Journal of Applied MathematicsVolume 2012 (2012), Article ID 902312, 14 pagesdoi:10.1155/2012/902312

Research Article

General Common Fixed Point Theorems and Applications

Shyam Lal Singh,¹ Swami Nath Mishra,² Renu Chugh,³ and Raj Kamal³

¹Pt. L. M. S. Goverment Autonomous Postgraduate College, Rishikesh 249201, India²Department of Mathematics, W. S. University, Mthatha 5117, South Africa³Department of Mathematics, M. D. University, Rohtak 124001, India

Received 31 October 2011; Revised 9 December 2011; Accepted 9 December 2011

Academic Editor: Yonghong Yao

Copyright © 2012 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.

The main result is a common fixed point theorem for a pair of multivalued maps on a complete metric space extending a recent result of orić and Lazović (2011) for a multivalued map on a metric space satisfying Ćirić-Suzuki-type-generalized contraction. Further, as a special case, we obtain a generalization of an important common fixed point theorem of Ćirić (1974). Existence of a common solution for a class of functional equations arising in dynamic programming is also discussed.

1. Introduction

Consistent with Nadler [1, page 620], will denote a metric space and , the collection of all nonempty closed subsets of . For and , The hyperspace is called the generalized 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 .

Let . Then is a fixed point of if and only if and a common fixed point of and if and only if .

Let and be maps to be defined specifically in a particular context, while and are the elements of a metric space :

Recently Suzuki [2] and Kikkawa and Suzuki [3] obtained interesting generalizations of the Banach’s classical fixed point theorem and other fixed point results by Nadler [4], Jungck [5], and Meir and Keeler [6]. These results have important outcomes (see, e.g., [7–14]). The following result, due to orić and Lazović [9], extends and generalizes fixed point theorems from Ćirić [15], Kikkawa and Suzuki [3], Nadler [4], Reich [16], Rus [17], and others.

Theorem 1.1. Define a nonincreasing function from onto by Let be a complete metric space and . Assume there exists such that for every , Then there exists such that .

We remark that, for every , the generalized contraction , , was first studied by Ćirić [15]. The following important common fixed point theorem is due to Ćirić [18].

Theorem 1.2. Let be a complete metric space and . Assume there exists such that for every , Then and have a unique common fixed point.

For an excellent discussion on several special cases and variants of Theorem 1.2, one may refer to Rus [17]. However, the generality of Theorem 1.2 may be appreciated from the fact that (1.6) in Theorem 1.2 cannot be replaced by Indeed, Sastry and Naidu [19, Example  5] have shown that maps and satisfying (1.7) need not have a common fixed point on a complete metric space. Notice that the condition (1.7) with is the quasicontraction due to Ćirić [20].

The main result of this paper (cf. Theorem 2.2) generalizes Theorems 1.1 and 1.2. Further, a corollary of Theorem 2.2 is used to obtain a unique common fixed point theorem for multivalued maps on a metric space with values in . As another application, we deduce the existence of a common solution for a general class of functional equations under much weaker conditions than those in [12, 14, 21–24].

2. Main Results

We shall need the following result essentially due to Nadler [4] (see also [15, 25], [26, page 4], [27], [17, page 76]).

Lemma 2.1. If and , then for each , there exists such that .

Theorem 2.2. Let be a complete metric space and . Assume there exists such that for every , Then there exists an element such that .

