Approximate Coincidence Point of Two Nonlinear Mappings

We study the approximate coincidence point of two nonlinear functions introduced by Geraghty in 1973 and Mizoguchi and Takahashi ( -function) in 1989. 1. Introduction Fixed point theory has been an important tool for solving various problems in nonlinear functional analysis as well as a useful tool for proving the existence theorems for nonlinear differential and integral equations. However, in many practical situations, the conditions in the fixed point theorems are too strong and so the existence of a fixed point is not guaranteed. In that situation, one can consider nearly fixed points what we call as approximate fixed points. By an approximate fixed point of a function we mean in a sense that is “near to” . The study of approximate fixed point of a function we mean in a sense that is “near to” . The study of approximate fixed point theorems is equally interesting to that of fixed point theorems. Motivated by the article of Tijs et al. [1], Berinde [2] established some fundamental approximate fixed point theorems in metric space. In a recent paper, Dey and Saha [3] studied the existence of approximate fixed point for the Reich operator [4] which in turn generalizes the results of Berinde [2]. Coincidence point theory has a vast literature, and many generalizations have been emerged so far (see [5–11]). The aim of this paper is to define approximate coincidence point for a pair of single valued self-mappings to obtain some important results on approximate coincidence point using two nonlinear functions by Geraghty [12] in 1973 and Mizoguchi and Takahashi [13] ( -function) in 1989. 2. Approximate Coincidence Point Definition 1. Let be a metric space and , , . Then is an -fixed point (approximate fixed point) of if . The set of all -fixed points of , for a given , is denoted by Definition 2. Let . Then has the approximate fixed point property if Definition 3. Let be a metric space, and let？？ be two single valued maps. The maps and are said to have coincidence point if say, is called a point of coincidence of and . If , then is called a common fixed point of and . 3. Approximate Coincidence Point Results for Two Nonlinear Maps In this section, we establish existence of some results concerning approximate coincidence point for various types of nonlinear contractive maps in the setting of general metric spaces. For this purpose, we first define approximate coincidence point for two self-maps in metric space and prove results on approximate coincidence point using the idea of the Geraghty-type contractive condition [12]. In 1973, Geraghty [12] (see also