Error bound analysis for split weak vector mixed quasi variational inequality problems in fuzzy environment

APPLICABLE ANALYSIS 2022, AHEAD-OF-PRINT, 1-15 https://elkssl0a75e822c6f3334851117f8769a30e1csfdafs.casb.nju.edu.cn:4443/10.1080/00036811.2021.2008374 Error bound analysis for split weak

APPLICABLE ANALYSIS

2022, AHEAD-OF-PRINT, 1-15

https://elkssl0a75e822c6f3334851117f8769a30e1csfdafs.casb.nju.edu.cn:4443/10.1080/00036811.2021.2008374

Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment

Nguyen Van Hung a , Vo Minh Tam b , and Donal O'Reganc

a Department of Scientific Fundamentals Posts and Telecommunications Institute of Technology, Ho Chi Minh City,

Vietnamb Department of Mathematics, Dong Thap University, Cao Lanh City, Vietnam c School of Mathematics, Statistics and Applied Mathematics National University of Ireland, Galway, Ireland

ABSTRACT

In this paper, we consider a new class of split weak vector mixed quasi-variational inequality problems in fuzzy environments (for short, SWQVIP) Our aim is to establish error bounds for the underlying problem SWQVIP via regularized gap functions We first propose some regularized gap functions of the problem SWQVIP using the method of nonlinear scalarization functions Then, error bounds for the problem SWQVIP are investigated based

on regularized gap functions with some suitable conditions without monotonicity Finally, some examples are given to illustrate our results The main results obtained in this paper are new and extend some corresponding known results in the literature.

ARTICLE HISTORY

Received 22 October 2021

Accepted 14 November 2021

KEYWORDS

Split weak vector mixed quasi-variational inequality problem, regularized gap function, fuzzy mapping, error bound

2020 MATHEMATICS SUBJECT CLASSIFICATIONS

49J53, 90C33, 47S40, 46S40

1 Introduction

Gap functions are useful in studying solution methods, existence conditions and stability of solutions for optimization-related problems in order to simplify the computational aspects The concept of a gap function was first introduced by Auslender [1] to transform a variational inequality into an equivalent optimization problem Based on the gap function of Auslender [1], Fukushima [2] extended the concept of a regularized gap function for a variational inequality Also, based on the idea of Fukushima [2], the regularized function

of Moreau-Yosida type was introduced by Yamashita and Fukushima [3] and they also considered the so-called error bounds for variational inequalities using regularized gap functions The notion of error bounds is known as an upper estimate of the distance between an arbitrary feasible point and the solution set of a certain problem It plays a vital role in analyzing the rate of convergence of some algorithms for solving solutions of some problems Motivated by Yamashita and Fukushima [3], regularized gap functions and error bounds were investigated for various kinds of optimization-related problems We refer the reader to [4–14] and the references therein for a more detailed discussion

In 1965, the theory of fuzzy sets was introduced by Zadeh [15] Applying some fuzzy mappings to variational inequality problems, in 1989, Chang and Zhu [16] introduced and studied a class of variational

CONTACT Vo Minh Tam vmtammath@gmail.com

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inequality problems with fuzzy mappings in abstract spaces Based on the cut sets of fuzzy sets [15], Chang and Zhu [16] considered the concept of cut sets of fuzzy mapping via membership functions There are many papers on optimization problems, complementarity problems, variational inequalities and equilibrium problems using techniques of fuzzy theory; see [17–25] and the references therein Recently, Hung et al [8] established regularized gap functions and error bounds for generalized mixed weak vector quasi-variational inequality problems in fuzzy environments using the nonlinear scalarization method

On the other hand, a new class of split variational inequality problems on Hilbert spaces (for short, (SVIP)) was introduced Censor et al [26] An iterative algorithm for solving the problem (SVIP) was constructed under reasonable conditions and they also introduced the split zero problem and the split feasibility problem This class of problems is also at the core of modeling in the study of many inverse problems arising for other real-world and phase retrieval problems, for example, in sensor networks in data compression and computerized tomography (see e.g [27–30] and the references therein) He [31] considered

a class of split equilibrium problems via nonlinear bifunctions (for short, (SEP)) which extended the class of split variational inequality problems in [26] Many authors presented some iterative algorithms for solving various kinds of the problems (SEP) and (SVIP) such as split equality problems, split equilibrium problems [32–35], split quasi-variational inequality problems [36] and split variational inclusions [37,38] Hu and Fang [39,40] established the well-posedness of split inverse variational inequality problems Hung et al [41] introduced and studied a new class of split general random variational inclusions with random fuzzy mappings and using the resolvent operator method, they proposed iterative algorithms for solving this class

of variational inclusions and some other special cases Hung et al [42] introduced and studied a new class of split mixed vector quasi-variational inequality problems of strong type They established some new results

on regularized gap functions and error bounds using the method of the nonlinear scalarization function under some conditions of monotonicity on the data However, to our best knowledge, up to now, there is no paper devoted to a class of split weak vector mixed quasi-equilibrium problems in fuzzy environments without monotonicity

