Research Article
Caristi Fixed Point Theorem in Metric Spaces with a Graph
M. R. Alfuraidan
1
and M. A. Khamsi
2
1
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2
Department of Mathematical Science, e University of Texas at El Paso, El Paso, TX 79968, USA
Correspondence should be addressed to M. R. Alfuraidan; monther@kfupm.edu.sa
Received 27 January 2014; Accepted 11 February 2014; Published 13 March 2014
Academic Editor: Qamrul Hasan Ansari
Copyright © 2014 M. R. Alfuraidan and M. A. Khamsi. is is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
We discuss Caristi’s fixed point theorem for mappings defined on a metric space endowed with a graph. is work should be seen as
a generalization of the classical Caristi’s fixed point theorem. It extends some recent works on the extension of Banach contraction
principle to metric spaces with graph.
Dedicated to Rashed Saleh Alfuraidan and Prof. Miodrag Mateljevi’c for his 65th birthday
1. Introduction
is work was motivated by some recent works on the
extension of Banach contraction principle to metric spaces
with a partial order [1] or a graph [2]. Caristi’s fixed point
theorem is maybe one of the most beautiful extensions of
Banach contraction principle [3, 4]. Recall that this theorem
states the fact that any map :→ has a fixed point
provided that is a complete metric space and there exists a
lower semicontinuous map : → [0, +∞) such that
(, ) ≤ () − () , (1)
for every ∈. Recall that ∈ is called a fixed point of
if () = . is general fixed point theorem has found many
applications in nonlinear analysis. It is shown, for example,
that this theorem yields essentially all the known inwardness
results [5] of geometric fixed point theory in Banach spaces.
Recall that inwardness conditions are the ones which assert
that, in some sense, points from the domain are mapped
toward the domain. Possibly, the weakest of the inwardness
conditions, the Leray-Schauder boundary condition, is the
assumption that a map points of anywhere except to
the outward part of the ray originating at some interior point
of and passing through .
e proofs given to Caristi’s result vary and use different
techniques (see [3, 6–8]). It is worth to mention that because
of Caristi’s result of close connection to the Ekeland’s [9]
variational principle, many authors refer to it as Caristi-
Ekeland fixed point result. For more on Ekeland’s variational
principle and the equivalence between Caristi-Ekeland fixed
point result and the completeness of metric spaces, the reader
is advised to read [10].
2. Main Results
Maybe one of the most interesting examples of the use of
metric fixed point theorems is the proof of the existence of
solutions to differential equations. e general approach is to
convert such equations to integral equations which describes
exactly a fixed point of a mapping. e metric spaces in which
such mapping acts are usually a function space. Putting a
norm (in the case of a vector space) or a distance gives us a
metric structure rich enough to use the Banach contraction
principle or other known fixed point theorems. But one
structure naturally enjoyed by such function spaces is rarely
used. Indeed we have an order on the functions inherited
from the order of R. In the classical use of Banach contraction
principle, the focus is on the metric behavior of the mapping.
e connection with the natural order is usually ignored.
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 303484, 5 pages
http://dx.doi.org/10.1155/2014/303484