Approximating Solutions of Fixed Point, Variational Inequality and Hammerstein Type Equation Problems

Abstract

Many physical problems in real life can be modeled as nonlinear problems. In general, finding the exact solutions of many nonlinear problems is difficult. Thus, development of different approximation methods for solving nonlinear problems has gained significant interest once the existence of these solutions has been established. In this thesis, we construct different iterative algorithms to approximate solutions of such non-linear problems. First, we introduce an iterative algorithm which solves the split equality fixed point problem involving uniformly continuous pseudo contractive mappings in the real Hilbert space settings. We also introduce a method for approximating solutions of split equality common f, g−fixed point problems involving uniformly continuous f,g−pseudocontractive mappings in reflexive real Banach spaces. We discuss their convergence analysis and give numerical examples to support each algorithm’s theoretical formulation. Next, we introduce the split equality Minty variational inequality problems and construct an iterative algorithm for approximating its solution, where the underlying mappings are Lipschitz quasimonotone in reflexive real Banach spaces and discuss its strong convergence analysis. Then, we construct an iterative algorithm for solving bilevel variational inequality problem with fixed point constraint, where the mapping in the lower level variational inequality problem is uniformly continuous pseudomonotone and the mapping for the fixed point problem is a general demimetric mapping. Moreover, we introduce a method which approximates solutions of the split equality of variational inequality and f, g− fixed point problems in reflexive real Banach spaces and discuss its convergence analysis. We prove strong convergence theorems and provide numerical examples for each algorithm. Finally, we introduce the split equality Hammerstein type equation problem and propose meth-ods for approximating its solution. We construct a method for approximating its solutions, where the underlying mappings are uniformly continuous monotone in reflexive real Banach spaces and its strong convergence analysis is discussed. Moreover, we introduce an iterative algorithm which involves continuous monotone mappings for solving split equality Hammerstein type equation problems in reflexive real Banach spaces. We discuss its convergence analysis and give examples for each algorithm to reveal their effectiveness.