Some fixed point theorems for a special case of the Ullah and Arshad iteration method

In this study, it is shown that the iterative sequence which is a simplified form of the iteration method introduced by Ullah and Arshad (Springer Plus (2016) 5(1), 1616), converges strongly to the solution of a nonlinear Volterra integral equation with delay. Also, it is proved the result of a data dependence for the solution of this integral equation. This thesis consists of five sections. Firstly, introductory section which it is introduced of problem in thesis is given. Afterwards, basic definitions and concepts, which are used in the study, are given in the second section, named as 'Basic Concepts'. In the third section, integral equations are introduced at first and the classification of integral equations is given. Then, it is discussed on the concept of fixed point and Banach Fixed Point Theorem is given. Finally, some iteration methods are examined. Some of them are Picard iteration method, Mann iteration method, Ishikawa iteration method, Noor iteration method, Picard-S iteration method, Vatan two-step iteration method and Ullah and Arshad iteration method. Third section is finished with a lemma to be used in research findings. In the fourth section, the strong convergence theorem of a iteration method to the solution of nonlinear Volterra integral equation is proven and the data dependence of the solution of this integral equation is researched. Then, an example is presented to support the results.