Application of fixed point theorems to differential equations

In this thesis, we used the Picard Successive Iteration Method and originally primarily defined in the literatüre Modified Krasnoselskii Iteration, Modified Ishakawa Iteration, New Modified Ishakawa Iteration, Ekstra Modifed Ishakawa Iteration, New Modified Krasnoselskii Iteration Methods are given in order to solve the ordinary lineer differential equation having initial condition. Additionaly, it is shown that the accuracy of new iteration methods are substantially improved by employing variable steps which adjust themselves to the solution of the differential equation for fixed point.In the first section, the basic definitions and theorems are given.In the second section, the approximate solution methods of the first degree ordinary differantial equations such as Euler, Runga-Kutta, 4th degree Runga-Kutta and Adomian Methods are introduced. And also the definition of fixed point mapping iterations are given such as Picard, Mann, Ishakawa, Krasnoselskii, Modified Krasnoselskii Iteration, Modified Ishakawa Iteration, New Modified Ishakawa Iteration, Ekstra Modifed Ishakawa Iteration, New Modified Krasnoselskii Iteration.In the third section, the methods which is mentioned above are applied to the different examples having tables and graphies.In the last section, the new methods are compared each other and also the best method are emphisized among the others. Additionally, some suggestion are presented for further study of fixed point.