Open Access

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 ω : [ 0 , 1 ) ( 1 2 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq1_HTML.gif is defined by
ω ( r ) = 1 1 + r for all  r [ 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ1_HTML.gif
(1.1)

In the sequel, the letters , R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq2_HTML.gif and 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 p : X × X R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq3_HTML.gif is said to be a partial metric if and only if for all x , y , z X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq4_HTML.gif the following conditions are satisfied:
  1. (P1)

    p ( x , x ) = p ( y , y ) = p ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq5_HTML.gif if and only if x = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq6_HTML.gif;

     
  2. (P2)

    p ( x , x ) p ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq7_HTML.gif;

     
  3. (P3)

    p ( x , y ) = p ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq8_HTML.gif;

     
  4. (P4)

    p ( x , z ) p ( x , y ) + p ( y , z ) p ( y , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq9_HTML.gif.

     

The pair ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif is called a partial metric space. If p ( x , y ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq11_HTML.gif, then (P1) and (P2) imply that x = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq6_HTML.gif. But the converse does not hold in general. A classical example of a partial metric space is the pair ( R + , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq12_HTML.gif, where p : X × X R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq3_HTML.gif is defined as p ( x , y ) = max { x , y } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq13_HTML.gif (see also [37]).

Example 1.2 [22]

If X = { [ a , b ] : a , b R , a b } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq14_HTML.gif, then
p ( [ a , b ] , [ c , d ] ) = max { b , d } min { a , c } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_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 T 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq15_HTML.gif topology τ p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq16_HTML.gif on X which has as a base the family of open balls (p-balls) { B p ( x , ε ) : x X , ε > 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq17_HTML.gif, where
B p ( x , ε ) = { y X : p ( x , y ) < p ( x , x ) + ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equb_HTML.gif
for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq18_HTML.gif and ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq19_HTML.gif. A sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq20_HTML.gif in a partial metric space ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif is called convergent to a point x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq18_HTML.gif with respect to τ p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq16_HTML.gif if and only if p ( x , x ) = lim n p ( x , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq21_HTML.gif (for details, see [22]). If p is a partial metric on X, then the mapping p S : X × X R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq22_HTML.gif given by p S ( x , y ) = 2 p ( x , y ) p ( x , x ) p ( y , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq23_HTML.gif defines a metric on X. Furthermore, a sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq20_HTML.gif converges in a metric space ( X , p S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq24_HTML.gif to a point x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq18_HTML.gif if and only if
p ( x , x ) = lim n p ( x , x n ) = lim n , m p ( x n , x m ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ2_HTML.gif
(1.2)

Definition 1.3 [22]

Let ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space, then
  1. (a)

    A sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq20_HTML.gif in X is called Cauchy if and only if lim n , m p ( x n , x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq25_HTML.gif exists and is finite.

     
  2. (b)

    A partial metric space ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif is said to be complete if every Cauchy sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq20_HTML.gif in X converges with respect to τ p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq16_HTML.gif to a point x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq18_HTML.gif such that p ( x , x ) = lim n , m p ( x n , x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq26_HTML.gif.

     

Lemma A [22, 35]

Let ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space, then
  1. (c)

    A sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq20_HTML.gif in X is Cauchy in ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif if and only if it is Cauchy in ( X , p S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq24_HTML.gif.

     
  2. (d)

    A partial metric space ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif is complete if and only if ( X , p S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq24_HTML.gif is complete.

     
Consistent with [29], let C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq27_HTML.gif be the family of all non-empty, closed and bounded subsets of the partial metric space ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif, induced by the partial metric p. Note that closedness is taken from ( X , τ p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq28_HTML.gif ( τ p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq16_HTML.gif is the topology induced by p) and boundedness is given as follows: A is a bounded subset in ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif if there exists an x 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq29_HTML.gif and M 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq30_HTML.gif such that for all a A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq31_HTML.gif, we have a B p ( x 0 , M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq32_HTML.gif, that is, p ( x 0 , a ) < p ( a , a ) + M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq33_HTML.gif. For A , B C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq34_HTML.gif and x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq18_HTML.gif, define δ p : C B p ( X ) × C B p ( X ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq35_HTML.gif and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equc_HTML.gif

It can be verified that p ( x , A ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq36_HTML.gif implies p S ( x , A ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq37_HTML.gif, where p S ( x , A ) = inf { p S ( x , a ) : a A } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq38_HTML.gif.

Lemma B [35]

Let ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space and A be a non-empty subset of X, then a A ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq39_HTML.gif if and only if p ( a , A ) = p ( a , a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq40_HTML.gif.

Proposition 1.4 [29]

Let ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space. For any A , B , C C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq41_HTML.gif, we have the following:
  1. (i)

    δ p ( A , A ) = sup { p ( a , a ) : a A } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq42_HTML.gif;

     
  2. (ii)

    δ p ( A , A ) δ p ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq43_HTML.gif;

     
  3. (iii)

    δ p ( A , B ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq44_HTML.gif implies A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq45_HTML.gif;

     
  4. (iv)

    δ p ( A , B ) δ p ( A , C ) + δ p ( C , B ) inf c C p ( c , c ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq46_HTML.gif.

