A Hybrid Proximal Point Three-Step Algorithm for Nonlinear Set-Valued Quasi-Variational Inclusions System Involving -Accretive Mappings

Abstract

The main purpose of this paper is to introduce and study a new class of generalized nonlinear set-valued quasi-variational inclusions system involving -accretive mappings in Banach spaces. By using the resolvent operator due to Lan-Cho-Verma associated with -accretive mappings and the matrix analysis method, we prove the convergence of a new hybrid proximal point three-step iterative algorithm for this system of set-valued variational inclusions and an existence theorem of solutions for this kind of the variational inclusions system. The results presented in this paper generalize, improve, and unify some recent results in this field.

1. Introduction

The variational inclusion, which was introduced and studied by Hassouni and Moudafi [1], is a useful and important extension of the variational inequality. It provides us with a unified, natural, novel, innovative, and general technique to study a wide class of problems arising in different branches of mathematical and engineering sciences. Various variational inclusions have been intensively studied in recent years. Ding and Luo [2], Verma [3, 4], Huang [5], Fang et al. [6], Fang and Huang [7], Fang et al. [8], Lan et al. [9], Zhang et al. [10] introduced the concepts of -subdifferential operators, maximal -monotone operators, -monotone operators, -monotone operators, -monotone operators, -accretive mappings, -monotone operators, and defined resolvent operators associated with them, respectively. Moreover, by using the resolvent operator technique, many authors constructed some approximation algorithms for some nonlinear variational inclusions in Hilbert spaces or Banach spaces. Verma has developed a hybrid version of the Eckstein-Bertsekas [11] proximal point algorithm, introduced the algorithm based on the -maximal monotonicity framework [12], and studied convergence of the algorithm. For the past few years, many existence results and iterative algorithms for various variational inequalities and variational inclusion problems have been studied. For details, please see [1–37] and the references therein.

On the other hand, some new and interesting problems for systems of variational inequalities were introduced and studied. Peng and Zhu [14], Cohen and Chaplais [15], Bianchi [16], and Ansari and Yao [17] considered a system of scalar variational inequalities. Ansari et al. [18] introduced and studied a system of vector equilibrium problems and a system of vector variational inequalities using a fixed point theorem. Allevi et al. [19] considered a system of generalized vector variational inequalities and established some existence results with relative pseudomonotonicity. Kassay and Kolumbán [20] introduced a system of variational inequalities and proved an existence theorem through the Ky Fan lemma. Kassay et al. [21] studied Minty and Stampacchia variational inequality systems with the help of the Kakutani-Fan-Glicksberg fixed point theorem. J. K. Kim and D. S. Kim [22] introduced a new system of generalized nonlinear quasi-variational inequalities and obtained some existence and uniqueness results on solutions for this system of generalized nonlinear quasi-variational inequalities in Hilbert spaces. Cho et al. [23] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces. They proved some existence and uniqueness theorems for solutions for the system of nonlinear variational inequalities. As generalizations of a system of variational inequalities, Agarwal et al. [24] introduced a system of generalized nonlinear mixed quasi-variational inclusions and investigated the sensitivity analysis of solutions for this system of generalized nonlinear mixed quasi-variational inclusions in Hilbert spaces. Kazmi and Bhat [25] introduced a system of nonlinear variational-like inclusions and gave an iterative algorithm for finding its approximate solution. Fang and Huang [26], Fang et al. [8] introduced and studied a new system of variational inclusions involving H-monotone operators and -monotone operators, respectively. Yan et al. [27] introduced and studied a system of set-valued variational inclusions which is more general than the model in [3].

Inspired and motivated by recent research work in this field, in this paper, a general set-valued quasi-variational inclusions system with -accretive mappings is studied in Banach spaces, which includes many variational inclusions (inequalities) as special cases. By using the resolvent operator associated with -accretive operator due to Lan, an existence theorem of solution for this class of variational inclusions is proved, and a new hybrid proximal point algorithm is established and suggested, and the convergence of iterative sequences generated by the algorithm is discussed in -uniformly smooth Banach spaces. The results presented in this paper generalize, and unify some recent results in this field.