Inspired by the work above, in this paper we continue the study of a new class of split weak vector mixed quasi-variational inequality problems in fuzzy environments (for short, SWQVIP) Then, using the method of nonlinear scalarization functions, we propose some regularized gap functions of the problem SWQVIP Furthermore, error bounds for the problem SWQVIP are established based on regularized gap functions and some suitable conditions without monotonicity on the data The analysis of error bounds for this new class of split weak vector mixed quasi-variational inequality problems in fuzzy environments is one

of the novelties of this paper Finally, to illustrate our main results, some examples are presented

The remainder of this paper is structured as follows In Section 2, we recall some definitions and we introduce the problem SWQVIP We also impose some hypotheses on the data of the problem SWQVIP In Section 3, we propose some regularized gap functions of the problem SWQVIP based on the method of nonlinear scalarization functions Then, some error bounds for the problem SWQVIP are studied in terms of regularized gap functions under some suitable conditions without monotonicity

2 Notation and preliminaries

First, we introduce a new class of split weak vector mixed quasi-variational inequality problems in fuzzy environments

Let be a collection of all fuzzy sets over a Hilbert space H, i.e

(i) A mapping is called a fuzzy mapping on ; (ii)

If is a fuzzy mapping on H, then (denoted by , in the sequel) is a fuzzy set on H and

is the membership function of y in ; (iii) For any and , the set

is called a α-cut set of W.

Throughout the paper, unless other specified, for each , let be a real Hilbert space with

be the space of all continuous linear mappings from to , and denote by the

on

In this paper, we consider the following split weak vector mixed quasi-variational inequality problem with fuzzy mappings (for short, SWQVIP):

Find a point such that

(1)

(2)

where and are the cut sets of fuzzy sets and , respectively, defined by

The solution set of the problem (1) (resp., (2)) is denoted by (resp., ) Then we denote the solution

set

Next, we recall some basic concepts and their properties

Definition 2.1 see [43] Let U and T be two Hausdorff topological spaces and be closed pointed and convex cone with A mapping is said to be -convex on a convex subset

if, for all and ,

Definition 2.2 see [44] Let U and T be two Hausdorff topological spaces A set-valued mapping

is said to be:

i lower semicontinuous at if, for some open subset implies the existence of a neighborhood V of such that for all ;

ii upper semicontinuous at if, for each open superset O of , there is a neighborhood V

of such that for all ;

The following result provide some properties of a nonlinear scalarization function which be useful in the next section

For each , we now impose the following hypotheses on the data of the problem SWQVIP

(4)

(5)

(6)

(7)

(8)

iii lower resp., upper semicontinuous on a subset A of U if it is lower (resp., upper) semicontinuous

at each ;

iv continuous on A if it is both lower and upper semicontinuous on A.

Lemma 2.1 see [43,45] Let T be a convex Hausdorff topological vector space and be closed pointed and convex cone with For any fixed , , the nonlinear scalarization function defined by

(3) for all has the following properties:

i is positively homogeneous, convex and continuous on T, especially,

ii.

iii.

Remark 2.1 Note that, if and is the nonnegative orthant of , then ( 3 ) becomes

where is the canonical base of the space

(10)

3 Main results

In this section, first, some gap functions of regularized type are established for the problem SWQVIP by using the method of the nonlinear scalarization functions Then, we propose some error bounds for the underlying problem SWQVIP based on regularized gap functions and some suitable conditions without monotonicity

(11) where

Remark 2.2

i It is easy to see that if for all , then for

ii The conditions ( 9 )(b), ( 10 )(c) are developed from the conditions and considered in Hung et al [ 8 ].

Definition 3.1 A function is said to be a gap function for the problem SWQVIP if it satisfies the following properties:

i for all

ii for any , if and only if is a solution of the problem SWQVIP.

and

(13)

Now, we show that is a gap function for the problem SWQVIP

Theorem 3.1 For each , suppose that conditions (4), (5)(a), (6), (7), (8)(a,b), (9)(a) and (10)

(a,b) hold In addition, for all Then for any , the function defined by (11) is a gap function for the problem SWQVIP.

