Abstract
Fixed point theory is one of the most important subjects in the setting of metric spaces since fixed point theorems can be used to determine the existence and the uniqueness of solutions of such mathematical problems. It is known that many problems in applied sciences and engineering can be formulated as functional equations. Such equations can be transferred to fixed point theorems in an easy manner. Moreover, we use the fixed point theory to prove the existence and uniqueness of solutions of such integral and differential equations. Let X be a non-empty set. A fixed point for a self-mapping Ton X is a point e is an element of X that satisfying T e = e. One of the most challenging problems in mathematics is to construct some iterations to faster the calculation or approximation of the fixed point of such problems. Some mathematicians constructed and generated some new iteration schemes to calculate or approximate the fixed point of such problems such as Mann et al. Mann (1953); Ishikawa (1974); Sintunavarat and Pitea (2016); Berinde (2004b); Agarwal, O'Regan and Sahu (2007)]. The main purpose of the present paper is to introduce and construct a new iteration scheme to calculate or approximate the fixed point within a fewer number of steps as much as we can. We prove that our iteration scheme is faster than the iteration schemes given by Sintunavarat et al. [Sintunavarat and Pitea (2016); Agarwal, O'Regan and Sahu (2007); Mann (1953); Ishikawa (1974)]. We give some numerical examples by using MATLAB to compare the efficiency and effectiveness of our iterations scheme with the efficiency of Mann et al. Mann (1953); Ishikawa (1974); Sintunavarat and Pitea (2016); Abbas and Nazir (2014); Agarwal, O'Regan and Sahu (2007)] schemes. Moreover, we introduce a problem raised from Newton's law of cooling as an application of our new iteration scheme. Also, we support our application with a numerical example and figures to illustrate the validity of our iterative scheme.