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Journal of Function Spaces
Volume 2016 (2016), Article ID 8404529, 9 pages
http://dx.doi.org/10.1155/2016/8404529
Research Article

A New Nonlinear Contraction Principle in Partial Metric Spaces

1School of Mathematics and Physics, XuZhou Institute of Technology, Xuzhou 221008, China
2School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

Received 15 August 2016; Revised 19 October 2016; Accepted 26 October 2016

Academic Editor: Nawab Hussain

Copyright © 2016 Yi Zhang and Jiang Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a new nonlinear contraction principle on partial metric spaces and prove the existence of common fixed point. We also give some examples to show our results and apply our results to study the existence of common bounded solution of the system of functional equations.

1. Introduction

There exist many generalizations of the well-known Banach contraction mapping principle in the literature. In particular, Matthews [1, 2] introduced the notion of a partial metric space as part of the study of denotational semantics of dataflow networks, showing that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. After that, fixed point results in partial metric spaces have been studied by many authors. For example, Matthews [2] proved a Banach fixed point theorem for a self-map of a partial metric spaces under the following contractive condition:where Altun and Erduran [3] improved (1) by the following contractive condition:where is a continuous nondecreasing function such that for all and proved the existence of the unique fixed point. Altun et al. [4] and Altun and Sadarangani [5] improved (2) by the following contractive condition:where is continuous nondecreasing function such that and the series converges for all and proved the existence of the unique fixed point. Romaguera [6] improved (3) by assuming that is a right upper semicontinuous function such that for all and proved the existence of the unique fixed point without assuming that the series converges for all . Abdeljawad et al. [7] proved the existence of fixed point for self-maps of under the following:where are continuous nondecreasing function such that if and only if Abdeljawad et al. [8] proved the existence of common fixed point for self-maps and of under the following:where , is a continuous, nondecreasing function with for all and

Recently, Haghi et al. made [9] a paper which stated that we should “be careful on partial metric fixed point results” along with giving some results. The authors noted that some fixed point theorems in partial metric spaces can be obtained from the corresponding results in metric spaces. Abdeljawad et al. [10] used the technique of Haghi et al. in [9] to partial -metric spaces. Reminded of the warning of Haghi et al. [9], one can see that the main characteristic of a partial metric space is that the self-distance of a point may be not zero. But the all above contraction conditions are not proper for the characteristics. For example, if is a partial metric space and satisfies , then the all above contraction conditions do not hold at any fixed point in this partial metric space . This problem also shows that the abovementioned contractive conditions are only metric type contractive conditions but do not reflect the structure of a partial metric space. On the other hand, Ilić et al. [11] proved a more reasonable contraction principle in the partial metric space in which they used self-distance terms. They gave the following linear contractive condition allowing use of self-distance in the contractive condition so that completeness, rather than the 0-completeness, of the partial metric space is needed:where . After that, Chi et al. [12] gave the following linear generalization of (6): , where and . Abdeljawad and Zaidan [13] gave the following linear generalization of (6): where is an increasing function such that is increasing with being right continuous at . Also assume for all (and hence and for ). The method of self-distance term is also used in [1416]. In [17], the authors formulated the concept of -metric spaces as a generalization of partial metric spaces, as well as a generalization of metric spaces. It is worth mentioning that the technique of Haghi et al. in [9] also is not applicable in -metric spaces and the contraction condition presented in this paper is also not reflecting the structure of a -metric space. After that, very recently, Abodayeh et al. [18] succeeded in studying -metric and partial metric spaces topologically and proved a Caristi type fixed point theorem there by using self-distance term in their contractive conditions. For more general fixed point theorems, one can refer to [19, 20]. In [19], the authors proved some common fixed point theorems for four mappings in partial metric spaces by using weakly compatible concept. In [20], the authors proved some fixed point theorems in more general ordered partial -metric spaces. But these results still have similar gap between the contractive conditions and the structure of the spaces.

Inspired by the above works, in the present manuscript, we will focus on presenting a new nonlinear contraction principle on partial metric spaces for two mappings by using self-distance term in contractive conditions and proving the existence of common fixed point. Our result generalize and improve the results in [28, 1114]. We also give some examples to show our results and apply our results to study the existence of common bounded solution of the system of functional equations arising in dynamic programing.

