A Modified Subgradient Extragradient Method for Solving Monotone Variational Inequalities in Banach Spaces

A Modified Subgradient Extragradient Method for Solving Monotone Variational Inequalities in Banach Spaces.

ABSTRACT

The subgradient extragradient method is considered an improvement of the extragradient method for variational inequality problem for the class of monotone and Lipschitz continuous mappings in the setting of Hilbert spaces.

In this Thesis, we proposed an improved sub- gradient extragradient method for variational inequality problem for the class of monotone and Lipschitz continuous mappings in the setting of real Banach spaces.

In this chapter, we will give a general introduction on variational inequality problem and then a brief review of existing results on variational inequality.

Background of study

The contributions of this thesis falls within the general area of nonlinear functional analysis and applications, in particular, nonlinear operator theory. We are interested in finding or approximating solution(s) of a variational inequality problem for a monotone Lipschitz- continuous map on a level set of convex function in Banach spaces.

Variational Inequality Problems

The problem of finding x^∗ ∈ C such that the variational inequality (V I)  (Ax^∗, x − x^∗) ≥ 0        ∀x ∈ C,                                (1.2.1) is called the variational inequality problem (V IP ). Where C is a nonempty closed convex subset in a real Hilbert space H, A : H H  is a single valued mapping,      ., .  and    . are inner product and norm in H, respectively.

Let SOL(C, A) be the solution of the variational inequality (1.2.1). Finding a solution for V IP is a fundamental problem in optimization theory, partial differential equation, mathematical modeling, image recovery and Data processing.

It is well Known that the problem (1.2.1) is equivalent to solving the fixed point problem

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