     

Proposition 1.5 [29]

Let ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space. For any A , B , C C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq41_HTML.gif, we have the following:
  1. (h1)

    H p ( A , A ) H p ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq47_HTML.gif;

     
  2. (h2)

    H p ( A , B ) = H p ( B , A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq48_HTML.gif;

     
  3. (h3)

    H p ( A , B ) H p ( A , C ) + H p ( C , B ) inf c C p ( c , c ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq49_HTML.gif;

     
  4. (h4)

    H p ( A , B ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq50_HTML.gif implies that A = B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq51_HTML.gif.

     

The mapping H p : C B p ( X ) × C B p ( X ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_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 ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space and A , B C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq34_HTML.gif and h > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq53_HTML.gif, then for any a A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq31_HTML.gif, there exists a b B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq54_HTML.gif such that p ( a , b ) h H p ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq55_HTML.gif.

Theorem 1.6 [29]

Let ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space. If T : X C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq56_HTML.gif is a multi-valued mapping such that for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq57_HTML.gif, we have H p ( T x , T y ) k p ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq58_HTML.gif, where k ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq59_HTML.gif. Then T has a fixed point.

Definition 1.7 Let ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space and f : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq60_HTML.gif and T : X C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq61_HTML.gif. A point x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq18_HTML.gif is said to be (i) a fixed point of f if x = f ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq62_HTML.gif, (ii) a fixed point of T if x T ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq63_HTML.gif, (iii) a coincidence point of a pair ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif if f x T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq65_HTML.gif, (iv) a common fixed point of the pair ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif if x = f x T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq66_HTML.gif.

We denote the set of all fixed points of f, the set of all coincidence points of the pair ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif and the set of all common fixed points of the pair ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif by F ( f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq67_HTML.gif, C ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq68_HTML.gif and F ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_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 ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space and f : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq60_HTML.gif and T : X C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq61_HTML.gif. The pair ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif is called (i) commuting if T f x = f T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq70_HTML.gif for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq18_HTML.gif, (ii) weakly compatible if the pair ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif commutes at their coincidence points, that is, f T x = T f x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq71_HTML.gif whenever x C ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq72_HTML.gif, (iii) IT-commuting [11] at x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq18_HTML.gif if f T x T f x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq73_HTML.gif.

Definition 1.9 Let ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space and Y be any non-empty set. Let f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq74_HTML.gif and T : Y C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq75_HTML.gif be single-valued and multi-valued mappings, respectively. Suppose that x 0 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq76_HTML.gif, then the set
O ( f , T ; x 0 ) = { y n : y n + 1 = f x n + 1 T x n  for  n = 0 , 1 , 2 , } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ3_HTML.gif
(1.3)

is called an orbit for the pair ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif at x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq77_HTML.gif. A partial metric space X is called ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif-orbitally complete if and only if every Cauchy sequence in the orbit for ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif at x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq77_HTML.gif converges with respect to τ p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq16_HTML.gif to a point x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq18_HTML.gif such that p ( x , x ) = lim n , m p ( x m , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_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 ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space, f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq74_HTML.gif and T : Y C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq75_HTML.gif be single-valued and multi-valued mappings, respectively. The hybrid pair ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif is said to satisfy Suzuki-Zamfirescu hybrid contraction condition if there exists r [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq79_HTML.gif such that ω ( r ) p ( f x , T x ) p ( f x , f y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq80_HTML.gif implies that
H p ( T x , T y ) r M p , f ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ4_HTML.gif
(1.4)
for all x , y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq81_HTML.gif and
M p , f ( x , y ) = max { p ( f x , f y ) , p ( f x , T x ) + p ( f y , T y ) 2 , p ( f x , T y ) + p ( f y , T x ) 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ5_HTML.gif
(1.5)

Lemma D Let ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space, f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq74_HTML.gif and T : Y C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq75_HTML.gif be single-valued and multi-valued mappings, respectively. Then the partial metric space ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif is ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif-orbitally complete if and only if ( X , p S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq24_HTML.gif is ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif-orbitally complete.

Proof Suppose that ( X , p S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq24_HTML.gif is ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif-orbitally complete and x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq77_HTML.gif is an arbitrary element of X. If { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq82_HTML.gif is a Cauchy sequence in O ( f , T ; x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq83_HTML.gif in ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif, then it is also Cauchy in ( X , p S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq24_HTML.gif. Therefore, by (1.2) we deduce that there exists y in X such that
p ( y , y ) = lim n p ( y , y n ) = lim n , m p ( y n , y m ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equd_HTML.gif
and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq82_HTML.gif converges to y in ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif. Conversely, let ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif-orbitally complete. If { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq82_HTML.gif is a Cauchy sequence in O ( f , T ; x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq84_HTML.gif in ( X , p S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq24_HTML.gif, then it is also a Cauchy sequence in ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif. Therefore,
p ( y , y ) = lim n p ( y , y n ) = lim n , m p ( y n , y m ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Eque_HTML.gif
For given ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq19_HTML.gif, there exists n ε N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq85_HTML.gif such that
| p ( y , y n ) p ( y , y ) | < ε 2 and | p ( y , y n ) p ( y n , y m ) | < ε 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equf_HTML.gif
for all m , n > n ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq86_HTML.gif. Consequently, we have
p S ( y , y n ) = 2 p ( y , y n ) p ( y , y ) p ( y n , y m ) | p ( y , y n ) p ( y , y ) + p ( y , y n ) p ( y n , y m ) | | p ( y , y n ) p ( y , y ) | + | p ( y , y n ) p ( y n , y m ) | < ε 2 + ε 2 = ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equg_HTML.gif

whenever m , n > n ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_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 ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space and Y be any non-empty set. Assume that a pair of mappings f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq74_HTML.gif and T : Y C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq75_HTML.gif satisfies Suzuki-Zamfirescu hybrid contraction condition with T ( Y ) f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq87_HTML.gif. If there exists u 0 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq88_HTML.gif such that f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq89_HTML.gif is ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif-orbitally complete at u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq90_HTML.gif, then C ( f , T ) ϕ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq91_HTML.gif. If Y = X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq92_HTML.gif and ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif is IT-commuting at coincidence points of ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif, then F ( f , T ) ϕ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq93_HTML.gif provided that fz is a fixed point of f for some z C ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq94_HTML.gif.

