Relaxed Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of Strictly Pseudocontractive Mappings

Abstract

We propose a relaxed composite implicit iteration process for finding approximate common fixed points of a finite family of strictly pseudocontractive mappings in Banach spaces. Several convergence results for this process are established.

1. Introduction and Preliminaries

Let be a real Banach space, and let be its dual space. Denote by the normalized duality mapping from into defined by

(1.1)

where is the generalized duality pairing between and . If is smooth, then is single valued and continuous from the norm topology of to the weak* topology of .

A mapping with domain and range in is called -strictly pseudocontractive in the terminology of Browder and Petryshyn [1], if there exists a constant such that

(1.2)

for all and all . Without loss of generality, we may assume . If denotes the identity operator, then (1.2) can be written in the form

(1.3)

for all and all . In (1.2) and (1.3), the positive number is said to be a strictly pseudocontractive constant.

The class of strictly pseudocontractive mappings has been studied by several authors (see, e.g., [1–10]). It is shown in [4] that a strictly pseudocontractive map is -Lipschitzian (i.e., for some ). Indeed, it follows immediately from (1.3) that

(1.4)

and hence where . It is clear that in Hilbert spaces the important class of nonexpansive mappings (mappings for which ) is a subclass of the class of strictly pseudocontractive maps.

Let be a nonempty convex subset of , and let be a finite family of nonexpansive self-maps of . In [11], Xu and Ori introduced the following implicit iteration process; for any initial and , the sequence is generated as follows:

(1.5)

The scheme is expressed in a compact form as

(1.6)

where . Moreover, they proved the following convergence theorem in a Hilbert space.

Theorem 1.1 ([11]).

Let be a Hilbert space, and let be a nonempty closed convex subset of . Let be nonexpansive self-maps of such that where . Let , and let be a sequence in , such that . Then the sequence defined implicitly by (1.6) converges weakly to a common fixed point of the mappings .

Subsequently, Osilike [12] extended their results from nonexpansive mappings to strictly pseudocontractive mappings and derived the following convergence theorems in Hilbert and Banach spaces.

Theorem 1.2 ([12]).

Let be a real Hilbert space, and let be a nonempty closed convex subset of . Let be strictly pseudocontractive self-maps of such that , where . Let , and let be a sequence in such that . Then the sequence defined by (1.6) converges weakly to a common fixed point of the mappings .

Theorem 1.3 ([12]).

Let be a real Banach space, and let be a nonempty closed convex subset of . Let be strictly pseudocontractive self-maps of such that , where , and let be a real sequence satisfying the conditions:

(i)

(ii);

(iii).

Let , and let be defined by (1.6). Then

(i) exists for all ;

(ii).

Let be a nonempty closed convex subset of a real Banach space . Very recently, Su and Li [13] introduced a new implicit iteration process for strictly pseudocontractive self-maps of :

(1.7)

that is,

(1.8)

where and . First, they established the following convergence theorem.

Theorem 1.4 ([13, Theorem ]).

Let be a real Banach space, and let be a nonempty closed convex subset of . Let be strictly pseudocontractive self-maps of such that , where , and let be two real sequences satisfying the conditions:

(i);

(ii);

(iii);

(iv), where is common Lipschitz constant of .

For , let be defined by (1.8). Then

(i) exists for all ;

(ii).

Second, they derived the following result by using Theorem 1.4.

Theorem 1.5 ([13, Theorem ]).

Let be a nonempty closed convex subset of a real Banach space , let be a semicompact strictly pseudocontractive self-map of such that , where , and let be a real sequence satisfying the conditions:

(i);

(ii).

Then for , the sequence defined by Mann iterative process,

(1.9)

converges strongly to a fixed point of .

On the other hand, Zeng and Yao [14] introduced a new implicit iteration scheme with perturbed mapping for approximation of common fixed points of a finite family of nonexpansive self-maps of a real Hilbert space and established some convergence theorems for this implicit iteration scheme. To be more specific, let be a finite family of nonexpansive self-maps of , and let be a mapping such that for some constants is a -Lipschitz and -strongly monotone mapping. Let and and take a fixed number . The authors proposed the following implicit iteration process with perturbed mapping .

For an arbitrary initial point , the sequence is generated as follows:

(1.10)

The scheme is expressed in a compact form as

(1.11)

It is clear that if , then the implicit iteration scheme (1.11) with perturbed mapping reduces to the implicit iteration process (1.6).

Theorem 1.6 ([14, Theorem ]).

Let be a real Hilbert space, and let be a mapping such that for some constants ; is -Lipschitz and -strongly monotone. Let be nonexpansive self-maps of such that . Let , let , , and let satisfying the conditions: and , for some . Then the sequence defined by (1.11) converges weakly to a common fixed point of the mappings .

The above Theorem 1.6 extends Theorem 1.1 from the implicit iteration process (1.6) to the implicit iteration scheme (1.11) with perturbed mapping.

Let be a real Banach space, and let be a nonempty convex subset of . Recall that a mapping is said to be -strongly accretive if there exists a constant such that

(1.12)

for all and all .

Proposition 1.7.

Let be a real Banach space, and let be a mapping:

(i)if is -strictly pseudocontractive, then is a Lipschitz mapping with constant .

(ii)if is both -strictly pseudocontractive and -strongly accretive with , then is nonexpansive.

