#### Abstract

Recently, a number of fixed point theorems for contraction type mappings in partial metric spaces have been obtained by various authors. Most of these theorems can be obtained from the corresponding results in metric spaces. The purpose of this paper is to present certain fixed point results for single and multivalued mappings in partial metric spaces which cannot be obtained from the corresponding results in metric spaces. Besides discussing some useful examples, an application to Volterra type system of integral equations is also discussed.

*Dedicated to Professor Shyam Lal Singh on his 75th birthday*

#### 1. Introduction and Preliminaries

Throughout this paper , , and denote the set of all natural numbers, the set of all real numbers, and the set of all nonnegative real numbers, respectively.

The well-known Banach contraction theorem (BCT) has been generalized and extended by many authors in various ways. In 1974, Ćirić [1] introduced the notion of* quasi-contraction* and obtained a forceful generalization of Banach contraction theorem.

*Definition 1. *A self-mapping of a metric space is a quasi-contraction if there exists a number such that, for all ,where ,

Theorem 2 (see [1]). *A quasi-contraction on a complete metric space has a unique fixed point.*

We remark that a quasi-contraction for a self-mapping on a metric space is considered as the most general among contractions listed by Rhoades [2].

In 2006, Proinov [3] established an equivalence between two types of generalizations of the BCT. The first type involves Meir-Keeler [4] type contraction conditions and the second type involves Boyd and Wong [5] type contraction conditions. Further, generalizing certain results of Jachymski [6] and Matkowski [7] he obtained the following general fixed point theorem, which extends Ćirić’s quasi-contraction.

Theorem 3 (see [3], Th. ). *Let be a continuous and asymptotically regular self-mapping on a complete metric space satisfying the following conditions: ** for all ,**, whenever ,**where , a function satisfying the following: for any there exists such that implies , , andThen has a unique fixed point.**Moreover if and is continuous with for all , then the continuity of can be dropped.*

A mapping satisfying the conditions (P1) and (P2) is called a Proinov contraction. The following example shows the generality of Proinov contraction over quasi-contraction.

*Example 4 (see [8]). *Let with the usual metric and such thatThe mapping does not satisfy the condition . However, satisfies the conditions (P1) and (P2) with , where .

On the other hand, in 1994, Matthews [9] introduced the notion of partial metric spaces to study the denotational semantics of dataflow networks. It is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation (see [10] and references thereof). Matthews also obtained the partial metric version of Banach contraction theorem. Subsequently, many authors studied partial metric spaces and their topological properties and obtained a number of fixed point theorems for single and multivalued mappings (cf. [9–27] and many others).

In [28], Haghi et al. pointed out that some fixed point generalizations to partial metric spaces can be obtained from the corresponding results in metric spaces. To demonstrate facts they considered certain cases. Motivated by Proinov’s results, in this paper, we present some fixed point theorems in partial metric spaces which cannot be obtained from the corresponding results in metric spaces. Indeed, we obtain some fixed and common fixed point theorems for single and multivalued mappings in the setting of partial metric spaces. Our results complement, extend, and generalize a number of fixed point theorems including some recent results in [10, 11, 14, 16, 23] and others. Besides discussing some useful examples, an application to Volterra type system of integral equations is also given. Finally, we show that fixed point problems discussed herein are well-posed and have limit shadowing property.

For the sake of completeness, we recall the following definitions and results from [9, 10, 14].

*Definition 5. *A partial metric on a nonempty set is a function such that for all (*p*_{1}) if and only if ;(*p*_{2});(*p*_{3});(*p*_{4}). The pair is called a partial metric space.

A partial metric on generates a -topology on with a base of the family of open -balls , whereIf is a partial metric on , then the function given byfor all is a metric on .

*Example 6 (see [10, 14]). *Let and given by for all . Then is a partial metric space.

*Example 7 (see [9, 14]). *Let . Then defines a partial metric on .

*Definition 8. *Let be a partial metric space. Then one has the following: (1)A sequence in converges to a point if and only if .(2)A sequence in is Cauchy if exists and is finite.(3) is complete if every Cauchy sequence in converges to a point , that is, .

