Auxiliary Principle for Generalized Strongly Nonlinear Mixed Variational-Like Inequalities

Abstract

We introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities, which includes several classes of variational inequalities and variational-like inequalities as special cases. By applying the auxiliary principle technique and KKM theory, we suggest an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality. The existence of solutions and convergence of sequence generated by the algorithm for the generalized strongly nonlinear mixed variational-like inequalities are obtained. The results presented in this paper extend and unify some known results.

1. Introduction

It is well known that the auxiliary principle technique plays an efficient and important role in variational inequality theory. In 1988, Cohen [1] used the auxiliary principle technique to prove the existence of a unique solution for a variational inequality in reflexive Banach spaces, and suggested an innovative and novel iterative algorithm for computing the solution of the variational inequality. Afterwards, Ding [2], Huang and Deng [3], and Yao [4] obtained the existence of solutions for several kinds of variational-like inequalities. Fang and Huang [5] and Liu et al. [6] discussed some classes of variational inequalities involving various monotone mappings. Recently, Liu et al. [7, 8] extended the auxiliary principle technique to two new classes of variational-like inequalities and established the existence results for these variational-like inequalities.

Inspired and motivated by the results in [1–13], in this paper, we introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities. Making use of the auxiliary principle technique, we construct an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality. Several existence results of solutions for the generalized strongly nonlinear mixed variational-like inequality involving strongly monotone, relaxed Lipschitz, cocoercive, relaxed cocoercive and generalized pseudocontractive mappings, and the convergence results of iterative sequence generated by the algorithm are given. The results presented in this paper extend and unify some known results in [9, 12, 13].

2. Preliminaries

In this paper, let , let be a real Hilbert space endowed with an inner product and norm , respectively, let be a nonempty closed convex subset of . Let and let be mappings. Now we consider the following generalized strongly nonlinear mixed variational-like inequality problem: find such that

(2.1)

where is a coercive continuous bilinear form, that is, there exist positive constants and such that

(C1)

(C2)Clearly,

Let satisfy the following conditions:

(C3) for each , is linear in the first argument;

(C4) is bounded, that is, there exists a constant such that

(C5);

(C6) for each , is convex in the second argument.

Remark 2.1.

It is easy to verify that

(m1)

(m2),

where (m2) implies that for each , is continuous in the second argument on .

Special Cases

(m3) If and for all where , then the generalized strongly nonlinear mixed variational-like inequality (2.1) collapses to seeking such that

(2.2)

which was introduced and studied by Ansari and Yao [9], Ding [11] and Zeng [13], respectively.

(m4) If for all where , then the problem (2.2) reduces to the following problem: find such that

(2.3)

which was introduced and studied by Yao [12].

In brief, for suitable choices of the mappings and , one can obtain a number of known and new variational inequalities and variational-like inequalities as special cases of (2.1). Furthermore, there are a wide classes of problems arising in optimization, economics, structural analysis and fluid dynamics, which can be studied in the general framework of the generalized strongly nonlinear mixed variational-like inequality, which is the main motivation of this paper.

Definition 2.2.

Let and be mappings.

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

(2.4)

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

(2.5)

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

(2.6)

is said to be relaxed-cocoercive with respect to in the first argument if there exist constants such that

(2.7)

is said to be Lipschitz continuous with constant if there exists a constant such that

(2.8)

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

(2.9)

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

(2.10)

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

(2.11)

is said to be strongly monotone with constant if there exists a constant such that

(2.12)

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

(2.13)

is said to be cocoercive with constant if there exists a constant such that

(2.14)

is said to be Lipschitz continuous with constant if there exists a constant such that

(2.15)

is said to be Lipschitz continuous in the first argument if there exists a constant such that

(2.16)

Similarly, we can define the Lipschitz continuity of in the second argument.

Definition 2.3.

Let be a nonempty convex subset of and let be a functional.

(d1) is said to be convex if for any and any ,

(2.17)

(d2) is said to be concave if is convex;

(d3) is said to be lower semicontinuous on if for any , the set is closed in ;

(d4) is said to be upper semicontinuous on , if is lower semicontinuous on .

