ISRN Mathematical AnalysisVolume 2011 (2011), Article ID 924396, 13 pagesdoi:10.5402/2011/924396

Research Article

Some Generalizations of Fixed Point Results for Multivalued Contraction Mappings

A. Azizi and H. P. Masiha

Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran

Received 18 August 2011; Accepted 26 September 2011

Academic Editors: A. Levy and G. Ólafsson

Copyright © 2011 A. Azizi and H. P. Masiha. 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.

We establish some fixed point results for multivalued contraction type mappings in terms of a -distance in a complete metric space. Our results generalize very recent results of some authors (Ćirić, 2008, 2009; Feng and Liu 2006; Klim and Wardowski 2007; and Latif and Abdou 2011).

1. Introduction

The Banach contraction principle [1] plays an important role in nonlinear analysis. Following the Banach contraction principle, Nadler [2] first initiated the study of fixed point theorems for multivalued contraction mappings and inspired by his results, the fixed point theory of multi-valued contraction has been further developed in different directions by many authors, in particular, by Reich, Mizoguchi-Takahashi, Feng-Liu, and many others.

The aim of this paper is to present results which are generalizations of the very recent results of Klim and Wardowski [3], Ćirić [4, 5], and Latif and Abdou [6], as well as of the result of Mizoguchi and Takahashi [7] and many others.

Let be a metric space. We denote the collection of all nonempty closed bounded subsets of by and the collectifon of all nonempty closed subsets and all nonempty compact subsets of by and , respectively. Throughout this paper, we assume that , , , and denote the sets of all real numbers, positive integers, even positive integers, and odd positive integers, respectively. Let be the Hausdorff metric induced by , that is, for all , where . An element is said to be a fixed point of a multi-valued mapping if . A map is called lower semicontinuous if for any sequence in and such that , we have .

The following theorem is an extension of Banach contraction principle for multi-valued mappings, which was obtained by Nadler in [2].

Theorem 1.1 (see [2]). Let be a complete metric space, and let be a multi-valued mapping. Assume that there exists such that for all, then there exists such that . A generalization of Theorem 1.1 was proved by Mizoguchi and Takahashi that is, in fact, a partial answer of question of Reich [8].

Theorem 1.2 (see [7]). Let be a complete metric space, and let be a multi-valued mapping. If there exists a function such that for each , and if for all , then there exists such that . Recently, an interesting result had been obtained by Feng and Liu [9]. They extended Theorem 1.1 in another direction different from that of Mizoguchi-Takahashi's theorem.

Theorem 1.3 (see [9]). Let be a complete metric space, and let be a multi-valued mapping. If there exist constants , , such that for any there is satisfying the following two conditions: (i),(ii), then there exists such that provided the function is lower semicontinuous.

By using the ideas of Mizoguchi-Takahashi and Feng-Liu, Klim and Wardowski proved the following two theorems that are different from Theorem 1.2.

Theorem 1.4 (see [3]). Let be a complete metric space, and let be a multi-valued mapping. Assume that the following conditions hold: (i)the map , defined by , , is lower semi-continuous; (ii)there exists a constant and a function satisfying , for each , and for any there is satisfying Then there exists such that .

Theorem 1.5 (see [3]). Let be a complete metric space, and let be a multi-valued mapping. Assume that the following conditions hold: (i)the map , defined by , , is lower semi-continuous;(ii)there exists a function satisfying , for each , and for any there is satisfying Then there exists such that .

In [4, 5], Ćirić proved a few interesting theorems which are generalizations of the above-mentioned theorems, one of which is as follows.

Theorem 1.6 (see [4]). Let be a complete metric space, and let be a multi-valued mapping. If there exist a function and a nondecreasing function , , such that and for any there is satisfying the following two conditions: then has a fixed point in provided is lower semi-continuous.

Very recently the fixed point theorems of Ćirić were extended by Liu et al. [10] and by Nicolae [11] in a new direction. Also Latif and Abdou [6] improve two theorems of Ćirić with respect to -distance. The concept of -distance was introduced by Kada et al. [12] on a metric space as follows.

Let be a metric space, then a function is called a -distance on if the following axioms are satisfied: (1), for any ; (2)for any , is lower semi-continuous; (3)for any , there exists such that and imply .

