Approximating Fixed Points of Non-Lipschitzian Mappings by Metric Projections

Abstract

We define and study a new iterative algorithm inspired by Matsushita and Takahashi (2008). We establish a strong convergence theorem of the proposed algorithm for asymptotically nonexpansive in the intermediate sense mappings in uniformly convex and smooth Banach spaces by using metric projections. This theorem generalizes and refines Matsushita and Takahashi's strong convergence theorem which was established for nonexpansive mappings.

1. Introduction and Preliminaries

Let be a real Banach space, and let be a nonempty subset of . A mapping is said to be asymptotically nonexpansive if there exists a sequence in with such that

(1.1)

for all and each . If , then is known as nonexpansive mapping. is said to be asymptotically nonexpansive in the intermediate sense [1] provided is uniformly continuous and

(1.2)

is said to be Lipschitzian if there exists a constant such that

(1.3)

for all .

It follows from the above definitions that every asymptotically nonexpansive mapping is asymptotically nonexpansive in the intermediate sense and Lipschitzian mapping but the converse does not hold such as in the following example.

Example 1.1.

Let , and . We define and for each

(1.4)

We see that is continuous on the compact interval and so it is uniformly continuous. Consider the function defined as

(1.5)

Then, for all and uniformly. On the other hand, compactness of gives that for each there exist such that

(1.6)

Therefore,

(1.7)

Thus, is asymptotically nonexpansive in the intermediate sense.

It is easy to see that is differentiable on and for all . Let there exist such that

(1.8)

for all . Now, choose such that . Then, for each with , it follows from (1.8) that

(1.9)

This contradiction shows that is not Lipschitzian mapping and so it is not asymptotically nonexpansive mapping. Another example of an asymptotically nonexpansive in the intermediate sense mapping which is not asymptotically nonexpansive can be found in [2].

It is known [3] that if is a uniformly convex Banach space and is asymptotically nonexpansive in the intermediate sense self-mapping of a bounded closed convex subset of , then , where denotes the set of all fixed points of . Let be the dual of . We denote the value of at by . When is a sequence in , we denote strong convergence of to by and weak convergence by . A Banach space is said to be strictly convex if for all with and . A Banach space is also said to be uniformly convex if for any two sequences and in such that and . A Banach space is said to have Kadec-Klee property if for every sequence in , and imply that . Every uniformly convex Banach space has the Kadec-Klee property [4]. Let be the unit sphere of . Then the Banach space is said to be smooth if

(1.10)

exists for each . The normalized duality mapping from to is defined by

(1.11)

for all . It is known that a Banach space is smooth if and only if the normalized duality mapping is single-valued. Some properties of duality mapping have been given in [4–6]. Let be a closed convex subset of a reflexive, strictly convex and smooth Banach space . Then for any there exists a unique point such that

(1.12)

The mapping defined by is called the metric projection from onto . Let and . Then, it is known that if and only if

(1.13)

for all (see [4, 6, 7]).

Fixed points of nonlinear mappings play an important role in solving systems of equations and inequalities that often arise in applied sciences. Approximating fixed points of asymptotically nonexpansive and nonexpansive mappings with implicit and explicit iterative schemes has been studied by many authors (see, e.g., [8–14]).

On the other hand, using the metric projection, Nakajo and Takahashi [15] introduced an iterative algorithm in the framework of Hilbert spaces and gave strong convergence theorem for nonexpansive mappings. Xu [16] extended Nakajo and Takahashi's theorem to Banach spaces by using the generalized projection. Recently, Matsushita and Takahashi [17] introduced an iterative algorithm for nonexpansive mappings in Banach spaces as follows.

Let be a nonempty convex bounded subset of a uniformly convex and smooth Banach space , and let be a nonexpansive self-mapping of . For a given , compute the sequence by the iterative algorithm

(1.14)

where denotes the convex closure of the set and is a sequence in with . They proved that generated by (1.14) converges strongly to a fixed point of .

