International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 405315, 17 pages

doi:10.1155/2012/405315

Research Article

Strong Convergence Theorems for a Common Fixed Point of a Finite Family of Pseudocontractive Mappings

O. A. Daman and H. Zegeye

Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana

Received 16 May 2012; Accepted 12 July 2012

Academic Editor: Billy Rhoades

Copyright © 2012 O. A. Daman and H. Zegeye. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

It is our purpose, in this paper, to prove strong convergence of Halpern-Ishikawa iteration method to a common fixed point of finite family of Lipschitz pseudocontractive mappings. There is no compactness assumption imposed either on C or on T. The results obtained in this paper improve most of the results that have been proved for this class of nonlinear mappings.

1. Introduction

Let be a nonempty subset of a real Hilbert space . The mapping is called Lipschitz if there exists such that

If , then is called nonexpansive, and if , then is called a contraction. It follows from (1.1) that every contraction mapping is nonexpansive and every nonexpansive mapping is Lipschitz.

A mapping is called -strictly pseudocontractive [1] if for all there exists such that

A mapping is called pseudocontractive if

We note that (1.2) and (1.3) can be equivalently written as

respectively.

We observe from (1.4) and (1.5) that every nonexpansive mapping is -strict pseudocontractive mapping and every -strict pseudocontractive mapping is pseudocontractive mapping, and hence class of pseudocontractive mappings is a more general class of mappings. Furthermore, pseudocontractive mappings are related with the important class of nonlinear monotone mappings, where a mapping with domain and range in is called monotone if the inequality

holds for every . We note that is pseudocontractive if and only if is monotone, and hence a fixed point of , is a zero of , . It is now well known (see, e.g., [2]) that if is monotone, then the solutions of the equation correspond to the equilibrium points of some evolution systems. Consequently, many researchers have made efforts to obtain iterative methods for approximating fixed points of , when is pseudocontractive (see, e.g., [3–10] and the references contained therein).

Let be a closed subset of a Hilbert space , and let be a contraction. Then the Picard iteration method given by

converges to the unique fixed point of . However, this Picard iteration method may not always converge to a fixed point of , when is nonexpansive mapping. We can take, for example, to be the anticlockwise rotation of the unit disk in (with the Euclidean norm) about the origin of coordinate of an angle, say, .

The scheme that has been used to approximate fixed points of nonexpansive mappings is the Mann iteration method [5] given by

where is a real sequence in the interval satisfying certain conditions. But it is worth mentioning that the Mann iteration process does not always converge strongly to a fixed point of nonexpansive mapping . One has to impose compactness assumption on (e.g., is compact) or on (e.g., is semicompact) to get strong convergence of Mann iteration method to a fixed point of nonexpansive self-map (see, e.g., [11, 12]).

We also note that efforts to approximate a fixed point of a Lipschitz pseudocontractive mapping defined even on a compact convex subset of a Hilbert space by Mann iteration method proved abortive. One can see an example of a Lipschitz pseudocontractive self-map of a compact convex subset of a Hilbert space with a unique fixed point for which no Mann sequence converges by Chidume and Mutangadura [13]. This leads now to our next concern.

Can we construct an iterative sequence for approximating fixed point of the Lipschitz pseudocontractive mappings?

In 1974, Ishikawa [14] introduced an iteration process which converges to a fixed point of Lipschitz pseudocontractive self-map of , when is compact. In fact, he proved the following theorem.

Theorem I. If is a compact convex subset of a Hilbert space is a Lipschitz pseudocontractive mapping and is any point of , then the sequence converges strongly to a fixed point of , where is defined iteratively for each integer by

here are sequences of positive numbers satisfying the conditions

We observe that Theorem I imposes compactness assumption on , and it is still an open problem whether or not scheme (1.9), known as the Ishikawa iterative method, can be used to approximate fixed points of Lipschitz pseudocontractive mappings without compactness assumption on or on .