2. Preliminaries

A partial metric space (see, e.g., [1]) is a pair (where denotes the set of all nonnegative real numbers) such that for all (PM1);(PM2) if and only if ;(PM3);(PM4).

Each partial metric on generates a topology on with a base of the family of open -balls , where for all and .

In a partial metric space, the concepts of convergence, the Cauchy sequence, completeness, and continuity are defined as follows.

Definition 1 (see [1, 2]). (i) A sequence in a partial metric space converges to if and only if .
(ii) A sequence in a partial metric space is called Cauchy if exists and is finite.
(iii) A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .

Example 2. Let and define , for all Then is a complete partial metric space. It is clear that is not a (usual) metric.

3. Main Results

Let be a partial metric space, , be self-maps of a set , and for any . For any , the sequence defined by and , for , is called an -iteration sequence of We first present the following contraction condition.

(C1) , for any , where is upper-semicontinuous (i.e., ), such that for any .

Theorem 3. Let , be self-mappings of a complete partial metric space such that (C1) holds; then , have a common fixed point .

Proof. Fix . Let be -iteration sequence of . For define
Step one: we prove that is a decreasing sequence; that is, for any ,(1) Assume that . Then , andIt follows from (C1) that If , we have that This implies that . If , we have that Thus, we have proved that .
(2) Assume that ; using a similar argument as in (1), one can verify that .
Combining the results of (1) and (2), we may conclude that (10) holds. Hence there exists , such that
Step two: we prove that Assume that Note that , and we can obtain that Since is decreasing, by the proof of Step one, we have that If there are infinite number of , such that then we have that . Thus, by letting we can obtain that . This implies that , so we have that . Thus, Therefore, if , then there exists , such that, for any , so we have that This has two cases. Case one: there exists , such that , for any , or , for any . In this case we have that and, thus, . Case two: if case one does not hold, then there are subsequences , such that , , and In this case we also have thatFrom this, we obtain that . By the conclusion of Step one, we know that for any . Then, by case one and case two, we obtain that there exists positive integer , such that for any .
Step three: we prove thatBy the conclusion of Step two, we get that for any . Thus, if (24) does not hold, then there exist and increasing sequence of integers and such that, for all integers , Since , we have that when is large enough. So we can assume that is the minimum positive integer such that (25) holds. Then we have thatUsing (25), (27), and the triangle inequality for partial metric we have Thus,Again, the triangle inequality gives us Thus,Letting , we getSimilarly, we can prove thatputtingBy use of (29), (32), and (33), we have And . But, by using (C1), we know that Thus, by letting , we get This yields a contradiction. Therefore, is a Cauchy sequence. By the completeness of is a convergent sequence. Then there exists such that Step four: we prove that For any , we have that This implies thatAgain note thatand we obtain that This implies that . Similarly, one can prove that .
Step five: let and , we prove that there exists , such that .
For any , we take , such that Assume that is the -iterative sequence of and converges to . By using the conclusions of Step one and Step four, we obtain that This shows that, by using the conclusion of Step two,Thus, we have that This implies that Sincewe obtain that This shows that ; that is, is a Cauchy sequence. Since the partial metric space is completed, there is , such thatIt follows from thatOn the other hand, we know that Then by letting we get that This implies that , so we have thatSimilarly, one can prove thatAssume that is the -iterative sequence of and converges to By using the conclusions of Step one and Step two, we get thatand, thus, .
Step six: we prove that , have unique common fixed point with .
By using the conclusions of Step one, Step two, and Step five, we know that there is , such that for any and Thus, we have thatIt follows from (61) that for and this shows that that is Then we have proved that is a common fixed point of , Assume that are two common fixed points of , , with the properties of . Clearly, , since the -iterative sequence of a fixed point is a constant sequence. By using (C1) we have that and this implies that . Thus, . The proof is completed.

Remark 4. If is a metric space, then contraction condition (C1) reduces to the corresponding contraction conditions on metric spaces. Thus, the corresponding results in the set of a metric space is a special case of our results. But, in a partial metric space, since the partial metric space topology is only topology, the limit of a sequence in a partial metric space is not unique (refer to Remark  7 in [18]), and this induced that the fixed point may not be unique. In Theorem 3, we can only get that , have unique common fixed point .

Remark 5. In Theorem 3, if , then the condition “ is upper-semicontinuous” can be replaced by “ is upper-semicontinuous from the right.” It is only needed to modify the proof of Theorem 3.

