Fixed point of Suzuki-Zamfirescu hybrid contractions in partial metric spaces via partial Hausdorff metric

Fixed Point Theory and Applications20132013:21

DOI: 10.1186/1687-1812-2013-21

Received: 9 October 2012

Accepted: 12 January 2013

Published: 31 January 2013

Abstract

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.

Keywords

coincidence point orbitally complete common fixed point partial metric space

1 Introduction and preliminaries

Fixed point theory plays a fundamental role in solving functional equations [1] 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 [2] and Nadler [3] proved a multi-valued version of the Banach contraction principle employing the notion of a Hausdorff metric. Afterwards, a number of generalizations (see [49]) 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 [1014]). Among several generalizations of the Banach contraction principle, Suzuki’s work [[15], Theorem 2.1] led to a number of results (for details, see [13, 1621]).

On the other hand, Matthews [22] 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, 2528]). Recently, Aydi et al.[29] initiated the concept of a partial Hausdorff metric and obtained an analogue of Nadler’s fixed point theorem [3] 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, 3134].

Throughout this work, a mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq1_HTML.gif is defined by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ1_HTML.gif
(1.1)

In the sequel, the letters ℝ, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq2_HTML.gif 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[22]

Let X be any non-empty set. A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq3_HTML.gif is said to be a partial metric if and only if for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq4_HTML.gif the following conditions are satisfied:
  • (P1) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq5_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq6_HTML.gif ;

  • (P2) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq7_HTML.gif ;

  • (P3) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq8_HTML.gif ;

  • (P4) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq9_HTML.gif .

The pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif is called a partial metric space. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq11_HTML.gif , then (P1) and (P2) imply that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq6_HTML.gif . But the converse does not hold in general. A classical example of a partial metric space is the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq12_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq3_HTML.gif is defined as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq13_HTML.gif (see also [37]).

Example 1.2[22]

If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq14_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equa_HTML.gif

defines a partial metric p on X.

For more interesting examples, we refer to [23, 27, 28, 35, 38, 39]. Each partial metric p on X generates a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq15_HTML.gif topology http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq16_HTML.gif on X which has as a base the family of open balls (p-balls) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq17_HTML.gif , where
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equb_HTML.gif
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq18_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq19_HTML.gif . A sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq20_HTML.gif in a partial metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif is called convergent to a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq18_HTML.gif with respect to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq16_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq21_HTML.gif (for details, see [22]). If p is a partial metric on X, then the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq22_HTML.gif given by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq23_HTML.gif defines a metric on X. Furthermore, a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq20_HTML.gif converges in a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq24_HTML.gif to a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq18_HTML.gif if and only if
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ2_HTML.gif
(1.2)

Definition 1.3[22]

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space, then
  • (a) A sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq20_HTML.gif in X is called Cauchy if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq25_HTML.gif exists and is finite.

  • (b) A partial metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif is said to be complete if every Cauchy sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq20_HTML.gif in X converges with respect to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq16_HTML.gif to a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq18_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq26_HTML.gif .

Lemma A[22, 35]

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space, then
  • (c) A sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq20_HTML.gif in X is Cauchy in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif if and only if it is Cauchy in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq24_HTML.gif .

  • (d) A partial metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif is complete if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq24_HTML.gif is complete.

Consistent with [29], let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq27_HTML.gif be the family of all non-empty, closed and bounded subsets of the partial metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif , induced by the partial metric p. Note that closedness is taken from http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq28_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq16_HTML.gif is the topology induced by p) and boundedness is given as follows: A is a bounded subset in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif if there exists an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq29_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq30_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq31_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq32_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq33_HTML.gif . For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq18_HTML.gif , define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq35_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equc_HTML.gif

It can be verified that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq36_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq37_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq38_HTML.gif .

Lemma B[35]

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space and A be a non-empty subset of X, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq39_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq40_HTML.gif .

Proposition 1.4[29]

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space. For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq41_HTML.gif , we have the following:
  1. (i)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq42_HTML.gif ;

     
  2. (ii)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq43_HTML.gif ;

     
  3. (iii)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq44_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq45_HTML.gif ;

     
  4. (iv)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq46_HTML.gif .

     

Proposition 1.5[29]

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space. For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq41_HTML.gif , we have the following:
  • (h1) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq47_HTML.gif ;

  • (h2) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq48_HTML.gif ;

  • (h3) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq49_HTML.gif ;

  • (h4) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq50_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq51_HTML.gif .

The mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq52_HTML.gif 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 [29]).

Lemma C[29]

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq53_HTML.gif , then for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq31_HTML.gif , there exists a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq54_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq55_HTML.gif .

Theorem 1.6[29]

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq56_HTML.gif is a multi-valued mapping such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq57_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq58_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq59_HTML.gif . Then T has a fixed point.

Definition 1.7 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq61_HTML.gif . A point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq18_HTML.gif is said to be (i) a fixed point of f if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq62_HTML.gif , (ii) a fixed point of T if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq63_HTML.gif , (iii) a coincidence point of a pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq65_HTML.gif , (iv) a common fixed point of the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq66_HTML.gif .

We denote the set of all fixed points of f, the set of all coincidence points of the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif and the set of all common fixed points of the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq67_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq68_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq69_HTML.gif , respectively. Motivated by the work of [4, 13], we give the following definitions in partial metric spaces.

Definition 1.8 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq61_HTML.gif . The pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif is called (i) commuting if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq70_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq18_HTML.gif , (ii) weakly compatible if the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif commutes at their coincidence points, that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq71_HTML.gif whenever http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq72_HTML.gif , (iii) IT-commuting [11] at http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq18_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq73_HTML.gif .

Definition 1.9 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space and Y be any non-empty set. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq75_HTML.gif be single-valued and multi-valued mappings, respectively. Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq76_HTML.gif , then the set
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ3_HTML.gif
(1.3)

is called an orbit for the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif at http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq77_HTML.gif . A partial metric space X is called http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif -orbitally complete if and only if every Cauchy sequence in the orbit for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif at http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq77_HTML.gif converges with respect to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq16_HTML.gif to a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq18_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq78_HTML.gif .

Singh and Mishra [13] introduced Suzuki-Zamfirescu type hybrid contractive condition in complete metric spaces. In the context of partial metric spaces, the condition is given as follows.

Definition 1.10 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq75_HTML.gif be single-valued and multi-valued mappings, respectively. The hybrid pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif is said to satisfy Suzuki-Zamfirescu hybrid contraction condition if there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq79_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq80_HTML.gif implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ4_HTML.gif
(1.4)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq81_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ5_HTML.gif
(1.5)

Lemma D Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq75_HTML.gif be single-valued and multi-valued mappings, respectively. Then the partial metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif -orbitally complete if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq24_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif -orbitally complete.

Proof Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq24_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif -orbitally complete and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq77_HTML.gif is an arbitrary element of X. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq82_HTML.gif is a Cauchy sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq83_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif , then it is also Cauchy in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq24_HTML.gif . Therefore, by (1.2) we deduce that there exists y in X such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equd_HTML.gif
and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq82_HTML.gif converges to y in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif . Conversely, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif -orbitally complete. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq82_HTML.gif is a Cauchy sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq84_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq24_HTML.gif , then it is also a Cauchy sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif . Therefore,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Eque_HTML.gif
For given http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq19_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq85_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equf_HTML.gif
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq86_HTML.gif . Consequently, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equg_HTML.gif

whenever http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq86_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space and Y be any non-empty set. Assume that a pair of mappings http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq75_HTML.gif satisfies Suzuki-Zamfirescu hybrid contraction condition with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq87_HTML.gif . If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq88_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq89_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif -orbitally complete at http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq90_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq91_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif is IT-commuting at coincidence points of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq93_HTML.gif provided that fz is a fixed point of f for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq94_HTML.gif .

