Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces

Syed Shakaib Irfan,Mijanur Rahaman,Rais Ahmad,Iqbal Ahmad,Saddam Husain

doi:10.3390/math7040345

Syed Shakaib Irfan, Mijanur Rahaman + Show 3 more

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https://doi.org/10.3390/math7040345

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Journal: Mathematics	Publication Date: Apr 10, 2019
Citations: 2	License type: CC BY 4.0

Affiliation: Qassim University, Aligarh Muslim University

Abstract

The aim of this article is to study new types of generalized nonsmooth exponential type vector variational-like inequality problems involving Mordukhovich limiting subdifferential operator. We establish some relationships between generalized nonsmooth exponential type vector variational-like inequality problems and vector optimization problems under some invexity assumptions. The celebrated Fan-KKM theorem is used to obtain the existence of solution of generalized nonsmooth exponential-type vector variational like inequality problems. In support of our main result, some examples are given. Our results presented in this article improve, extend, and generalize some known results offer in the literature.

Highlights

The vector variational inequality has been introduced and studied in [1] in finite-dimensionalEuclidean spaces
We introduce generalized nonsmooth exponential-type vector variational like inequality problems involving Mordukhovich limiting subdifferential in Asplund spaces
Suppose that K 6= ∅ is a subset of Asplund space X, C = Rn+ and f = ( f 1, f 2, · · ·, f n ) : K −→

Summary

Introduction

The vector variational inequality has been introduced and studied in [1] in finite-dimensionalEuclidean spaces. The vector variational inequality has been introduced and studied in [1] in finite-dimensional. Vector variational inequalities have emerged as an efficient tool to provide imperative requirements for the solution of vector optimization problems. For more details on vector variational inequalities and their generalizations, see the references [2,3,4,5,6,7,8]. In 1998, Giannessi [9] proved a necessary and sufficient condition for the existence of an efficient solution of a vector optimization problem for differentiable and convex mappings by using a Minty type vector variational inequality problem. Many researchers have studied vector optimization problems by using different types of Minty type vector variational inequality problems. Yang et al [8]

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