The metric is a -distance on . One can see other examples of -distances in [12]. One of theorems, which was proved by Latif and Abdou, is as follows.

Theorem 1.7 (see [6]). Let be a complete metric space with a -distance . Let be a multi-valued map. Assume that the following conditions hold: (i)there exist a constant and a function such that , for each ; (ii)the map , defined by , , is lower semi-continuous; (iii)for any , there is satisfying Then there exists such that . Further, if , then .

For the proof of the main results, we need the following crucial lemma [13].

Lemma 1.8. Let (X,d) be a metric space, and let be a -distance on . Let and be two sequences in , let and be sequences in converging to zero, and let . Then the following hold: (1)if and for any , then converges to ; (2)if for any with , then is a Cauchy sequence.

In the present paper, using the concept of -distance, in the same direction which has been used in [10, 11], we introduce some new contraction conditions for multi-valued mappings in complete metric spaces and three fixed point theorems for such contractions are proved. Our results generalize Theorems 1.6 and 1.7, and many other theorems.

2. Main Results

Recall that a sequence in is called an orbit of at if , for all , and for any and .

Now, we shall prove a theorem which is a generalization of Theorem 1.7.

Theorem 2.1. Let be a complete metric space with a -distance , and let be a multi-valued mapping from into . Assume that the following conditions hold: (i)the map , defined by , , is lower semi-continuous;(ii)there exist the functions , which satisfy (iii)for any , there is satisfying Then, there exists such that .

Proof. Let . Then there exists such that Then, from (2.3), we have Continuing this process, we can define an orbit of in , such that for all , which imply that We can assume that for all , since for some , then from (2.5), , and so for every , there exists such that ; consequently, ; so by Lemma 1.8, and thus , that is, the assertion of the theorem is proved. Since for all , then we have Thus is a decreasing sequence of positive real numbers and hence converges to a nonnegative number , . Let . Then, for , we can take such that Then, thus, . Therefore, by we can take such that Now, from (2.5), (2.9), and (2.10), we have for all , where and . Then for any , , we have Hence, by Lemma 1.8,   is a Cauchy sequence so there exists such that . Since is lower semi-continuous, we obtain and thus, Now, we clime that . Notice that the function is lower semi-continuous for all . Since , then by (2.12), for all . On the other hand, from (2.14), for every , there exists such that . Then, for all , we have Therefore, from (2.15), (2.16), we can find the sequences and in converging to zero, such that and for any ; then by Lemma 1.8  . The closedness of implies .

Now, we shall prove a theorem which is different from Theorem 2.1 and is a generalization of Theorem 1.6.

Theorem 2.2. Let be a complete metric space with a -distance , and let be a multi-valued mapping from into . Assume that the following conditions hold: (i)the map , defined by , , is lower semi-continuous;(ii)there exist the functions , which satisfy and is nondecreasing; (iii)for any , there is satisfying Then, there exists such that .

Proof. By following the lines in the proof of Theorem 2.1, one can construct an orbit of in such that for all , which imply that Thus, is a decreasing sequence, and so there exists such that . From (2.19) and (2.20), we have for all . Now we clime that is a nonincreasing sequence. Suppose not. Then there exists such that . Since is nondecreasing, then from (2.22), we get that which is a contradiction. Then, is a nonincreasing sequence and so is convergent. Now by using the same argument as in the proof of Theorem 2.1, we obtain the existence of a real number and such that thus . On the other hand, since is nondecreasing, then by (2.19), we have For the rest of the proof, we can go on as in the proof of Theorem 2.1.

In the same manner, we can present the following theorem.

Theorem 2.3. Let be a complete metric space with a -distance , and let be a multi-valued mapping from into . Assume that the following conditions hold: (i)the map , defined by , , is lower semi-continuous,(ii)there exist the functions , which satisfy and also one of and is nondecreasing; (iii)for any , there is satisfying Then, there exists such that .