Proof Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq95_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq88_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq96_HTML.gif . By the given assumption, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq97_HTML.gif . So, there exists a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq98_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq99_HTML.gif . As http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq53_HTML.gif , so by Lemma C, there exists a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq100_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equh_HTML.gif
Using the fact that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq101_HTML.gif , we obtain a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq102_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq103_HTML.gif . Therefore,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equi_HTML.gif
Since
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equj_HTML.gif
we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equk_HTML.gif
If
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equl_HTML.gif
then
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equm_HTML.gif
If
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equn_HTML.gif
then we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equo_HTML.gif
As http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq104_HTML.gif , we choose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq105_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq106_HTML.gif . Using the fact that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq107_HTML.gif , we obtain a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq108_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq109_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equp_HTML.gif
Since
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equq_HTML.gif
so we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equr_HTML.gif
Following the arguments similar to those given above, we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equs_HTML.gif
which further implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equt_HTML.gif
Continuing this process, we obtain a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq110_HTML.gif such that for any integer http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq111_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq112_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equu_HTML.gif
for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq113_HTML.gif . This shows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq114_HTML.gif . Since
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equv_HTML.gif
so we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equw_HTML.gif
Now, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq115_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equx_HTML.gif
It follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq82_HTML.gif is a Cauchy sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq116_HTML.gif . By Lemma A, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq82_HTML.gif is a Cauchy sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq117_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq118_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif -orbitally complete at http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq90_HTML.gif , so again by Lemma D, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq119_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif -orbitally complete at http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq90_HTML.gif . Hence, there exists an element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq120_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq121_HTML.gif . This implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ6_HTML.gif
(2.1)
Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq122_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq123_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq124_HTML.gif . Now,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equy_HTML.gif
give
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equz_HTML.gif
Similarly, we can show that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equaa_HTML.gif
Now, we will claim that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ7_HTML.gif
(2.2)
If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq125_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq126_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq127_HTML.gif . This gives http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq128_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq65_HTML.gif and we are done. Now from (2.1), there exists a positive integer http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq129_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq130_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equab_HTML.gif
So, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq131_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equac_HTML.gif
Hence, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq131_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equad_HTML.gif
This implies
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equae_HTML.gif
On taking limit as n tends to ∞, we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equaf_HTML.gif
If
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equag_HTML.gif
then we are done. If
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equah_HTML.gif
then we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equai_HTML.gif
and hence (2.2) holds. Next, we show that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ8_HTML.gif
(2.3)
for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq132_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq125_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq126_HTML.gif , and the claim follows from (2.2). Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq133_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq134_HTML.gif . As f is a non-constant single-valued mapping, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equaj_HTML.gif
This implies
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equak_HTML.gif
Therefore,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equal_HTML.gif
Hence, (2.3) holds for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq132_HTML.gif . Note that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equam_HTML.gif
On taking limit as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq135_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equan_HTML.gif

We obtain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq136_HTML.gif , which further implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq137_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq138_HTML.gif . Further if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq139_HTML.gif , then due to IT-commutativity of the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq140_HTML.gif . This shows that fz is a common fixed point of the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif . □

Corollary A Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space and Y be any non-empty set. Assume that here exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq79_HTML.gif such that the mappings http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq75_HTML.gif satisfy
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equao_HTML.gif

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq81_HTML.gif , with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq87_HTML.gif . If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq88_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq89_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif -orbitally complete at http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq90_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq141_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif is IT-commuting at coincidence points of the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq93_HTML.gif provided that fz is a fixed point of f for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq94_HTML.gif .

Example 2.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq142_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq143_HTML.gif . Define a mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq3_HTML.gif as follows:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equap_HTML.gif
Then p is a partial metric on X. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq144_HTML.gif be as given in Theorem 2.1 and the mappings http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq75_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq145_HTML.gif be given as
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equaq_HTML.gif
Note that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equar_HTML.gif
If we take http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq146_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq147_HTML.gif , then for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq81_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equas_HTML.gif
holds. If we consider http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq148_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq149_HTML.gif . Then, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq150_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq151_HTML.gif , hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq152_HTML.gif is satisfied trivially. Now consider
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equat_HTML.gif
Hence, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq81_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equau_HTML.gif
implies
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equav_HTML.gif

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq153_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq154_HTML.gif . As http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq155_HTML.gif , there exists a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq156_HTML.gif in Y such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq157_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq158_HTML.gif , we obtain a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq159_HTML.gif in Y such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq160_HTML.gif . Continuing this way, we construct an orbit http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq161_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif at http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq153_HTML.gif . Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq89_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif -orbitally complete at http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq162_HTML.gif . So, all the conditions of Corollary A are satisfied. Moreover, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq163_HTML.gif .

On the other hand, the metric http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq164_HTML.gif induced by the partial metric p is given by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equaw_HTML.gif
Now, we show that Corollary A is not applicable (in the case of a metric induced by a partial metric p) in this case. Since
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equax_HTML.gif
is satisfied for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq79_HTML.gif , x and y in X, so it must imply http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq165_HTML.gif . But
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equay_HTML.gif
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equaz_HTML.gif
Hence, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq166_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equba_HTML.gif
Corollary B Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq10_HTML.gif be a partial metric space, Y be any non-empty set and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq167_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq87_HTML.gif . Suppose that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq88_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq89_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif -orbitally complete at http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq90_HTML.gif . Assume further that there exists an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq166_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equbb_HTML.gif
implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equbc_HTML.gif

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq81_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq91_HTML.gif . Further, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq92_HTML.gif and the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq168_HTML.gif is commuting at x where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq72_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq69_HTML.gif is a singleton.