2. Preliminaries

Let be a real Banach space with dual space , be the dual pair between and , denote the family of all the nonempty subsets of , and denote the family of all nonempty closed bounded subsets of . The generalized duality mapping is defined by

(21)

where is a constant.

The modulus of smoothness of is the function defined by

(22)

A Banach space is called uniformly smooth if

(23)

is called -uniformly smooth if there exists a constant such that

(24)

Remark 2.1.

In particular, is the usual normalized duality mapping. It is known that, for all , is single-valued if is strictly convex [10], or is uniformly smooth (Hilbert space and space are uniformly Banach space), and if , the Hilbert space, then becomes the identity mapping on . In what follows we always denote the single-valued generalized duality mapping by in real uniformly smooth Banach space unless otherwise states.

Let us recall the following results and concepts.

Definition 2.2.

A single-valued mapping is said to be -Lipschitz continuous if there exists a constant such that

(25)

Definition 2.3.

A single-valued mapping is said to be

(i)accretive if

(26)

(ii)strictly accretive if is accretive and if and only if for all ;

(iii)-strongly -accretive if there exists a constant such that

(27)

(iv)-Lipschitz continuous if there exists a constant such that

(28)

Definition 2.4.

A set-valued mapping is said to be

(i)-Lipschitz continuous if there exists a constant such that

(29)

where is the Hausdorff metric on

(ii)-strongly -accretiveif there exists a constant such that

(210)

(iii)-relaxed cocoercive if there exist two constants such that

(211)

(iv)-strongly -accretive with respect to the first argument of the mapping , if there exists a constant such that

(212)

where .

Definition 2.5.

Let be a single-valued mapping and be a set-valued mapping . For , a single-valued mapping is said to be

(i)-Lipschitz continuous if there exist three constants such that

(213)

(ii)-relaxed cocoercive with respect to in the first argument, if there exist constants such that

(214)

In a similar way, we can define Lipschitz continuity and -relaxed cocoercive with respect to of in the second, or the three argument.

Definition 2.6.

Let and be single-valued mappings. A set-valued mapping is said to be

• (i) accretive if

  

  (215)

(ii)-accretive if

(216)

(iii)-relaxed -accretive, if there exists a constant such that

(217)

(iv)-accretive if is accretive and for all ;

(v)-accretive if is -relaxed -accretive and for every .

Based on [9], we can define the resolvent operator as follows.

Definition 2.7 (see [9]).

Let be a single-valued mapping be a strictly -accretive single-valued mapping and be a ()-accretive mapping. The resolvent operator is defined by

(218)

where is a constant.

Remark 2.8.

The -accretive mappings are more general than -monotone mappings, -accretive mappings, -monotone operators, -subdifferential operators, and -accretive mappings in Banach space or Hilbert space, and the resolvent operators associated with -accretive mappings include as special cases the corresponding resolvent operators associated with -monotone operators, -accretive mappings, -accretive mappings, -monotone operators, -subdifferential operators [5, 6, 11, 14, 15, 26, 27, 35–37].

Lemma 2.9 (see [9]).

Let be -Lipschtiz continuous mapping, be an -strongly -accretive mapping, and be set-valued -accretive mapping, respectively. Then the generalized resolvent operator is -Lipschitz continuous, that is,

(219)

where , , and .

In the study of characteristic inequalities in -uniformly smooth Banach spaces , Xu [29] proved the following result.

Lemma 2.10 (see [29]).

Let be a real uniformly smooth Banach space. Then is -uniformly smooth if andonly if there exists a constant such that for all

(220)

Theorem 2.11.

Let the function where and , then

(221)

Proof.

Let , where , , , . Then by the . We can obtain

(222)

Let , and , where . It follows that

(223)

This completes the proof.

Corollary 2.12.