Proof Let h = 1 / r https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq95_HTML.gif and u 0 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq88_HTML.gif be such that y 0 = f u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq96_HTML.gif. By the given assumption, we have T u 0 f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq97_HTML.gif. So, there exists a point u 1 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq98_HTML.gif such that y 1 = f u 1 T u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq99_HTML.gif. As h > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq53_HTML.gif, so by Lemma C, there exists a point y 2 T u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq100_HTML.gif such that
p ( f u 1 , y 2 ) h H p ( T u 0 , T u 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equh_HTML.gif
Using the fact that T u 1 f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq101_HTML.gif, we obtain a point u 2 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq102_HTML.gif such that y 2 = f u 2 T u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq103_HTML.gif. Therefore,
p ( f u 1 , f u 2 ) h H p ( T u 0 , T u 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equi_HTML.gif
Since
ω ( r ) p ( f u 0 , T u 0 ) ω ( r ) p ( f u 0 , f u 1 ) p ( f u 0 , f u 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equj_HTML.gif
we have
p ( f u 1 , f u 2 ) h H p ( T u 0 , T u 1 ) h r max { p ( f u 0 , f u 1 ) , p ( f u 0 , T u 0 ) + p ( f u 1 , T u 1 ) 2 , p ( f u 0 , T u 1 ) + p ( f u 1 , T u 0 ) 2 } 1 r r max { p ( y 0 , y 1 ) , p ( y 0 , y 1 ) + p ( y 1 , y 2 ) 2 , p ( y 0 , y 2 ) + p ( y 1 , y 1 ) 2 } r max { p ( y 0 , y 1 ) , p ( y 0 , y 1 ) + p ( y 1 , y 2 ) 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equk_HTML.gif
If
max { p ( y 0 , y 1 ) , p ( y 0 , y 1 ) + p ( y 1 , y 2 ) 2 } = p ( y 0 , y 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equl_HTML.gif
then
p ( y 1 , y 2 ) h H p ( T u 0 , T u 1 ) r p ( y 0 , y 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equm_HTML.gif
If
max { p ( y 0 , y 1 ) , p ( y 0 , y 1 ) + p ( y 1 , y 2 ) 2 } = p ( y 0 , y 1 ) + p ( y 1 , y 2 ) 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equn_HTML.gif
then we obtain
p ( y 1 , y 2 ) r 2 r p ( y 0 , y 1 ) r p ( y 0 , y 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equo_HTML.gif
As f u 2 T u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq104_HTML.gif, we choose y 3 T u 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq105_HTML.gif such that p ( f u 2 , y 3 ) h H ( T u 1 , T u 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq106_HTML.gif. Using the fact that T u 2 f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq107_HTML.gif, we obtain a point u 3 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq108_HTML.gif such that y 3 = f u 3 T u 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq109_HTML.gif and
p ( f u 2 , f u 3 ) h H p ( T u 1 , T u 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equp_HTML.gif
Since
ω ( r ) p ( f u 1 , T u 1 ) ω ( r ) p ( f u 1 , f u 2 ) p ( f u 1 , f u 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equq_HTML.gif
so we have
p ( f u 2 , f u 3 ) h H p ( T u 1 , T u 2 ) h r max { p ( f u 1 , f u 2 ) , p ( f u 1 , T u 1 ) + p ( f u 2 , T u 2 ) 2 , p ( f u 1 , T u 2 ) + p ( f u 2 , T u 1 ) 2 } 1 r r max { p ( y 1 , y 2 ) , p ( y 1 , y 2 ) + p ( y 2 , y 3 ) 2 , p ( y 1 , y 3 ) + p ( y 2 , y 2 ) 2 } r max { p ( y 1 , y 2 ) , p ( y 1 , y 2 ) + p ( y 2 , y 3 ) 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equr_HTML.gif
Following the arguments similar to those given above, we obtain
p ( y 2 , y 3 ) r p ( y 1 , y 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equs_HTML.gif
which further implies that
p ( y 2 , y 3 ) ( r ) 2 p ( y 0 , y 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equt_HTML.gif
Continuing this process, we obtain a sequence { y n } Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq110_HTML.gif such that for any integer n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq111_HTML.gif, y n + 1 = f u n + 1 T u n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq112_HTML.gif and
p ( y n , y n + 1 ) ( r ) n p ( y 0 , y 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equu_HTML.gif
for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq113_HTML.gif. This shows that lim n p ( y n , y n + 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq114_HTML.gif. Since
p ( y n , y n ) + p ( y n + 1 , y n + 1 ) 2 p ( y n , y n + 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equv_HTML.gif
so we obtain
lim n p ( y n , y n ) = 0 and lim n p ( y n + 1 , y n + 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equw_HTML.gif
Now, for m > n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq115_HTML.gif, we have
p S ( y n , y n + m ) = 2 p ( y n , y n + m ) p ( y n , y n ) p ( y n + m , y n + m ) 2 p ( y n , y n + 1 ) + 2 p ( y n + 1 , y n + 2 ) + + 2 p ( y n + m 1 , y n + m ) 2 ( ( r ) n + ( r ) n + 1 + + ( r ) n + m 1 ) p ( y 0 , y 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equx_HTML.gif
It follows that { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq82_HTML.gif is a Cauchy sequence in ( f ( Y ) , p S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq116_HTML.gif. By Lemma A, we have { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq82_HTML.gif is a Cauchy sequence in ( f ( Y ) , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq117_HTML.gif. Since ( f ( Y ) , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq118_HTML.gif is ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif-orbitally complete at u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq90_HTML.gif, so again by Lemma D, ( f ( Y ) , p S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq119_HTML.gif is ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif-orbitally complete at u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq90_HTML.gif. Hence, there exists an element u f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq120_HTML.gif such that lim n p S ( y n , y ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq121_HTML.gif. This implies that
