Meir-Keeler Type Multidimensional Fixed Point Theorems in Partially Ordered Metric Spaces

ROLDÁN, Antonio
ROLDÁN, Concepcion
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Fixed point theory plays a crucial role in nonlinear functional analysis. In particular, fixed point results are used to prove the existence (and also uniqueness) when solving various type of equations. On the other hand, fixed point theory has a wide application potential in almost all positive sciences, such as Economics, Computer Science, Biology, Chemistry, and Engineering. One of the initial results in this direction (given by S. Banach), which is known as Banach fixed point theorem or Banach contraction mapping principle [1] is as follows. Every contraction in a complete metric space has a unique fixed point. In fact, this principle not only guarantees the existence and uniqueness of a fixed point, but it also shows how to get the desired fixed point. Since then, this celebrated principle has attracted the attention of a number of authors (e.g., see [1–39]). Due to its importance in nonlinear functional analysis, Banach contraction mapping principle has been generalized in many ways with regards to different abstract spaces. One of the most interesting results on generalization was reported by Guo and Lakshmikantham [18] in 1987. In their paper, the authors introduced the notion of coupled fixed point and proved some related theorems for certain type mappings. After this pioneering work, Gnana Bhaskar and Lakshmikantham [10] reconsidered coupled fixed point in the context of partially ordered sets by defining the notion of mixed monotone mapping. In this outstanding paper, the authors proved the existence and uniqueness of coupled fixed points for mixed monotone mappings and they also discussed the existence and uniqueness of solution for a periodic boundary value problem. Following these initial papers, a significant number of papers on coupled fixed point theorems have been reported (e.g., see [6, 11, 13, 19, 22, 23, 29, 31–33, 36, 38, 40]). Following this trend, Berinde and Borcut [8] extended the notion of coupled fixed point to tripled fixed point. Inspired by this interesting paper, Karapınar [24] improved this idea by defining quadruple fixed point (see also [25– 28]). Very recently, Roldan et al. [ ´ 35] generalized this idea by introducing the notion of Φ-fixed point, that is to say, the multidimensional fixed point. Another remarkable generalization of Banach contrac tion mapping principle was given by Meir and Keeler [34]. In the literature of this topic, Meir-Keeler type contraction has been studied densely by many selected mathematicians (e.g., see [2–4, 9, 20, 21, 36, 39]). In this paper, we prove the existence and uniqueness of fixed point of multidimensional Meir-Keeler contraction in a complete partially ordered metric space. Our results improve, extend, and generalize the existence results on the topic in fixed point theory.