Abstract
We introduce and study variational-like inequalities for generalized pseudomonotone set-valued mappings in Banach spaces. By using KKM technique, we obtain the existence of solutions for variational-like inequalities for generalized pseudomonotone set-valued mappings in reflexive Banach spaces. The results presented in this paper are generalizations and improvements of the several well-known results in the literature.
1. Introduction and Preliminaries
Variational inequality theory plays an important role in many fields, such as optimal control, mechanics, economics, transportation equilibrium, and engineering science. It is well known that monotonicity plays an important role in the study of variational inequality theory. In recent years, a number of authors have proposed many generalizations of monotonicity such as pseudomonotonicity, relaxed monotonicity, relaxed - monotonicity, quasimonotonicity, and semimonotonicity, -monotonicity. For details, refer to [1ā11] and the references therein.
Verma [11] studied a class of variational inequalities with relaxed monotone operators. In 2003, Fang and Huang [4] introduced a new concept of relaxed - monotonicity and obtained the existence of solution for variational-like inequalities in reflexive Banach spaces. B. S. Lee and B. D. Lee [12] defined weakly relaxed -semipseudomonotone set-valued variational-like inequalities and generalize the result of Fang and Huang [4]. Bai et al. [1] defined relaxed --pseudomonotone concepts for single valued mappings. For set-valued mappings, Kang et al. [13] defined relaxed - pseudomonotone concepts which generalize monotone concepts for single valued mapping in Fang and Huang [4] and Bai et al. [1]. Recently Sintunavarat [14] established the existence of solution of mixed equilibrium problem with the weakly relaxed -monotone bifunction in Banach spaces.
In 2013, Kutbi and Sintunavarat [15] introduce two new concepts of weakly relaxed - monotone mappings and weakly relaxed - semimonotone mappings and obtained the existence of solution for variational-like inequality problems in reflexive Banach spaces.
Inspired and motivated by the results of Fang and Huang [4] and Kutbi and Sintunavarat [15], in this paper, we introduce the concept of weakly relaxed pseudomonotone mapping and by using Knaster Kuratowski Mazurkiewicz (KKM) technique [16], we study some existence of solution for variational-like inequality for set-valued pseudomonotone mapping.
In this paper, we suppose that is a reflexive Banach space with dual space , and denotes the pairing between and . Let be a nonempty closed convex subset of and denote the family of all the nonempty subset of .
The following definitions and results will be useful in our work.
Definition 1. A mapping is said to be weakly relaxed pseudomonotone if there exist a mapping and functions , with for , where is a function with , such that implying
Remark 2. (i) If in Definition 1 then we have the following pseudomonotone concept defined in Kang et al. [13]: implying (ii) If is single valued mappings, , and for , then we have the following relaxed - monotone concepts defined in Fang and Huang [4] and the following - pseudomonotone concepts, defined in Bai et al. [1]:(a)For any (b)For any
Definition 3 (see [17]). Let and be two mappings; is said to be -hemicontinuous for any , if the mapping defined by defined byis continuous at .
Definition 4 (see [4]). Let and be two mappings and be a proper functional. Then is said to be -coercive with respect to first argument of , if there exists such that whenever , for all .
If then there exists such that whenever , for all .
If , where is the indicator function of , then Definition 4 coincides with the definition of -coercivity in the sense of Yang and Chen [18].
Definition 5. A multivalued mapping is said to be relaxed - monotone if there exists a function and with for all , and such that
Remark 6. (i) If is single valued then (10) becomes and then is said to be relaxed - monotone [4].
(ii) If is single valued and then (10) becomes and then is called relaxed -monotone [4].
(iii) If for all and , where and , then (12) becomes and is called -monotone [5, 10, 11].
Definition 7. A mapping is said to be weakly relaxed - monotone if there exists a function and withfor all and such that
Remark 8. If is single valued mapping then Definition 7 reduces to Definition 9 of [15].
Remark 9. If is weakly relaxed - monotone, then is weakly relaxed - pseudomonotone mapping but the converse is not true.
Definition 10 (see [16]). A mapping is said to be KKM mapping if, for any , , where denote the convex hull of .
Lemma 11 (see [19]). Let be a nonempty subset of a Hausdorff topological vector space and let be a KKM mapping. If is closed in for all and compact for some , then
2. Existence Results
In this section, we discuss the existence of the following variational-like inequality: where is a nonempty closed convex subset of a reflexive Banach space .
Theorem 12. Suppose that is -hemicontinuous and weakly relaxed - pseudomonotone mapping. Let be a proper convex function and be a mapping. Suppose that the following conditions hold:(i)(ii)For any fixed , , the mapping is convex.(iii) and are convex.Then problems (16) and (17) are equivalent as follows:
Proof. Suppose that (16) has a solution. So there exist Since is weakly relaxed - pseudomonotone, we have Therefore is a solution of (17).