lim n p ( y n , u ) = lim n p ( y n , y n ) = p ( u , u ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ6_HTML.gif
(2.1)
Let z f 1 u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq122_HTML.gif, then z Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq123_HTML.gif and u = f z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq124_HTML.gif. Now,
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equy_HTML.gif
give
lim n p ( f u n + 1 , T x ) = p ( f z , T x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equz_HTML.gif
Similarly, we can show that
lim n p ( f u n , T x ) = p ( f z , T x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equaa_HTML.gif
Now, we will claim that
p ( f z , T x ) r p ( f z , f x ) for any  f x f ( Y ) { f z } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ7_HTML.gif
(2.2)
If x = z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq125_HTML.gif or f x = f z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq126_HTML.gif, then p ( f x , T x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq127_HTML.gif. This gives p S ( f x , T x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq128_HTML.gif, which implies that f x T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq65_HTML.gif and we are done. Now from (2.1), there exists a positive integer n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq129_HTML.gif such that for all n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq130_HTML.gif,
p ( f z , f u n + 1 ) 1 3 p ( f z , f x ) and p ( f z , f u n ) 1 3 p ( f z , f x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equab_HTML.gif
So, for any n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq131_HTML.gif, we have
ω ( r ) p ( f u n , T u n ) p ( f u n , T u n ) p ( f u n , f u n + 1 ) p ( f u n , f z ) + p ( f z , f u n + 1 ) p ( f z , f z ) 2 3 p ( f z , f x ) p ( f z , f x ) 1 3 p ( f z , f x ) p ( f z , f x ) p ( f z , f u n ) p ( f u n , f x ) p ( f u n , f u n ) p ( f u n , f x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equac_HTML.gif
Hence, for any n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq131_HTML.gif, we obtain
ω ( r ) p ( f u n , T u n ) p ( f u n , f x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equad_HTML.gif
This implies
p ( f u n + 1 , T x ) H p ( T u n , T x ) r max { p ( f u n , f x ) , p ( f u n , T u n ) + p ( f x , T x ) 2 , p ( f u n , T x ) + p ( f x , T u n ) 2 } r max { p ( y n , f x ) , p ( y n , y n + 1 ) + p ( f x , T x ) 2 , p ( y n , T x ) + p ( f x , y n + 1 ) 2 } r max { p ( y n , u ) + p ( u , f x ) p ( u , u ) , p ( y n , y n + 1 ) + p ( f x , T x ) 2 , p ( y n , u ) + p ( u , T x ) p ( u , u ) + p ( f x , u ) + p ( u , y n + 1 ) p ( u , u ) 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equae_HTML.gif
On taking limit as n tends to ∞, we obtain
p ( f z , T x ) r max { p ( u , f x ) , p ( f x , T x ) 2 , p ( u , T x ) + p ( f x , u ) 2 } = r max { p ( f z , f x ) , p ( f x , T x ) 2 , p ( f z , T x ) + p ( f x , , f z ) 2 } r max { p ( f z , f x ) , p ( f z , T x ) + p ( f x , , f z ) 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equaf_HTML.gif
If
max { p ( f z , f x ) , p ( f z , T x ) + p ( f x , , f z ) 2 } = p ( f z , f x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equag_HTML.gif
then we are done. If
max { p ( f z , f x ) , p ( f z , T x ) + p ( f x , , f z ) 2 } = p ( f z , T x ) + p ( f x , , f z ) 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equah_HTML.gif
then we obtain
p ( f z , T x ) r 2 r p ( f x , f z ) r p ( f x , f z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equai_HTML.gif
and hence (2.2) holds. Next, we show that
H p ( T z , T x ) r max { p ( f z , f x ) , p ( f x , T x ) + p ( f z , T z ) 2 , p ( f x , T z ) + p ( f z , T x ) 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ8_HTML.gif
(2.3)
for any x Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq132_HTML.gif. If x = z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq125_HTML.gif, then f x = f z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq126_HTML.gif, and the claim follows from (2.2). Suppose that x z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq133_HTML.gif, then f x f z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq134_HTML.gif. As f is a non-constant single-valued mapping, we have
p ( f x , T x ) p ( f x , f z ) + p ( f z , T x ) p ( f z , f z ) p ( f x , f z ) + r p ( f x , f z ) ( 1 + r ) p ( f x , f z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equaj_HTML.gif
This implies
ω ( r ) p ( f x , T x ) p ( f x , f z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equak_HTML.gif
Therefore,
H p ( T z , T x ) r max { p ( f z , f x ) , p ( f x , T x ) + p ( f z , T z ) 2 , p ( f x , T z ) + p ( f z , T x ) 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equal_HTML.gif
Hence, (2.3) holds for any x Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq132_HTML.gif. Note that
p ( T z , f u n + 2 ) H p ( T z , T u n + 1 ) r max { p ( f z , f u n + 1 ) , p ( f u n + 1 , T u n + 1 ) + p ( f z , T z ) 2 , p ( f u n + 1 , T z ) + p ( f z , T u n + 1 ) 2 } r max { p ( f z , y n + 2 ) , p ( y n + 2 , y n + 2 ) + p ( f z , T z ) 2 , p ( y n + 2 , f z ) + p ( f z , T z ) p ( f z , f z ) + p ( f z , y n + 2 ) 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equam_HTML.gif
On taking limit as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq135_HTML.gif, we obtain
p ( f z , T z ) r 2 p ( f z , T z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equan_HTML.gif

