الفهرس | Only 14 pages are availabe for public view |
Abstract The problem of approximating a fixed point of some operator (satises the conditions of Schauder fixed-point theorem) is of paramount importance since it corresponds to a solution of a dierential equation (satises 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 fixed 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 fixed 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. |