Results of variational inequality involving monotone operator

This thesis is an attempt to establish existence findings for certain variational inequality problems involving set-valued mappings in reflexive and non-reflexive Banach spaces. More precisely, the thesis presents new types of monotonicity in Banach space, all with the aim of obtaining results on the existence of findings of VI(ωηκ) in the case of convex subsets by using a KKM mapping, compactness and convex function. All this is done through three tasks. Task one presents a host of essential definitions along with some Theorems. The aim of this task is to help the reader to get some understanding of main results grounded in this thesis. Task two is designed to establish the existence of the solution of VI(ωηκ) by generalizing a new class of function in the setting of Banach space. Task three is intended to prove that the concept of well-posedness for VI(ωηκ) is equivalent to the existence of its solution. In so aiming, the results presented in this thesis develop and improve some corresponding solutions of several authors. The most important results we obtained are: Let K be a closed convex bounded subset of a reflexive Banach space E, and T:K→2^(E^* )be an η-H.C and (ηκαβ-p)-monotone operator with respect to the second argument of a mapping ω:K×E^*→E^* Under some assumptions, there exists at least one solution for the problem VI(ωηκ). Consider that K and A are nonempty, closed subsets of the real Banach space E and W, respectively, and that VI(ωηκ)is s-w-p in the sense of generalization, then Γ(c)≠∅,∀c,(lim)┬(c→0)⁡μ [Γ(c)]=0. Non-empty and convex subsets of finite dimensional Banach spaces are referred to as "K and A." when certain conditions are met. A well-posed issue.