`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 926017, 15 pageshttp://dx.doi.org/10.1155/2012/926017`
Research Article

Minimum-Norm Fixed Point of Pseudocontractive Mappings

1Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana
2Department of Mathematics, King Abdulaziz University, P.O. Box. 80203, Jeddah 21589, Saudi Arabia

Received 7 May 2012; Accepted 14 June 2012

Copyright © 2012 Habtu Zegeye et al. 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

Let K be a closed convex subset of a real Hilbert space H and let be a continuous pseudocontractive mapping. Then for and each , there exists a sequence satisfying which converges strongly, as , to the minimum-norm fixed point of T. Moreover, we provide an explicit iteration process which converges strongly to a minimum-norm fixed point of T provided that T is Lipschitz. Applications are also included. Our theorems improve several results in this direction.

1. Introduction

Let be a nonempty subset of a real Hilbert space . A mapping is called Lipschitz if there exists such that If , then is called a contraction; if then is called a nonexpansive. It is easy to see from (1.1) that every contraction mapping is nonexpansive, and every nonexpansive mapping is Lipschitz.

A mapping is called strongly pseudocontractive if there exists such that inequality holds for all . is called pseudocontractive if the inequality holds for all . Note that inequality (1.3) can be equivalently written as

It is easy to see that nonexpansive and strongly pseudocontractive mappings are pseudocontractive mappings. However, the converse may not be true (see [1, 2] for details).

Interest in pseudocontractive mappings stems mainly from their firm connection 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 monotone if and only if is pseudocontractive, and hence a zero of , is a fixed point of , .

Let be a nonempty closed convex subset of a real Hilbert space and a pseudcontractive mapping. Assume that the set of fixed points of is nonempty. It is known from [3] that is closed and convex.

Let the variational inequality be given as finding a point with the property that Then, is the minimum-norm fixed point of which exists uniquely and is exactly the (nearest point or metric) projection of the origin onto , that is, . We also observe that the minimum-norm fixed point of pseudocontractive is the minimum-norm solution of a monotone operator equation , where .

It is quite often to seek the minimum-norm solution of a given nonlinear problem. In an abstract way, we may formulate such problems as finding a point with the property In other words, is the projection of the origin onto , that is,

A typical example is the split feasibility problem (SFP), formulated as finding a point with the property that where and are nonempty closed convex subsets of the infinite-dimension real Hilbert spaces and , respectively, and is bounded linear mapping from to . Equation (1.9) models many applied problems arising from image reconstructions and learning theory (see, e.g., [4]). Some works on the finite dimensional setting with relevant projection methods for solving image recovery problems can be found in [57]. Defining the proximity function by we consider the convex optimization problem: It is clear that is a solution to the split feasibility problem (1.9) if and only if and which is the minimum-norm solution of the minimization problem (1.11).

Motivated by the above split feasibility problem, we study the general case of finding the minimum-norm fixed point of a pseudocontractive mapping , that is, we find minimum norm fixed point of which satisfies Let be a nonexpansive self-mapping on closed convex subset of a Banach space . For a given and for a given define a contraction by By Banach contraction principle, it yields a fixed point of , that is, is the unique solution of the equation: Browder [8] proved that as , converges strongly to a fixed point of which is closer to , that is, the nearest point projection of onto . In 1980, Reich [9] extended the result of Browder to a more general Banach spaces. Furthermore, Takahashi and Ueda [10] and Morales and Jung [11] improved results of Reich [9] to the class of continuous pseudocontractive mappings. For other results on pseudocontractive mappings, we refer to [1215].

We note that the above methods can be used to find the minimum-norm fixed point of if . However, if neither Browder's, Reich's, Takahashi and Ueda's, nor Morales and Jung's method works to find minimum-norm fixed point of .

Our concern is now the following: is it possible to construct a scheme, implicit or explicit, which converges strongly to the minimum-norm fixed point of for any closed convex domain of ?

In this direction, Yang et al. [4] introduced an implicit and explicit iteration processes which converge strongly to the minimum-norm fixed point of nonexpansive self-mapping , in real Hilbert spaces. In fact, they proved the following theorems.

Theorem YLY1 (see [4]). Let be a nonempty closed convex subset of a real Hilbert space and a nonexpansive mapping with . For and each , let be defined as the unique solution of fixed point equation: Then the net converges strongly, as , to the minimum-norm fixed point of .