In order to gain our results, we need the following assumption.

Assumption 2.4.

The mappings satisfy the following conditions:

(d5)

(d6) for given the mapping is concave and upper semicontinuous on .

Remark 2.5.

It follows from (d5) and (d6) that

(m5)

(m6) for any given , the mapping is convex and lower semicontinuous on .

Proposition 2.6 (see [9]).

Let be a nonempty convex subset of . If is lower semicontinuous and convex, then is weakly lower semicontinuous.

Proposition 2.6 yields that if is upper semicontinuous and concave, then is weakly upper semicontinuous.

Lemma 2.7 (see [10]).

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

3. Auxiliary Problem and Algorithm

In this section, we use the auxiliary principle technique to suggest and analyze an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality (2.1). To be more precise, we consider the following auxiliary problem associated with the generalized strongly nonlinear mixed variational-like inequality (2.1): given , find such that

(3.1)

where is a constant, is a mapping. The problem is called a auxiliary problem for the generalized strongly nonlinear mixed variational-like inequality (2.1).

Theorem 3.1.

Let be a nonempty closed convex subset of the Hilbert space . Let be a coercive continuous bilinear form with (C1) and (C2), and let be a functional with (C3)–(C6). Let be Lipschitz continuous and relaxed Lipschitz with constants and , respectively. Let be Lipschitz continuous with constant , and let satisfy Assumption 2.4. Then the auxiliary problem (3.1) has a unique solution in .

Proof.

For any , define the mappings by

(3.2)

We claim that the mappings and satisfy all the conditions of Lemma 2.7 in the weak topology. Note that

(3.3)

and for any Since is convex in the second argument and is a coercive continuous bilinear form, it follows from Remark 2.1 and Assumption 2.4 that for each , is weakly upper semicontinuous on . It is easy to show that the set is a convex set for each fixed Let be fixed and put

(3.4)

Clearly, is a weakly compact subset of . From Assumption 2.4, the continuity of and , and the properties of and , we gain that for any

(3.5)

Thus the conditions of Lemma 2.7 are satisfied. It follows from Lemma 2.7 that there exists a such that for any , that is,

(3.6)

Let and . Replacing by in (3.6) we gain that

(3.7)

Letting in (3.7), we get that

(3.8)

which means that is a solution of (3.1).

Suppose that are any two solutions of the auxiliary problem (3.1). It follows that

(3.9)

(3.10)

Taking in (3.9) and in (3.10) and adding these two inequalities, we get that

(3.11)

Since is relaxed Lipschitz, we find that

(3.12)

which implies that . That is, the auxiliary problem (3.1) has a unique solution in . This completes the proof.

Applying Theorem 3.1, we construct an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality (2.1).

Algorithm 3.2.

• (i) At step , start with the initial value .

• (ii) At step , solve the auxiliary problem (3.1) with . Let denote the solution of the auxiliary problem (3.1). That is,

  

  (3.13)

where is a constant.

• (iii) If, for given , stop. Otherwise, repeat (ii).

4. Existence of Solutions and Convergence Analysis

The goal of this section is to prove several existence of solutions and convergence of the sequence generated by Algorithm 3.2 for the generalized strongly nonlinear mixed variational-like inequality (2.1).

Theorem 4.1.

Let be a nonempty closed convex subset of the Hilbert space . Let be a coercive continuous bilinear form with (C1) and (C2), and let be a functional with (C3)–(C6). Let be Lipschitz continuous with constants in the first and second arguments, respectively. Let and be Lipschitz continuous with constants respectively, let be cocoercive with constant with respect to in the first argument, let be relaxed Lipschitz with constant and let be strongly monotone with constant . Assume that Assumption 2.4 holds. Let

(4.1)

If there exists a constant satisfying

(4.2)

and one of the following conditions:

(4.3)

(4.4)

(4.5)

(4.6)

then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence defined by Algorithm 3.2 converges to

Proof.