In order to obtain a strong convergence theorem for pseudocontractive mappings without the compactness assumption, Zhou [15] established the hybrid Ishikawa algorithm for Lipschitz pseudocontractive mappings as follows:

He proved that the sequence defined by (1.11) converges strongly to , where is the metric projection from into .

Recently, several authors (see, e.g., [16–18]) also used the hybrid Mann and hybrid Ishikawa algorithm methods to obtain strong convergence to a fixed point of Lipschitz pseudocontractive mappings. But it is worth mentioning that the hybrid schemes are not easy to compute. They involve computation of and for each .

Another iteration scheme was introduced and studied by Chidume and Zegeye [19] with which they approximated fixed point of Lipschitz pseudocontractive mapping in a more general real Banach space.

Let be a convex nonempty subset of real Banach space , and let be a mapping. From arbitrary , define by

where and are real sequences in satisfying the following conditions: , . Examples of real sequences which satisfy these conditions are and , where and . They proved the following theorem.

Theorem CZ. Let be a nonempty closed convex subset of a reflexive real Banach space with a uniformly Gâteaux differentiable norm. Let be a Lipschitz pseudocontractive mapping with Lipschitz constant and . Suppose every closed convex and bounded subset of has the fixed point property for nonexpansive self-mappings. Let a sequence be generated iteratively by (1.12). Then converges strongly to a fixed point of .

Theorem CZ solves the open problem of approximating fixed point of Lipschitz pseudocontractive mappings that has been in the air for many years. However, it is still an open problem whether or not this scheme can be used to approximate a common fixed point of a family of Lipschitz pseudocontractive mappings. Moreover, we observe that the conditions on the real sequences and excluded the natural choice, and .

Our concern now is the following: can we construct an iterative sequence for a common fixed point of a family of Lipschitz pseudocontractive mappings?

For a sequence of real numbers in and an arbitrary , let the sequence in be iteratively defined by

The recursion formula (1.13) known as Halpern scheme was first introduced in 1967 by Halpern [20] in the framework of Hilbert spaces. He proved that convergs strongly to a fixed point of nonexpansive self-mapping of .

Recently, considerable research efforts have been devoted to developing iterative methods for approximating a common fixed point of a family of several nonlinear mappings (see, e.g., [4, 21, 22]). In 1996, Bauschke [3] introduced the following Halpern-type iterative process for approximating a common fixed point for a finite family of nonexpansive self-mappings. In fact, he proved the following theorem.

Theorem B. Let C be a nonempty closed convex subset of a Hilbert space , and let be a finite family of nonexpansive mappings of into itself with . Let be a real sequence in which satisfies certain mild conditions. Given points, , let be generated by

where . Then converges strongly to , where is the metric projection.

But it is worth mentioning that it is still an open problem whether or not this scheme can be used to approximate a common fixed points of Lipschitz pseudocontractive mappings?

In 2008, Zhou [22] studied weak convergence of an implicit scheme to a common fixed point of finite family of pseudocontractive mappings. More precisely, he proved the following theorem.

Theorem Z. Let be a real uniformly convex Banach space with a Frêchet differentiable norm. Let be a closed convex subset of , and let be a finite family of Lipschitzian pseudocontractive self-mappings of such that . Let be defined by

where . If is chosen so that with , then converges weakly to a common fixed point of the family .

Here, we remark that the scheme in Theorem Z is implicit, and the convergence is weak convergence.

More recently, Zegeye et al. [23] proved the following strong convergence of Ishikawa iterative process for a common fixed point of finite family of Lipschitz pseudocontractive mappings.

Theorem ZSA (see [23]). Let be a nonempty, closed and convex subset of a real Hilbert space . Let , be a finite family of Lipschitz pseudocontractive mappings with Lipschitzian constants , for , respectively. Assume that the interior of is nonempty. Let be a sequence generated from an arbitrary by

where and satisfying certain appropriate conditions. Then, converges strongly to a common fixed point of .

