%0 Journal Article
%T Unified Metrical Common Fixed Point Theorems in 2-Metric Spaces via an Implicit Relation
%A Sunny Chauhan
%A Mohammad Imdad
%A Calogero Vetro
%J Journal of Operators
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/186910
%X We prove some common fixed point theorems for two pairs of weakly compatible mappings in 2-metric spaces via an implicit relation. As an application to our main result, we derive Bryant's type generalized fixed point theorem for four finite families of self-mappings which can be utilized to derive common fixed point theorems involving any finite number of mappings. Our results improve and extend a host of previously known results. Moreover, we study the existence of solutions of a nonlinear integral equation. 1. Introduction and Preliminaries In 1963, G£¿hler [1] initiated the concept of 2-metric space as a natural generalization of a metric space. The topology induced by 2-metric space is called 2-metric topology which is generated by the set of all open spheres with two centers (see [2, 3]). In this course of development, Is¨¦ki [4] studied the fixed point theorems in 2-metric spaces. For more references on the recent development of common fixed point theory in 2-metric spaces, we refer readers to [5¨C19]. In metric fixed point theory, implicit relations are often utilized to cover several contraction conditions in one go rather than proving a separate theorem for each contraction condition. The first ever attempt to coin an implicit function can be traced back to Popa [20]. Recently, Popa et al. [21] proved some interesting fixed point results for weakly compatible mappings in 2-metric spaces satisfying an implicit relation. In this paper, utilizing the implicit function due to Popa et al. [21], we prove some common fixed point theorems for two pairs of weakly compatible mappings employing common limit range property. In process, many known results (especially the ones contained in Popa et al. [21, 22]) are enriched and improved. Some related results are also derived. Finally, we study the existence of solutions of a nonlinear integral equation using the presented results. A 2-metric space is a set equipped with a real valued function on which satisfies the following conditions: For distinct points , there exists a point such that . if at least two of , , are equal. , for all . , for all . The function is called a 2-metric on the set whereas the pair stands for 2-metric space. Geometrically a 2-metric represents the area of a triangle with vertices , , and . It has been known since G£¿hler [1] that a 2-metric is a nonnegative continuous function in any one of its three arguments but it does not need that to be continuous in two arguments. A 2-metric is said to be continuous if it is continuous in all of its arguments. Throughout this paper stands for a
%U http://www.hindawi.com/journals/joper/2013/186910/