We obtain p ( f z , T z ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq136_HTML.gif, which further implies that p S ( f z , T z ) 2 p ( f z , T z ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq137_HTML.gif. Hence, f z T z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq138_HTML.gif. Further if Y = X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq92_HTML.gif and f f z = f z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq139_HTML.gif, then due to IT-commutativity of the pair ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif, we have f z = f f z f T z T f z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq140_HTML.gif. This shows that fz is a common fixed point of the pair ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif. □

Corollary A Let ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space and Y be any non-empty set. Assume that here exists r [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq79_HTML.gif such that the mappings f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq74_HTML.gif and T : Y C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq75_HTML.gif satisfy
ω ( r ) p ( f x , T x ) p ( f x , f y ) H p ( T x , T y ) r p ( f x , f y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equao_HTML.gif

for all x , y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq81_HTML.gif, with T ( Y ) f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq87_HTML.gif. If there exists u 0 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq88_HTML.gif such that f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq89_HTML.gif is ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif-orbitally complete at u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq90_HTML.gif, then C ( f , T ) ϕ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq141_HTML.gif. If Y = X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq92_HTML.gif and ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif is IT-commuting at coincidence points of the pair ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif, then F ( f , T ) ϕ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq93_HTML.gif provided that fz is a fixed point of f for some z C ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq94_HTML.gif.

Example 2.2 Let X = { 0 , 1 , 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq142_HTML.gif and Y = { 0 , 1 , 2 , 3 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq143_HTML.gif. Define a mapping p : X × X R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq3_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equap_HTML.gif
Then p is a partial metric on X. Let ω ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq144_HTML.gif be as given in Theorem 2.1 and the mappings T : Y C B p ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq75_HTML.gif and f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq145_HTML.gif be given as
T x = { { 0 } when  x 2 , { 0 , 1 } when  x = 2 , and f x = { 0 , if  x { 0 , 1 } , 2 , if  x = 2 , 1 , if  x = 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equaq_HTML.gif
Note that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equar_HTML.gif
If we take r 3 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq146_HTML.gif and ω ( r ) 5 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq147_HTML.gif, then for all x , y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq81_HTML.gif,
ω ( r ) p ( f x , T x ) p ( f x , f y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equas_HTML.gif
holds. If we consider r = 5 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq148_HTML.gif, then ω ( r ) = 1 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq149_HTML.gif. Then, for x , y { 0 , 1 , 3 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq150_HTML.gif, we have H p ( T x , T y ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq151_HTML.gif, hence H p ( T x , T y ) r p ( f x , f y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq152_HTML.gif is satisfied trivially. Now consider
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equat_HTML.gif
Hence, for all x , y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq81_HTML.gif,
ω ( r ) p ( f x , T x ) p ( f x , f y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equau_HTML.gif
implies
H p ( T x , T y ) r p ( f x , f y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equav_HTML.gif

Let u 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq153_HTML.gif, y 0 = f ( 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq154_HTML.gif. As T ( 0 ) f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq155_HTML.gif, there exists a point u 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq156_HTML.gif in Y such that y 1 = f ( 1 ) = 0 T ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq157_HTML.gif and T ( 0 ) = { 0 } f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq158_HTML.gif, we obtain a point u 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq159_HTML.gif in Y such that y 2 = 0 = f ( 1 ) T ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq160_HTML.gif. Continuing this way, we construct an orbit { y 0 = y 1 = y 2 = = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq161_HTML.gif for ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif at u 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq153_HTML.gif. Also, f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq89_HTML.gif is ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif-orbitally complete at u 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq162_HTML.gif. So, all the conditions of Corollary A are satisfied. Moreover, C ( f , T ) = { 0 , 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq163_HTML.gif.

