On the System of Nonlinear Mixed Implicit Equilibrium Problems in Hilbert Spaces

Abstract

We use the Wiener-Hopf equations and the Yosida approximation notions to prove the existence theorem of a system of nonlinear mixed implicit equilibrium problems (SMIE) in Hilbert spaces. The algorithm for finding a solution of the problem (SMIE) is suggested; the convergence criteria and stability of the iterative algorithm are discussed. The results presented in this paper are more general and are viewed as an extension, refinement, and improvement of the previously known results in the literature.

1. Introduction and Preliminaries

Let be a real Hilbert space whose inner product and norm are denoted by and respectively. Let be given two bi-functions satisfying for all and . Let be a nonlinear mapping. Let be a nonempty closed convex subset of . In this paper, we consider the following problem.

Find such that

(1.1)

The problem of type (1.1) is called the system of nonlinear mixed implicit equilibrium problems.

We denote by SMIE the set of all solutions of the problem (1.1).

Some examples of the problem (1.1) are as follows.

• (I) If where is a maximal monotone mapping for then the problem (1.1) becomes the following problem.

Find such that

(1.2)

which is called the system of variational inclusion problems. In particular, when and the problem (1.2) is reduced to the problem, so-called the generalized variational inclusion problem, which was studied by Kazmi and Bhat [1].

It is worth noting that the variational inclusions and related problems are being studied extensively by many authors and have important applications in operations research, optimization, mathematical finance, decision sciences, and other several branches of pure and applied sciences.

• (II) If for all , where is a real valued function for each Then the problem (1.1) reduces to the following problem.

Find such that

(1.3)

Some corresponding results to the problem (1.3) were considered by Kassay and Kolumbán [2] when .

• (III) For each , let be a nonlinear mapping and fixed positive real numbers. If and for all , then the problem (1.3) reduces to the following problem.

Find such that

(1.4)

which is called the system of nonlinear mixed variational inequalities problems. A special case of the problem (1.4), when and , has been studied by He and Gu [3].

• (IV) If for all , where is the indicator function of defined by

(1.5)

then the problem (1.4) reduces to the following problem.

Find such that

(1.6)

which is called the system of nonlinear variational inequalities problems. Some corresponding results to the problem (1.6) were studied by Agarwal et al. [4], Chang et al. [5], Cho et al. [6], J. K. Kim and D. S. Kim, [7] and Verma [8, 9].

For the recent trends and developments in the problem (1.6) and its special cases, see [3, 8–11] and the references therein, for examples.

• (V) If , and is a univariate mapping, then the problem (1.6) reduces to the following problem.

Find such that

(1.7)

which is known as the classical variational inequality introduced and studied by Stampacchia [12] in 1964. This shows that a number of classes of variational inequalities and related optimization problems can be obtained as special cases of the system (1.1) of mixed equilibrium problems.

Inspired and motivated by the recent research going on in this area, in this paper, we use the Wiener-Hopf equations and the Yosida approximation notion to suggest and prove the existence and uniqueness of solutions for the problem (1.1). We also discuss the convergence criteria and stability of the iterative algorithm. The results presented in this paper improve and generalize many known results in the literature.

In the sequel, we need the following basic concepts and lemmas.

Definition 1.1 (Blum and Oettli [13]).

A real valued bifunction is said to be:

monotone if

(1.8)

strictly monotone if

(1.9)

upper-hemicontinuous if

(1.10)

Definition 1.2.

A function is said to be lower semicontinuous at if, for all there exists a constant such that

(1.11)

where denotes the ball with center and radius , that is,

(1.12)

The function is said to be lower semicontinuous on if it is lower semicontinuous at every point of .

Lemma 1.3 (Combettes and Hirstoaga [14]).

Let be a nonempty closed convex subset of and be a bifunction of into satisfying the following conditions:

(C1) is monotone and upper hemicontinuous;

(C2) is convex and lower semi-continuous for all .

For all and , define a mapping as follows:

(1.13)

Then is a single-valued mapping.

Definition 1.4.

Let be a positive number. For any bi-function the associated Yosida approximation over and the corresponding regularized operator are defined as follows:

(1.14)

in which is the unique solution of the following problem:

(1.15)

Remark 1.5.

Definition 1.4 is an extension of the Yosida approximation notion introduced in [15]. The existence and uniqueness of the solution of the problem (1.15) follow from Lemma 1.3.

Definition 1.6.

Let be a set-valued mapping.

  is said to be monotone if, for any ,

(1.16)

A monotone operator is said to bemaximal if is not properly contained in any other monotone operators.

Example 1.7 (Huang et al. [16]).

Let where is a maximal monotone mapping. Then it directly follows that

(1.17)

where is the Yosida approximation of and we recover the classical concepts.

Using the idea as in Huang et al. [16], we have the following result.

Lemma 1.8.

If is a monotone function, then the operator is a nonexpansive mapping, that is,

(1.18)

Proof.