It follows from (3.13) that

(4.7)

Adding (4.7), we obtain that

(4.8)

Since is relaxed Lipschitz and Lipschitz continuous with constants and , and is strongly monotone and Lipschitz continuous with constants and , respectively, we get that

(4.9)

Notice that is Lipschitz continuous in the first and second arguments, and are both Lipschitz continuous, and is cocoercive with constant with respect to in the first argument. It follows that

(4.10)

Let

(4.11)

It follows from (4.8)–(4.10) that

(4.12)

From (4.2) and one of (4.3)–(4.6), we know that . It follows from (4.12) that is a Cauchy sequence in . By the closedness of there exists satisfying . In term of (3.13) and the Lipschitz continuity of , we gain that

(4.13)

By Assumption 2.4, we deduce that

(4.14)

Since as and is bounded, it follows that

(4.15)

which implies that

(4.16)

In light of (C3) and (m2), we get that

(4.17)

which means that as . Similarly, we can infer that as . Therefore,

(4.18)

This completes the proof.

Theorem 4.2.

Let and be as in Theorem 4.1. Assume that , are Lipschitz continuous with constants and , respectively, is relaxed Lipschitz with constant , and is relaxed Lipschitz with constant with respect to in the second argument. Let

(4.19)

If there exists a constant satisfying

(4.20)

and one of (4.3)–(4.6), then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence defined by Algorithm 3.2 converges to

Proof.

As in the proof of Theorem 4.1, we deduce that

(4.21)

Because is relaxed Lipschitz and Lipschitz continuous, is relaxed Lipschitz with respect to in the second argument and Lipschitz continuous, and is Lipschitz continuous in the second argument, we conclude that

(4.22)

The rest of the argument is the same as in the proof of Theorem 4.1 and is omitted. This completes the proof.

Theorem 4.3.

Let and be as in Theorem 4.1, and let be as in Theorem 4.2. Assume that is -cocoercive with constant with respect to in the first argument and Lipschitz continuous with constant . Let

(4.23)

If there exists a constant satisfying (4.2) and one of (4.3)–(4.6), then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence defined by Algorithm 3.2 converges to

Proof.

As in the proof of Theorem 4.1, we derive that

(4.24)

Because is Lipschitz continuous, is Lipschitz continuous in the first argument, and is -cocoercive with with respect to in the first argument and Lipschitz continuous, we gain that

(4.25)

The rest of the proof is identical with the proof of Theorem 4.1 and is omitted. This completes the proof.

Theorem 4.4.

Let and be as in Theorem 4.1. Let and be as in Theorems 4.2 and 4.3, respectively. Assume that , are Lipschitz continuous with constants , and , respectively, is g-generalized pseudocontractive with constant with respect to in the second argument, and is cocoercive with constant . Let

(4.26)

If there exists a constant satisfying

(4.27)

and one of (4.3), (4.4), and (4.6), then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence defined by Algorithm 3.2 converges to

Proof.

By a similar argument used in the proof of Theorem 4.1, we conclude that

(4.28)

Since is -generalized pseudocontractive with respect to in the second argument and Lipschitz continuous, is Lipschitz continuous and is Lipschitz continuous in the second argument, is cocoercive and Lipschitz continuous, it follows that

(4.29)

The rest of the argument follows as in the proof of Theorem 4.1 and is omitted. This completes the proof.

Theorem 4.5.

Let and be as in Theorem 4.1. Assume that are Lipschitz continuous with constants , , respectively, is relaxed -cocoercive with respect to in the first argument, is -relaxed Lipschitz with constant with respect to in the second argument. Let

(4.30)

If there exists a constant satisfying

(4.31)

and one of (4.3)–(4.6), then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence defined by Algorithm 3.2 converges to

Proof.

Notice that

(4.32)

The rest of the proof is similar to the proof of Theorem 4.1 and is omitted. This completes the proof.

Remark 4.6.

Theorems 4.1–4.5 extend, improve, and unify the corresponding results in [9, 12, 13].