Proof. Obviously iff is a common fixed point of and . So, we may take without any loss of generality that for distinct . Let be such that . Let and . Then by Lemma 2.1, their exists such that Similarly, their exists such that Continuing in this manner, we find a sequence in such that Now, we consider two cases and show that for any , Case 1. If , then Therefore by the assumption, Case 2. If , then So by the assumption, Hence in either case we obtain by (2.7) and (2.9), This yields (2.5). Analogously, we obtain , and conclude that for any , Therefore is a Cauchy sequence and has a limit in . Call it . Now we show that for any, Since , there exists (natural numbers) such that Then as in [2, page 1862], Therefore Now either or .So in either case by (2.16), Hence by the assumption (2.1), Making , This yields (2.12). Similarly, we can show (2.13). Now, we show that .For , the following cases arise.Case 1. Suppose and . Then as in [8, page 6], let be such that and be such that .Since implies , we have from (2.12) and (2.13), On the other hand, since , Therefore by the assumption (2.1), This gives . So by (2.22), . Thus This contradicts . Consequently . Similarly .Case 2. Let and. Then as in the previous case, let be such that Since , we have from (2.13), On the other hand, Since , Therefore by the assumption (2.1), This gives .So by (2.22), . Thus This contradicts . Consequently .Case 3. and . As in the previous case, it follows that . Now we consider the case .First we show that Assume that . Then for every , there exists such that Therefore Using (2.13) with , (2.33) implies If then (2.34) gives Making , Thus .Then and by the assumption (2.1), If then (2.34) gives that is, .

Making , Then , and by the assumption, we get (2.37).

Taking in (2.37) and passing to the limit, we obtain This gives . Analogously, .

The following result generalizes Theorem 1.2.

Corollary 2.3. Let be a complete metric space and maps from into . Suppose there exists such that for every , Then and have a unique common fixed point.

Proof. For single-valued maps and , it comes from Theorem 2.2 that they have a common fixed point. The uniqueness of the common fixed point follows easily.

Remark 2.4. Theorem 1.1 is obtained as a particular case of Theorem 2.2 when .

Now we derive the following result due to orić and Lazović [9, Corollary 2.3].

Corollary 2.5. Let be a complete metric space and a map from into . Suppose there exists such that for every , Then has a unique fixed point.

Proof. It comes from Corollary 2.3 when .

The following example shows the generality of our results.

Example 2.6. Let be endowed with the metric defined by Let and be such that Then and do not satisfy the condition (1.6) of Theorem 1.2 at . However, this is readily verified that all the hypotheses of Corollary 2.3 are satisfied for the maps and .

Theorem 2.7. Let be a complete metric space and . Assume there exists such that for every , implies Then there exsits a unique point such that .

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, (2.45) gives and this implies (2.46). Therefore So (2.50), namely, implies where .Hence by Theorem 2.2, and have a unique point such that . This implies .

Corollary 2.8. Let be a complete metric space and . Assume there exists such that for every , Then there exists a unique point such that .

Proof. It comes from Theorem 2.7 when .

3. Applications

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

In the multistage process, some functional equations arise in a natural way (cf. [22, 23]; see also [21, 24, 28, 29]). In this section, we study the existence of common solution of the functional equations (3.1) arising in dynamic programming.

Let denotes 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) There exists such that for every and , implies where are defined as follows:

Theorem 3.1. Assume the conditions (DP-1) and (DP-2). Then the functional equations (3.1), , have a unique common solution in .

Proof. For any , let . Then is a complete metric space.Let be any arbitrary positive number and . Pick and choose such that where .Further, Therefore, the first inequality in (DP-2) becomes and this together with (3.5) and (3.7) implies Similarly, (3.5), (3.6), and (3.8) imply So, from (3.10) and (3.11), we obtain Since this inequality is true for any , and is arbitrary, on taking supremum, we find from (3.8) and (3.11) that implies Therefore, Corollary 2.3 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 (3.1), .

The following result generalizes a recent result of Singh and Mishra [12, Corollary  4.2] which in turn extends certain results from [21, 23, 24].

Corollary 3.2. Suppose that the following conditions hold. (i) and are bounded.(ii)There exists such that for every and , where is defined as Then the functional equation (3.1) with and possesses a unique bounded solution in .

Proof. It comes from Theorem 3.1 when and .

Acknowledgments

The authors are grateful to all the three referees for their appreciation and valuable suggestions to improve upon the paper. They also thank Professor Yonghong Yao for his suggestions in this paper. The first author (S. L. Singh) acknowledges the support of the University Grants Commission, New Delhi under Emeritus Fellowship.

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