Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 180698, 14 pages
http://dx.doi.org/10.1155/2014/180698
Research Article
A Unification of -Metric, Partial Metric, and -Metric Spaces
Nawab Hussain,
1
Jamal Rezaei Roshan,
2
Vahid Parvaneh,
3
and Abdul Latif
1
1
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2
Department of Mathematics, Islamic Azad University, Qaemshahr Branch, Qaemshahr, Iran
3
Young Researchers and Elite Club, Islamic Azad University, Kermanshah Branch, Kermanshah, Iran
Correspondence should be addressed to Vahid Parvaneh; vahid.parvaneh@kiau.ac.ir
Received 9 October 2013; Accepted 18 January 2014
Academic Editor: Mohamed Amine Khamsi
Copyright © 2014 Nawab Hussain et al. his 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.
Using the concepts of G-metric, partial metric, and b-metric spaces, we deine a new concept of generalized partial b-metric space.
Topological and structural properties of the new space are investigated and certain ixed point theorems for contractive mappings
in such spaces are obtained. Some examples are provided here to illustrate the usability of the obtained results.
1. Introduction and Mathematical
Preliminaries
he concept of a -metric space was introduced by Czerwik in
[1, 2]. Ater that, several interesting results about the existence
of ixed point for single-valued and multivalued operators in
(ordered) -metric spaces have been obtained (see, e.g., [3–
13]).
Deinition 1 (see [1]). Let be a (nonempty) set and ≥1 a
given real number. A function :×→ R
+
is a -metric
on if, for all ,,∈, the following conditions hold:
(b
1
)(,)=0 if and only if =,
(b
2
)(,)=(,),
(b
3
)(,)≤[(,)+(,)].
In this case, the pair (,) is called a -metric space.
he concept of a generalized metric space, or a -metric
space, was introduced by Mustafa and Sims [14].
Deinition 2 (see [14]). Let be a nonempty set and :×
×→ R
+
a function satisfying the following properties:
(G1) (,,)=0 if ==;
(G2) 0<(,,), for all ,∈ with ̸ =;
(G3) (,,)≤(,,), for all ,,∈ with ̸ =;
(G4) (,,)=({,,}), where is any permutation
of ,, (symmetry in all the three variables);
(G5) (,,)≤(,,)+(,,), for all ,,,∈
(rectangle inequality).
hen, the function is called a -metric on and the
pair (,) is called a -metric space.
Aghajani et al. in [15] introduced the class of generalized
-metric spaces (
-metric spaces) and then they presented
some basic properties of
-metric spaces.
he following is their deinition of
-metric spaces.
Deinition 3 (see [15]). Let be a nonempty set and ≥1 a
given real number. Suppose that a mapping :××→
R
+
satisies
(G
b
1) (,,)=0 if ==,
(G
b
2) 0<(,,) for all ,∈ with ̸ =,
(G
b
3) (,,)≤(,,) for all ,,∈ with ̸ =,
(G
b
4) (,,) = ({,,}), where is a permutation
of ,, (symmetry),
(G
b
5) (,,)≤[(,,)+(,,)] for all ,,,∈
(rectangle inequality).
hen is called a generalized -metric and the pair (,)
is called a generalized -metric space or a G
b
-metric space.