On the other hand, the metric p S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq164_HTML.gif induced by the partial metric p is given by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_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
ω ( r ) p S ( f 1 , T 1 ) = ω ( r ) p S ( 0 , 0 ) = 0 p S ( f x , f y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equax_HTML.gif
is satisfied for any r [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq79_HTML.gif, x and y in X, so it must imply H p S ( T 1 , T 2 ) r p ( f 1 , f 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq165_HTML.gif. But
H p S ( T 1 , T 2 ) = H p S ( { 0 } , { 0 , 1 } ) = 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equay_HTML.gif
and
p S ( f 1 , f 2 ) = p S ( 0 , 2 ) = 7 15 < 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equaz_HTML.gif
Hence, for any r [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq166_HTML.gif,
H p S ( T 1 , T 2 ) r p ( f 1 , f 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equba_HTML.gif
Corollary B Let ( X , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq10_HTML.gif be a partial metric space, Y be any non-empty set and f , T : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq167_HTML.gif be such that T ( Y ) f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq87_HTML.gif. Suppose that there exists u 0 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq88_HTML.gif such that f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq89_HTML.gif is ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif-orbitally complete at u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq90_HTML.gif. Assume further that there exists an r [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq166_HTML.gif such that
ω ( r ) p ( f x , T x ) p ( f x , f y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equbb_HTML.gif
implies that
p ( T x , T y ) r max { p ( f x , f y ) , p ( f x , T x ) + p ( f y , T y ) 2 , p ( f x , T y ) + p ( f y , T x ) 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equbc_HTML.gif

for all x , y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq81_HTML.gif. Then C ( f , T ) ϕ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq91_HTML.gif. Further, if Y = X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq92_HTML.gif and the pair ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq168_HTML.gif is commuting at x where x C ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq72_HTML.gif, then F ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq69_HTML.gif is a singleton.

Proof It follows from Theorem 2.1, that C ( f , T ) ϕ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq91_HTML.gif. If u C ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq169_HTML.gif, then f u = T u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq170_HTML.gif. Further, if Y = X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq92_HTML.gif and ( f , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq64_HTML.gif is commuting at u, then f f u = f T u = T f u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq171_HTML.gif. Now,
ω ( r ) p ( f u , T f u ) p ( f u , T f u ) = p ( f u , f f u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equbd_HTML.gif
implies that
p ( f u , f f u ) = p ( T u , T f u ) r M p , f ( u , f u ) r max { p ( f u , f f u ) , p ( f f u , T f u ) + p ( f u , T u ) 2 , p ( f f u , T u ) + p ( f u , T f u ) 2 } r max { p ( f u , f f u ) , p ( f f u , f f u ) + p ( f u , T u ) 2 , p ( f f u , f u ) + p ( f u , f f u ) 2 } r max { p ( f u , f f u ) , p ( f f u , f u ) + p ( f u , f f u ) 2 , p ( f f u , f u ) + p ( f u , f f u ) 2 } r p ( f u , f f u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Eqube_HTML.gif

As r < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq172_HTML.gif, we obtain p ( f u , f f u ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq173_HTML.gif, which further implies that p S ( f u , f f u ) 2 p ( f u , f f u ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq174_HTML.gif. Hence, fu is a common fixed point of f and T.

For uniqueness, assume there exist z 1 z 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq175_HTML.gif, such that z 1 = f z 1 = T z 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq176_HTML.gif and z 2 = f z 2 = T z 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq177_HTML.gif. Then
ψ ( r ) p ( f z 1 , T z 1 ) p ( f z 1 , T z 1 ) = p ( f z 1 , f z 1 ) p ( f z 1 , f z 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equbf_HTML.gif
which implies
p ( z 1 , z 2 ) = p ( T z 1 , T z 2 ) r max { p ( f z 1 , f z 2 ) , p ( f z 1 , T z 1 ) + p ( f z 2 , T z 2 ) 2 , p ( f z 2 , T z 1 ) + p ( f z 1 , T z 2 ) 2 } r max { p ( z 1 , z 2 ) , p ( z 1 , z 2 ) , p ( z 1 , z 2 ) } r p ( z 1 , z 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equbg_HTML.gif

We obtain p ( z 1 , z 2 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq178_HTML.gif, which further implies that p S ( z 1 , z 2 ) 2 p ( z 1 , z 2 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq179_HTML.gif. Hence, z 1 = z 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq180_HTML.gif. □

3 An application

In this section, we assume that U and V are Banach spaces, W U https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq181_HTML.gif and D V https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq182_HTML.gif. Suppose that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_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:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ9_HTML.gif
(3.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ10_HTML.gif
(3.2)
Then equations (3.1) and (3.2) can be reformulated as
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ11_HTML.gif
(3.3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_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 B ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq183_HTML.gif denote the set of all bounded real-valued functions on W. For an arbitrary h B ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq184_HTML.gif, define h = sup x W | h ( x ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq185_HTML.gif. Then ( B ( W ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq186_HTML.gif is a Banach space endowed with the metric d defined as d ( h , k ) = sup x W | h x k x | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq187_HTML.gif. Now, consider
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ13_HTML.gif
(3.5)

where h , k B ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq188_HTML.gif, b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq189_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq190_HTML.gif is a partial metric on B ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq183_HTML.gif. Let ω ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq144_HTML.gif be defined as in Section 1. Suppose that the following conditions hold:

(C1): G, F, g, and g https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq191_HTML.gif are bounded.

