Viscosity Projection Algorithms for Pseudocontractive Mappings in Hilbert Spaces

Abstract

An explicit projection algorithm with viscosity technique is constructed for finding the fixed points of the pseudocontractive mapping in Hilbert spaces. Strong convergence theorem is demonstrated. Consequently, as an application, we can approximate to the minimum-norm fixed point of the pseudocontractive mapping.

1. Introduction

Let be a real Hilbert space with inner product and norm , respectively. Let be a nonempty closed convex subset of .

Recall that a mapping is said to be(i)Lipschitz there exists a constant such that for all ; if , then is said to be contractive; if , then is said to be nonexpansive;(ii)pseudocontractive ; ; ,  for all .

Interest in pseudocontractive mappings stems mainly from their firm connection with the class of nonlinear accretive operators. It is a classical result, see Deimling [1], that if is an accretive operator, then the solutions of the equations correspond to the equilibrium points of some evolution systems. This explains the importance, from this point of view, of the improvement brought by the Ishikawa iteration which was introduced by Ishikawa [2] in 1974. The original result of Ishikawa is stated in the following.

Theorem 1. Let be a convex compact subset of a Hilbert space and let be an -Lipschitzian pseudocontractive mapping with . For any , define the sequence iteratively by for all , where and    satisfy the conditions: and . Then the sequence generated by (1) converges strongly to a fixed point of .

The iteration (1) is now referred to as the Ishikawa iterative sequence. However, we note that is compact subset. Now, we know that strong convergence has not been achieved without compactness assumption on the involved operation or the underlying spaces. A counter example can be found in Chidume and Mutangadura [3].

In order to obtain a strong convergence result for pseudocontractive mappings without the compactness assumption, in [4], Zhou coupled the Ishikawa algorithm with the hybrid technique and presented the following algorithm for Lipschitz pseudocontractive mappings: Zhou proved that the sequence generated by (2) converges strongly to the fixed point of . Further, in [5], Yao et al. introduced the hybrid Mann algorithm as follows.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence in . Let . For and , define a sequence of as follows: Note that, in iterations (2) and (3), we need to compute the half-spaces (and/or ). Very recently, Zegeye et al. [6] further studied the convergence analysis of the Ishikawa iteration (1). They proved ingeniously the strong convergence of the Ishikawa iteration (1). However, we have to assume that the interior of is nonempty. This appears very restrictive since even in with the usual norm, Lipschitz pseudocontractive maps with finite number of fixed points do not enjoy this condition that . For some related works, please refer to [7–19].

On the other hand, we notice that it is quite often to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. For instance, given a closed convex subset of a Hilbert space and a bounded linear operator , where is another Hilbert space. The -constrained pseudoinverse of , , is then defined as the minimum-norm solution of the constrained minimization problem which is equivalent to the fixed point problem where is the adjoint of , is a constant, and is such that .

It is, therefore, an interesting problem to invent iterative algorithms that can generate sequences which converge strongly to the minimum-norm solution of a given fixed point problem. The purpose of this paper is to solve such a problem for pseudocontractions. More precisely, we will introduce an explicit projection algorithm with viscosity technique for finding the fixed points of a Lipschitzian pseudocontractive mapping. Strong convergence theorem is demonstrated. Consequently, as an application, we can find the minimum-norm fixed point of the pseudocontractive mapping.

2. Preliminaries

Recall that the metric projection satisfies . The metric projection is a typical firmly nonexpansive mapping. The characteristic inequality of the projection is for all , .

In the sequel we will use the following expressions:(i) denotes the set of fixed points of ;(ii) denotes the weak convergence of to ;(iii) denotes the strong convergence of to .

The following lemmas will be useful for the next section.

Lemma 2 (see [20]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping with . Then,

Lemma 3 (see [21]). Let be a nonempty closed convex subset of a real Hilbert space . Assume that a mapping is monotone and weakly continuous along segments (i.e., weakly, as , whenever for ). Then the variational inequality is equivalent to the dual variational inequality

Lemma 4 (see [22]). Assume that the sequence satisfies and where is a sequence in and is a sequence such that and (or ). Then .

3. Main Results

In order to prove our main result, we need the following proposition.

Proposition 5. Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping with . Let be a -contraction.
For each , let the net be defined by Then, as , the net converges strongly to a point which solves the following variational inequality:

Proof. For , define a mapping by For any , we have Hence, is a -contraction on with as its unique fixed point. So, is well defined.
Let . From (9), we have It follows that Thus, and are bounded.
Again from (9), we get Let be a sequence such that as . Put . From (15), we have From (9), we obtain It follows that where is a constant such that In particular, we have Noting that is bounded, without loss of generality, we assume that . It is obvious that . From (16) and Lemma 2, we deduce . Substitute for in (20) to get Since , we deduce from (21) that . The net is, therefore, relatively compact, as , in the norm topology.
In (20), letting , we get Therefore, solves the variational inequality By Lemma 3, (23) equals its dual variational inequality This indicates that . That is, is the unique fixed point in of the contraction . So, the entire net converges in norm to as . This completes the proof.

Corollary 6. Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping with .
For each , let the net be defined by Then, as , the net converges strongly to the minimum-norm fixed point of .

Proof. As a matter of fact, in (9), we choose , and then (9) reduces to (25). From Proposition 5, (24) is reduced to Equivalently, This implies that Therefore, is the minimum-norm fixed point of . This completes the proof.

Algorithm 7. Let be a nonempty closed subset of a real Hilbert space . Let be a pseudocontraction and let be a -contraction. Let and be two real number sequences. For , we define a sequence iteratively by

Theorem 8. Let be a nonempty closed convex subset of a real Hilbert space . Let be an -Lipschitzian and pseudocontraction with and a -contraction. Suppose the following conditions are satisfied: (C1) and ;(C2);(C3).Then the sequence generated by the algorithm (29) converges strongly to .

Proof. First, we prove that the sequence is bounded. We will show this fact by induction. According to conditions (C1) and (C2), there exists a sufficiently large positive integer such that Fix a and take a constant such that
Next, we show that .
Set Then we have Since is pseudocontractive, is monotone. So, we have From (29), (33), and (34), we obtain It follows that By (29), we have Substitute (37) into (36) to obtain that is, By induction, we get which implies that is bounded and so is . Now, we take a constant such that
Set (i.e., is a resolvent of the monotone operator ). We then have that is a nonexpansive self mapping of and .
By Proposition 5, we know that, whenever and , the sequence defined by converges strongly to the fixed point of (and of as ). Without loss of generality, we may assume that for all .
It suffices to prove that as (for some ). To this end, we rewrite (42) as By using the property of the metric projection, we have Note that Hence, we get From (42), we have It follows that Set By condition (C2), and , for large enough. Hence, by (46) and (48), we have By (29), we have Next, we estimate . Since , . Then, we have
It follows that Thus, where the finite constant is given by Set Then (54) can be rewritten as By the conditions (C1)–(C3), we deuce that From Lemma 4 and (57), we get as . This completes the proof.

Algorithm 9. Let be a nonempty closed subset of a real Hilbert space . Let be a pseudocontraction. Let and be two real number sequences. For , we define a sequence iteratively by

Corollary 10. Let be a nonempty closed convex subset of a real Hilbert space . Let be an Lipschitzian and pseudocontraction with . Suppose the following conditions are satisfied:(C1) and ;(C2);(C3).Then the sequence generated by the algorithm (59) converges strongly to the minimum-norm fixed point of .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The first author was supported by the National Natural Science Foundation of China, Grant no. 11226125.