Theorem YLY2 (see [4]). Let be a nonempty closed convex subset of a real Hilbert space , and let be a nonexpansive mapping with . For a given , define a sequence iteratively by where and , satisfying certain conditions. Then the sequence converges strongly to the minimum-norm fixed point of .

A natural question arises whether the above theorems can be extended to a more general class of pseudocontractive mappings or not.

Let be a closed convex subset a real Hilbert space and let be continuous pseudocontractive mapping.

It is our purpose in this paper to prove that for and each , there exists a sequence satisfying which converges strongly, as , to the minimum-norm fixed point of . Moreover, we provide an explicit iteration process which converges strongly to the minimum-norm fixed point of provided that is Lipschitz. Our theorems improve Theorem YLY1 and Theorem YLY2 of Yang et al. [4] and Theorems  3.1, and  3.2 of Cai et al. [16].

2. Preliminaries

In what follows, we shall make use of the following lemmas.

Lemma 2.1 (see [11]). Let be a real Hilbert space. Then, for any given , the following inequality holds:

Lemma 2.2 (see [17]). Let be a closed and convex subset of a real Hilbert space . Let . Then if and only if

Lemma 2.3 (see [18]). Let , , and be sequences of nonnegative numbers satisfying the conditions: , , and , as . Let the recursive inequality: be given where is a strictly increasing function such that it is positive on and . Then , as .

Lemma 2.4 (see [3]). Let be a real Hilbert space, be a closed convex subset of and 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 [19]). Let be a real Hilbert space. Then for all and , the following equality holds:

3. Main Results

Theorem 3.1. Let be a nonempty closed and convex subset of a real Hilbert space . Let be a continuous pseudocontractive mapping with . Then for and each , there exists a sequence satisfying the following condition: and the net converges strongly, as , to the minimum-norm fixed point of .

Proof. For and each let . Then using nonexpansiveness of and pseudocontractivity of , for , we have that This implies that is strongly pseudocontractive on . Thus, by Corollary 1 of [20] has a unique fixed point, , in . This means that the equation: has a unique solution for each . Furthermore, since , for , we have that which implies that and hence . Therefore, and hence is bounded.
Furthermore, from (3.3) and using nonexpansiveness of we get that which implies that
Furthermore, from (3.3), convexity of , (1.4), and (3.7), we get that This implies that Now, for , as , let be a subsequence of such that . Then, we have from (3.7) and Lemma 2.4 that . Furthermore, replacing by in (3.9) and the fact that imply that which implies that Thus, from (3.9) and (3.11), we have that which is equivalent to the inequality: If there is another subsequence of such that , similar argument gives that , which implies, by uniqueness of , that . Therefore, the net which is the minimum-norm of fixed point of . The proof is complete.

We now state and prove a convergence theorem for the minimum-norm zero of a monotone mapping .

Theorem 3.2. Let be a real Hilbert space. Let be a continuous monotone mapping with . Then for and each , there exists a sequence satisfying the following condition: and the net converges strongly, as , to the minimum-norm zero of .

Proof. Let . Then, we get that is continuous pseudocontractive mapping with . Moreover, since is an identity mapping on , when is replaced with scheme (3.14) reduces to scheme (3.1), and hence the conclusion follows from Theorem 3.1.

If in Theorem 3.1, we consider such that , and , the method of proof of Theorem 3.1 provides the following corollary.

Corollary 3.3. Let be a nonempty closed and convex subset of a real Hilbert space . Let be continuous pseudocontractive mapping with . Then the sequence defined by where such that , , as , converges strongly, as , to the minimum-norm fixed point of .

The following proposition and lemma play an important role in proving the next theorem.

Proposition 3.4. Let be a nonempty closed and convex subset of a real Hilbert space . Let be continuous pseudocontractive mapping. Then the sequence in (3.15) satisfies the following inequality: where for decreasing sequence.

Proof. If we put , (3.15) reduces to Thus, using pseudocontractivity of and nonexpansiveness of we get that which implies, using the fact that is decreasing, that and hence
The proof is complete.

For the rest of this paper, let , (decreasing) and be real sequences in satisfying the following conditions: (i) ; (ii) , , ; (iii) and . Examples of real sequences which satisfy these conditions are , and .

