M.Sc Thesis
M.Sc Student Shemen Lital Iterative Methods for Solving Nonlinear Problems Department of Mathematics Professor Emeritus Simeon Reich

Abstract

The main objective of this thesis is to study iterative methods for solving certain nonlinear problems in Banach spaces and in the Hilbert ball. We examine, in particular, certain algorithms for approximating fixed points of nonexpansive mappings and prove strong convergence theorems for them.

Many pracrical problems can be formulated as a fixed point problem, i.e., the problem of finding the solutions of an equation x=Tx, where in our case T is a nonlinear operator. The solutions are called fixed points of T. In the case where T is a strict contraction, the classic Banach Contraction Principle asserts that there exists a unique fixed point of T, and the sequence of iterations {Tnx} converges to it. We consider the case where T is not a strict contraction, but only nonexpansive.

In the first part of the thesis we study two algorithms, one of them is implicit and the other is explicit. This algorithms have been studied in Hilbert spaces. We ask when they strongly converge to a fixed point of a nonexpansive mapping in a Banach space.

Another natural question to ask, since the mapping T is nonlinear, is whether we can drop the linearity assumption on the space. We ask whether a fixed point theory can be developed in hyperbolic spaces, for example, in Hadamard manifolds. In the second part of the thesis we consider Halpern's algorithm. This algorithm has been studied in linear spaces and in Hadamard manifolds of finite dimension. We ask if analogous results can be proved in an example of an infinite dimensional manifold, namely, the Hilbert ball. We examine the analogue of Halpern's algorithm in this space and prove a strong convergence theorem for it.