Proof It follows from Theorem 2.1, that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq91_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq169_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq170_HTML.gif . Further, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq64_HTML.gif is commuting at u, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq171_HTML.gif . Now,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equbd_HTML.gif
implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Eqube_HTML.gif

As http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq172_HTML.gif , we obtain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq173_HTML.gif , which further implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq174_HTML.gif . Hence, fu is a common fixed point of f and T.

For uniqueness, assume there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq175_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq176_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq177_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equbf_HTML.gif
which implies
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equbg_HTML.gif

We obtain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq178_HTML.gif , which further implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq179_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq180_HTML.gif . □

3 An application

In this section, we assume that U and V are Banach spaces, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq181_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq182_HTML.gif . Suppose that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equbh_HTML.gif
Considering W and D as the state and decision spaces respectively, the problem of dynamic programming reduces to the problem of solving the functional equations:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ9_HTML.gif
(3.1)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ10_HTML.gif
(3.2)
Then equations (3.1) and (3.2) can be reformulated as
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ11_HTML.gif
(3.3)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ12_HTML.gif
(3.4)

For more on the multistage process involving such functional equations, we refer to [23, 3134]. 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.

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq183_HTML.gif denote the set of all bounded real-valued functions on W. For an arbitrary http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq184_HTML.gif , define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq185_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq186_HTML.gif is a Banach space endowed with the metric d defined as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq187_HTML.gif . Now, consider
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ13_HTML.gif
(3.5)

where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq188_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq189_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq190_HTML.gif is a partial metric on http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq183_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq144_HTML.gif be defined as in Section 1. Suppose that the following conditions hold:

(C1): G, F, g, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq191_HTML.gif are bounded.

(C2): For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq192_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq184_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq189_HTML.gif , define
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ14_HTML.gif
(3.6)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ15_HTML.gif
(3.7)
Moreover, assume that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq166_HTML.gif such that for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq193_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq188_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq194_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ16_HTML.gif
(3.8)
implies
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ17_HTML.gif
(3.9)
where
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equbi_HTML.gif
(C3): For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq184_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq195_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq192_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equbj_HTML.gif
(C4): There exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq184_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equbk_HTML.gif

Theorem 3.1 Assume that the conditions (C1)-(C4) are satisfied. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq196_HTML.gif is a closed convex subspace of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq183_HTML.gif , then the functional equations (3.3) and (3.4) have a unique, common and bounded solution.

Proof Note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq197_HTML.gif is a complete partial metric space. By (C1), J, K are self-maps of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq183_HTML.gif . The condition (C3) implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq198_HTML.gif . It follows from (C4) that J and K commute at their coincidence points. Let λ be an arbitrary positive number and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq199_HTML.gif . Choose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq192_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq200_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ18_HTML.gif
(3.10)
where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq201_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq202_HTML.gif . Further, from (3.5) and (3.6), we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ19_HTML.gif
(3.11)
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ20_HTML.gif
(3.12)
Therefore, (3.8) in (C2) becomes
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ21_HTML.gif
(3.13)
Then (3.13) together with (3.10) and (3.12) implies
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ22_HTML.gif
(3.14)
Now, (3.10), (3.11) and (3.13) imply
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ23_HTML.gif
(3.15)
From (3.14) and (3.15), we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ24_HTML.gif
(3.16)
As the above inequality is true for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq192_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq203_HTML.gif is taken arbitrarily, so from (3.13) we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ25_HTML.gif
(3.17)
implies
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_Equ26_HTML.gif
(3.18)

Therefore, by Corollary B, the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq204_HTML.gif has a common fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq205_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_355_IEq206_HTML.gif is a unique, bounded and common solution of (3.3) and (3.4). □

Declarations

Acknowledgements

The authors would like to thank the editor and anonymous reviewers for their useful comments that helped to improve the presentation of this paper.

Authors’ Affiliations

(1)
Department of Mathematics, Lahore University of Management Sciences
(2)
Department of Mathematics, University of Management and Technology

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