Lemma 3.5. Let be a nonempty closed convex subset of a real Hilbert space . Let be a Lipschitz pseudocontractive mapping with Lipschitz constant and . Let a sequence be generated from arbitrary by for all positive integers . Then is bounded.

Proof. We follow the method of proof of Chidume and Zegeye [21]. Since , there exists such that , for all . Let and be sufficiently large such that and . Now, we show by induction that belongs to for all integers . By construction, we have . Assume that for any . Then, we prove that . Suppose is not in . Then , and thus from the recursion formula (1.2) and Lemma 2.1 we get that Since is pseudocontractive we have . Thus, (3.22) gives which implies that Since , from (3.24) we get that and hence , since , and . But this is a contradiction. Therefore, for all positive integers , and hence the sequence is bounded.

For the next theorem, let denotes the sequence defined by , guaranteed by Corollary 3.3 (which reduces to .

Theorem 3.6. Let be a nonempty closed convex subset of a real Hilbert space . Let be a Lipschitz pseudocontractive mapping with Lipschitz constant and . Let a sequence be generated from arbitrary by for all positive integers . Then converges strongly to the minimum-norm fixed point of , as .

Proof. By Lemma 3.5, we have that the sequence is bounded. Now, we show that it converges strongly to a minimum-norm fixed point of . But from (3.26) and Lemma 2.1, we have that Observe that by the property of and pseudocontractivity of we have (see (3.17)) and for all . Thus, we have from (3.27) that But by Corollary 3.3, we have that is bounded. Therefore, there exists such that . Thus from (3.28), we get that But using triangle inequality and Proposition 3.4, we have that for some , and for all . Now, substituting (3.30) in (3.29) we obtain that for some constant . Now, by Lemma 2.3 and the conditions on , , and we get . Consequently, as .
Therefore, since by Corollary 3.3 we have that , where is with the minimum-norm in , we get that converges strongly to the minimum-norm of fixed point of .

Corollary 3.7. Let be a real Hilbert space. Let be a Lipschitz monotone mapping with Lipschitz constant and . Let a sequence be generated from arbitrary by for all positive integers . Then converges strongly to the minimum-norm solution of the equation .

Proof. Let . Then is a Lipschitz pseudocontractive mapping with Lipschitz constant , and the minimum-norm solution of the equation is the minimum-norm fixed point of . Moreover, if we replace by in (3.26), then the equation reduces to (3.32). Thus, the conclusion follows from Theorem 3.6.

4. Applications

For the rest of this paper, let be a Hilbert space and a bounded linear operator. Consider the convexly constrained linear inverse problem, which has extensively been discussed in the literature (see, e.g., [22]), given by: where is closed and convex subset of and , which is a special case of the SFP problem (1.9). Set The least-square solution of (4.1) is the least-norm minimizer of the minimization problem (4.2). Let denote the solution set of (4.2). It is known that is nonempty if and only if . In this case, has a unique element with minimum norm which is a least-square solution of (4.1), that is, there exists a unique point such that We note that is a quadratic function with gradient: where is adjoint of . Let and . Thus, is the minimum-norm solution of the minimization problem (4.2) if and only if a solution of Now, we state applications of our theorems.

Theorem 4.1. Assume that the solution set of convexly constrained linear inverse problem (4.1) with , a real Hilbert space, is nonempty and that is monotone. Then for and each , there exists a sequence satisfying the following condition: where is adjoint of , and the net converges strongly, as , to the minimum-norm solution of the split feasibility problem (4.1).

Proof. We note that is continuously differentiable function with gradient: where is adjoint of , which is Lipschitz (see Lemma 8.1 of [5]) and monotone (by hypothesis). Thus, the conclusion follows from Theorem 3.2.

Theorem 4.2. Assume that the solution set of split feasibility problem (4.1) is nonempty and that with , a real Hilbert space, is monotone. Let a sequence be generated from arbitrary by for all positive integers , where and is adjoint of . Then, converges strongly to the minimum-norm solution of the split feasibility problem (4.1).

Remark 4.3. Theorem 3.1 improves Theorem YLY1 and Theorem  3.1 of Cai et al. [16] to a more general class of pseudocontractive mappings. Moreover, Theorem 3.6 improves Theorem YLY1 and Theorem  3.2 of Cai et al. [16] in the sense that our scheme provides a minimum-norm fixed point of pseudocontractive mapping .

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