For a maximal monotone operator T in a Banach space an iterative solution of 0 ∈ Tx has been found through weak and strong convergence of resolvents of these operators. Identity mapping in the definition of resolvents has been replaced by the duality mapping. Solution after finite steps has also been established.

In this paper, we present an iterative method for fixed point problems and variational inequality problems. Our method is based on the so-called extragradient method and viscosity approximation method. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for monotone mapping.

In this paper, the generalized variational inequality with multi-valued mapping (GVI) is considered. To solve the problem, we first establish a global error bound estimation for GVI with the underlying multi-valued mapping being   strict monotone and Holder continuous. Based on this, we propose a new type of method to solve the GVI, and its global convergence is also established. Keywords-GVI...

We study distributed algorithms for seeking a Nash equilibrium in class of convex networked games with strongly monotone mappings. Each player has access to her own smooth local cost function and can communicate neighbors some undirected graph. To deal fast learning equilibria under such settings, we introduce so called augmented game mapping provide conditions which this is monotone. consider ...

Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization. It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood and analyzed from the alternative viewpoint of firmly nonexpansive mappings, which were found to be precisely the resolvents of monotone operators. For examp...

In this paper we prove coupled fixed point theorem for mapping having the mixed monotone property in partially ordered metric space. We give an example to support of our result.

Equilibrium problems have many uses in optimization theory and convex analysis and which is why different methods are presented for solving equilibrium problems in different spaces, such as Hilbert spaces and Banach spaces. The purpose of this paper is to provide a method for obtaining a solution to the equilibrium problem in Banach spaces. In fact, we consider a hybrid proximal point algorithm...

and Applied Analysis 3 2. Preliminaries Now, we list some of the knowledge we need in the sequel. Let X be a real Banach space with a strictly convex dual space X∗. We use ·, · to denote the generalized duality pairing between X and X∗. We use “→ ” to denote strong convergence. Let “X ↪→ Y” denote the space X embedded continuously in space Y . For any subset G of X, we denote by intG its interi...