Remark 6. If , the conclusion of Theorem 3 is a generalization and improvement for the results in [26, 1113].

Remark 7. In Theorem 3, if is replaced by , where are continuous function such that if and only if , then the conclusion of Theorem 3 in this case is a generalization and improvement for the results in [7, 8].

Remark 8. Let be a complete partial -metric space and be a self-mapping on (see [14]), if we define , , then is a complete partial metric space, and the contraction conditionof Theorem  2.1 in [14] will become The contraction condition of Theorem  2.2 in [14] will become Therefore, our Theorem 3 is a generalization and improvement for the results in [14].

Remark 9. Some nice results for more general fixed point theorems were proved in [19, 20]. In [19], the authors proved some common fixed point theorems for four mappings in partial metric spaces by using weakly compatible concept and some control functions, but these control functions are different from upper-semicontinuous ones. In [20], the authors proved some fixed point theorems in more general ordered partial -metric spaces. However, there is still the problem mentioned in the Introducion. That is, the contractive conditions presented in these papers do not reflect the structure of the partial metric space or partrial metric space. Thus, our results can not be deduced from the results in [19, 20]. But the results in [19, 20] also show that there is still more greater research space to prove some contraction principles with the contractive conditions includeing self-distance terms.

The following simple example is an illustration of our extension.

Example 10. Suppose that and . Then is a partial metric space. Suppose that such that for all and and , for any . Then, for all with , we have that Therefore, the conditions of Theorem 3 are satisfied. Indeed and are the fixed point. But Thus, we cannot apply the results of [28] to this example.

Example 11. Suppose that and . Then is a partial metric space. Define by and for all and define by , for all . Then, for all , we have If , then . If and , we have that Thus, we cannot apply the results of [28] to this example. But in both cases and , and we always have that holds. Thus, the conditions of Theorem 3 are satisfied and , have a common fixed point .

4. An Application to a Dynamical Process

Generally, a dynamical process consists of a state space and a decision space. The state space is the set of the initial state, actions, and transition model of the process; the decision space is the set of possible actions that are allowed for the process.

In this section we assume that and are Banach spaces; is a state space and is a decision space. It is well-known that the dynamic programing provides useful tools for mathematical optimization and computer programing as well. In particular, the problem of dynamic programing related to multistage process reduces to the problem of solving the system of functional equationswhere and signify the state and decision vectors, , , and () represent the transformations of the process, and , denote the optimal return functions with the initial state However, for the detailed background of the problem, the reader can refer to [21, 22]. Here, we study the existence of the common bounded solution of the system of functional equations (76).

Let denote the set of all bounded real-valued functions on and, for an arbitrary , define . Clearly, endowed with the metric defined by is a Banach space. Indeed, the convergence in the space with respect to is uniform. Thus, if we consider a Cauchy sequence in , then converges uniformly to a function, for example, , that is bounded and so Define for all and a fixed positive number . Then is a complete partial metric space.

We define byfor all and . Obviously, if the functions and , are bounded then , are well-defined.

We will prove the following theorems.

Theorem 12. Letwhere , . Suppose the following conditions are satisfied:() for all , for all and , where is upper-semicontinuous, such that for any . Then system (76) has a common bounded solution.

Proof. Let be an arbitrary positive number, , and . From (79), there exist such that Then, from (82) and (85), it follows easily that and, similarly, from (83) and (84) we have Therefore, we have We haveSince the above inequality does not depend on and is taken arbitrary, then we conclude immediately that Then all the hypotheses of Theorem 3 are satisfied with , . Thus we deduce that , have a common fixed point That is, the system of functional equations (76) has a common bounded solution.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All authors read and approved the final manuscript.

Acknowledgments

The authors are very grateful to Professor N. Mlaiki for sending his papers: “N. Mlaiki, A Partially Alpha-Contractive Principle, J. Adv. Math. Stud. Vol 7 (1), 121–126 (2014)” and “Thabet Abdeljawad, Kamal Abo Dayeh, Nabil Mlaiki, On Fixed Point Generalizations to Partial -Metric Spaces, Journal of Computational Analysis and Application, vol. 19 (5) 883–891 (2015),” to us for helping us in modifying the present manuscript. Finally, this work was supported by the National Natural Science Foundation of China (11171286).

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