On Solvability of a Generalized Nonlinear Variational-Like Inequality

Abstract

A new generalized nonlinear variational-like inequality is introduced and studied. By applying the auxiliary principle technique and KKM theory, we construct a new iterative algorithm for solving the generalized nonlinear variational-like inequality. By means of the Banach fixed-point theorem, we establish the existence and uniqueness of solution for the generalized nonlinear variational-like inequality. The convergence of the sequence generated by the iterative algorithm is also discussed.

1. Introduction

It is well known that variational inequality theory has become a very effective and powerful tool for studying a wide range of problems arising in many diverse fields. One of the most interesting and important problems in the variational inequality theory is the development of an efficient iterative algorithm to compute approximate solutions. The researchers in [1–10] suggested a lot of iterative algorithms for solving various variational inequalities and variational-like inequalities. By using the auxiliary principle technique, Ding and Yao [3], Ding et al. [4], Huang and Deng [5], Liu et al. [8, 9], and others studied several classes of nonlinear variational inequalities and variational-like inequalities in reflexive Banach and Hilbert spaces, respectively, suggested some iterative algorithms to compute approximate solutions for these nonlinear variational inequalities and variational-like inequalities, and proved the existence of solutions for the nonlinear variational inequalities and variational-like inequalities involving different monotone mappings.

Motivated and inspired by the research work in [1–10], we introduce and study a generalized nonlinear variational-like inequality. By using the auxiliary principle technique and KKM theorem due to Zhang and Xiang [2], we suggest a new iterative scheme for solving the generalized nonlinear variational-like inequality. Utilizing the Banach fixed-point theorem, we prove the existence and uniqueness of solution for the generalized nonlinear variational-like inequality. Under certain conditions, we discuss the convergence of the iterative sequence generalized by the iterative algorithm.

2. Preliminaries

Throughout this paper, let be a real Hilbert space endowed with an inner product and a norm , respectively, and . Let be a nonempty closed convex subset of . Assume that is a coercive continuous bilinear form, that is, there exist positive constants such that

(a1)

(a2).

Remark 2.1.

It follows from (a1) and (a2) that

Let be nondifferentiable and satisfy the following conditions:

(b1) is linear in the first argument,

(b2) is convex in the second argument,

(b3) is bounded, that is, there exists a constant satisfying

(2.1)

(b4)

Remark 2.2.

It follows that

(2.2)

which implies that is continuous in the second argument.

Let be mappings and . Now we consider the following generalized nonlinear variational-like inequality.

Find such that

(2.3)

It is clear that for appropriate and suitable choices of the mappings and , the generalized nonlinear variational-like inequality (2.3) includes some variational inequalities and variational-like inequalities in [1–10] as special cases.

Recall the following concepts and results.

Definition 2.3.

Let and be mappings.

• (1)
  

  is said to beLipschitz continuous if there exists a constant such that

(2.4)

• (2)
  

  is said to be -relaxed Lipschitz if there exists a constant such that

(2.5)

• (3)
  

  is said to be -strongly monotone with respect to in the first argument if there exists a constant such that

(2.6)

• (4)
  

  is said to be -monotone with respect to in the second argument if

(2.7)

• (5)
  

  is said to be -relaxed cocoercive with respect to in the second argument if there exists a constant such that

(2.8)

• (6)
  

  is said to beLipschitz continuous in the second argument if there exists a constant such that

(2.9)

• (7)
  

  is said to beLipschitz continuous if there exists a constant such that

(2.10)

• (8)
  

  and are said to be-hemicontinuous with respect to and in if for any, the mapping is continuous on

Lemma 2.4 (see [2]).

Let be a nonempty closed convex subset of a Hausdorff linear topological space , and let be mappings satisfying the following conditions:

(a) and

(b)for each is upper semicontinuous on

(c)for each the set is a convex set;

(d)there exists a nonempty compact set and such that

Then there exists such that

Lemma 2.5 (see [11]).

Let and be nonnegative sequences satisfying

(2.11)

where

(2.12)

Then .

Assumption 2.6.

Let satisfy that

(1)

(2)for any the mapping is convex and lower semicontinuous in

3. Auxiliary Problem and Algorithm

Now we consider the following auxiliary problem with respect to the generalized nonlinear variational-like inequality (2.3). For each find such that

(3.1)

where is a constant.

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space ,, and let be Lipschitz continuous with constant . Assume that is a coercive continuous bilinear form satisfying (a1) and (a2), satisfies (b1)–(b4). Let and be mappings such that is Lipschitz continuous with constant , is -strongly monotone with respect to in the first argument with constant , -relaxed cocoercive with respect to and Lipschitz continuous in the second argument with constants and , respectively, is -relaxed Lipschitz with constant , and and are -hemicontinuous with respect to and in . Assume that Assumption 2.6 holds and there exists a positive constant satisfying

(3.2)

Then for each , the auxiliary problem (3.1) has a unique solution in

Proof.

Let be in Define two functionals and by

(3.3)

for all .

