subset. Let Open image in new window be a continuous quasi-nonexpansive map of Open image in new window into itself satisfying Condition 1. If Open image in new window is as in (1.4), where Open image in new window , then Open image in new window converges to a fixed point of Open image in new window .

Next we establish strong convergence of Ishikawa iterates of a generalized nonexpansive map.

Theorem 2.4.

Let Open image in new window and Open image in new window be as in Theorem 2.3. Let Open image in new window be a continuous generalied nonexpansive map of Open image in new window into itself with at least one fixed point. If Open image in new window is as in (1.3), where Open image in new window and Open image in new window then Open image in new window converges to a fixed point of Open image in new window .

Proof.

Thus Open image in new window is quasi-nonexpansive.

For any Open image in new window we also observe that

If Open image in new window where Open image in new window then

where Open image in new window Thus Open image in new window satisfies Condition 4 (and hence Condition 3). The result now follows from Theorem 2.2.

Remark 2.5.

In the above theorem, we have assumed that the generalied nonexpansive map Open image in new window has a fixed point. It remains an open questions: what conditions on Open image in new window , and Open image in new window in (*) are sufficient to guarantee the existence of a fixed point of Open image in new window even in the setting of a metric space.

Choose Open image in new window for all Open image in new window in Theorem 2.4 to get the following Mann type convergence result.

Theorem 2.6.

Let Open image in new window and Open image in new window be as in Theorem 2.4. If Open image in new window is as in (1.4), where Open image in new window then Open image in new window converges to a fixed point of Open image in new window .

Proof.

Thus Open image in new window satisfies Condition 2 (and hence Condition 1) and so the result follows from Theorem 2.3.

The analogue of Kannan result in uniformly convex metric space can be deduced from Theorem 2.6 (by taking Open image in new window , and Open image in new window for all Open image in new window ) as follows.

Theorem 2.7.

Let Open image in new window be a uniformly convex complete metric space with continuous convex structure and let Open image in new window be its nonempty closed convex subset. Let Open image in new window be a continuous map of Open image in new window into itself with at least one fixed point such that Open image in new window Open image in new window for all Open image in new window . Then the sequence Open image in new window where Open image in new window and Open image in new window converges to a fixed point of Open image in new window

Next we give sufficient conditions for the existence of fixed point of a Open image in new window -Lipschitz map in terms of the Ishikawa iterates.

Theorem 2.8.

Let Open image in new window be a convex metric space and let Open image in new window be its nonempty convex subset. Let Open image in new window be a Open image in new window -Lipschitz selfmap of Open image in new window Let Open image in new window be the sequence as in (1.3), where Open image in new window and Open image in new window satisfy (i) Open image in new window for all Open image in new window (ii) Open image in new window and (iii) Open image in new window If Open image in new window and Open image in new window then Open image in new window is a fixed point of Open image in new window

Proof.

Taking Open image in new window on both the sides in the above inequality and using the condition Open image in new window , we have Open image in new window

Finally, using a generalized nonexpansive map Open image in new window on a metric space Open image in new window , we provide a necessary and sufficient condition for the convergence of an arbitrary sequence Open image in new window in Open image in new window to a fixed point of Open image in new window in terms of the approximating sequence Open image in new window

Theorem 2.9.

for all Open image in new window Then a sequence Open image in new window in Open image in new window converges to a fixed point of Open image in new window if and only if Open image in new window

Proof.

That is,

Since Open image in new window and Open image in new window therefore from the above inequality, it follows that Open image in new window is a Cauchy sequence in Open image in new window In view of closedness of Open image in new window this sequence converges to an element Open image in new window of Open image in new window Also Open image in new window gives that Open image in new window Now using the continuity of Open image in new window we have Open image in new window Hence Open image in new window is a fixed point of Open image in new window

Conversely, suppose that Open image in new window converges to a fixed point Open image in new window of Open image in new window Using the continuity of Open image in new window we have that Open image in new window Thus Open image in new window

Remark 2.10.

Theorem 2.8 improves Lemma 2 in [3] from real line to convex metric space setting. Theorem 2.9 is an extension of Theorem 4 in [21] to metric spaces. If we choose Open image in new window in Theorem 2.9, it is still an improvement of [21, Theorem 4].

Remark 2.11.

We have proved our results (2.1)–(2.8) in convex metric space setting. All these results, in particular, hold in Banach spaces if we set Open image in new window

## Notes

### Acknowledgment

The author A. R. Khan is grateful to King Fahd University of Petroleum & Minerals for support during this research.

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