Common Fixed Point Theorems for Conversely Commuting Mappings Using Implicit Relations

The object of this paper is to utilize the notion of conversely commuting mappings due to Lü (2002) and prove some common fixed point theorems in Menger spaces via implicit relations. We give some examples which demonstrate the validity of the hypotheses and degree of generality of our main results. 1. Introduction In 1986, Jungck [1] introduced the notion of compatible mappings in metric space. Most of the common fixed point theorems for contraction mappings invariably require a compatibility condition besides continuity of at least one of the mappings. Later on, Jungck and Rhoades [2] studied the notion of weakly compatible mappings and utilized it as a tool to improve commutativity conditions in common fixed point theorems. Many mathematicians proved several fixed point results in Menger spaces (see, e.g., [3–9]). In 2002, Lü [10] presented the concept of the converse commuting mappings as a reverse process of weakly compatible mappings and proved common fixed point theorems for single-valued mappings in metric spaces (also see [11]). Recently, Pathak and Verma [12, 13], Chugh et al. [14], and Chauhan et al. [15] proved some interesting common fixed point theorems for converse commuting mappings. In this paper, we prove some unique common fixed point theorems for two pairs of converse commuting mappings in Menger spaces by using implicit relations. 2. Preliminaries Definition 1 (see [16]). A？？ -norm is a function satisfying(T1) ,？？ ;(T2) ;(T3) for ,？？ ;(T4) for all ,？？ ,？？ in . Examples of -norms are , , and . Definition 2 (see [16]). A real valued function on the set of real numbers is called a distribution function if it is nondecreasing, left continuous with and . We shall denote by the set of all distribution functions defined on , while will always denote the specific distribution function defined by If is a nonempty set, is called a probabilistic distance on and the value of at is represented by . Definition 3 (see [17]). A probabilistic metric space is an ordered pair , where is a nonempty set of elements and is a probabilistic distance satisfying the following conditions: for all and , (1) for all if and only if ;(2) ;(3) ;(4)if and , then for all and . Every metric space can always be realized as a probabilistic metric space by considering defined by for all and . So probabilistic metric spaces offer a wider framework than that of metric spaces and are better suited to cover even wider statistical situations; that is, every metric space can be regarded as a probabilistic metric space of a special kind. Definition 4 (see [16]). A Menger space