(C2): For x W https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq192_HTML.gif, h B ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq184_HTML.gif and b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq189_HTML.gif, define
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ14_HTML.gif
(3.6)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ15_HTML.gif
(3.7)
Moreover, assume that there exists r [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq166_HTML.gif such that for every ( x , y ) W × D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq193_HTML.gif, h , k B ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq188_HTML.gif and t W https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq194_HTML.gif,
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ16_HTML.gif
(3.8)
implies
| G ( x , y , h ( t ) ) G ( x , y , k ( t ) ) | r M p B , J ( h ( t ) , k ( t ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ17_HTML.gif
(3.9)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equbi_HTML.gif
(C3): For any h B ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq184_HTML.gif, there exists k B ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq195_HTML.gif such that for x W https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq192_HTML.gif,
K h ( x ) = J k ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equbj_HTML.gif
(C4): There exists h B ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq184_HTML.gif such that
K h ( x ) = J h ( x ) implies that J K h ( x ) = K J h ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equbk_HTML.gif

Theorem 3.1 Assume that the conditions (C1)-(C4) are satisfied. If J ( B ( W ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq196_HTML.gif is a closed convex subspace of B ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq183_HTML.gif, then the functional equations (3.3) and (3.4) have a unique, common and bounded solution.

Proof Note that ( B ( W ) , p B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq197_HTML.gif is a complete partial metric space. By (C1), J, K are self-maps of B ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq183_HTML.gif. The condition (C3) implies that K ( B ( W ) ) J ( B ( W ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_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 h 1 , h 2 B ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq199_HTML.gif. Choose x W https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq192_HTML.gif and y 1 , y 2 D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq200_HTML.gif such that
K h j < g ( x , y j ) + G ( x , y j , h j ( x j ) b + λ , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ18_HTML.gif
(3.10)
where x j = τ ( x , y j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq201_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq202_HTML.gif. Further, from (3.5) and (3.6), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ19_HTML.gif
(3.11)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ20_HTML.gif
(3.12)
Therefore, (3.8) in (C2) becomes
ω ( r ) p B ( K h 1 ( x ) , J h 1 ( x ) ) p B ( J h 1 ( x ) J h 2 ( x ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ21_HTML.gif
(3.13)
Then (3.13) together with (3.10) and (3.12) implies
K h 1 ( x ) K h 2 ( x ) < G ( x , y 1 , h 1 ( x 1 ) ) G ( x , y 1 , h 2 ( x 2 ) ) b + λ | G ( x , y 1 , h 1 ( x 1 ) ) G ( x , y 1 , h 2 ( x 2 ) ) | b + λ r M p B , J ( h ( t ) , k ( t ) ) b + λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ22_HTML.gif
(3.14)
Now, (3.10), (3.11) and (3.13) imply
K h 2 ( x ) K h 1 ( x ) G ( x , y 1 , h 2 ( x 2 ) ) G ( x , y 1 , h 1 ( x 1 ) ) b | G ( x , y 1 , h 1 ( x 1 ) ) G ( x , y 1 , h 2 ( x 2 ) ) | b r M p B , J ( h ( t ) , k ( t ) ) b . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ23_HTML.gif
(3.15)
From (3.14) and (3.15), we have
| K h 1 ( x ) K h 2 ( x ) | + b r M p B , J ( h ( t ) , k ( t ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ24_HTML.gif
(3.16)
As the above inequality is true for any x W https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq192_HTML.gif and λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq203_HTML.gif is taken arbitrarily, so from (3.13) we obtain
ω ( r ) p B ( K h 1 , J h 2 ) p B ( J h 1 , I h 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ25_HTML.gif
(3.17)
implies
p B ( K h 1 , K h 2 ) r M p B , J ( h ( t ) , k ( t ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_Equ26_HTML.gif
(3.18)

Therefore, by Corollary B, the pair ( K , J ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq204_HTML.gif has a common fixed point h https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_355_IEq205_HTML.gif, that is, h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-21/MediaObjects/13663_2012_Article_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

