A Study on Fixed Point Iterative Procedures

Abstract

Fixed point theory has fascinated hundreds of researchers since 1922 with the celebrated Banach’s fixed point theorem. Fixed point iterative procedures are one of the early achievements of fixed point theory for their usefulness to construct the solving technique of different nonlinear problems. Most of the physical problems of applied sciences and engineering are usually formulated as functional equations. Such equations can be written in the form of fixed point equations in an easy manner. It is always desired to develop an iterative procedure which approximates the solution of these equations in fewer numbers of steps. From this point of view, the main objective of our research is to fit a best fixed point iterative procedure whose working ability (rate of convergence) is better than that of the analogous fixed point iterative procedures. There exist a numeral number of fixed point iterative procedures in literature. But there raised a natural question that, “Which is the best fixed point iterative procedure under the equivalent situation?”. To find the answer of that question already many works have been completed by various renowned researchers; see for instance [12, 27, 37, 39, 47, 49] and their references. By the inspiration of these works here we have proposed a new three-step fixed point iterative procedure whose rate of convergence is better than that of analogous fixed point iterative procedures in case of contraction mapping. Using our new fixed point iterative procedure we have also established some weak and strong convergence theorems for non-expansive mapping and we apply these results to find the solutions of constrained minimization problems and feasibility problems. In the last part of our research, we have studied the fixed point iterative procedures with errors and proveda convergence theorem of multi-step Noor fixed point iterative procedure with errors for Zamfirescu operator, which generates the convergence theorems of rest fixed point iterative procedures with errors for the same operator.