Proof. As in the proof of Theorem 2.1, one can construct an orbit of in such that (2.19), (2.20), (2.21), and (2.22) hold. Then, is a decreasing sequence and so there exists such that . Now we clime that is a nonincreasing sequence. Suppose not. Then there exists such that . Since one of and is nondecreasing, it follows from (2.22) that which is a contradiction. Thus is a nonincreasing sequence and so is convergent. Now as in the proof of Theorem 2.1, we obtain the existence of a real number and such that thus . If is nondecreasing, then from assumption and (2.19), we have and if is nondecreasing then from (2.19) and (2.20), we have for all . Therefore, in all cases we have shown and so, by following as in the proof of Theorem 2.1, we can take such that for all , where . The rest of the proof is similar to that of Theorem 2.1.

Remark 2.4. Theorem 2.1 is a generalization of Theorem 1.7. In fact, if we consider and , then the assumptions of Theorem 2.1 are satisfied. Also, one can see that Theorem 2.1 generalizes Theorem 2.2 of Nicolae [11].

Remark 2.5. Theorem 2.2 essentially generalizes Theorem 1.6. Indeed, if we consider and , then all assumptions of Theorem 2.2 are satisfied.

The following example shows that there are mappings which satisfy the assumptions of Theorem 2.1 but do not satisfy the assumptions of Theorem 1.7.

Example 2.6. Consider , for , and . Then is a bounded complete subset of . Let , for all . Define a mapping from into by It is easy to verify that is lower semi-continuous in . Define and by Since then, we have For , there exists such that and for , , if , there exists satisfying and, if , there exists satisfying Therefore, all assumptions of Theorem 2.1 are satisfied and , are two fixed points of . Let us observe that does not satisfy the assumptions of Theorem 1.7 provided that , for all . Indeed, for any function , , there exists , , such that for , if , we have and if , we have that is, the assumptions of Theorem 1.7 are not satisfied. The next example is an application of Theorem 2.3.

Example 2.7. Let be as in the Example 2.6, and let , for all . Note that is a -distance on . Define a mapping from into by Clearly, is lower semi-continuous in . Define and by Note that is nondecreasing and , for each . Since then, For , there exists such that and for , , if , there exists satisfying and, if , there exists satisfying Then, all assumptions of Theorem 2.3 are satisfied and , are two fixed points of . Note that .

Acknowledgment

The authors would like to thank the referees for their valuable and useful comments.

References

S. Banach, “Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.

S. B. Nadler, “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969.

D. Klim and D. Wardowski, “Fixed point theorems for set-valued contractions in complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 132–139, 2007.

L. B. Ćirić, “Fixed point theorems for multi-valued contractions in metric spaces,” Journal of Mathematical Analysis and Applications, vol. 348, pp. 499–507, 2008.

L. Ćirić, “Multi-valued nonlinear contraction mappings,” Nonlinear Analysis, Theory, Methods and Applications, vol. 71, no. 7-8, pp. 2716–2723, 2009.

A. Latif and A. A. N. Abdou, “Multivalued generalised nonlinear contractive maps and fixed points,” Nonlinear Analysis, vol. 74, pp. 1436–1444, 2011.

N. Mizoguchi and W. Takahashi, “Fixed point theorems for multivalued mappings on complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 141, no. 1, pp. 172–188, 1989.

S. Reich, “Some problems and results in fixed point theory,” Contemporary Mathematics, vol. 21, pp. 179–187, 1983.

Y. Feng and S. Liu, “Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings,” Journal of Mathematical Analysis and Applications, vol. 317, no. 1, pp. 103–112, 2006.

Z. Liu, W. Sun, S. M. Kang, and J. S. Ume, “On fixed point theorems for multi-valued contractions,” Fixed Point Theory and Applications, vol. 2010, Article ID 870980, 18 pages, 2010.

A. Nicolae, “Fixed point theorems for multi-valued mappings of Feng-Liu type,” Fixed Point Theory, vol. 12, pp. 145–154, 2011.

O. Kada, T. Suzuki, and W. Takahashi, “Nonconvex minimization theorems and fixed point theorems in complete metric spaces,” Mathematica Japonica, vol. 44, pp. 381–391, 1996.

T. Suzuki and W. Takahashi, “Fixed point theorems and characterizations of metric completeness,” Topological Methods in Nonlinear Analysis, vol. 8, pp. 371–382, 1996.