A class of Yosida inclusion and graph convergence on Yosida approximation mapping with an application

Abstract

The proposed work is presented in two folds. The first aim is to deals with the new notion called generalized αiβj-Hp(., ., ...)-accretive mappings that is the sum of two symmetric accretive mappings. It is extension of C_n-monotone mapping introduced by Nazemi. The proximal point mapping linked with generalized generalized αiβj-Hp(., ., ...)-accretive mapping is defined and demonstrate the aspects single-valued property as well as Lipschitz continuity. The graph convergence of generalized generalized αiβj-Hp(., ., ...)-accretive mappings is discussed.

 Second aim is to introduce and study the generalized Yosida approximation mapping and Yosida inclusion problem. Next, we obtain the convergence on generalized Yosida approximation mappings by using the graph convergence of generalized generalized αiβj-Hp(., ., ...)-accretive mappings without using the convergence of its proximal-point mapping. As an application, we consider the Yosida inclusion problem in q-uniformly smooth Banach spaces and proposed an iterative scheme connected with generalized Yosida approximation mapping of generalized αiβj-Hp(., ., ...)-accretive mapping to find the solution of Yosida inclusion problem and discuss its convergence criteria with appropriate assumptions. Some examples are constructed and demonstrate few graphics for the convergence of proximal-point mapping as well as generalized Yosida approximation mapping linked with generalized generalized αiβj-Hp(., ., ...)-accretive mappings.