Common Fixed Point Results for Six Mappings via Integral Contractions with Applications

Bahadur Zada, Mian; Sarwar, Muhammad; Mehmood, Nayyar

doi:https://doi.org/10.1155/2016/7480469

International Journal of Analysis

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Research Article | Open Access

Volume 2016 | Article ID 7480469 | https://doi.org/10.1155/2016/7480469

Common Fixed Point Results for Six Mappings via Integral Contractions with Applications

Mian Bahadur Zada,¹Muhammad Sarwar,¹and Nayyar Mehmood²

Academic Editor: Remi Léandre

Received03 Jul 2016

Accepted20 Sept 2016

Published19 Oct 2016

Abstract

Common fixed point theorems for six self-mappings under integral type inequality satisfying and properties in the context of complex valued metric space (not necessarily complete) are established. The derived results are new even for ordinary metric spaces. We prove existence result for optimal unique solution of the system of functional equations used in dynamical programming with complex domain.

1. Introduction and Preliminaries

Metric fixed point theory is the most impressive and active branch of modern mathematics that has vast applications in applied functional and numerical analysis. Banach contraction principle [1] is one of the best known results in this theory. This principle can be considered as the launch of metric fixed point theory that guarantees the existence and uniqueness of fixed points of mappings. In the following years, various efforts have been done to further generalize Banach contraction principle in different direction for a single map.

The exploration of common fixed point theory is an active field of research activity since 1976. The work of Jungck [2] is considered as major achievement in the field of common fixed point theory. Jungck presented the notion of commuting maps to introduce the common fixed point results for two self-maps on complete metric space. To improve common fixed point theorems, researchers began to utilize weaker conditions than commuting mappings such as weakly commuting maps, compatible mappings, compatible mappings of type , compatible mappings of type , compatible mappings of type , and compatible mappings of type . In the study of common fixed point results of weakly compatible mappings we often require the assumption of the continuity of mappings or the completeness of underlying space. As a consequence a natural question arises as to whether there exist common fixed point theorems, which do not enforce such conditions. Regarding this Aamri and El Moutawakil [3] relaxed these conditions by introducing the notion of property and it was marked that property does not require the condition of continuity of mappings and completeness of the underlying space. However, property tolerates the condition of closeness of the range subspaces of the involved mappings. In 2011, the new notion of Common Limit in the range property (shortly property) was given by Sintunavarat and Kumam [4] that does not enforce the above-mentioned conditions. Moreover, the significance of property reveals that closeness of range subspaces is not essential. Using these two important notions many fixed point theorems were established [3–6].

One of the most pleasant generalizations of Banach principle is the Branciari [7] fixed point theorem for a single mapping satisfying an integral type inequality. After that, serval researchers ([8–11], etc.) generalize the result of Branciari in ordinary metric spaces.

On the other hand Azam et al. [12] studied complex valued metric space and proved common fixed point theorems for two self-mappings satisfying a rational type inequality. Manro et al. [13] generalized the theorem of Branciari [7] for two self-maps under contractive condition of integral type satisfying property and property in the setting of complex valued metric spaces. Bahadur Zada et al. [6] generalized the results of [13] for four self-maps in the context of complex valued metric spaces.

The aim of this paper is to prove common fixed point theorems for six self-maps, satisfying integral type contractive condition using property and property in complex valued metric spaces, which extends and generalizes many results of the existing literature.

Throughout the paper , stand for or . and are Banach spaces, is the state space, is the decision space, is a Lebesgue integrable mapping which is summable on each compact subset of , nonnegative and nondecreasing such that, for each , , and is a complex valued Lebesgue integrable mapping, which is summable and nonvanishing on each measurable subset of , such that, for each ,

Definition 1 (see [12]). Let be the set of complex numbers and . Define a partial order on as follows: Note that(i) and for all ;(ii) for all ;(iii) and for all .

Definition 2 (see [14]). The “” function for the partial order relation “” is defined by the following:(1).(2)if , then or .(3) or .

Definition 3 (see [12]). Let be a nonempty set and be the mapping satisfying the following axioms:(1), for all and if and only if .(2), for all .(3), for all .Then pair is called a complex valued metric space.

Example 4. Let and define the mapping by Then is a complex valued metric space.

Definition 5 (see [12]). Let be a sequence in complex valued metric and . Then is called the limit of if for every , with , there is such that for all and one writes .

Lemma 6 (see [12]). Any sequence in complex valued metric space converges to if and only if as .

Definition 7 (see [4]). Let be a nonempty set and be two self-maps. Then(i) is called a fixed point of if ;(ii) is called a coincidence point of and if ;(iii) is called a common fixed point of and if .Jungck [2] initiated the concept of commuting maps in the following way.

Definition 8. Two self-maps and of nonempty set are commuting if , for all .

Jungck [15] initiated the concept of weakly compatible maps in ordinary metric spaces while Bhatt et al. [16] refined this notion in the complex valued metric space in the following way.

