Chapter 4 - Viscosity Methods for Some Applied Nonlinear Analysis Problems

Abstract

The aim of this chapter is to present a useful viscosity approach for solving some applied nonlinear analysis problems which were solved through translations into split fixed point problems. The underlying principle is to operate a certain nonlinear quasi-nonexpansive mapping iteratively and generate a convergent sequence to its fixed point. However, such a mapping often has infinitely many fixed points, meaning that a selection from the fixed point set should be of great importance. Nevertheless, most fixed point methods can only return an unspecified solution. Therefore, based on a viscosity idea used in solving optimization problems which are not well posed (namely, to consider a family of regularized problems such that each of them is well posed, then a viscosity approximation method is applied to seek a particular solution of the original problem as a limit of the solutions of the regularized problems), we accomplish this challenging task by viscosity approximation methods and obtain a particular solution from the fixed point set that solves a variational inequality criterion. We present the mathematical idea of the proposed approach in the context of fixed point theory together with examples in linear programming, semicoercive elliptic problems and in the area of finance as well as elegant techniques of analysis to overcome the no-Fejer monotonicity of the generated sequences. Split common fixed point problems are then investigated, the convergence is studied with different step sizes, convergence results in some specific cases when the projections onto the involved sets are not available are proposed and numerical experiments on an inverse heat equation are provided. Special attention is also given to split equilibrium problems by making a link with the hybrid steepest descent approach. We then conclude with a reference to viscosity methods for hierarchical fixed point problems which permit the design of many powerful schemes with the strong support of both viscosity approximation and the Mann iterative process.