Abstract and Applied Analysis

Volume 2014 (2014), Article ID 642378, 11 pages

http://dx.doi.org/10.1155/2014/642378

Research Article

A New Type of Coincidence and Common Fixed Point Theorem with Applications

Shyam Lal Singh,¹ Raj Kamal,² Manuel De la Sen,³ and Renu Chugh²

¹Gurukula K. Vishwavidyalaya, Haridwar 249404, India

²Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, India

³Institute of Research and Development Processes, University of Basque Country, Campus of Leioa (Bizkaia), Aptdo. 644-Bilbao, 48080-Bilbao, Spain

Received 26 September 2013; Accepted 24 March 2014; Published 30 April 2014

Academic Editor: Douglas R. Anderson

Copyright © 2014 Shyam Lal Singh 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.

Abstract.

Coincidence and common fixed point theorems for a class of Ćirić-Suzuki hybrid contractions involving a multivalued and two single-valued maps in a metric space are obtained. Some applications including the existence of a common solution for certain class of functional equations arising in a dynamic programming are also discussed.

1. Introduction

Consistent with [1] (see also [2, 3]), denotes an arbitrary nonempty set, a metric space, and (resp., ), the collection of all nonempty closed (resp., closed bounded) subsets of . The hyperspace (resp., ) is called the generalized Hausdorff (resp., the Hausdorff) metric space induced by the metric on .

For nonempty subsets , of , denotes the gap between the subsets and , while

As usual, we write (resp., ) for (resp., ) when .

For the sake of brevity, we follow the following notations, wherein , , and are maps to be defined specifically in a particular context, while and are elements of some specific domain:

The Banach contraction principle (Bcp) plays an important role in nonlinear analysis and has numerous generalizations and several applications (see, e.g., [1–21] and others). Nadler Jr. [1] (see also [22]) initiated the study of multivalued Banach contractions in metric spaces. In view of its numerous applications, the Nadler multivalued contraction theorem received enormous attention (see, e.g., [2, 3, 7, 8, 11–15, 17–21, 23–36] and references thereof).

The following result [13, p. 250] extends and generalizes many results due to Fisher [37], Goebel [38], Kubiak [29], and others.

Theorem 1. Let and be such that , and one of , or is a complete subspace of . Assume there exists such that for every ,

Then

(i)and have a coincidence point in , (ii) and have a coincidence point in . Further, if  , then(iii) and have a common fixed point provided that is a fixed point of , and and commute at ;(iv) and have a common fixed point provided that is a fixed point of , and and commute at ;(v), , and have a common fixed point provided that (iii) and (iv) both are true.

We remark that certain contractive conditions studied for and by Ćirić [5], Covitz and Nadler Jr. [16], Czerwik [6], Fisher [37], Goebel [38], Jungck [17], Kubiak [29], Naimpally et al. [8], Pathak [15], Pathak et al. [9], Petrusel and Rus [10], Reich [11], and Rus [3] are included in the following condition:

for every , where .

In particular, (4) with and the identity map on was studied by Ćirić [5].

Recently, Suzuki [39, Th. 2] obtained a remarkable generalization of the Bcp. The same has been extended to multivalued maps by Kikkawa and Suzuki [30] in the following manner.

Theorem 2. Define a strictly decreasing function by

Let be a complete metric space and . Assume there exists such that for every ,

Then there exists such that .

Subsequently, some interesting extensions and generalizations of Theorem 2 were obtained among others by Abbas et al. [23], Dhompongsa and Yingtaweesittikul [24], Dorić and Lazović [25], Kamal et al. [18], Moţ and Petruşel [26], Singh and Mishra [27, 31, 36], and Singh et al. [28, 32, 33].

The importance of Suzuki contraction theorem [39, Th. 2] and subsequently obtained coincidence and fixed point theorems (cf. [23–28, 30–33, 36] and others) for maps in metric spaces satisfying Suzuki-type contractive conditions is that the contractive conditions are required to be satisfied not for all points of the domain.

In all that follows we take a nonincreasing function from onto defined by

Recently, Singh et al. [33] obtained the following coincidence and common fixed point theorem which is a generalization of a result of Dorić and Lazović [25].

Theorem 3. Let and be such that . Assume there exists such that for every ,

If one of or is a complete subspace of , then there exists a point such that .

Further, if and is a fixed point of , then is a fixed point of provided that is IT-commuting with at .

In this paper, we obtain a coincidence and common fixed point theorem (cf. Theorem 6) extending and generalizing Theorems 1, 2, 3, and several others. We also deduce the existence of common solution for a certain class of functional equations arising in dynamic programming. Examples are given to justify theorems and applications.

2. Main Results

The following definition is due to Itoh and Takahashi [19] (see also [27]).

