Fixed Point Theorems for Geraghty Type Contractive Mappings and Coupled Fixed Point Results in 0-Complete Ordered Partial Metric Spaces

Abstract

We establish new fixed point theorems in 0-complete ordered partial metric spaces. Also, we give remark on coupled generalized Banach contraction. Some examples illustrate the usability of our results. The theorems presented in this paper are generalizations and improvements of the several well known results in the literature.

1. Introduction and Preliminaries

Henceforward, the letters , , and will indicate the set of real numbers, the set of nonnegative real numbers, and the set of positive integer numbers, respectively.

Definition 1 (see [1]). A partial metric on a nonempty set is a function such that, for all , () , () , () , and () . The pair is called a partial metric space.

If is a partial metric on , then the function given by is a metric on . Each partial metric on introduces a topology on which has as a base the family of open balls for all and

Let be a partial metric space, and let be any sequence in and . Then (i) a sequence is convergent to with respect to , if as , (ii) a sequence is a Cauchy sequence in if exists and is finite; (iii) is called complete if for every Cauchy sequence in there exists such that as

Romaguera [2] introduced the notion of 0-Cauchy sequence, 0-complete partial metric spaces and proved some characterizations of partial metric spaces in terms of completeness and 0-completeness. After that many authors extended the results of [2] and studied fixed point theorems in 0-complete partial metric space (see [2–10]).

Definition 2 (see [2]). Let be a partial metric space. A sequence in is called a 0-Cauchy sequence if as . The space is said to be 0-complete if every 0-Cauchy sequence in converges with respect to to a point such that

Remark 3 (see [11, 12]). (1) Let be a partial metric space. If as , then as for all .
(2) If is a continuous at , then for each sequence in , we have as as (see [5]).

Let be the class of functions with implying Amini-Harandi and Emami [13] presented the following results.

Theorem 4 (see [13]). Let be an ordered set endowed with a metric and let be a given mapping. Suppose that the following conditions hold:(i) is complete.(ii) (1) is continuous or(2)if a nondecreasing sequence in converges to some point , then for all .(iii) is nondecreasing.(iv)There exists such that .(v)There exists such that for all with ,Then has a fixed point. Moreover, if for all there exists a such that and , we obtain uniqueness of the fixed point.

In this paper, we establish new fixed point theorems in 0-complete ordered partial metric spaces (briefly 0-COPMS). Also, we give remark on coupled generalized Banach contraction. Some examples illustrate the usability of our results. The theorems presented in this paper are generalizations and improvements of the several well known results in the literature.

2. Main Results

Theorem 5. Let be a 0-COPMS. Let be a nondecreasing mapping such thatfor all with and . Also suppose that there exists such that . One assumes (1) is continuous or(2)if a nondecreasing sequence in converges to some point , then for all .Then has a fixed point .

Proof. By assumption there exists such that . Define as . Then we have . In a similar manner, we get as . In that case, . Continuing this procedure we have in such thatIf for some , then the proof is completed. Suppose farther that for each . Consider, as is nondecreasing, we obtain thatFrom (2), (3), and (4), for all , we get thatThen is a monotone decreasing. Hence as . Assume . Then by (2) we haveEquation (6) yields as . By virtue of , this implies thatNow we claim that is a 0-Cauchy sequence. Conversely, suppose thatBy and (2), we have, for ,Owing to (7) and (8), we get thatfrom which we have which implies . Since , we obtain . It is a contradiction. Thus is a 0-Cauchy sequence. As is 0-complete, it follows that there exists such that in and . Furthermore,We will show that . Consider two cases.
Case 1. If is continuous, thenhence .
Case 2. If (2) holds, then,In view of as , then we have

The following is example which illustrate Theorem 5 and that the generalizations are proper.

Example 6. Let , and let be defined by for all . Then is a 0-COPMS. Yet it is not complete partial metric space. We endow with the partial orderLet for all . Then it is clear that . Define asAssume that . Then we have two cases.
Case 1. If , thenTherefore, we haveHence, for , .
Case 2. If , thenHence, we getwhereThus, for , .
Moreover, by Cases 1 and 2, it is clear that both assumptions (1) and (2) of Theorem 5 are satisfied, and for , we have . Hence, all assumptions of Theorem 5 are satisfied, and has a fixed point
On the contrary, consider Example 6 in the standard metric . If and , thenand soThus, is not satisfied.

3. Remark on Coupled Generalized Banach Contraction

The following result generalizes and extends Theorem 2.1. in [14]. When making the proof of the theorem, Radenović’s technique [15] is used.

Theorem 7. Let be a 0-COPMS and let be a mapping. Suppose that, for all and , the following conditionholds. Then has a fixed point.

Proof. Consider the metrics defined byIf is complete, then is complete (resp. 0-complete), too. Now, define the operator byLet be a metric on defined by for all
From (23), for all , with and , we getThis implies that for all , with and ,that is,which is in fact condition (2), for all , with . Hence, all conditions of Theorem 5 are satisfied. In this case, applying Theorem 5, we have that has a fixed point. From the definition of , we have and ; that is to say, is a coupled fixed point of

Remark 8. If ,   in the inequality (23), then we obtain results of Bhaskar and Lakshmikantham [16] in 0-COPMS.

The following example illustrates the case when Theorem 7 is applicable, while Theorem 2.1. in [14] is not.

Example 9. Let , and let be defined by for all . Then is a 0-COPMS. Yet it is not complete partial metric space. We consider the following order relation on : Let for all . Then it is clear that. Define as for all
We have the following cases.
Case 1. For or or and , we have . Thus, (23) holds.
Case 2. For and , we haveThus, (23) holds.
Case 3. For and , we haveThus, (23) holds.
Therefore, all the conditions of Theorem 7 are satisfied and is a coupled fixed point of .

Competing Interests

The author declares that she has no competing interests.