On the Convergence for an Iterative Method for Quasivariational Inclusions

Abstract

We introduce an iterative algorithm for finding a common element of the set of solutions of quasivariational inclusion problems and of the set of fixed points of strict pseudocontractions in the framework Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others.

1. Introduction and Preliminaries

Throughout this paper, we always assume that is a real Hilbert space with the inner product and the norm . Let be a nonlinear mapping. In this paper, we use to denote the fixed point set of

Recall the following definitions.

(1)The mapping is said to becontractive with the coefficient if

(1.1)

(2)The mapping is said to benonexpansive if

(1.2)

(3)The mapping is said to bestrictly pseudocontractive with the coefficient if

(1.3)

(4)The mapping is said to bepseudocontractive if

(1.4)

Clearly, the class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. See, for example, [1–6] and the references therein.

A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space :

(1.5)

where is a linear bounded and strongly positive operator and is a potential function for (i.e., for ).

Recently, Marino and Xu [2] studied the following iterative scheme:

(1.6)

They proved that the sequence generated in the above iterative scheme converges strongly to the unique solution of the variational inequality:

(1.7)

which is the optimality condition for the minimization problem (1.5).

Next, let be a nonlinear mapping. Recall the following definitions.

(1)The mapping is said to bemonotone if for each , we have

(1.8)

(2) is said to be -strongly monotone if

(1.9)

(3)The mapping is said to be -inverse-strongly monotone if there exists a constant such that

(1.10)

(4)The mapping is said to berelaxed-cocoercive if there exists a constant such that

(1.11)

(5)The mapping is said to berelaxed-cocoercive if there exist two constants such that

(1.12)

(6)Recall also that a set-valued mapping is called monotone if for all , and imply The monotone mapping is maximal if the graph of of is not properly contained in the graph of any other monotone mapping.

The so-called quasi-variational inclusion problem is to find a for a given element such that

(1.13)

where and are two nonlinear mappings. See, for example, [7–12]. A special case of the problem (1.13) is to find an element such that

(1.14)

In this paper, we use to denote the solution of the problem (1.14). A number of problems arising in structural analysis, mechanics, and economic can be studied in the framework of this class of variational inclusions.

Next, we consider two special cases of the problem (1.14).

If , where is a proper convex lower semicontinuous function and is the subdifferential of , then the variational inclusion problem (1.14) is equivalent to finding such that

(1.15)

which is said to be themixed quasi-variational inequality. See, for example, [7, 8] for more details.

If is the indicator function of then the variational inclusion problem (1.14) is equivalent to the classical variational inequality problem, denoted by , to find such that

(1.16)

For finding a common element of the set of fixed points of a nonexpansive mapping and of the set of solutions to the variational inequality (1.16), Iiduka and Takahashi [13] proved the following theorem.

Theorem IT

Let be a closed convex subset of a real Hilbert space . Let be an -inverse-strongly monotone mapping of into and let be a nonexpansive mapping of into itself such that . Suppose that and is given by

(1.17)

for every where is a sequence in and is a sequence in . If and are chosen so that for some with ,

(1.18)

then converges strongly to

Recently, Zhang et al. [11] considered the problem (1.14). To be more precise, they proved the following theorem.

Theorem ZLC

Let be a real Hilbert space, an -inverse-strongly monotone mapping, a maximal monotone mapping, and a nonexpansive mapping. Suppose that the set , where is the set of solutions of variational inclusion ( 1.14 ). Suppose that and is the sequence defined by

(1.19)

where and is a sequence in satisfying the following conditions:

(a)

(b)

Then converges strongly to

In this paper, motivated by the research work going on in this direction, see, for instance, [2, 3, 7–21], we introduce an iterative method for finding a common element of the set of fixed points of a strict pseudocontraction and of the set of solutions to the problem (1.14) with multivalued maximal monotone mapping and relaxed -cocoercive mappings. Strong convergence theorems are established in the framework of Hilbert spaces.

In order to prove our main results, we need the following conceptions and lemmas.

Definition 1.1 (see [11]).

Let be a multivalued maximal monotone mapping. Then the single-valued mapping defined by for all , is called the resolvent operator associated with , where is any positive number and is the identity mapping.

Lemma 1.2 (see [4]).

Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that

(a)

(b) or

Then

Lemma 1.3 (see [22]).

Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose that for all and Then

Lemma 1.4 (see [11]).

is a solution of variational inclusion (1.14) if and only if for all , that is,

(1.20)

Lemma 1.5 (see [11]).

The resolvent operator associated with is single-valued and nonexpansive for all .

Lemma 1.6 (see [23]).

Let be a closed convex subset of a strictly convex Banach space . Let and be two nonexpansive mappings on . Suppose that is nonempty. Then a mapping on defined by , where , for is well defined and nonexpansive and holds.

Lemma 1.7 (see [24]).

Let be a real Hilbert space, let be a nonempty closed convex subset of , and let be a nonexpansive mapping. Then is demiclosed at zero.

Lemma 1.8 (see [25]).

Let be a nonempty closed convex subset of a real Hilbert space and a -strict pseudocontraction. Define by for each . Then, as , is nonexpansive such that .

2. Main Results

Theorem 2.1.

Let be a real Hilbert space and a maximal monotone mapping. Let be a relaxed -cocoercive and -Lipschitz continuous mapping, and a -strict pseudocontraction with a fixed point. Define a mapping by . Let be a contraction of into itself with the contractive coefficient and a strongly positive linear bounded self-joint operator with the coefficient . Assume that and . Let and be a sequence generated by

(2.1)

where and are sequences in . Assume that , . If the control consequences and satisfy the following restrictions:

(C1), for all

(C2) and ,

then converges strongly to which solves uniquely the following variational inequality:

(2.2)

Equivalently, one has

Proof.

The uniqueness of the solution of the variational inequality (2.2) is a consequence of the strong monotonicity of . Suppose that and both are solutions to (2.2); then and Adding up the two inequalities, we see that

(2.3)

The strong monotonicity of (see [2, Lemma ]) implies that and the uniqueness is proved. Below we use to denote the unique solution of (2.2).

Next, we show that the mapping is nonexpansive. Indeed, for all , one see from the condition that

(2.4)

which implies that the mapping is nonexpansive. Taking we have It follows from Lemma 1.5 that

(2.5)

Note that from the conditions (C1) and (C2), we may assume, without loss of generality, that for all . Since is a strongly positive linear bounded self-adjoint operator, we have . Now for with , we see that

(2.6)

that is, is positive. It follows that

(2.7)

Set . From Lemma 1.8, we see that is nonexpansive. It follows from (2.5) that

(2.8)

From (2.7) and (2.8), we arrive at

(2.9)

By simple inductions, one obtains that which gives that the sequence is bounded, so are and .

On the other hand, we see from the nonexpansivity of the mappings that

(2.10)

It follows that

(2.11)

Setting

(2.12)

we see that

(2.13)

It follows that

(2.14)

which combines with (2.11) yields that

(2.15)

It follows from the conditions (C1) and (C2) that Hence, from Lemma 1.3, one obtains From (2.12), one has Thanks to the condition (C1), we see that

(2.16)

On the other hand, we have

(2.17)

It follows that

(2.18)

From the conditions (C1) and (C2) and (2.16), we see that

(2.19)

Next, we prove that where To see this, we choose a subsequence of such that

(2.20)

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that Next, we show that . Define a mapping by

(2.21)

In view of Lemma 1.6, we see that is nonexpansive such that

(2.22)

From (2.19), we obtain It follows from Lemma 1.7 that That is, Thanks to (2.20), we arrive at

(2.23)

Finally, we show that as Indeed, we have

(2.24)

which implies that

(2.25)

From the condition (C2), (2.23), and using Lemma 1.2, we see that This completes the proof.

Letting and , the identity mapping, we can obtain from Theorem 2.1 the following result immediately.

Corollary 2.2.

Let be a real Hilbert space and a maximal monotone mapping. Let be a relaxed -cocoercive and -Lipschitz continuous mapping, and a -strict pseudocontraction with a fixed point. Define a mapping by . Let be a contraction of into itself with the contractive coefficient Assume that . Let and be a sequence generated by

(2.26)

where and are sequences in . Assume that , . If the control consequences and satisfy the following restrictions:

(C1), for all

(C2) and ,

then converges strongly to

Remark 2.3.

Corollary 2.2 improves Theorem 2.1 of Zhang et al. [11] in the following sense:

(1)from nonexpansive mappings to strict pseudocontractions;

(2)the analysis technique used in this paper is different from [11]'s: the proof is also more concise than [11]'s;

(3)the restriction imposed on the parameter is relaxed.