Proof.

It is easy to see that statement (i) immediately follows from the definition of strict pseudocontraction. Now utilizing the definitions of strict pseudocontraction and strong accretivity, we obtain

(1.13)

Since ,

(1.14)

and hence is nonexpansive.

Let be a real Banach space, and let be a nonempty convex subset of such that . Let be strictly pseudocontractive self-maps of , and let be a perturbed mapping which is both -strongly accretive and -strictly pseudocontractive with . In this paper we introduce a general implicit iteration process as follows:

(1.15)

where , and . In particular, whenever , it is easy to see that (1.15) reduces to (1.8).

Let denote common Lipschitz constant of strictly pseudocontractive self-maps of . Since is a nonempty convex subset of such that , for each , the operator

(1.16)

maps into itself.

Utilizing Proposition 1.7, we have

(1.17)

for all . Thus, is strongly pseudocontractive, if for each . Since is also Lipschitz mapping, it follows from [12, 15, 16] that has a unique fixed point , that is, for each

(1.18)

Therefore, if , then the composite implicit iteration process (1.15) with perturbed mapping can be employed for the approximation of common fixed points of strictly pseudocontractive self-maps of .

The purpose of this paper is to investigate the problem of approximating common fixed points of strictly pseudocontractive mappings of Browder-Petryshyn in an arbitrary real Banach space by this general implicit iteration process (1.15). To this end, we need the following lemma and definition.

Lemma 1.8 ([8]).

Let , and be sequences of nonnegative real numbers satisfying the inequality

(1.19)

If

(1.20)

then exists.

The following definition can be found, for example, in [13].

Definition 1.9.

Let be a closed subset of a real Banach space , and let be a mapping. is said to be semicompact if, for any bounded sequence in such that , there must exist a subsequence such that .

2. Main Results

We are now in a position to prove our main results in this paper.

Theorem 2.1.

Let be a real Banach space, and let be a nonempty closed convex subset of such that . Let be a perturbed mapping which is both -strongly accretive and -strictly pseudocontractive with . Let be strictly pseudocontractive self-maps of such that , where , and let , and be three real sequences in satisfying the conditions:

(i);

(ii);

(iii);

(iv);

(v), where is the common Lipschitz constant of .

For , let be defined by

(2.1)

where , then

(i) exists for all ;

(ii).

Proof.

First, since each strictly pseudocontractive mapping is a Lipschitz mapping, there exists a constant such that

(2.2)

It is now well known (see, e.g., [15]) that

(2.3)

for all and all . Take arbitrarily. Then it follows from (2.1) that

(2.4)

Utilizing (2.3), we obtain

(2.5)

Since each , is strictly pseudocontractive, there exists such that

(2.6)

Thus, utilizing Proposition 1.7(ii) we know from (2.5) that

(2.7)

From (2.1), we also have that

(2.8)

Since is a Lipschitz mapping with constant , we have

(2.9)

Substituting (2.8) and (2.9) into (2.7), we deduce that

(2.10)

and hence

(2.11)

Setting

(2.12)

we conclude from (2.11) that

(2.13)

Thus

(2.14)

Since

(2.15)

and , we get

(2.16)

Setting , it follows from condition (ii) that and so there must exist a natural number such that for all ,

(2.17)

Therefore, it follows from (2.14) that

(2.18)

In order to consider the second term on the right-hand side of (2.18), we will prove that is bounded. Indeed, utilizing (2.8) and (2.9) and simplifying these inequalities, we have

(2.19)

and hence

(2.20)

This implies that

(2.21)

Now, we consider the second term on the right-hand side of (2.21). Since , we have

(2.22)

Since , there exists a natural number such that for all ,

(2.23)

Again, it follows from condition that

(2.24)

Therefore, it follows from (2.21) that

(2.25)

According to conditions (ii)–(iv), we can readily see that

(2.26)

Thus, in terms of Lemma 1.8 we deduce that exists, and hence is bounded.

Now, we consider the second term on the right-hand side of (2.18). Since is bounded, and , there exists a constant and a natural number such that for all ,

(2.27)

Thus, it follows from (2.18) that

(2.28)

Since is bounded, there exists a constant such that . It follows from (2.28) that

(2.29)

and hence

(2.30)

Utilizing conditions (ii)–(iv), we know from (2.30) that

(2.31)

Since , we have

(2.32)

This completes the proof of Theorem 2.1.

The iterative scheme (1.15) becomes the explicit version as follows, whenever :

(2.33)

In the case when , (2.33) is the Mann iteration process as follows:

(2.34)

The conclusion of Theorem 2.1 remains valid for the iteration processes (2.33) and (2.34). Furthermore, we have the following result.

Theorem 2.2.

Let be a real Banach space, and let be a nonempty closed convex subset of such that . Let be a perturbed mapping which is both -strongly accretive and -strictly pseudocontractive with . Let be a semicompact strictly pseudocontractive self-map of such that , where , and let and be two real sequences in satisfying the conditions:

(i);

(ii);

(iii).

Then Mann iteration process (2.34) converges strongly to a fixed point of .

Proof.

Since

(2.35)

there exists a subsequence of such that

(2.36)

By the semicompactness of , there must exist a subsequence of such that

(2.37)

It follows from (2.36) that , and hence . Since exists, we have

(2.38)

This completes the proof of Theorem 2.2.