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Abstract In this thesis some fixed point theorems are given. The importance of metrical fixed point theory and its application to differential equations are studied. The fixed point theory for solving the functional equations depends on reformulating the functional equation: = 0 to the equivalent fixed point problem = from which, we can write the iteration form = , = 0, 1, 2,⋯ from this iteration form, we construct the sequence which in general and subject to some conditions approximates the solution. There is some questions about the convergence of this sequence. If this sequence is convergent what about its limit, is this limit satisfies the original equation what about the conditions which must be imposed and what we can say about the rate of convergence, is it possible to accelerate the convergence process! The difficulty in applying Picard fixed point depends on the choice of appropriate iteration form, the situation has more complications in differential equations because of the included integrations! These difficulties has restricted the use of Picard fixed point theorem as a method of solution. For a long time Picard method was used as a method for proving the existence or existence and uniqueness of a solution. In comparing the series solution methods with Picard method, we can treat equations with more general forms not only polynomial functions by Picard method. It is important to note that the point of solution in this case represent a function ”a point in the solution space”. In this work, we tried to introduce the basic concepts that help in the use of fixed point theorem as a method of solution as well as accelerate the convergence of series of approximations. The thesis contains summary, motivations and four chapters. Chapter One: ”Definitions and basic theorems for Contractions”, this chapter contains four sections. In section 1: introduction. In section 2: iii contraction mapping principle. In section 3: the convergence and the stability of a fixed point of a contraction operator. In section 4: the relation between Picard iteration and other iterations ”Mann and Ishikawa iterations”. Chapter two: ”Successive approximations of differential and integral equations” the chapter contains three sections. In section 1: Introduction. In section 2: the fundamental theorems of existence and uniqueness. In section 3: Construction of Green functions. Chapter three: ”On the Effect of Using Integrating Factors Approach with Picard Iteration”, the chapter contains five sections. In section 1: Introduction. In section 2: the Gauss-Seidel modification of Picard iterations. In section 3: the integrating factor modification of Picard iteration. In section 4: the general linear two dimensional systems In section 5: differential equations with critical points. Chapter four: ”Accelerating the Modified Picard Iteration by using Green’s Function Approach”, the chapter contains four sections. In section 1: Introduction. In section 2: modification of Picard iteration with Green’s function integral, In section 3: second order non-linear equation of Bernoulli type. In section 4: Application to the Lotka- Volterra model. It is worth to mention that: - all calculations were done by the use Mathematica 6. - the results of chapter three has been accepted for publication in the Egyptian Journal of Pure and Applied Science 2011 with title ”Picard Fixed Point Iteration Combined with Integrating Factor Approach” - the content of chapter four is in the preparation for submission. |