Lemma 9 (see [9]). *Let be a partial metric space. Then is complete if and only if the metric space is complete. Furthermore, if and only if *

In [25], Romaguera introduced the following notions of -Cauchy sequence and -complete partial metric spaces. He obtained a characterization of completeness for partial metric space using the notion of -completeness.

*Definition 10. *A sequence in a partial metric space is -Cauchy if . The partial metric space is -complete if each -Cauchy sequence in converges to a point such that

Notice that every -Cauchy sequence in is Cauchy in and every complete partial metric space is -complete. However, a -complete partial metric space need not be complete (cf. [29] and [25]).

A subset of is closed (resp., compact) in if it is closed (resp., compact) with respect to the topology induced by on . The subset is bounded in if there exist and such that for all ; that is,Let be the collection of all nonempty, closed, and bounded subsets of with respect to the partial metric . For , one definesFor ,

Proposition 11 (see [14]). *Let be a partial metric space. For any , one has **;**;** implies that ;**.*

Proposition 12 (see [14]). *Let be a partial metric space. For any , one has *(H1)*;*(H2)*;*(H3)*;*(H4)*. But the converse is not true.*

In view of Propositions 11 and 12, is a partial Hausdorff metric induced by the partial metric .

#### 2. Auxiliary Results

Hitzler and Seda [19] obtained the following result to establish a relation between a partial metric and the corresponding metric on a nonempty set .

Proposition 13 (see [19, 28]). *Let be a partial metric space. Then the function defined by whenever and whenever is a metric on such that . Moreover, is complete if and only if is -complete.*

The following lemma is the key result in [28].

Lemma 14 (see [28]). *Let be a partial metric space, a self-mapping, the metric constructed in Proposition 13, and . DefineThen for all with .*

Using Proposition 13 and Lemma 14 above, we obtain the following result.

Lemma 15. *Let be a partial metric space and a self-mapping. Suppose is the constructed metric in Proposition 13 and . DefineThen *(a)* for all ;*(b)* for all .*

*Proof. *To prove (a), we shall consider three cases and the rest of the cases will follow in the same manner.*Case 1* (). One has*Case 2* (). One has *Case 3* (). One has The proof of (b) follows easily form [8, page 3300].

#### 3. Single Valued Mappings

For the sake of brevity, in this section, we shall use the following denotations:(1) the class of functions such that is continuous nondeceasing function satisfying and the series converges for all ;(2) the class of functions such that is upper semicontinuous from the right satisfying for all .Let be mappings. (3),;(4).

*Remark 16. *It can be easily seen that (i) for all (see [8]);(ii).

Browder and Petryshyn [30] introduced the notion of* asymptotic regularity* for a single valued mapping in a metric space (see also [3], page 547).

*Definition 17. *A self-mapping of a metric space is* asymptotically regular* at a point ifIf is asymptotically regular at each point of then one says that is asymptotically regular on

Sastry et al. [31] and Singh et al. [32] extended the above definition to three mappings as follows.

*Definition 18. *Let , , and be self-mappings of a metric space . The pair is asymptotically regular with respect to at a point if there exists a sequence in such thatfor all and

If then one gets the definition of asymptotic regularity of with respect to (see, for instance, Rhoades et al. [33]). Further, if is the identity mapping on , then one gets the usual definition of asymptotic regularity for a mapping .

We extend the above notion to four self-mappings on a partial metric space as follows.

*Definition 19. *Let , , , and be self-mappings of a partial metric space . The mappings , , , and will be called asymptotically regular at if there exist sequences and in such that and

The following theorem is the main result in [10].

Theorem 20. *Suppose , and are self-mappings of a complete partial metric space such that , , andfor all , where .**If one of , , , and is a closed subset of , then *(i)* and have a coincidence point;*(ii)* and have a coincidence point.**Moreover, if the pairs and are weakly compatible, then , , , and have a unique common fixed point.*

Now we present a more general result than Theorem 20.