Definition 9. Two self-maps and on complex valued metric space are weakly compatible if there exists point such that whenever .

Aamri and El Moutawakil [3] initiated the concept of property in ordinary metric spaces while Verma and Pathak [14] defined this concept in complex valued metric space as follows.

Definition 10. Two self-maps and on a complex valued metric space satisfy property if there exists sequence in such that

Sintunavarat and Kumam [4] introduced the notion of property in ordinary metric spaces, in a similar mode. Verma and Pathak [14] defined this notion in a complex valued metric space in the following way.

Definition 11. Two self-maps and on a complex valued metric space satisfy if there exists sequence in such that

Remark 12 (see [6]). Let , such that , and is a sequence in converges to , and then

Lemma 13 (see [6]). Let , such that , and is a sequence in , and then if and only if , as

2. Main Results

Let be the class of all functions that satisfy the following properties:(1) is nondecreasing on .(2) is upper semicontinuous on .(3) and for every .

Now, we present our first result.

Theorem 14. Let be a complex valued metric space and be six self-mappings satisfying the following conditions:(1)One of pairs and satisfies property such that and .(2). where andIf one of and is closed subspace of such that pairs and are weakly compatible, then each pair of pairs and has a coincidence point in . Moreover, if , , , and are commuting pairs, then , and have a unique common fixed point in .

Proof. Let pair satisfy property, so there exists sequence in such thatSince , there exists in such that and thus, from (7), we getWe assert that If , then, upon putting and in condition () of Theorem 14, we havewhereTaking upper limit as in (9), we havewhich contradict with our assumption; thus and . Therefore (8) becomesAlso, since is closed subspace of , there exists such that and, using (12), we getNow, we claim that . To support the claim, let . Then, using condition () of Theorem 14 with and , one can getwhereTaking upper limit as in (14), we have which is a contradiction. Thus, and henceSince , there exists such that and it follows from (17) thatWe show that . Let on contrary ; then, using condition () of Theorem 14 with and , we havewhereTherefore, which is a contradiction to our assumption that . Thus and hence, from (18), we getNow, using the weak compatibility of pairs , , and (22), we haveHence is the coincident point of each pair and .
Next, we have to show that is the common fixed point of , and . For this, we claim that . If , then upon putting in condition () of Theorem 14 and using (22) and (23) we have whereTherefore, which is impossible. Thus and hence, in view of (23), we getSimilarly, we can show thatHence, from (28) and (29), we getNow, by commuting conditions of pairs and and using (28) and (30), we have and ; from here it follows thatAlso, by commuting conditions of pairs and and taking (29) and (30), we have and ; from here it follows thatFurther, assume the . Then upon putting in condition () of Theorem 14 and using (29) and (31), we have where Therefore, which is a contradiction; thus . Also as , so from (30) it follows thatSimilarly, using condition () of Theorem 14 with and and taking (28) and (32), one can easily obtain that . Also as . Hence, from (36), we getThat is is a common fixed point of , and in .
Similarly, if satisfies property and is closed subspace of , then we can prove that is a common fixed point of , and in in the same arguments as above.
Uniqueness. For the uniqueness of common fixed point, let be another fixed point of , and . Then, using condition () of Theorem 14, we have whereThus, which is a contradiction; hence is a unique common fixed point of , and in .

Now we present some corollaries; their proofs are easily followed from Theorem 14, so we omit the proofs.

Corollary 15. Let be a complex valued metric space and be five self-mappings satisfying the following conditions:(1)One of pairs and satisfies property such that and .(2). where andIf one of and is closed subspace of such that pairs and are weakly compatible, then each pair of pairs and has a coincidence point in . Moreover, if , , , and are commuting pairs, then , and have a unique common fixed point in .

Corollary 16. Let be a complex valued metric space and be four self-mappings satisfying the following conditions:(1)One of the pairs and satisfies property such that and .(2). where andIf one of and is closed subspace of , then pairs and have a coincidence point in . Moreover, if and are weakly compatible, then , and have a unique common fixed point in .

Corollary 17. Let be a complex valued metric space and be three self-mappings satisfying the following conditions:(1)One of the pairs and satisfies property such that and .(2). where andIf is closed subspace of , then pairs and have a coincidence point in . Moreover, if and are weakly compatible, then , and have a unique common fixed point in .

Corollary 18. Let be a complex valued metric space and be two self-mappings satisfying the following conditions:(1)Pair satisfies property .(2). where andIf is closed subspace of , then pair has a coincidence point in . Moreover, if is weakly compatible, then mappings and have a unique common fixed point in .

Similar to the arguments of Theorem 14, we conclude the following result and omit their proof.

Theorem 19. Let be a complex valued metric space and be six self-mappings satisfying the following conditions:(1)One of pairs and satisfies property such that and .(2). where and If one of and is closed subspace of such that pairs and are weakly compatible, then each pair of pairs and has a coincidence point in . Moreover, if , , , and are commuting pairs, then , and have a unique common fixed point in .

