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Abstract
The problem of approximating a xed point of some operator (satis
es the conditions of Schauder xed-point theorem) is of paramount
importance since it corresponds to a solution of a di erential equation
(satis es Peano local existence theorem) which is needed for numerous
applications in science.
This Ph. D. thesis is organized as follows:
1. In chapter #1, we introduced a brief history and a motivation
for the problem of approximating a xed point;\assuming that it
exists”; for:
self maps in Banach and Hilbert spaces.
non-self maps in Banach spaces and how that problem was
tackled by using the notions of both the metric and the generalized
projection operators.
2. In chapter #2, we explore almost of the details needed in this
thesis and it contains an exposition of the most important de -
nitions, examples, theorems and results obtained in various real
Banach spaces. We also prove some basic lemmas that will be
used in the sequel and study several properties and formulas of
the metric and generalized projection operators.
3. In chapter #3, we study iterative methods for approximating xed
points of new classes of nonlinear operators recently introduced
by Y. Alber [3, 6] and C. Chidume [20]. Then, assuming the
existence of xed points for maps in these classes of operators,
and using several results of Alber and Guerre-Delabriere [6], we
prove convergence theorems with estimates of convergence rates.