Conversely, suppose that is a solution of (17) and is any point with . From (17) we know that . For let ; then we have . Since is a solution of problem (17), it follows that The convexity of and condition (ii) of Theorem 12 imply that It follows from (21) that for all and . Taking in the previous inequality and using -hemicontinuity of , we get for all and with . In case of the relation is trivial. Therefore is solution of (16).
Theorem 13. Let be a nonempty bounded closed convex subset of a real reflexive Banach space and the dual space of . Suppose that is an -hemicontinuous and weakly relaxed - pseudomonotone mapping. Let be a proper convex lower semicontinuous function and be a mapping. Assume that(i),(ii)for any fixed and the mapping is convex and lower semicontinuous function,(iii) and are convex and lower semicontinuous,(iv) is weakly lower semicontinuous; that is, for any net , converges to in implying that .Then problem (17) is solvable.
Proof. Define two set-valued mappings as follows: We claim that is a KKM mapping. If is not a KKM mapping, then there exist such that This implies that there exist such that , where , and , but . From the definition of , we have and it follows that which is a contradiction. This implies that is a KKM mapping. Now we prove that , for all
For any given and letting , we have Since is weakly relaxed - pseudomonotone, we get It follows that and so .
This implies that is also KKM mapping. From the assumption, we know that is weakly closed for all . In fact since and are two convex lower semicontinuous functions, from the definition of and weakly semilower continuity of , it is easy to see that is weakly closed for all . Since is bounded closed and convex, we know that is weakly compact and so is weakly compact in for each in . From Lemma 11 and Theorem 12, we obtain that Hence, there exists such that that is, problem (16) has a solution.
Theorem 14. Let be a nonempty unbounded closed convex subset of a real reflexive Banach space and the dual space of . Suppose that is an -hemicontinuous and weakly relaxed - pseudomonotone mapping. Let be a proper convex lower semicontinuous function and be a mapping. Assume that(i),(ii)for any fixed and the mapping is convex and lower semicontinuous function,(iii) and are convex and lower semicontinuous,(iv) is weakly lower semicontinuous,(v) is -coercive with respect to ; that is, there exist such that āwhenever .Then problem (16) is solvable.
Proof. LetConsider the following problem, , such that By Theorem 13, we know that (26) has a solution ; choose with as in the coercivity conditions. Then we have MoreoverNow if for all , we may choose large enough so that the above inequality and -coercivity of with respect to imply that which contradicts Hence there exist such that . For any , we can choose small enough so that and It follows from (ii) that By the assumption of , we have So is a solution of (16).
It is easy to see that weakly relaxed - monotonicity implies weakly relaxed - pseudomonotonicity. So Theorems 12, 13, and 14 are deduced to the following corollaries.
Corollary 15. Suppose that is -hemicontinuous and weakly relaxed - monotone mapping. Let be a proper convex function and be a mapping. Suppose that the following conditions hold:(i)(ii)For any fixed , , the mapping is convex.(iii) and are convex.Then problems and are equivalent as follows:
Corollary 16. Let be a nonempty bounded closed convex subset of a real reflexive Banach space and the dual space of . Suppose that is an -hemicontinuous and weakly relaxed - monotone mapping. Let be a proper convex lower semicontinuous function and be a mapping. Assume that(i),(ii)for any fixed and the mapping is convex and lower semicontinuous function,(iii) and are convex and lower semicontinuous,(iv) is weakly lower semicontinuous; that is, for any net , converges to in implying that .Then problem (17) is solvable.
Corollary 17. Let be a nonempty unbounded closed convex subset of a real reflexive Banach space and the dual space of . Suppose that is an -hemicontinuous and weakly relaxed - monotone mapping. Let be a proper convex lower semicontinuous function and be a mapping. Assume that(i),(ii)for any fixed and the mapping is convex and lower semicontinuous function,(iii) and are convex and lower semicontinuous,(iv) is weakly lower semicontinuous,(v) is -coercive with respect to ; that is, there exist such that āwhenever .Then problem (16) is solvable.
Remark 18. Theorems 12, 13, and 14, improve Theorems 11, 12, and 15 of results of Kutbi and Sintunavarat [15] and also Fang and Huang [4]. These results are also the extensions of the known results of Bai et al. [1] and Hartman and Stampacchia [7] and corresponding results of Goeleven and Motreanu [5], B. S. Lee and B. D. Lee [12], Siddiqi et al. [8], and Verma [9].
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.