Research Article
Existence of Fixed Points of Four Maps for a New Generalized
-Contraction and an Application
Muhammad Nazam,
1
Hassen Aydi ,
2,3
Mohd Salmi Noorani,
4
and Haitham Qawaqneh
4
1
Department of Mathematics, Allama Iqbal Open University, H-8, Islamabad, Pakistan
2
Universit´ e de Sousse, Institut Sup´ erieur d’Informatique et des Techniques de Communication, Hammam Sousse 4000, Tunisia
3
China Medical University Hospital, China Medical University, 40402 Taichung, Taiwan
4
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM),
43600 Selangor Darul Ehsan, Malaysia
Correspondence should be addressed to Hassen Aydi; hmaydi@iau.edu.sa
Received 20 February 2019; Accepted 25 March 2019; Published 7 April 2019
Academic Editor: Giuseppe Marino
Copyright © 2019 Muhammad Nazam et al. Tis 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 initiate the concept of a new generalized -contraction satisfying some contractive conditions involving four maps on a partial
metric space. We set up an example to elucidate our main result. An application is derived where a system of elliptic boundary value
equations has a common solution.
1. Introduction
Te Banach Contraction Principle (BCP) has many useful
applications in various directions like diferential, integral,
functional, and matrix equations (both linear and nonlinear).
Tis contraction principle is being generalized extensively
in diferent distance spaces. One of its best generalizations
is the -contraction presented by Wardowski [1]. Aydi el
al. [2] (see also [3]) modifed the concept of -contractions
via -admissibility. Many authors have formed diferent
generalizations of Wardowski [1] result (see [4–6]). Recently,
in [7] authors provided a fxed point result for a
́
Ciri ́ c type
-contraction. Berinde [8] and Berinde and Vetro [9] have
established some common fxed point results for mappings
satisfying compatible conditions (resp., implicit contractions)
in a complete metric space.
Here, we discuss a new generalized -contraction based
on four self-mappings related to elliptic boundary value
problem; in particular, we collect a common fxed point result
for four self-mappings to show the existence of a common
solution for operators satisfying an elliptic boundary value
problem. It is remarked that the notion of the -contraction
in partial metric spaces is more general with respect to the
metric space.
Defnition 1 (see [10]). Let I be a nonempty set. If the
function : I × I → [0,∞) satisfes the following prop-
erties:
(p
1
) = ⇐⇒ (, ) = (, ) = (, )
(p
2
) (, ) ≤ (, )
(p
3
) (,) = (,)
(p
4
) (,)≤(,)+(,)−(,),
for all ,,∈ I, then is called a partial metric on I
Example 2. Let I =[0,∞). Defne the classical partial metric
: I × I → I as (, ) = max{,}. Note that is not a
metric on I. Indeed (,)=>0 for each >0.
Matthews [10] explored the following aspects of a partial
metric on I.
(1) Te function : I × I → [0,∞) defned by
(,)=2(,)−(,)−(,) (1)
for all , ∈ I, defnes a metric on I (called the
induced metric by )
Hindawi
Journal of Function Spaces
Volume 2019, Article ID 5980312, 8 pages
https://doi.org/10.1155/2019/5980312