Theorem 21. *Let , , , and be self-mappings of a partial metric space such that *(A)* and ;*(B)*, , , and are asymptotically regular at ;**for all , where .**If one of , , , and is a -complete subspace of , then *(a)* and have a coincidence point;*(b)* and have a coincidence point;*(c)* and have a common fixed point provided that the pair is commuting at one of their coincidence points;*(d)* and have a common fixed point provided that the pair is commuting at one of their coincidence points.**Moreover, the mappings , and have a common fixed point provided that () and () are true.*

*Proof. *Let be such that , and are asymptotically regular at . Since , there exists such that . Also since , there exists such that . Continuing this process, we construct sequences and in defined by where . Since , and are asymptotically regular at , we have We claim that is a -Cauchy sequence. Suppose is not -Cauchy. Then there exist and increasing sequences and of positive integers such thatfor all . By the triangle inequality, we have Thus Now, by (20), we have Since is upper semicontinuous from the right, we deduce thata contradiction. Therefore .

Suppose that is a -complete subspace of . Then the subsequence being contained in has a limit in . Call it . Let . Then . Note that the subsequence also converges to . By (20), we have Since is upper semicontinuous from the right, making implies , a contradiction, unless . Therefore and is a coincidence point of and .

Since , . Hence there exists such that . Again, by (20), we have Thus , a contradiction, unless . Therefore and is a coincidence point of and .

If the pairs and are commuting at and , respectively, thenNow, in view of (20), it follows that a contradiction. Therefore and is a common fixed point of and . Similarly, is a common fixed point of and . Since , we conclude that is a common fixed point of , and . The proof is similar when is a complete subspace of . The cases in which or is a complete subspace of are also similar since and .

When and (the identity mapping) in Theorem 21, we get the following result which extends a result of Romaguera [23].

Corollary 22. *Let be an asymptotically regular self-mapping of a partial metric space such thatfor all , where . Then has a fixed point.*

The following example shows the generality of our results.

*Example 23. *Let endowed with the partial metric for all Then is a -complete partial metric space. Define the mappings byDefine sequences and bywhere . Then the mappings , and are asymptotically regular at

Further,for all , where and . So the assumptions of Theorem 21 are fulfilled and , and have fixed points and .

On the other hand, for any . Therefore the mappings , , , and do not satisfy the requirements of Theorem 20.

*Example 24. *Let be endowed with the partial metric for all Then is a -complete partial metric space. Define the mappings byThen and Now define the function by We now show thatfor all , where For this, let with ThenThenNow we show that the mappings , and are asymptotically regular. For this, by Definition 19, we have for all Solving the above two equations we getSo for given we can define a sequence by using (40) and then we can easily define sequence by (39). It can be easily seen that

On the other hand, for and , we haveTherefore for any Therefore the mappings , and do not satisfy the requirement of Theorem 20.

Now we give an example in which the mapping is not asymptotically regular at each point of interval but satisfies condition (4) for one mapping.

*Example 25. *Let be endowed with the partial metric for all Then is a -complete partial metric space. Define the mapping byNow define the function by It can be easily seen that is asymptotically regular at each point of interval and not asymptotically regular at any point of interval Now we show thatfor all , where For this we distinguish the following cases.*Case 1*. If with , we have*Case 2*. If and , we have *Case 3*. If with , we have

#### 4. Multivalued Mappings

Rhoades et al. [33] and Singh et al. [32] extended the concept of* asymptotic regularity* from single valued to multivalued mappings in metric spaces. We extend it to partial metric spaces.

*Definition 26. *Let be a partial metric space and . The mapping is* asymptotically regular* at if, for any sequence in and each sequence in such that ,for

Aydi et al. [14] obtained the following equivalent to the well-known multivalued contraction theorem due to Nadler Jr. [34].

Theorem 27. *Let be a complete partial metric space. If is a multivalued mapping such that, for all and , one hasthen has a fixed point.*

In [24], Romaguera pointed out that if and defined by for all , then and the approach used in Theorem 27 and elsewhere has a disadvantage that the fixed point theorems for self-mappings may not be derived from it, when . To overcome this problem he introduced the concept of mixed multivalued mappings and obtained a different version of Nadler Jr.’s theorem in a partial metric space.

