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

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

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##### Date

2014-12-10

##### Authors

KARAPINAR, Erdal

ROLDÁN, Antonio

MARTÍNEZ, Juan

ROLDÁN, Concepcion

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##### Abstract

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.

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##### Keywords

mathematics