Let be real, for any real , if , then

(224)

Proof.

The proof directly follows from the (i) in the Theorem 2.11.

Definition 2.13 (see [38]).

Let is a real be a real matrix set, then the mappings

(225)

is called the -norm, and -norm, respectively.

Obviously, may be a Banach space on real field , which is called the real matrix-Banach space.

Definition 2.14 (see [38]).

Let is a real be a real matrix-Banach Space with the matrix-norm (, or ). If

(226)

then the matrix is called the limit matrix of matrix sequence , noted by , where is a real sequence, , and , .

Lemma 2.15 (see [38]).

, if and only if

(227)

Hence, if , then

In this paper, the matrix norm symbol is noted by .

Definition 2.16.

Let be real numbers, and and be two real vectors, then if and only if .

3. Quasi-Variational Inclusions System Problem and Hybrid Proximal Point Algorithm

Let be a real -uniformly smooth Banach space with dual space , be the dual pair between and , denote the family of all the nonempty subsets of , and denote the family of all nonempty closed bounded subsets of . The generalized duality mapping is defined by

(31)

where is a constant. Now, we consider the following generational nonlinear set-valued quasi-variational inclusions system problem with -accretive mappings (GNSVQVIS) problem.

Let , , and be single-valued mappings for . Let be a set-valued -accretive mapping and be set-valued mappings for .

For any , finding such that , , , , , and

(32)

where

Remark 3.1.

Some special cases of problem (3.2) are as follows.

• (i) If , , , , and is a Hilbert space, then the problem (3.2) reduces to the problem associated with the system of variational inclusions with -monotone operators, which is finding such that , , , , , , , , , and

  

  (33)

where

Problem (3.3) contains the system of variational inclusions with -monotone operators in Peng and Zhu [14], and the system of variational inclusions with -monotone operators in [8] as special cases.

• (ii) If , , (Hilbert space) and, , where is a proper, -subdifferentiable functional and denotes the -subdifferential operator of , then problem(3.3) changes to the problem associated with the following system of variational-like inequalities, which is finding such that

  

  (34)

where

• (iii) If , , , (Hilbert space) and , where is a proper, convex, lower semicontinuous functional and denotes the subdifferential operator of , then problem (3.3) changes to the problem associated with the following system of variational inequalities, which is finding such that

  

  (35)

where

• (iv) If , (Hilbert space), and , where is a nonempty, closed, and convex subsets and denotes the indicator of , then problem (3.5) reduces to the problem associated with the following system of variational inequalities, which is finding such that

  

  (36)

where

• (v) If , and is a Hilbert space, is a nonempty, closed and convex subset, for all , , where is a mapping on , is a constant, then problem (3.6) changes to the following problem: find such that

  

  (37)

where

Moreover, if , problem (3.7) becomes the problem introduced and studied by Verma [31].

We can see that problem (3.2) includes a number of known classes of system of variational inequalities and variational inclusions as special cases (see, e.g., [2–9, 11–27, 29, 32–37]). It is worth noting that problems (3.2)–(3.7) are all new mathematical models.

Theorem 3.2.

Let be a Banach space, be -Lipschtiz continuous mapping, be an -strongly -accretive mapping, and be a set-valued -accretive mapping for . Then the following statements are mutually equivalent.

(i)An element is a solution of problem (3.2), .

(ii)For and , , , , the following relations hold:

(38)

where is a constant and , respectively.

Proof.

This directly follows from definition of and the problem (3.2) for .

Algorithm 3.3.

Let , and be three nonnegative sequences such that

(39)

Step 1.

For arbitrarily chosen initial points , , , , , , Set

(310)

where the satisfies

(311)

By using [39], we can choose suitable , , and such that

(312)

for .

Step 2.

The sequences , and are generated by an iterative procedure:

(313)

(314)

where

(315)

Thus, we can choose suitable , , and such that

(316)

for and .

Remark 3.4.