From (1.15), for all , we can obtain

(1.19)

(1.20)

By adding (1.19) with (1.20) and using the monotonicity of , we have

(1.21)

and so

(1.22)

This implies that is a nonexpansive mapping. This completes the proof.

Now, for solving the problem (1.1), we consider the following equation: let and be fixed positive real numbers. Find such that

(1.23)

Lemma 1.9.

is a solution of the problem (1.1) if and only if the problem (1.23) has a solution where

(1.24)

that is,

(1.25)

Proof.

The proof directly follows from the definitions of and .

In this paper, we are interested in the following class of nonlinear mappings.

Definition 1.10.

A mapping is said to be -strongly monotone if there exists a constant such that

(1.26)

A mapping is said to be -Lipschitz if there exist constants such that

(1.27)

2. Existence of Solutions of the Problem (1.1)

In this section, we give an existence theorem of solutions for the problem (1.1). Firstly, in view of Lemma 1.9, we can obtain the following, which is an important tool, immediately.

Lemma 2.1.

Let . Then if and only if there exist positive real numbers such that is a fixed point of the mapping defined by

(2.1)

where are defined, respectively, by

(2.2)

Now, we are in position to prove the existence theorem of solutions for the problem (1.1).

Theorem 2.2.

For each , let be a monotone bi-function. Let be a -strongly monotone with respect to the first argument and -Lipschitz mapping and be a -strongly monotone with respect to the second argument and -Lipschitz mapping. Suppose that there are positive real numbers such that

(2.3)

Then is a singleton.

Proof.

Notice that, in view of Lemma 2.1, it is sufficient to show that the mapping defined in Lemma 2.1 has the unique fixed point. Since is nonexpansive, we have the following estimate:

(2.4)

Since is a -Lipschitz mapping and, for all , the mapping is a -strongly monotone, we obtain

(2.5)

Consequently, from (2.4)-(2.5), it follows that

(2.6)

Next, we have the following estimate:

(2.7)

From (2.6) and (2.7), we have

(2.8)

where

(2.9)

Now, define the norm on by

(2.10)

Notice that is a Banach space and

(2.11)

By the condition (2.3), we have , which implies that is a contraction mapping. Hence, by Banach contraction principle, there exists a unique such that This completes the proof.

3. Convergence and Stability Analysis

In view of Lemma 2.1, for the fixed point formulation of the problem (2.1), we suggest the following iterative algorithm.

3.1. Mann Type Perturbed Iterative Algorithm (MTA)

For any , compute approximate solution given by the iterative schemes:

(3.1)

where is a sequence of real numbers such that and

In order to consider the convergence theorem of the sequences generated by the algorithm (MTA), we need the following lemma.

Lemma 3.1.

Let and be two nonnegative real sequences satisfying the following conditions. There exists a positive integer such that

(3.2)

where with and . Then

Now, we prove the convergence theorem for a solution for the problem (1.1).

Theorem 3.2.

If all the conditions of the Theorem 2.2 hold, then the sequence in generated by the algorithm (3.1) converges strongly to the unique solution for the problem (1.1).

Proof.

It follows from Theorem 2.2 that there exists which is the unique solution for the problem (1.1). Moreover, in view of Lemma 2.1, we have

(3.3)

Since is nonexpansive, from the iterative sequences (3.1) and (3.3), it follows that

(3.4)

Next, we have the following estimate:

(3.5)

Substituting (3.5) into (3.4) yields that

(3.6)

Similarly, we have

(3.7)

Thus, from (3.6) and (3.7), we have

(3.8)

where and are givenin (2.9). Setting

(3.9)

From the condition (2.3), it follows that and so . Moreover, since , we have . Hence all the conditions of Lemma 3.1 are satisfied and so as that is,

(3.10)

Thus the sequence in converges strongly to a solution for the problem (1.1). This completes the proof.

3.2. Stability of the Algorithm (MTA)

Consider the following definition as an extension of the concept of stability of the iterative procedure given by Harder and Hicks [17].

Definition 3.3 (Kazmi and Khan [18]).

Let be a Hilbert space and be nonlinear mappings. Let be defined as for all and . Assume that defines an iterative procedure which yields a sequence in . Suppose that and the sequence converges to some . Let be an arbitrary sequence in and

(3.11)

If implies that , then the iterative procedure is said to be -stable or stable with respect to .

Theorem 3.4.

Assume that all the conditions of Theorem 2.2 hold. Let be an arbitrary sequence in and define by

(3.12)

where

(3.13)

where is a sequence defined in (3.1). If is defined as in (2.1), then the iterative procedure generated by (3.1) is -stable.

Proof.

Assume that . Let be the unique fixed point of the mapping This means that

(3.14)

Now, from (3.12) and (3.13), it follows that

(3.15)

Notice that for each , which implies that

(3.16)

Using (3.16) and the assumption , it follows from (3.15) that

(3.17)

This completes the proof.