Fixed Points of -Admissible Mappings on Partial Metric Spaces

Abstract

In this paper, a general class of -admissible contractions on partial metric spaces is introduced. Fixed point theorems for these contractions on partial metric spaces and their consequences are stated and proved. Illustrative example is presented.

1. Introduction and Preliminaries

A rapid progress in the fixed point theory has been observed in the last few decades. This is a consequence of the fact that fixed point theory is a major tool in nonlinear analysis and has application in almost all branches of mathematics and natural sciences.

In 1992 Matthews ([1, 2]) introduced a new type of a metric called partial metric and a corresponding space called partial metric space (PMS), which have been defined due to a need in computer sciences. Partial metric spaces have been studied extensively since then; see [3–11] and references therein.

Improvement and generalization of the contractive conditions on the mappings are main concerns of most of the studies in fixed point theory. Such improvements and generalizations are usually done by means of auxiliary functions. Altering distance functions defined by Khan et al. [12] have been widely used for this reason both alone and combined with other functions.

In what follows, we employ two types of functions to define a class of contractions on partial metric spaces and investigate the existence and uniqueness of fixed points for these maps.

First, we introduce some basic concepts and notations to be used throughout the paper. We will denote by the set of natural numbers, denote by the set of nonnegative integers, and denote by the set of nonnegative real numbers.

Definition 1 (see [12]). An altering distance function is a function which satisfies the following.(1)is continuous and nondecreasing.(2).

Partial metric space has been defined by Matthews as follows (See [1]).

Definition 2. Let be a nonempty set and let satisfy for all , , and . Then the pair is called a partial metric space and is called a partial metric on .

One can easily see that the function , defined by is a metric on . Moreover, every partial metric on generates a topology on , whose base is a family of open -balls where , for all and .

Topological concepts such as convergence, Cauchy sequence, completeness, and continuity on PMS have also been defined in [1] as follows.

Definition 3. (1) A sequence in the PMS converges to the limit if and only if .
(2) A sequence in the PMS is called a Cauchy sequence if exists and is finite.
(3) A PMS is called complete if every Cauchy sequence in converges with respect to , to a point such that .
(4) A mapping is said to be continuous at if for every , there exists such that .

Remark 4. The limit of a sequence in a partial metric space may not be unique.

We give next some basic results in PMS.

Lemma 5 (see [1, 2, 6]). (1) A sequence is a Cauchy sequence in the PMS if and only if it is a Cauchy sequence in the metric space .
(2) A PMS is complete if and only if the metric space is complete.
Moreover,

Lemma 6 (see [7, 9]). Assume as in a PMS such that . Then for every .

Lemma 7 (see [7, 9]). Let be a complete PMS.(A)If , then ;(B)If , then .

Admissible mappings have been defined recently by Samet et al. [13] and employed quite often in order to generalize the results on various contractions, see [14–17]. We state next the definitions of -admissible mapping and triangular -admissible mappings.

Definition 8. A mapping is called -admissible if for all we have where is a given function.

Definition 9. A mapping is called triangular -admissible if it is -admissible and satisfies where and is a given function.

In [16], Alsulami et al. defined the following weaker condition which is sufficient in the proof of existence and uniqueness theorems.

Definition 10. A mapping is said to be weak triangular -admissible if it is -admissible and satisfies where and is a given function.

Weak triangular -admissible mappings satisfy a property stated in the following Lemma the proof of which easily follows from the definition and can be found in [15].

Lemma 11 (see [15]). Let be a weak triangular -admissible mapping. Assume that there exists such that . If , then for all with .

2. Fixed Point Theorems on Complete Partial Metric Spaces

Our main results include theorems on existence and uniqueness of fixed points for a class of weak triangular -admissible mappings defined on partial metric spaces. Inspired by a recent study of Alsulami et al. [16] and Yan et al. [18], we define a class of -admissible contractions on a PMS via auxiliary functions and discuss the existence and uniqueness of their fixed points.

Our main theorem is stated below.

Theorem 12. Let be a complete partial metric space. Let be a continuous, weak triangular -admissible mapping such that where is an altering distance functions, is a continuous function satisfying , for all , and If there exists such that , then has a fixed point.