Proof.

(14)

for all Since is -convex and

, we have

(15)

which implies that

(16) Moreover, it follows from the -convexity of and the assumption

that

(17)

In view of (15)–(17) and the convexity of the cone , we have

(18) From the positive homogeneousness of and Lemma 2.1(iv), (18) implies

(19) Combining the relations of (14) and (19), we have

or

(20)

Taking in (20), we have

The following result provides some sufficient conditions for the gap function

to be continuous

is a gap function for SWQVIP This completes the proof □

Lemma 3.2 For each , suppose that conditions (4), (5)(a), (6)(b), (7)(b), (8)(a), (9)(a) and (10)

compact values, the gap function defined by

is continuous on By a similar argument, we also show that defined by (13) is continuous on In

addition to the continuity of C, is continuous on Therefore, the gap function

defined by

is continuous on This completes the proof □

Now, we establish the regularized gap function of Moreau-Yosida type introduced in [3,6,13] of the gap

(21) where

Now, we prove that is a gap function for the problem SWQVIP

Theorem 3.3 Suppose that all conditions of Theorem 3.1 hold and assume further that the condition (8)(c)

is satisfied Then, for any , defined by (21) is a gap function for the problem SWQVIP.

Proof.

Since is a gap function, Thus, we conclude that for all By a similar argument, we verify that

and so

(22)

Next, we establish error bounds for the problem SWQVIP based on the regularized gap functions studied above

Hence we have

Therefore, we have □

Theorem 3.4 For each , suppose that all conditions (4)–(10) hold If, for each ,

and satisfy

then, for any with ,

(23)

Proof. For any , it follows from the condition (8)(d) that and

Then, from the definitions of , and in (11)–(13), we have

(24)

It follows from the condition (10)(c) that

(25) From the condition (9)(b), we obtain

(26) From (25), (26) and the convexity of , we have

(27) Using the result of Lemma 2.1(ii), (27) implies

(28)

By a similar way, since , we have and so

(29)

In view of (24), (28) and (29), we get

Since C is ξ-strongly nonexpanding, we have

Therefore, we have

Thus,

This completes the proof □

Theorem 3.5 Suppose that all hypotheses of Theorem 3.4 hold Then, for any with

, we have

(30)

Proof. From Theorem 3.4, we get that

Now, we give the following example to illustrate the main results established above.

Hence we have

This completes the proof □

, be fuzzy mappings defined by

and , Then we have

Let be the mappings defined by

for all Let and be the mappings defined by

Let be the mapping defined by for all By direct calculation, it follows that Since , the solution of the problem SWQVIP is

It is easy to show that the assumptions of Theorems 3.1 and 3.3 hold Thus, for any , the functions defined by 11 and defined by ( 21 ) are gap functions for the problem SWQVIP Indeed, for example, taking and and

, for any and , we have

and so

Hence is a gap function for SWQVIP.

We also have

The following example show that Theorems 3.4 and 3.5 are applicable without using the assumption of strong monotonicity

This implies that

Hence is a gap function for the problem SWQVIP.

Moreover, all the conditions of Theorems 3.4 and 3.5 are satisfied with , , ,

which implies that ( 23 ) holds Next, if then we get

Thus the inequality ( 30 ) holds By similar arguments, we also verify that the inequality ( 30 ) holds in the case

.

Example 3.2 For each , let 4, , , C as in Example 3.1.

4 Conclusions

In this paper, we study a new class of split weak vector mixed quasi-variational inequality problems in fuzzy environments Using the method of nonlinear scalarization functions, we propose some regularized gap functions of the problem SWQVIP (Theorems 3.1 and 3.3) Furthermore, error bounds for the problem SWQVIP are provided based on regularized gap functions under some suitable conditions without monotonicity (Theorems 3.4 and 3.5) The analysis of error bounds for this new class of split weak vector mixed quasi-variational inequality problems in fuzzy environments is one of the novelties of this paper Finally, some examples are given to illustrate our main results

Acknowledgements

The authors are grateful to the editor and anonymous referees for their valuable remarks which improved the results and presentation of this article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

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Then both SWQVIP and (SMVIP) reduce to the problem of finding such that

It follows from a direct computation that , and It is easy to see that the assumptions of Theorems 3.4 and 3.5 are fulfilled.

However, the functions and are not strongly monotone with modulus Indeed, for example, for all such that Then, we have

Hence, is not strongly monotone with any modulus