Definition 4. Let and . Then the hybrid pair is IT-commuting at if .

We remark that IT-commuting maps are more general than commuting maps [34, p. 2]. However, a pair of maps are IT-commuting (also called weakly compatible by Jungck and Rhoades [20]) at if when .

We will need the following lemma essentially due to Nadler Jr. [1] (see also [5], [2, p. 61], [35, p. 4], [3, p. 76]).

Lemma 5. If and , then for each , there exists such that .

Let denote the collection of all coincidence points of and ; that is, when and ; and when .

The following is the main result of this section.

Theorem 6. Let and be such that . Assume there exists such that for every ,

implies

If one of , , or is a complete subspace of , then

(I) is nonempty; that is, there exists a point such that .(II) is nonempty; that is, there exists a point such that . Further if, , then(III) and have a common fixed point provided that the maps and are IT-commuting just at coincidence point and is fixed point of ;(IV) and have a common fixed point provided that the maps and are IT-commuting just at coincidence point and is fixed point of ;(V), , and have a common fixed point provided that both (III) and (IV) are true.

Proof. Without loss of generality, we may take and , nonconstant maps.

Let be such that . We construct two sequences in and in as follows.

Let and . By Lemma 5, there exists such that

Similarly, there exists such that

Continuing in this manner, we find a sequence in such that

Now, we show that for any ,

Suppose if , then

Therefore, by the assumption,

This yields (13).

Suppose if , then

Therefore, by the assumption,

yielding (13). So, in both cases, we obtain (13). In an analogous manner, we show that

We conclude from (13) and (18) that for any ,

Therefore the sequence is Cauchy. Assume that the space is complete. Notice that the sequence is contained in and has a limit in . Call it . Let . Then and . The subsequence also converges to . Let . Then

Now we show that for any ,

and for any ,

Since , there exists (naturals) such that

Also, since , there exists such that

Then, as in [39, p. 1862] (see also [25]),

Therefore,

Now, either or .

In each case, by (26) and the assumption,

Making ,

This yields (21); that is,

Analogously, we can prove (22); that is,

Now, we show that is nonempty.

We first consider the case .

Suppose . Then as in [24, p. 6], let be such that .

Since implies , we have from (21) and (22),

On the other hand, since ,

Therefore, by the assumption (13),

This gives .

So by (31), . Thus, by the assumption,

This contradicts . Consequently, , and is nonempty.

In an analogous manner, we can prove in the case that is nonempty.

We now consider the case . We first show that

Assume that . Then for every , there exists such that

Therefore,

So using (31), the inequality (37) implies

If , then (38) gives

Making ,

Thus,

Then

and by the assumption,

If , then (38) gives

that is, .

Making , .

Then , and by the assumption, we get (43).

Since , taking in (43) and passing to the limit, we obtain

This gives ; that is, is a coincidence point of and . Analogously, . Thus, (I) and (II) are completely proved.

Further, if , is a fixed point of , and and are IT-commuting at , then . Therefore, implies , so . This proves that is a common fixed point of and . This proves (III). Analogously, and have a common fixed point . Therefore (20) implies that is a common fixed point of and . This proves (IV). Now (V) is immediate.

Remark 7. In Theorem 6, the hypothesis “ is a fixed point of ” is essential for the existence of a common fixed point of and (see also [8]). Similarly, the hypothesis “ is a fixed point of ” is essential for the existence of a common fixed point of and . Further, the contractive condition for three maps and studied by Abbas et al. [23] are included in the assumptions of Theorem 6.

Corollary 8. Theorem 2.

Proof. It comes from Theorem 6 when .

The following result due to Dorić and Lazović [25] generalizing many fixed point theorems is obtained as a special case from Theorem 6 when and and are the identity map on .

Corollary 9. Let be a complete metric space and . Assume there exists such that for every ,

Then there exists an element such that .

The following result extends and generalizes coincidence and fixed point theorems of Fisher [37], Goebel [38], Jungck [17], and others.

Corollary 10. Let be such that . Let or or be a complete subspace of . Assume there exists such that for every ,

implies

Then and are nonempty. Further, if   and if commutes with and at a common coincidence point, then , , and have a unique common fixed point; that is, there exists a unique point such that .

Proof. Set for every . Then it easily comes from Theorem 6 that and are nonempty. Further, if and commutes with and at , then and .

Also , and this implies

This says that is fixed point of and . Analogously is fixed point of and . The uniqueness of the common fixed point follows easily.

Corollary 11. Let be a complete metric space and let be an onto maps. Assume there exists such that for every ,

Then and have a unique common fixed point.

Proof. It comes from Corollary 10 when and is the identity map on .

Corollary 12. Let be a complete metric space and let be onto maps. Assume there exists such that for every ,

Then has a unique fixed point.

