08:57, 31/03/2022 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 vector mixed quasi-variational inequality problems in fuzzy environment Nguyen Van Hung a, Vo Minh Tam b, and Donal O'Regan c a Department of Scientific Fundamentals Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam b 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 CONTACT Vo Minh Tam vmtammath@gmail.com COMMUNICATED BY R P Gilbert © 2022 Informa UK Limited, trading as Taylor & Francis Group 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 socalled 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 https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true 1/17 08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment 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 Notation and preliminaries First, we introduce a new class of split weak vector mixed quasi-variational inequality problems in fuzzy environments Let If be a collection of all fuzzy (i) A mapping is a fuzzy mapping on H, then (denoted by sets over a Hilbert space H, is called a fuzzy mapping on , in the sequel) is a fuzzy set on H and https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true i.e ; (ii) 2/17 08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment 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 the norm , be the distance from an element x to a subset Let be the space of all continuous linear mappings from to , and denote by the value of a continuous linear operator at Let be fuzzy mappings and Let , where is the interior of , let be nonlinear mappings and , be a closed pointed , be an operator and convex cone with , 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) and such that solves (2) where and are the cut sets of fuzzy sets The solution set of the problem (1) (resp., (2)) is denoted by set of the problem SWQVIP by set Next, we recall some basic concepts and their properties and , respectively, defined by (resp., ) Then we denote the solution We always assume that Π is a nonempty 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 existence of a neighborhood V of ii upper semicontinuous at of such that if, such that for some open subset for all ; if, for each open superset O of for all ; https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true implies the , there is a neighborhood V 3/17 08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment 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 The following result provide some properties of a nonlinear scalarization function which be useful in the next section Lemma 2.1 see [43,45] Let T be a convex Hausdorff topological vector space and pointed and convex cone with For any fixed , , scalarization function defined by be closed the nonlinear (3) for all i has the following properties: is positively homogeneous, convex and continuous on T, especially, ii iii iv If , then Remark 2.1 Note that, if where For each and is the nonnegative orthant of is the canonical base of the space , then (3) becomes , we now impose the following hypotheses on the data of the problem SWQVIP (4) (5) (6) (7) (8) https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true 4/17 08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment (9) (10) Remark 2.2 i It is easy to see that if all ; for all ii The conditions (9)(b), (10)(c) are developed from the conditions Hung et al [8] , then and for considered in 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 Definition 3.1 A function the following properties: i ii for any For any is said to be a gap function for the problem SWQVIP if it satisfies for all , if and only if is a solution of the problem SWQVIP , we now consider the following function defined by (11) where https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true 5/17 08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment (12) and (13) for any fixed Now, we show that for each is a gap function for the problem SWQVIP Theorem 3.1 For each (a,b) hold In addition, function , suppose that conditions (4), (5)(a), (6), (7), (8)(a,b), (9)(a) and (10) for all Then for any , the defined by (11) is a gap function for the problem SWQVIP Proof i For any and Since for all , we have and so we also obtain , Then It follows from For any convex set, we have Thus ii Suppose that there exists for all and and and and that , let and so Since is a (14) https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true 6/17 08:57, 31/03/2022 for Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment all Since is -convex and , we have (15) From the -convexity of , we get which implies that (16) Moreover, it follows from the that -convexity of and the assumption (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 for all for all that Conversely, if in (20), we have and and implies , then In view of Lemma 2.1(iii), that is Hence, Therefore, and Since By a similar argument, we prove https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true , 7/17 08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment for all and From Lemma 2.1(iii), we have for all and This implies that for all and Thus, we obtain Hence Similarly, we verify that Combining with the result of implies , we conclude Thus we have Therefore, is a gap function for SWQVIP This completes the proof □ The following result provides some sufficient conditions for the gap function to be continuous Lemma 3.2 For each (b) hold Then, for any Proof Since , suppose that conditions (4), (5)(a), (6)(b), (7)(b), (8)(a), (9)(a) and (10) , is continuous on , continuous Moreover, since compact values, the gap function and are continuous, we conclude that and is are continuous with defined by is continuous on By a similar argument, we also show that addition to the continuity of C, is continuous on defined by (13) is continuous on Therefore, the gap function In defined by is continuous on This completes the proof □ https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true 8/17 08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment Now, we establish the regularized gap function of Moreau-Yosida type introduced in [3,6,13] of the gap functions and for the problem SWQVIP For any , and , the function is defined by (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 i For any Since for all for all Thus, for all ii Let Then and so Since and This implies that , we conclude that Conversely, let Then, for each , and , by condition (8)(c) we have is a gap function, Thus, we conclude that By a similar argument, we verify that and , i.e , there are and such that (22) https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true 9/17 08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment Hence we have as In particular, we have are continuous, we conclude that Therefore, we have and , as and Since This and implies □ Next, we establish error bounds for the problem SWQVIP based on the regularized gap functions studied above Theorem 3.4 For each and satisfy then, for any , suppose that all conditions (4)–(10) hold If, for each 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) for all and 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 https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true 10/17 08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment or (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 , we have with (30) Proof From Theorem 3.4, we get that https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true 11/17 08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment Hence we have This completes the proof □ Now, we give the following example to illustrate the main results established above Example 3.1 Let and , For each , let , be fuzzy mappings defined by https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true 12/17 08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment and , Then we have By a similar argument, we obtain , Let be the mappings defined by for all Let , and and , be the mappings defined by and for all 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 , for any and and and , we have and so Hence is a gap function for SWQVIP We also have https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true 13/17 08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment 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 , Then, for which implies that (23) holds Next, if and , , we have , , and then we get Thus the inequality (30) holds By similar arguments, we also verify that the inequality (30) holds in the case The following example show that Theorems 3.4 and 3.5 are applicable without using the assumption of strong monotonicity Example 3.2 For each , let 4, Let be the mappings defined by , , C as in Example 3.1 https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true 14/17 08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment for all Then both SWQVIP and (SMVIP) reduce to the problem of finding and such that solves It follows from a direct computation that , and assumptions of Theorems 3.4 and 3.5 are fulfilled However, the functions and are not strongly monotone with modulus all such that Then, we have Hence, such that is not strongly monotone with any modulus It is easy to see that the Indeed, for example, for 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 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https://elksslsfdafs.casb.nju.edu.cn:4443/doi/epub/10.1080/00036811.2021.2008374?needAccess=true 17/17 ...08:57, 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment inequality problems with fuzzy mappings in abstract spaces Based... 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment variational inclusion problems and general split equality optimization problems. .. 31/03/2022 Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment for all and From Lemma 2.1(iii), we have for all and This implies that for all