From Theorem ZSA, we observe that the assumption that the interior of  is nonempty is severe restriction.

Motivated by Halpern [20] and Zegeye et al. [23], it is our purpose, in this paper, to prove strong convergence of Halpern-Ishikawa algorithm (3.3) to a common fixed point of a finite family of Lipschitz pseudocontractive mappings. No compactness assumption is imposed either on one of the mappings or on . The assumption that interior of    is nonempty is dispensed with. Moreover, computation of closed and convex set for each is not required. The results obtained in this paper improve and extend the results of Theorems I and ZSA, Zhou [15], Yao et al. [17], and Tang et al. [16].

2. Preliminaries

In what follows we will make use of the following lemmas.

Lemma 2.1. Let be a real Hilbert space. Then for any given , the following inequality holds:

Lemma 2.2 (see [24]). Let be a convex subset of a real Hilbert space . Let . Then if and only if

Lemma 2.3 (see [25]). Let be a sequence of nonnegative real numbers satisfying the following relation:

where and satisfying the following conditions: , and . Then, .

Lemma 2.4 (see [18]). Let be a real Hilbert space, let be a closed convex subset of , and let be a continuous pseudocontractive mapping; then

(i) is closed convex subset of ;  (ii) is demiclosed at zero; that is, if is a sequence in such that and , as , then .

Lemma 2.5 (see [26]). Let be sequences of real numbers such that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence such that ,and the following properties are satisfied by all (sufficiently large) numbers :

In fact, .

Lemma 2.6 (see [27]). Let be a real Hilbert space. Then for all and for such that the following equality holds:

3. Main Result

We now prove the following lemma and theorems.

Lemma 3.1. Let be a nonempty convex subset of a real Hilbert space . Let , be a finite family of Lipschitz pseudocontractive mappings with constants , respectively. Let , where . Then is Lipschitz pseudocontractive mapping on .

Proof. Let . Then

Hence is pseudocontractive. Moreover, since

where , we get that is -Lipschitz. The proof is complete.

Let be a finite family of pseudocontractive mappings. The family is said to satisfy condition if , for .

Theorem 3.2. Let be a nonempty, closed and convex subset of a real Hilbert space . Let be a finite family of Lipschitz pseudocontractive mappings with Lipschitz constants , respectively, satisfying condition . Assume that is nonempty. Let a sequence be a sequence generated from an arbitrary by

where , for such that , for all and satisfying the following conditions:  , for all such that and ;  , for all , for . Then, converges strongly to a common fixed point of nearest to .

Proof. Let . Then from (3.3), Lemma 2.6, (1.5), and Lemma 3.1 we have the following:

In addition, we have that

Substituting (3.5) and (3.6) into (3.4) we obtain that

Since from , we have that and for all , (3.7) implies that

Thus, by induction,

which implies that and hence are bounded.

Furthermore, from (3.3), Lemma 2.1, and following the methods used in (3.7) we get that

On the other hand, using Lemma 2.6 and condition (), we get that

Thus, substituting (3.11) into (3.10) we obtain that

Now, we consider the following two cases.

Case  1. Suppose that there exists such that is nonincreasing. Then, we get that is convergent. Thus, from (3.12) and the fact that , as , we have that

for each . Let . Then from (3.3) we obtain that

Furthermore, from (3.3) and (3.14) we get that

as , and hence (3.16) and the fact that is -Lipschitz imply that

Now, (3.15) and (3.17) imply that

Moreover, since is bounded and is reflexive, we choose a subsequence of such that and . This implies from (3.18) that . Then, from (3.14) and Lemma 2.4 we have that , for each . Hence, . Therefore, by Lemma 2.2, we immediately obtain that

Then, since from (3.13) we have that

It follows from (3.20), (3.19), and Lemma 2.3 that , as . Consequently, .

