On the convergence of the sequence of iterates

Abstract

The main objective of this thesis is to study the convergence of sequences of various iterates to a fixed point in certain Banach spaces. -- The preliminaries of metric spaces, normed linear spaces, and basic fixed point concepts are covered in the first chapter. -- In the second chapter, the first section gives a brief statement of the problem of convergence of sequences of iterates. -- The second section traces the development of some of the more important results on the convergence of sequences of iterates. The concept of a densifying mapping is then introduced and a similar development of theorems for these mappings is given. -- The last section covers a more general sequence of iterates. An extension of a theorem by Senter [19] is presented and a corrected version of his original theorem is given. The corrected version is then shown to be a corollary of the new theorem. It is also shown that a condition Z of Rhoades [17] is a subset of one of the conditions of the new theorem. The final theorem in this section is an extension of one by Petryshyn and Williamson [15] to this more general sequence of iterates.