If we choose suitable some operators , and space , then Algorithm 3.3 can be degenerated to a number of known algorithms for solving the system of variational inequalities and variational inclusions (see, e.g., [2–9, 11–27, 29, 31–35, 38, 39]).

4. Existence and Convergence

In this section, we prove the existence of solutions for problem (3.2) and the convergence of iterative sequences generated by Algorithm 3.3.

Theorem 4.1.

Let be a -uniformly smooth Banach space, be a -Lipschtiz continuous mapping, and be a -strongly -accretive mapping and -Lipschitz continuous. Let be a set-valued mappings of -Lipschitz continuous with constants , and be -relaxed cocoercive, respectively. Let be Lipschitz continuous with constants and for all , and -relaxed cocoercive with respect to in the first, second and third arguments, respectively. Let be Lipschitz continuous with constants and for all . Let , , be some set-valued mappings such that for each given , , , and be an -accretive mapping, respectively. Suppose that , and are three nonnegative sequences with

(41)

(42)

where is the same as in Lemma 2.10, , and . Then the problem (3.2) has a solution .

Proof.

Let

(43)

for . Then it follows from (3.13) that

(44)

Since is -Lipschitz continuous with constants and -relaxed cocoercive,

(45)

By (3.15), we have

(46)

Since and , by Lemma 2.9, we have

(47)

Since is Lipschitz continuous with constants , and -relaxed cocoercive with respect to in the first arguments, be Lipschitz continuous with constants , respectively, Lemma 2.10, we have

(48)

By (3.13), we know that . Since is Lipschitz continuous with constants , and is -Lipschitz continuous with constants , respectively, combing (4.4)–(4.8) and using Corollary 2.12, we have

(49)

and so

(410)

For the sequences , we have

(411)

Since is -Lipschitz continuous with constant and -relaxed cocoercive, we have

(412)

It follows from (3.15) that

(413)

Since and , by using Lemma 2.9, we obtain

(414)

Since is Lipschitz continuous with constants , and -relaxed cocoercive with respect to in the first arguments, is Lipschitz continuous with constants , respectively, it follows from Lemma 2.10 that

(415)

By (3.13), we know that . Since is Lipschitz continuous with constants , and is -Lipschitz continuous with constants , respectively, combing (4.11)–(4.20) and using Corollary 2.12, we have

(416)

and so

(417)

Using the same as the method, we can obtain

(418)

Let

(419)

Letting , then combining (4.10),(4.17)–(4.19), we have , where

(420)

which is called the iterative matrix for Hybrid proximal point three-step algorithm of nonlinear set-valued quasi-variational inclusions system involving -Accretive mappings. Using (4.20), , , , we have

(421)

where , and

(422)

By using [38], we have

(423)

Letting

(424)

It follows from (4.22) and assumption condition (4.2) that and hence there exists and such that for all . Therefore, by (4.23), we have

(425)

Without loss of generality we assume

(426)

By the property of the matrix norm [38], for , we have

(427)

Hence, for any and , we have

(428)

It follows that , as , and so that the is a Cauchy sequence in . Let as . By the Lipschitz continuity of , we can obtain

(429)

It follows that , , , and are also Cauchy sequences in . We can assume that , , , and , respectively. Noting that , we have

(430)

Hence and therefore . Similarly, we can prove that , , and . By the Lipschitz continuity of , , , and , we have

(431)

for , where is a constant. Thus, by Theorem (3.3), we know that is solution of problem (3.2). This completes the proof.

Corollary 4.2.

Let be a -uniformly smooth Banach space, be a -Lipschtiz continuous mapping, and be an -strongly -accretive mapping and -Lipschitz continuous. Let be the same as in Theorem 4.1. If

(432)

where is the same as in Lemma 2.10, , and . Then problem (3.2) has a solution .

Remark 4.3.

For a suitable choice of the mappings , we can obtain several known results in [2–5, 9, 11–27, 29, 32–37] as special cases of Theorem 4.1 and Corollary 4.2.