Approximating Fixed Points of Some Maps in Uniformly Convex Metric Spaces

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.

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. [1] and the references therein) if

(x2a)

for all where with

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, 3–6]. 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 be a nonempty, bounded, closed, and convex subset of a uniformly convex Banach space. Let be a map of into itself such that

(i) for all ,

(ii) where is any nonempty convex subset of which is mapped into itself by and is the diameter of

Then the sequence defined by converges to the fixed point of where is any arbitrary point of

Theorem 1.2 (see [1]).

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

(1.1)

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

In Theorem 1.2, taking , and for all it becomes Theorem 1.1 without requiring condition (ii).

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

(1.2)

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 [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 [13–15] 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].

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

(1.3)

where and are control sequences in

If we choose then (1.3) reduces to the following Mann iteration scheme:

(1.4)

where is a control sequence in

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

(1.5)

where and are control sequences in

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

(1.6)

whenever and

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:

Letbe a map with nonempty fixed point set and . Thenis said to satisfy the following Condotions.

Condition 1.

If there is a nondecreasing function with and for all such that for .

Condition 2.

If there exists a real number such that for .

Condition 3.

If there is a nondecreasing function with and for all such that for and all corresponding where .

Condition 4.

If there exists a real number such that for and all corresponding where

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 [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

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 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]).

Let be a uniformly convex metric space with a continuous convex structure Then for arbitrary positive numbers and , there exists such that

(1.7)

for all and

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

Proof.

For , we consider

(2.1)

This implies that the sequence is nonincreasing and bounded below. Thus exists. We may assume that

For any , we have that

(2.2)

Since exists, so is bounded and hence exists. We show that . Assume that

Then

(2.3)

Hence by Theorem 1.3, there exists such that

(2.4)

That is,

(2.5)

Taking and summing up the terms on the both sides in the above inequality, we have

(2.6)

Let . Then, we have

(2.7)

This is contradiction and hence

In the light of above result, we can construct subsequences and of and , respectively, such that and hence

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

Theorem 2.2.

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 .

Proof.

In Lemma 2.1, we have shown that Therefore . This implies that the sequence is nonincreasing and bounded below. Thus exists. Now by Condition 3, we have

(2.8)

Using the properties of we have . As exists, therefore

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

(2.9)

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

Finally, we prove that is a fixed point of

Since

(2.10)

therefore As is closed, so

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

Theorem 2.3.

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.

Theorem 2.4.

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

Proof.

Let be any fixed point of Then setting in (*), we have

(2.11)

which implies

(2.12)

Thus is quasi-nonexpansive.

For any we also observe that

(2.13)

If where then

(2.14)

(2.15)

Using (2.14) in (2.13), we have

(2.16)

Also it is obvious that

(2.17)

Combining (2.16) and (2.17), we get that

(2.18)

Now inserting (2.15) in (2.18), we derive

(2.19)

That is,

(2.20)

where Thus 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 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.

Choose for all in Theorem 2.4 to get the following Mann type convergence result.

Theorem 2.6.

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

Proof.

For for all , the inequality (2.20) in the proof of Theorem 2.4 becomes

(2.21)

Thus 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 , and for all ) as follows.

Theorem 2.7.

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

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

Theorem 2.8.

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

Proof.

Let Then

(2.22)

That is,

(2.23)

Since therefore there exists such that for all This implies that

(2.24)

Taking on both the sides in the above inequality and using the condition , we have

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

Theorem 2.9.

Suppose that is a closed subset of a complete metric space and is a continuous map such that for some , the following inequality holds:

(2.25)

for all Then a sequence in converges to a fixed point of if and only if

Proof.

Suppose that First we show that is a Cauchy sequence in To acheive this goal, consider:

(2.26)

That is,

(2.27)

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

Conversely, suppose that converges to a fixed point of Using the continuity of we have that Thus

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 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