Theorem 20. Let be a complex valued metric space and be six self-mappings satisfying condition (2) of Theorem 14 and either pair satisfies property or pair satisfies property such that and . If pairs and are weakly compatible, then each pair of pairs and has a coincidence point in . Moreover, if , , , and are commuting pairs, then , and have a unique common fixed point in .

Proof. Suppose that pair satisfies property, then there exists sequence in such thatSince , there exists such that .
We claim that . To support the claim, let . Then on using condition () of Theorem 14, with setting and , we havewhereTaking upper limit as in (52) and using (51), we getwhich is a contradiction. Thus and henceAlso, since , there exists such that . Thus (55) becomesNow, we assert that . Let on contrary ; then setting and , in condition () of Theorem 14, we get whereUsing (56), we havewhich is impossible. Thus and henceTherefore, from (56) and (60), we getFinally, following the lines in the proof of Theorem 14 we can show that is the coincident point of pairs and and is a unique common fixed point of the mappings , and .

Similar to the arguments of Theorem 20, we conclude the following results and omit their proofs.

Theorem 21. Let be a complex valued metric space and be six self-mappings satisfying the following conditions:(1)Either pair satisfies property or pair satisfies property such that and .(2). where and If pairs and are weakly compatible, then each pair of pairs and has a coincidence point in . Moreover, if , , , and are commuting pairs, then , and have a unique common fixed point in .

Corollary 22. Let be a metric space and be six self-mappings satisfying the following conditions:(1)Either pair satisfies property or pair satisfies property such that and .(2). where andIf pairs and are weakly compatible, then each pair and has a coincidence point in . Moreover, if , , , and are commuting pairs, then , and have a unique common fixed point in .

Similarly to Theorem 14 one can derive variant of corollaries from Theorems 19, 20, and 21.

Remark 23. The conclusions of Theorems 14, 19, 20, and 21 are still valid if we replace with , where

Remark 24. Theorems 14 and 20 and Corollary 15 extends Theorem of [11] in complex valued metric space. Corollary 16 generalizes the results of [8–11] in complex valued metric space. Moreover, the real valued metric space version of our main results generalizes the results of [8–11].

To support Theorem 21, we present the following example.

Example 25. Let be a complex valued metric space with metric defined by Define self-maps , and on by , , , , , and .
Then, Also, we define by and by .
Clearly and .
Now, we construct sequence in such thatthat is, there exists sequence in such that Hence satisfies property.
Next, check the following conditionwhereSincethereforeThus, from (71), (73), and (74) and by using the value of , we have where Also pairs and are weakly compatible and , , , and are commuting pairs. Hence from Theorem 21, is a unique common fixed point of , and .

3. Applications

Many researchers study the applications of common fixed point theorems in complex valued metric spaces; see for instance [17, 18] and the references therein. On the other hand, Liu et al. [19] and Sarwar et al. [20] study the existence and uniqueness of common solution for the system of functional equations arising in dynamic programming with real domain. We apply Corollary 22 for the existence and uniqueness of a common solution for the following system of functional equations arising in dynamic programming with complex domain (see [21]).where and signify the state and decision vectors, respectively, denotes the optimal return functions with initial state , , and .

Let be the space of all continuous real valued functions on possibly complex domain with metric

We prove the following result.

Theorem 26. Let and , , be bounded functions and let be six operators defined asfor all and Assume that the following conditions hold:(i)There exist such that , for some .(ii) such that pairs and are weakly compatible.(iii)Pairs , , , and are commuting.(iv)For . where where , and is a nonnegative summable Lebesgue integrable function such that for each Then the system of functional equations (77) has a unique bounded solution.

Proof. Notice that the system of functional equations (77) has a unique bounded solution if and only if the system of operators (79) have a unique common fixed point. Now since , and are bounded, there exists positive number such that Now, by using properties of the theory of integration and definition of , we conclude that, for each positive number , there exists positive , such thatfor all with , where is the Lebesgue measure of
Now, we consider two possible cases.
Case 1. Suppose that . Let and ; then for there exist such that From inequalities (85) and (88) it follows that which gives Similarly, using inequalities (86) and (87) we obtain Therefore from (90) and (91) we get where and .
Case 2. Suppose that . By following the procedure in Case , one can check that (92) holds.
Now, from , we have And, by condition (iv) of Theorem 26, we get and using (84) we get Since above inequality is true for each and is taken arbitrarily, we deduce that where Also, from condition of Theorem 26 pair satisfies property. Thus all hypothesis of Corollary 22 are satisfied. Consequently operators (79) have a unique common fixed point, that is, system (77) of functional equations has a unique bounded solution.

Competing Interests

The authors declare that they have no competing interests regarding this manuscript.

Authors’ Contributions

All authors read and approved the final version.

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Copyright © 2016 Mian Bahadur Zada et al. 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.

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