Proof. Take which satisfies and define the sequence as for .
If for some , then obviously, a fixed point of . Suppose that , for all .
Since is -admissible and , we deduce and continuing in this way, we get Due to (10) we can put and in (7) which gives where If for some , then the inequality (11) becomes which is not possible since for . Then, we should have for all and, thus, which gives since is a nondecreasing function. Therefore, the sequence is a decreasing sequence bounded below by and hence converges to a limit; say . Taking limit as in (11), we get However, since by definition of and we have for , the above inequality is possible only for , that is, On the other hand, by (PM2), we have or upon letting , that is, We prove next that is a Cauchy sequence in the metric space , where is the metric defined in (2) associated with the partial metric . Assume that is not Cauchy. Then, for some there exist subsequences and of with for all , where corresponding to each , we choose to be smallest integer for which (21) holds. Then Note that from we have Using triangle inequality and regarding (21) and (22), we obtain Letting in the above inequality and using (24), we get On the other hand, we also have Again by letting and using (24) and (26), we get From (26) and (28) and using (20) it is easy to see that or
Thus, the limit of as , is calculated as due to (17) and (30). Recall that is weak triangular -admissible. Then, from Lemma 11 we have . Therefore, we can apply condition (7) with and to obtain Letting and taking into account (30) and (32), we have Note however that the condition , for implies that the above inequality holds only if , or, equivalently, which contradicts the assumption that is not a Cauchy sequence. Thus, must be a Cauchy sequence in the metric space . By Lemma 5, the sequence is also a Cauchy sequence in the PMS which is a complete PMS. Again by Lemma 5, is a complete metric space. Therefore, there exists such that Notice that from Lemma 5 we also have Finally, the continuity of gives that is, is a fixed point of , which completes the proof.

The continuity condition on -admissible mappings is not required for the existence of a fixed point if the space under consideration has the following property.

(I) If is a sequence in such that then there exists a subsequence of for which

Under this condition, we can state another existence theorem as follows.

Theorem 13. Let be a complete PMS on which the condition (I) holds. Let be a weak triangular -admissible mapping such that where is an altering distance functions, is a continuous function satisfying for all and If there exists such that , then has a fixed point.

Proof. As in the proof of Theorem 12, we take which satisfies and define the sequence as for . The proof of convergence of this sequence to a limit is exactly the same as the proof of Theorem 12. Since , then the condition (I) implies , for all . Applying the inequality (40) with and we get where Taking limit as and regarding the continuity of and , we get Again, using the fact that , for , we conclude that and hence, from Lemma 7, , which completes the proof.

For the uniqueness of fixed points of -admissible contractions we need an extra condition. This condition reads as follows: We prove the uniqueness of a fixed point for a subclass of contractions defined in Theorems 12 and 13. The reason for this is that the condition (I) is not sufficient for the uniqueness of fixed points of maps defined in these two theorems.

Theorem 14. Let be a complete partial metric space satisfying the condition (II). Let be a weak triangular -admissible mapping such that where is an altering distance functions, is a continuous function satisfying for all . Assume that either is continuous or satisfies the condition (I). If there exists such that , then has a unique fixed point.

Proof. The existence proof is similar to that of Theorem 12 (resp., Theorem 13) and hence we omit the details. To show the uniqueness, we assume that has two different fixed points; say . From the condition (II), there exists , such that Then, since is -admissible, we have from (47) for all . Define the sequence as . If for some , then, , that is, the sequence converges to the fixed point . Assume that for all . Applying (46) with and we get Since is nondecreasing, then for all . Thus, the sequence is a positive non increasing sequence and hence, converges to a limit say . Taking limit as in (49), and regarding continuity of and , we deduce which is possible only if . Hence, we conclude that
In a similar way, we obtain By Lemma 6 and (51) and (52), it follows that and using the fact that , the condition (PM1) implies , which completes the proof of uniqueness.

3. Consequences and an Example

The class of contractions defined in Theorems 12 and 13 is quite general and many particular results can be concluded from these theorems. Some of these conclusions are stated below.

Corollary 15. Let be a complete PMS. Let be weak triangular -admissible mapping such that where and Assume that either is continuous or satisfies the condition (I). If there exists such that , then has a fixed point.

Proof. Proof is obvious by choosing and in Theorem 12.

Corollary 16. Let be a complete PMS. Let be a weak triangular -admissible mapping such that for all , where . Assume that either is continuous or satisfies the condition (I). If there exists such that , then has a fixed point.

Proof. Due to the fact that proof follows from Corollary 15.

Last, we give the following example to illustrate our results.

Example 1. Let , and define and on as respectively. Let and and be defined as Clearly, is continuous. Then for with we have and similarly, for with , For , , or the contractive condition of Theorem 12 is already satisfied since in this case . In addition, for we have Since all conditions of Theorem 12 hold, then has a fixed point which clearly is .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.