Case  2. Suppose that there exists a subsequence of such that

for all . Then, by Lemma 2.5, there exists a nondecreasing sequence such that , and for all . Now, from (3.12) and the fact that , we get that , as , for each . Thus, as in Case 1, we obtain that and that

Now, from (3.13) we have that

and hence, since , (3.23) implies that

But noting that , we obtain that

Then, from (3.22) we get that , as . This together with (3.23) gives that , as . But , for all ; thus we obtain that . Therefore, from the previous two cases, we can conclude that converges strongly to an element of , and the proof is complete.

If, in Theorem 3.2, we consider single Lipschitz pseudocontractive mapping, then the assumption of condition is not required. In fact, we have the following corollary.

Corollary 3.3. Let be a nonempty, closed and convex subset of a real Hilbert space . Let be a Lipschitz pseudocontractive mapping with Lipschitz constants . Assume that is nonempty. Let a sequence be a sequence generated from an arbitrary by

where satisfying the following conditions: , for all such that and ;  , for all . Then, converges strongly to a fixed point of nearest to .

Proof. Putting in (3.3) the scheme reduces to scheme (3.26), and following the method of proof of Theorem 3.2 we get that (see, (3.10))

Now, considering cases as in the proof of Theorem 3.2 we obtain the required result.

We now state and prove a convergence theorem for a common zero of finite family of monotone mappings.

Corollary 3.4. Let be a real Hilbert space. Let be a finite family of Lipschitz monotone mappings with Lipschitz constants , respectively, satisfying , for all .

Assume that is nonempty. Let a sequence be generated from an arbitrary by

where , for such that , for all and satisfying the following conditions:  , for all such that and ;  , for all , for . Then, converges strongly to a common zero point of nearest to .

Proof . Let , for . Then we get that every for all is Lipschitz pseudocontractive mapping with Lipschitz constants and . Moreover, when is replaced with , for each , we get that scheme (3.28) reduces to scheme (3.3), and hence the conclusion follows from Theorem 3.2.

If, in Corollary 3.4 we consider a single Lipschitz monotone mapping, then we obtain the following corollary.

Corollary 3.5. Let be a real Hilbert space. Let be Lipschitz monotone mappings with Lipschitz constant . Assume that is nonempty. Let a sequence be generated from an arbitrary by

where satisfying the following conditions:  , for all such that and ;  , . Then, converges strongly to a zero point of nearest to .

We now give examples of Lipschitz pseudocontractive mappings satisfying condition . Let and . Let be defined by

Then we observe that ,and , and hence common fixed point of and is which is nonempty. Now, we show that and are pseudocontractive mappings. But, since

are monotone, we have that and are pseudocontractive mappings.

Now, we show that and are Lipschitzian mappings. First, we show that is Lipschitzian with constant . Let , . If , then we have that

If ,then we have that

If and then we get that

Therefore, from (3.32), (3.33), and (3.34), we obtain that is Lipschitz.

Next, we show that is Lipschitz with constant .

Let , . If , then we have that

If ,then we get that

If and then we have that

and for (3.37) implies that

For inequality (3.37) gives that

Therefore, from (3.35), (3.36), and (3.39) we obtain that is Lipschitz. Furthermore, we show that and satisfy condition . If ,then we have that , and if we get that . Therefore, and satisfy property .

Remark 3.6. Theorem 3.2 provides convergence sequence to a common fixed point of finite family of Lipschitzian pseudocontractive mappings whereas Corollary 3.4 provides convergence sequence to a common zero of finite family of monotone mappings in Hilbert spaces. No compactness assumption is imposed either on or . This provides affirmative answer to the question raised.

Remark 3.7. Theorem 3.2 improves Theorem I, Theorem 3.1 of Zhou [15], Theorem 3.1 of Yao et al. [17], and Theorem 3.1 of Tang et al. [16] in the sense that either our convergence does not require compactness of or computation of from for each .

Remark 3.8. Theorem 3.2 improves Theorems I and ZSA in the sense that our convergence is for a fixed point of a finite family of Lipschitz pseudocontractive mappings. The condition that interior of is nonempty is dispensed with.

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