*Definition 28. *Let be a partial metric space. A mapping is called a mixed multivalued mapping on if is a multivalued mapping on such that for each either or .

A self-mapping and a multivalued mapping both are mixed multivalued mappings (see also [35]).

Motivated by Proinov’s theorem and the above facts, we obtain the following result, which extends Theorem 27 above and Corollary in [16].

Theorem 29. *Let be a -complete partial metric space and a continuous mixed multivalued mapping such that *(S1)* for all ;*(S2)* whenever ,**where is as in Theorem 3, is a collection of all nonempty compact subsets of , , andIf is asymptotically regular at , then has a fixed point.**Moreover, if and is continuous and satisfies for all , then the continuity of can be dropped.*

*Proof. *We construct a sequence in in the following way. Let such that is asymptotically regular at Let be any element of . If or , then is a fixed point of and there is nothing to prove. Assume that and is not singleton. Then and by compactness of we can choose such thatIf is a singleton, then obviouslyTherefore, in either case, we haveAgain, since is compact, we choose a point such thatContinuing in the same manner we getFollowing largely [3, 8], we show that the sequence is a -Cauchy. Fix . Since is as in Theorem 3, there exists such that, for any ,Without loss of generality we may assume that . Since is asymptotically regular at ,So, there exists an integer such thatfor all . By induction we shall show thatfor all with

Let be fixed. Obviously, (58) holds for . Assuming (58) to hold for an integer , we shall prove it for . By the triangle inequality, we getWe claim thatTo prove the above claim, we consider two cases.*Case 1* (). By (S2) it follows that , and (60) holds.*Case 2* (). By (S1), we haveBy the definition of , we obtain Using (57) and (58) in this inequality, we getThus . Hence (55) implies that . Now (61) implies (60). By (60), (59), and (57), it follows thatThis proves (58). Since , (58) implies that for all integers and with and hence is a -Cauchy sequence. Since is -complete, has a limit. Call it . We note thatIf is continuous, then obviously is a fixed point of .

Moreover, if and is continuous and satisfies for all , then it follows from (S2) that Making ,a contradiction, unless .

Now we present a slightly modified version of Theorem 29 to obtain a new result.

Theorem 30. *Let be a partial metric space and and such that and *(i)* for all ,* *where , and is as in Theorem 3 and is continuous;*(ii)* whenever **If is asymptotically regular and either or is a complete subspace of , then and have a coincidence point **Further, and have a common fixed point provided that and and are commuting at their coincidence point .*

*Proof. *It can be completed using the proofs of Theorem 29 above and Theorem in [8].

The following example illustrates our results.

*Example 31 (see [36]). *Let and defined by Clearly, is a -complete partial metric space. Now, define the mapping such thatIt can be easily seen that is asymptotically regular at andwith and . Therefore all the assumptions of Theorem 29 are fulfilled and is a fixed point of

The following example shows the generality of our results.

*Example 32. *Let be endowed with the partial metric defined byClearly, is a -complete partial metric space. Now, define the mapping such thatIt can be easily seen that is continuous and asymptotically regular and for all ,with and . Therefore the assumptions of Theorem 29 are fulfilled.

On the other hand, for all ; Theorem 27 and Corollary in [16] are not satisfied by the mapping .

*Example 33. *Let be endowed with the partial metric defined byNow define the mapping such thatIt can be easily seen that and is a complete metric space. Now, by Lemma 9, is a complete partial metric space and hence -complete. Now we show thatfor all with and For this we distinguish the following cases.*Case 1*. If with , we have Therefore, for all and , we have Similar conclusion can be done for .*Case 2*. If and , then we have Therefore, for all and , we have *Case 3*. If with , then we haveHence, for all and , we have Similar conclusion can be done for

On the other hand, at and , we have Now for all and Hencefor all

*Remark 34. *We remark that the mappings satisfying Theorems 2, 27, and 20 are also asymptotically regula