Now we prove that the functionals and satisfy all the conditions of Lemma 2.4 in the weak topology. It is easy to see for all

(3.4)

which imply that and satisfy condition (a) of Lemma 2.4. Since is a coercive continuous bilinear form, is convex and continuous in the second argument, and for given the mapping is convex and lower semicontinuous in , it follows that for each is weakly upper semicontinuous in the second argument and the set is convex for each That is, the conditions (b) and (c) of Lemma 2.4 hold. Let

(3.5)

Clearly, is a weakly compact subset of . For each we infer that

(3.6)

which means that the condition (d) of Lemma 2.4 holds. Thus Lemma 2.4 ensures that there exists such that for all that is,

(3.7)

Put for and Replacing by in (3.7), we obtain that

(3.8)

Notice that is convex in the second argument. It follows from Assumption 2.6 and (3.8) that

(3.9)

which implies that

(3.10)

Letting in (3.10), we conclude that

(3.11)

That is, is a solution of the auxiliary problem (3.1).

Now we prove the uniqueness of solution for the auxiliary problem (3.1). Suppose that are two solutions of the auxiliary problem (3.1) with respect to . It follows that

(3.12)

(3.13)

Taking in (3.12) and in (3.13), we get that

(3.14)

(3.15)

Adding (3.14) and (3.15), we deduce that

(3.16)

which yields that by (3.2). That is, is the unique solution of the auxiliary problem (3.1). This completes the proof.

The proof of the below result is similar to that of Theorem 3.1 and is omitted.

Theorem 3.2.

Let be a nonempty closed convex subset of a real Hilbert space , and Lipschitz continuous with constant . Assume that is a coercive continuous bilinear form satisfying (a1) and (a2), satisfies (b1)–(b4). Let and be mappings such that is -strongly monotone with respect to in the first argument with constant , -monotone with respect to in the second argument, is Lipschitz continuous with constant and and are -hemicontinuous with respect to and in . Assume that Assumption 2.6 holds and there exists a positive constant satisfying

(3.17)

Then for each , the auxiliary problem (3.1) has a unique solution in

Based on Theorems 3.1 and 3.2, we suggest the following iterative algorithm with errors for solving the generalized nonlinear variational-like inequality (2.3).

Algorithm 3.3.

For given compute sequence by the following iterative scheme:

(3.18)

where is a constant and is a sequence in introduced to take into account possible inexact computation and satisfies that

(3.19)

4. Existence and Convergence

In this section, we prove the existence of solution for the generalized nonlinear variational-like inequality (2.3) and discuss the convergence of the sequence generated by Algorithm 3.3.

Theorem 4.1.

Let be a nonempty closed convex subset of a real Hilbert space , and Lipschitz continuous with constant . Assume that is a coercive continuous bilinear form satisfying (a1) and (a2), satisfies (b1)–(b4). Let and be mappings such that is Lipschitz continuous with constant , is -strongly monotone with respect to in the first argument with constant , -relaxed cocoercive with respect to and Lipschitz continuous in the second argument with constants and , respectively, is -relaxed Lipschitz with constant , and and are -hemicontinuous with respect to and in . Assume that Assumption 2.6 holds and

(4.1)

Then the generalized nonlinear variational-like inequality (2.3) possesses a unique solution and the iterative sequence generated by Algorithm 3.3 converges strongly to

Proof.

Note that (4.1) implies that (3.2) holds. It follows from Theorem 3.1 that there exists a mapping such that for each is the unique solution of the auxiliary problem (3.1). Next we show that is a contraction mapping in . Let and be arbitrary elements in and a constant. Using (3.1), we deduce that

(4.2)

(4.3)

Letting in (4.2) and in (4.3), and adding these inequalities, we arrive at

(4.4)

that is,

(4.5)

where

(4.6)

by (4.1). Therefore, is a contraction mapping. It follows from the Banach fixed-point theorem that has a unique fixed point In light of (3.1), we get that

(4.7)

which implies that

(4.8)

that is, is a solution of the generalized nonlinear variational-like inequality (2.3).

Now we prove the uniqueness. Suppose that the generalized nonlinear variational-like inequality (2.3) has two solutions . It follows that

(4.9)

(4.10)

for all Taking in (4.9) and in (4.10), we obtain that

(4.11)

Adding (4.11), we deduce that

(4.12)

which together with (4.1) implies that That is, the generalized nonlinear variational-like inequality (2.3) has a unique solution in .

Next we discuss the convergence of the iterative sequence generated by Algorithm 3.3. Taking in (4.7) and in (3.18), and adding these inequalities, we infer that

(4.13)

That is,

(4.14)

where is defined by (4.6). It follows from (3.19), (4.1), (4.14), and Lemma 2.5 that the iterative sequence generated by Algorithm 3.3 converges strongly to This completes the proof.

As in the proof of Theorem 4.1, we have the following theorem.

Theorem 4.2.

Let be a nonempty closed convex subset of a real Hilbert space , and Lipschitz continuous with constant . Assume that is a coercive continuous bilinear form satisfying (a1) and (a2), satisfies (b1)–(b4). Let and be mappings such that is -strongly monotone with respect to in the first argument with constant , -monotone with respect to in the second argument, is Lipschitz continuous with constant and and are -hemicontinuous with respect to and in . Assume that Assumption 2.6 holds and

(4.15)

Then the generalized nonlinear variational-like inequality (2.3) possesses a unique solution and the iterative sequence generated by Algorithm 3.3 converges strongly to

Remark 4.3.

The conditions of Theorems 3.1, 3.2, 4.1, and 4.2 are different from the conditions of the results in [1–10]. In particular, the mappings and with respect to the second argument in Theorems 3.2 and 4.2 are Lipschitz continuous, but other mappings in Theorems 3.2 and 4.2 are not Lipschitz continuous.