Proof. It comes from Corollary 11 when .

The following example shows that Theorem 6 is indeed more general than Theorem 1.

Example 13. Consider a metric space , where is defined by

Let , and be such that

It is readily verified that

for all except for with .

For , condition (3) yields , which contradicts . Therefore, the condition (3) of Theorem 1 is not satisfied. So, in order to see that the maps , , and satisfy the assumption of Theorem 6, we notice that the condition (9a) of Theorem 6 does not hold for . Indeed, for ,

That is, .

This violates (9a) when (as ). Similarly (9a) is also not true for . It is easily seen that all other hypotheses of Theorem 6 are also true.

Now we give an application of Corollary 10.

Theorem 14. Let and be such that , and let one of , , or be a complete subspace of . Assume there exists such that for every ,

implies

Then and are nonempty.

Proof. Choose . Define single-valued maps as follows. For each , let be a point of which satisfies

Similarly, for each , let be a point of such that

Since and ,

So (56) gives

and this implies (57). Therefore,

So (61), namely, , implies

where .

Hence, by Corollary 10, there exist such that and . This implies that is a coincidence point of and , and is a coincidence point of and .

Corollary 15. Let and be such that , and let or be a complete subspace of . Assume there exists such that for every ,

implies

Then there exists such that .

Proof. It comes from Theorem 14 when .

Corollary 16. Let be a complete metric space and let . Assume there exists such that for every ,

implies

Then there exists a unique point such that .

Proof. It comes from Theorem 14 that has a fixed point when is the identity map on . The uniqueness of the fixed point follows easily.

3. Applications

Throughout this section, we assume that and are Banach spaces, , and . Let denote the field of reals, , , and . Considering and as the state and decision spaces, respectively, the problem of dynamic programming reduces to the problem of solving the functional equations:

Indeed, in the multistage process, some functional equations arise in a natural way (cf. Bellman [40] and Bellman and Lee [41]; see also [6, 9, 15, 28, 33, 42–45]). In this section, we study the existence of a common solution of the functional equations (68a) and (68b) arising in the dynamic programming.

Let denote the set of all bounded real-valued functions on . For an arbitrary , define . Then is a Banach space. Suppose that the following conditions hold:

(DP-1) , , , , and are bounded.

(DP-2) Let be considered as in the previous sections. Assume that there exists such that for every , and ,

implies

where

and , , and are defined as follows:

(DP-3) For any , there exists such that

(DP-4) There exists such that

Theorem 17. Assume the conditions (DP-1)–(DP-4). Let be a closed convex subspace of . Then the functional equations (68a) and (68b), , have a unique bounded common solution in .

Proof. For any , let . Then is a complete metric space. By virtue of (DP-3) and (DP-4), and the map is IT-commuting with and at coincidence points.

Let be an arbitrary positive number and . Pick , and choose such that

where . Further,

Therefore, the first inequality in (DP-2) becomes

and this together with (75), (77), and (78) implies

Similarly, (75), (76), and (78) imply

So, from (79) and (80), we obtain

As is arbitrary and (81) is true for any , taking supremum, we find from (78) and (81) that

implies

Therefore, Corollary 10 applies, wherein , and correspond, respectively, to the maps , , and . So and have a unique common fixed point ; that is, is the unique bounded common solution of the functional equations (68a) and (68b), .

Now we furnish an example in support of Theorem 17.

Example 18. Let be a Banach space endowed with the standard norm defined by , for all .

Suppose be the state space and the decision space. Define by

For any and , define by

Define by

Notice that , , , , and are bounded. Also

Now

Thus,

and this implies

Finally, for any with , we have

that is, , and with , we have ; that is, .

Thus, all the hypotheses of Theorem 17 are satisfied. So the system of (68a) and (68b) has a unique solution in .

Corollary 19. Suppose that the following conditions hold.

(i) , , , and are bounded.

(ii) Assume there exists such that for every , and ,

implies

where and are defined as follows:

(iii) For any , there exists such that

(iv) There exists such that

Then the functional equations (68a) and (68b) with possess a unique bounded common solution in .

Proof. It comes from Theorem 17 when .

Now we derive the the following result due to Dorić and Lazović [25], which in turn extends certain results from [41, 42].

Corollary 20. Suppose that the following conditions hold.

(i) and are bounded.

(ii) There exists such that for every , and ,

implies

where is defined as

Then the functional equation (68a) with possesses a unique bounded solution in .

Proof. It comes from Corollary 19 when , and as the assumption (DP-3) becomes redundant in this context.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for their appreciation and suggestions to improve upon the original typescript. The authors are grateful to the Spanish Government for its support of this research through Grant DPI2012-30651 and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. They are also grateful to the University of Basque Country for its financial support through Grant UFI 2011/07.

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