On -Tupled Coincidence Point Results in Metric Spaces

We prove some n-tupled coincidence point results whenever n is even. We give here several new definitions like n-tupled fixed point, n-tupled coincidence point, and so forth. The main result is supported with the aid of an illustrative example. 1. Introduction and Preliminaries The classical Banach Contraction Principle proved in complete metric spaces continues to be an indispensable and effective tool in theory as well as applications which guarantees the existence and uniqueness of fixed points of contraction self-mappings besides offering a constructive procedure to compute the fixed point of the underlying map. There already exists an extensive literature on this topic, but keeping in view the relevance of this paper, we merely refer to [1–14]. In 2006, Gnana Bhaskar and Lakshmikantham initiated the idea of coupled fixed point in partially ordered metric spaces and proved some interesting coupled fixed point theorems for mapping satisfying a mixed monotone property. In recent years, many authors obtained important coupled fixed point theorems (e.g., [15–20]). In this continuation, Lakshmikantham and Ciri？ [21] proved coupled common fixed point theorems for nonlinear -contraction mappings in partially ordered complete metric spaces which indeed generalize the corresponding fixed point theorems contained in Gnana Bhaskar and Lakshmikantham [22]. As usual, this section is devoted to preliminaries which include basic definitions and results on coupled fixed point for nonlinear contraction mappings defined on partially ordered complete metric spaces. In Section 2, we introduce the concepts of -tupled coincidence point and -tupled fixed point for mappings satisfying different contractive conditions and utilize these two definitions to obtain -tupled coincidence point theorems for nonlinear -contraction mappings in partially ordered complete metric spaces. Now, we present some basic notions and results related to coupled fixed point in metric spaces. Definition 1 (see [22]). Let be a partially ordered set equipped with a metric such that is a metric space. Further, equip the product space with the following partial ordering: Definition 2 (see [22]). Let be a partially ordered set and . One says that enjoys the mixed monotone property if is monotonically nondecreasing in and monotonically nonincreasing in ; that is, for any , Definition 3 (see [22]). Let be a partially ordered set and . One says that is a coupled fixed point of the mapping if Theorem 4 (see [22]). Let be a partially ordered set equipped with a metric such that is a complete metric space.