On generalized vector variational-like inequalities
Introduction
Giannessi [5] was the first to introduce the vector variational inequality in a finite-dimensional Euclidean space, which is the vector-valued version of the variational inequality of Hartman and Stampacchia [8]. Later on, many authors see [2], [3], [6], [7], [10], [12], [13], [19] have investigated vector variational inequalities in abstract spaces.
On the other hand, Minty's lemma [1], [9] has been shown to be an important tool in the variational field including variational inequality problems, obstacle problems, confined plasmas, free boundary problems, and stochastic optimal control problems when the operator is monotone and the domain is convex.
In 1999, Lee et al. [15] obtained a vector version of Minty's lemma using Nadler's result [16], and with their result they considered two kinds of vector variational-like inequalities for multifunctions under certain new pseudomonotonicity condition and certain new hemicontinuity condition, respectively, different from the conditions that in [11], [14], [17].
In this paper, a vector version of Minty's lemma is obtained and two kinds of vector variational-like inequalities for multifunctions are considered. We show the existence of solutions to a kind of vector variational like inequality for a multifunction under certain pseudomonotonic condition different from that in [11], [14], [17].
By using the vector version of Minty's lemma and the vector variational-like inequality, we prove the existence of solutions for another type of vector variational-like inequality for a compact-valued multifunction under certain hemicontinuity condition different from that in [11], [14], [17]. Further, in last section, an appropriate fixed point theorem is used to obtain an existence result of solution for vector variational-like inequality problems.
Section snippets
Preliminaries
In this section, we give some definitions and results, which are essential for our main results. Definition 2.1 Let be a subset of a topological vector space . Then a multifunction : 2 is called KKM if for each nonempty finite subset of , , where Co denotes the convex hull and . Definition 2.2 A multivalued mapping is called convex if for any , , Lemma 2.1 Let D be an arbitrary nonempty subset of a Hausdorff topological vector space. Let the multivaluedYuan [20]
Fan [4]
Main results
In this section, we state and prove the following vector version Minty's lemma under the conditions different from that in [2], [3], [10], [19]. Theorem 3.1 Let X and Y be real Banach space, D be a nonempty closed convex subset of X, and a family of closed convex solid cones of Y. Let be a nonempty compact-valued multifunction such that for any . where H is Hausdorff metric defined on and an operator, suppose that the following conditions
Fixed point theorem
In this section, we used the vector version Minty lemma and an appropriate fixed point theorem to obtain an existence results of solution for a vector variational-like inequality. Theorem 4.1 Let D be a nonempty closed and convex subset of real Banach space X. Let be a nonempty compact-valued multifunction such that for any , where H is a Hausdorff metric defined on an an operator, suppose that the following hold: is a continuous
References (20)
- et al.
On the generalized vector variational inequality problem
J. Math. Anal. Appl.
(1997) - et al.
Existence of solutions for vector optimization problems
J. Math. Anal. Appl.
(1998) - et al.
A vector version of Minty's Lemma and Application
Appl. Math. Lett.
(1999) A fixed point theorem equivalent to the Fan–Knaster–Kuratowski–Mazurkiewicz theorem
J. Math. Anal. Appl.
(1987)- et al.
Variational and Quasi-variational Inequalities, Applications to Free Boundary Problems
(1984) Existence of solution for a vector variational inequalityan extension of the Hartman–Stampacchia theorem
J. Optim. Theory Appl.
(1992)- et al.
The vector complementarily problem and its equivalence with the weak minimal element in ordered sets
J. Math. Anal. Appl.
(1990) A generalization of Tychonoff's fixed point theorem
Math. Ann.
(1961)Theorems of alternative, quadratic programmes and complementarity problems
- F. Giannessi, On connections among separation, penalization and regularization for variational inequalities with...
Cited by (28)
Existence of vector quasi-variational-like inequalities for fuzzy mappings
2013, Fuzzy Sets and SystemsExistence of solutions for generalized vector quasi-variational-like inequalities without monotonicity
2009, Computers and Mathematics with ApplicationsOn generalized implicit vector equilibrium problems in Banach spaces
2009, Computers and Mathematics with ApplicationsCitation Excerpt :The vector-valued version of the variational inequality of Hartman and Stampacchia (i.e., the vector variational inequality) was first introduced and studied by Giannessi [1] in a finite-dimensional Euclidean space in 1980. Later on, vector variational inequalities were investigated by many authors in abstract spaces, and extended to vector equilibrium problems, which include as special cases various problems, for example, vector complementarity problems, vector optimization problems, abstract economical equilibria and saddle-point problems; see [2–17]. In 1999, Lee et al. [12] first established a vector version of Minty’s lemma [18] by using Nadler’s result [19].
Generalized vector variational-type inequalities
2008, Computers and Mathematics with ApplicationsCitation Excerpt :A variational-like inequality, which is useful and important generalization of the ordinary variational inequality has been extensively studied by many authors, see [9–11] and the references therein. The problem of vector variational-like inequalities is also one of the recent generalization of vector variational inequalities studied by many authors, see [12–15] and the references therein. On the other hand, recently in [16], the authors considered scalar variational-type inequalities for pseudomonotone-type set-valued mappings in nonreflexive Banach spaces.
Existence of solutions for generalized implicit vector variational-like inequalities
2007, Nonlinear Analysis, Theory, Methods and ApplicationsAn existence result for generalized vector equilibrium problems without pseudomonotonicity
2006, Applied Mathematics LettersCitation Excerpt :By using their result they considered vector variational-like inequalities for multifunctions under certain new pseudomonotonicity conditions and certain new hemicontinuity conditions. Subsequently Khan and Salahuddin [8] also established a vector version of Minty’s lemma and applied it to obtain an existence theorem for a class of vector variational-like inequalities involving a compact-valued multifunction under certain similar pseudomonotonicity conditions and similar hemicontinuity conditions. On the other hand, the generalized vector equilibrium problems have been extensively studied by many authors (see, e.g., [1–3,6,10,14]) since they include as special cases generalized vector variational inequality problems, generalized vector variational-like inequality problems, generalized vector complementarity problems and vector equilibrium problems.