Fixed point of Suzuki-Zamfirescu hybrid contractions in partial metric spaces via partial Hausdorff metric
© Abbas and Ali; licensee Springer 2013
Received: 9 October 2012
Accepted: 12 January 2013
Published: 31 January 2013
Coincidence point theorems for hybrid pairs of single-valued and multi-valued mappings on an arbitrary non-empty set with values in a partial metric space using a partial Hausdorff metric have been proved. As an application of our main result, the existence and uniqueness of common and bounded solutions of functional equations arising in dynamic programming are discussed.
MSC:47H10, 54H25, 54E50.
1 Introduction and preliminaries
Fixed point theory plays a fundamental role in solving functional equations  arising in several areas of mathematics and other related disciplines as well. The Banach contraction principle is a key principle that made a remarkable progress towards the development of metric fixed point theory. Markin  and Nadler  proved a multi-valued version of the Banach contraction principle employing the notion of a Hausdorff metric. Afterwards, a number of generalizations (see [4–9]) were obtained using different contractive conditions. The study of hybrid type contractive conditions involving single-valued and multi-valued mappings is a valuable addition to the metric fixed point theory and its applications (for details, see [10–14]). Among several generalizations of the Banach contraction principle, Suzuki’s work [, Theorem 2.1] led to a number of results (for details, see [13, 16–21]).
On the other hand, Matthews  introduced the concept of a partial metric space as a part of the study of denotational semantics of dataflow networks. He obtained a modified version of the Banach contraction principle, more suitable in this context (see also [23, 24]). Since then, results obtained in the framework of partial metric spaces have been used to constitute a suitable framework to model the problems related to the theory of computation (see [22, 25–28]). Recently, Aydi et al.  initiated the concept of a partial Hausdorff metric and obtained an analogue of Nadler’s fixed point theorem  in partial metric spaces.
The aim of this paper is to obtain some coincidence point theorems for a hybrid pair of single-valued and multi-valued mappings on an arbitrary non-empty set with values in a partial metric space. Our results extend, unify and generalize several known results in the existing literature (see [13, 15, 21, 30]). As an application, we obtain the existence and uniqueness of a common and bounded solution for Suzuki-Zamfirescu class of functional equations under contractive conditions weaker than those given in [1, 31–34].
In the sequel, the letters ℝ, and ℕ will denote the set of all real numbers, the set of all non-negative real numbers and the set of all positive integers, respectively. Consistent with [22, 29, 35, 36], the following definitions and results will be needed in the sequel.
Definition 1.1 
if and only if ;
The pair is called a partial metric space. If , then (P1) and (P2) imply that . But the converse does not hold in general. A classical example of a partial metric space is the pair , where is defined as (see also ).
Example 1.2 
defines a partial metric p on X.
Definition 1.3 
A sequence in X is called Cauchy if and only if exists and is finite.
A partial metric space is said to be complete if every Cauchy sequence in X converges with respect to to a point such that .
A sequence in X is Cauchy in if and only if it is Cauchy in .
A partial metric space is complete if and only if is complete.
It can be verified that implies , where .
Lemma B 
Let be a partial metric space and A be a non-empty subset of X, then if and only if .
Proposition 1.4 
Proposition 1.5 
implies that .
The mapping is called a partial Hausdorff metric induced by a partial metric p. Every Hausdorff metric is a partial Hausdorff metric, but the converse is not true (see Example 2.6 in ).
Lemma C 
Let be a partial metric space and and , then for any , there exists a such that .
Theorem 1.6 
Let be a partial metric space. If is a multi-valued mapping such that for all , we have , where . Then T has a fixed point.
Definition 1.7 Let be a partial metric space and and . A point is said to be (i) a fixed point of f if , (ii) a fixed point of T if , (iii) a coincidence point of a pair if , (iv) a common fixed point of the pair if .
We denote the set of all fixed points of f, the set of all coincidence points of the pair and the set of all common fixed points of the pair by , and , respectively. Motivated by the work of [4, 13], we give the following definitions in partial metric spaces.
Definition 1.8 Let be a partial metric space and and . The pair is called (i) commuting if for all , (ii) weakly compatible if the pair commutes at their coincidence points, that is, whenever , (iii) IT-commuting  at if .
is called an orbit for the pair at . A partial metric space X is called -orbitally complete if and only if every Cauchy sequence in the orbit for at converges with respect to to a point such that .
Singh and Mishra  introduced Suzuki-Zamfirescu type hybrid contractive condition in complete metric spaces. In the context of partial metric spaces, the condition is given as follows.
Lemma D Let be a partial metric space, and be single-valued and multi-valued mappings, respectively. Then the partial metric space is -orbitally complete if and only if is -orbitally complete.
whenever . The result follows. □
2 Coincidence points of a hybrid pair of mappings
In the following theorem, the existence of coincidence points of a hybrid pair of single-valued and multi-valued mappings that satisfy Suzuki-Zamfirescu hybrid contraction condition in partial metric spaces is established.
Theorem 2.1 Let be a partial metric space and Y be any non-empty set. Assume that a pair of mappings and satisfies Suzuki-Zamfirescu hybrid contraction condition with . If there exists such that is -orbitally complete at , then . If and is IT-commuting at coincidence points of , then provided that fz is a fixed point of f for some .
We obtain , which further implies that . Hence, . Further if and , then due to IT-commutativity of the pair , we have . This shows that fz is a common fixed point of the pair . □
for all , with . If there exists such that is -orbitally complete at , then . If and is IT-commuting at coincidence points of the pair , then provided that fz is a fixed point of f for some .
Let , . As , there exists a point in Y such that and , we obtain a point in Y such that . Continuing this way, we construct an orbit for at . Also, is -orbitally complete at . So, all the conditions of Corollary A are satisfied. Moreover, .
for all . Then . Further, if and the pair is commuting at x where , then is a singleton.
As , we obtain , which further implies that . Hence, fu is a common fixed point of f and T.
We obtain , which further implies that . Hence, . □
3 An application
For more on the multistage process involving such functional equations, we refer to [23, 31–34]. Now, we study the existence and uniqueness of a common and bounded solution of the functional equations (3.3)-(3.4) arising in dynamic programming in the setup of partial metric spaces.
(C1): G, F, g, and are bounded.
Theorem 3.1 Assume that the conditions (C1)-(C4) are satisfied. If is a closed convex subspace of , then the functional equations (3.3) and (3.4) have a unique, common and bounded solution.
Therefore, by Corollary B, the pair has a common fixed point , that is, is a unique, bounded and common solution of (3.3) and (3.4). □
The authors would like to thank the editor and anonymous reviewers for their useful comments that helped to improve the presentation of this paper.
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