References

  1. Baskaran R, Subrahmanyam PV: A note on the solution of a class of functional equations. Appl. Anal. 1986, 22(3–4):235–241. 10.1080/00036818608839621MathSciNetView Article
  2. Markin J: A fixed point theorem for set valued mappings. Bull. Am. Math. Soc. 1968, 74: 639–640. 10.1090/S0002-9904-1968-11971-8MathSciNetView Article
  3. Nadler SB: Multi-valued contraction mappings. Pac. J. Math. 1969, 30: 475–488. 10.2140/pjm.1969.30.475MathSciNetView Article
  4. Ćirić L: Fixed points for generalized multi-valued contractions. Mat. Vesn. 1972, 9: 265–272.
  5. Ćirić L: Multi-valued nonlinear contraction mappings. Nonlinear Anal. 2009, 71: 2716–2723. 10.1016/j.na.2009.01.116MathSciNetView Article
  6. Covitz H, Nadler SB: Multi-valued contraction mappings in generalized metric spaces. Isr. J. Math. 1970, 8: 5–11. 10.1007/BF02771543MathSciNetView Article
  7. Daffer PZ, Kaneko H: Fixed points of generalized contractive multi-valued mappings. J. Math. Anal. Appl. 1995, 192: 655–666. 10.1006/jmaa.1995.1194MathSciNetView Article
  8. Reich S: Fixed points of contractive functions. Boll. Unione Mat. Ital. 1972, 5: 26–42.
  9. Semenov PV: Fixed points of multi-valued contractions. Funct. Anal. Appl. 2002, 36: 159–161. 10.1023/A:1015682926496MathSciNetView Article
  10. Naimpally SA, Singh SL, Whitfield JHM: Coincidence theorems for hybrid contractions. Math. Nachr. 1986, 127: 177–180. 10.1002/mana.19861270112MathSciNetView Article
  11. Singh SL, Mishra SN: Nonlinear hybrid contractions. J. Natur. Phys. Sci. 1991/1994, 5/8: 191–206.MathSciNet
  12. Singh SL, Mishra SN: On a Ljubomir Ćirić fixed point theorem for nonexpansive type maps with applications. Indian J. Pure Appl. Math. 2002, 33: 531–542.MathSciNet
  13. Singh SL, Mishra SN: Coincidence theorems for certain classes of hybrid contractions. Fixed Point Theory Appl. 2010., 2010: Article ID 898109
  14. Singh SL, Mishra SN: Remarks on recent fixed point theorems. Fixed Point Theory Appl. 2010. doi:10.1155/2010/452905
  15. Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 2008, 136: 1861–1869.View Article
  16. Ali B, Abbas M: Suzuki type fixed point theorem for fuzzy mappings in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 9
  17. Ćirić L, Abbas M, Rajović M, Ali B: Suzuki type fixed point theorems for generalized multi-valued mappings on a set endowed with two b-metrics. Appl. Math. Comput. 2012, 219: 1712–1723. 10.1016/j.amc.2012.08.011MathSciNetView Article
  18. Dhompongsa S, Yingtaweesittikul H: Fixed points for multi-valued mappings and the metric completeness. Fixed Point Theory Appl. 2009., 2009: Article ID 972395
  19. Kikkawa M, Suzuki T: Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Anal. 2008, 69: 2942–2949. 10.1016/j.na.2007.08.064MathSciNetView Article
  20. Kikkawa M, Suzuki T: Some similarity between contractions and Kannan mappings. Fixed Point Theory Appl. 2008., 2008: Article ID 649749
  21. Moţ G, Petruşel A: Fixed point theory for a new type of contractive multi-valued operators. Nonlinear Anal. 2009, 70: 3371–3377. 10.1016/j.na.2008.05.005MathSciNetView Article
  22. Matthews SG: Partial metric topology. Ann. New York Acad. Sci. 728. Proc. 8th Summer Conference on General Topology Appl. 1994, 183–197.
  23. Bari CD, Vetro P: Fixed points for weak φ -contractions on partial metric spaces. Int. J. Eng. Contemp. Math. Sci. 2011, 1: 5–13.
  24. Paesano D, Vetro P: Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces. Topol. Appl. 2012, 159: 911–920. 10.1016/j.topol.2011.12.008MathSciNetView Article
  25. Ćirić L, Samet B, Aydi H, Vetro C: Common fixed points of generalized contractions on partial metric spaces and an application. Appl. Math. Comput. 2011, 218: 2398–2406. 10.1016/j.amc.2011.07.005MathSciNetView Article
  26. Heckmann R: Approximation of metric spaces by partial metric spaces. Appl. Categ. Struct. 1999, 7: 71–83. 10.1023/A:1008684018933MathSciNetView Article
  27. Romaguera S: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 493298
  28. Schellekens MP: The correspondence between partial metrics and semivaluations. Theor. Comput. Sci. 2004, 315: 135–149. 10.1016/j.tcs.2003.11.016MathSciNetView Article
  29. Aydi H, Abbas M, Vetro C: Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces. Topol. Appl. 2012, 159: 3234–3242. 10.1016/j.topol.2012.06.012MathSciNetView Article
  30. Zamfirescu T: Fixed point theorems in metric spaces. Arch. Math. 1972, 23: 292–298. 10.1007/BF01304884MathSciNetView Article
  31. Bellman R Mathematics in Science and Engineering 61. In Methods of Nonlinear Analysis. Vol. II. Academic Press, New York; 1973.
  32. Bellman R, Lee ES: Functional equations in dynamic programming. Aequ. Math. 1978, 17: 1–18. 10.1007/BF01818535MathSciNetView Article
  33. Bhakta PC, Mitra S: Some existence theorems for functional equations arising in dynamic programming. J. Math. Anal. Appl. 1984, 98: 348–362. 10.1016/0022-247X(84)90254-3MathSciNetView Article
  34. Pathak HK, Cho YJ, Kang SM, Lee BS: Fixed point theorems for compatible mappings of type P and applications to dynamic programming. Matematiche 1995, 50: 15–33.MathSciNet
  35. Altun I, Simsek H: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud. 2008, 1: 1–8.MathSciNet
  36. Altun I, Sola F, Simsek H: Generalized contractions on partial metric spaces. Topol. Appl. 2010, 157: 2778–2785. 10.1016/j.topol.2010.08.017MathSciNetView Article
  37. Abbas M, Nazir T: Fixed point of generalized weakly contractive mappings in ordered partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 1
  38. Bukatin MA, Shorina SY, et al.: Partial metrics and co-continuous valuations. Lecture Notes in Comput. Sci. 1378. In Foundations of Software Science and Computation Structure. Edited by: Nivat M. Springer, Berlin; 1998:125–139.View Article
  39. Romaguera S, Valero O: A quantitative computational model for complete partial metric spaces via formal balls. Math. Struct. Comput. Sci. 2009, 19: 541–563. 10.1017/S0960129509007671MathSciNetView Article

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