Existence of solutions for generalized nonlinear vector quasi-variational-like inequalities with set-valued mappings

Husain, Shamshad; Gupta, Sanjeev

doi:10.2298/fil1205909h

Cited by 8 publications

(3 citation statements)

References 16 publications

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“…(3) For more details on variational inequalities and quasi-variational inequalities, completely generalized multi-valued co-variational inequalities, generalized nonlinear vector quasi-variational-like inequalities with set-valued mappings and others, refer to [1], [16] and [17].…”

Section: Existence Theorems For Generalized Bi-complementarity Problementioning

confidence: 99%

Generalized bi-quasi-variational inequalities for quasi-pseudo-monotone type II operators in non-compact settings

Chowdhury

Cho

2016

Filomat

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In this paper, we introduce a new class of generalized bi-quasi-variational inequalities for quasipseudo-monotone type II operators in non-compact settings of locally convex Hausdorff topological vector spaces and show the existence results of solutions for generalized bi-quasi-variational inequalities. Our results improve, extend and generalized the corresponding results given by some authors. The generalized bi-quasi-variational inequality problem was first introduced by Shih and Tan [19] in 1989. The following is the definition due to Shih and Tan [19]. Definition 1.1. Let E and F be vector spaces over Φ, •, • : F × E → Φ be a bilinear functional and be X a nonempty subset of E. If S : X → 2 X and M, T : X → 2 F are set-valued mappings, then the generalized bi-quasi variational inequality problem for the triple (S, M, T) is to findŷ ∈ X satisfying the following properties: (1)ŷ ∈ S(ŷ); (2) inf w∈T(ŷ) Re f − w,ŷ − x ≤ 0 for all x ∈ S(ŷ) and f ∈ M(ŷ). In this definition, for any nonempty set X, 2 X denote the class or family of all nonempty subsets of X. Also, we use F (X) to denote the family of all nonempty finite subsets of X. Moreover, throughout this paper, Φ denotes either the real field R or the complex field C.

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Section: Existence Theorems For Generalized Bi-complementarity Problementioning

confidence: 99%

Generalized bi-quasi-variational inequalities for quasi-pseudo-monotone type II operators in non-compact settings

Chowdhury

Cho

2016

Filomat

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“…In recent years, the system of generalized vector quasivariational-like inequality, which is a unified model for the system of vector quasi-variational-like inequalities, the system of vector variational-like inequalities, the system of vector variational inequalities, the system of vector equilibrium problems and the system of variational inequalities etc., has been studied (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein).…”

Section: A Andmentioning

confidence: 99%

An Existence Theorem of Solutions for the System of Generalized Vector Quasi-Variational-Like Inequalities

Husain¹,

Gupta²,

Mishra³

2013

AJOR

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In this paper, we introduce and study the system of generalized vector quasi-variational-like inequalities in Hausdorff topological vector spaces, which include the system of vector quasi-variational-like inequalities, the system of vector variational-like inequalities, the system of vector quasi-variational inequalities, and several other systems as special cases. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the g-diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for the system of generalized vector quasi-variational-like inequalities. The results of this paper can be seen as extensions and generalizations of several known results in the literature.

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“…(see [5,33] and the reference therein for more on the connection between quasi-variational inequalities and Nash equilibria). For more on the existence theory of solutions to quasi-variational and quasi-variational-like inequalities we refer the reader to [16,23,26,27,34,35,39,40] and the references therein. For methods of solving general quasi-variational inequalities, we refer to [32] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Nonsmooth quasi-variational-like inequalities with applications to nonsmooth vector quasi-optimization

Alshahrani

2017

Filomat

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We consider nonsmooth vector quasi-variational-like inequalities and nonsmooth vector quasioptimization problems. We utilize the method of scalarization to define nonsmooth quasi-variational-like inequalities by means of Clarke generalized directional derivative. We then study their relations with the problem of vector quasi-optimization and its scalarized version. Under the assumption of pseudomonotonicity and then densely pseudomonotonicity, we present some existence results for solutions to nonsmooth quasi-variational-like inequalities. To the best of our knowledge, the results we obtained are new in the sense of utilizing the scalarization method.

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Existence of solutions for generalized nonlinear vector quasi-variational-like inequalities with set-valued mappings

Cited by 8 publications

References 16 publications

Generalized bi-quasi-variational inequalities for quasi-pseudo-monotone type II operators in non-compact settings

Generalized bi-quasi-variational inequalities for quasi-pseudo-monotone type II operators in non-compact settings

An Existence Theorem of Solutions for the System of Generalized Vector Quasi-Variational-Like Inequalities

Nonsmooth quasi-variational-like inequalities with applications to nonsmooth vector quasi-optimization

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