Inspired and motivated by these facts, we introduce a new iterative algorithm to find fixed points of asymptotically nonexpansive in the intermediate sense mappings in a uniformly convex and smooth Banach space. Let , and compute the sequence by the iterative algorithm

(1.15)

where is a sequence in with and is the metric projection from onto .

In the sequel, the following lemmas are needed to prove our main convergence theorem.

Lemma 1.2 (see [18, Lemma  1.5]).

Let be a nonempty bounded closed convex subset of a uniformly convex Banach space and be a mapping which is asymptotically nonexpansive in the intermediate sense. For each , there exist integers and such that if is any integer, , and if for , then

(1.16)

for any numbers with .

Lemma 1.3 (see [18, Lemma  1.6]).

Let be a real uniformly convex Banach space, let be a nonempty closed convex subset of , and let be a mapping which is asymptotically nonexpansive in the intermediate sense. If is a sequence in converging weakly to and if

(1.17)

then is demiclosed at zero; that is, for each sequence in , if for some and , then .

2. Main Results

In this section, we study the iterative algorithm (1.15) to find fixed points of asymptotically nonexpansive in the intermediate sense mappings in a uniformly convex and smooth Banach space. We first prove that the sequence generated by (1.15) is well-defined. Then, we prove that converges strongly to , where is the metric projection from onto .

Lemma 2.1.

Let be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space , and let be a mapping which is asymptotically nonexpansive in the intermediate sense. If , then the sequence generated by (1.15) is well-defined.

Proof.

It is easy to check that is closed and convex and for each . Moreover and so . Suppose . Since , it follows from (1.13) that

(2.1)

for all and so for all , that is . Thus, . By mathematical induction, we obtain that for all . Therefore, is well-defined.

In order to prove our main result, the following lemma is needed.

Lemma 2.2.

Let be a nonempty bounded closed convex subset of a uniformly convex and smooth Banach space , and let be a mapping which is asymptotically nonexpansive in the intermediate sense. If is the sequence generated by (1.15), then

(2.2)

for all integers .

Proof.

Let be fixed, and let be arbitrary. We take for simplicity. Since , we have . Since , there exist elements in and numbers with such that

(2.3)

(2.4)

for all . We put , , and . The inequality (2.4) implies that

(2.5)

for all . Now, let , and choose an integer and with as in Lemma 1.2. Since and , we may choose an integer such that for all

(2.6)

This together with (2.5) implies that

(2.7)

for all and all . Thus,

(2.8)

and so by Lemma 1.2 we have

(2.9)

where . It follows from (2.3)–(2.9) that

(2.10)

for all ; that is,

(2.11)

This completes the proof.

Now, we state and prove the strong convergence theorem of the iterative algorithm (1.15).

Theorem 2.3.

Let be a nonempty bounded closed convex subset of a uniformly convex and smooth Banach space , let be a mapping which is asymptotically nonexpansive in the intermediate sense and let be the sequence generated by (1.15). Then converges strongly to the element of .

Proof.

Put . Since and , we have

(2.12)

for all . On the other hand, we observe that

(2.13)

and so by uniform continuity of and Lemma 2.2 we have

(2.14)

Since is bounded, there exists a subsequence of such that . It follows from (2.14) and Lemma 1.3 (demiclosedness of ) that . From the weakly lower semicontinuity of norm and (2.12), we obtain

(2.15)

This together with the uniqueness of implies that , and hence . This gives that . By using the same argument as in proof of (2.15), we have

(2.16)

Since is uniformly convex, by Kadec-Klee property, we obtain that . It follows that . This completes the proof.

Since every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is asymptotically nonexpansive in the intermediate sense, we have the following result which generalizes and refines the strong convergence theorem of Matsushita and Takahashi [17, Theorem  3.1].

Corollary 2.4.

Let be a nonempty bounded closed convex subset of a uniformly convex and smooth Banach space , let be a nonexpansive self-mapping of , and let be the sequence generated by (1.15). Then converges strongly to the element of .