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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 938724, 9 pages
http://dx.doi.org/10.1155/2013/938724
Research Article

The Study of Fixed Point Theory for Various Multivalued Non-Self-Maps

1Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan
2Department of Mathematics, Atilim University, İncek, 06836 Ankara, Turkey
3Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21859, Saudi Arabia

Received 24 April 2013; Accepted 19 July 2013

Academic Editor: Abdellah Bnouhachem

Copyright © 2013 Wei-Shih Du 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 basic motivation of this paper is to extend, generalize, and improve several fundamental results on the existence (and uniqueness) of coincidence points and fixed points for well-known maps in the literature such as Kannan type, Chatterjea type, Mizoguchi-Takahashi type, Berinde-Berinde type, Du type, and other types from the class of self-maps to the class of non-self-maps in the framework of the metric fixed point theory. We establish some fixed/coincidence point theorems for multivalued non-self-maps in the context of complete metric spaces.

1. Introduction

During the last few decades, the celebrated Banach contraction principle, also known as the Banach fixed point theorem [1], has become one of the core topics of applied mathematical analysis. As a consequence, a number of generalizations, extensions, and improvement of the praiseworthy Banach contraction principle in various direction have been explored and reported by various authors; see, for example, [230] and the references therein. In parallel with the Banach contraction principle, Kannan [5] and Chatterjea [6] created, respectively, different type, fixed point theorems as follows.

Theorem 1 (Kannan). Let be a complete metric space, is a single-valued map, and . Assume that Then has a unique fixed point in .

Theorem 2 (Chatterjea). Let be a complete metric space, is a single-valued map, and . Assume that Then has a unique fixed point in .

The characterization of the renowned Banach fixed point theorem in the setting of multivalued maps is one of the most outstanding ideas of research in fixed point theory. The remarkable examples in this trend were given by Nadler [2], Mizoguchi and Takahashi [3], and M. Berinde and V. Berinde [4]. On the other hand, investigation of the existence of a fixed point of non-self-maps under certain condition is an interesting research subject of metric fixed point theory, see, for example, [1927], and references therein.

The following attractive result was reported by M. Berinde and V. Berinde [4] in 2007.

Theorem 3 (M. Berinde and V. Berinde). Let be a complete metric space, a multivalued map, an -function (i.e., for all ), and . Assume that Then has a fixed point in .

If we take in Theorem 3, then we conclude the remarkable result of Mizoguchi and Takahashi [3] which is a partial answer of problem 9 in [8].

Theorem 4 (Mizoguchi and Takahashi). Let be a complete metric space, a multivalued map, and an -function. Assume that Then has a fixed point in .

Recently, Du [12] established the following theorem which is an extension of Theorem 3 and hence Theorem 4.

Theorem 5 (Du). Let be a complete metric space, a multivalued map, an -function and a function. Assume that Then has a fixed point in .

The basic objective of this paper is to investigate the existence of coincidence and fixed points of multivalued non-self-maps under the certain conditions in the setting of metric spaces. The presented results generalize, improve, and extend several crucial and notable results that examine the existence of the coincidence/fixed point of well-known maps such as Kannan type, Chatterjea type, Mizoguchi-Takahashi type, Berinde-Berinde type, Du type, and other types in the context of complete metric spaces.

2. Preliminaries

Let be a metric space. For each and , let . Denote by the class of all nonempty subsets of and the family of all nonempty closed and bounded subsets of . A function defined by is said to be the Hausdorff metric on induced by the metric on . It is also known that is a complete metric space whenever is a complete metric space.

Let be a nonempty subset of , a single-valued map, and a multivalued map. A point in is a coincidence point of and if . If is the identity map, then and call a fixed point of . The set of fixed points of and the set of coincidence point of and are denoted by and , respectively. In particular, if , we use and instead of and , respectively. Throughout this paper, we denote by , and , the set of positive integers and real numbers, respectively.

Let be a real-valued function defined on . For , we recall that

Definition 6 (see [918]). A function is said to be an -function (or -function) if for all .

It is evident that if is a nondecreasing function or a nonincreasing function, then is an -function. So the set of -functions is a rich class. An example which is not an -function is given as follows. Let be defined by We note that is not an -function, since .

In what follows that, we recall some characterizations of -functions proved first by Du [12].

Theorem 7 (see [12]). Let be a function. Then the following statements are equivalent.(a)is an -function.(b)For each , there exist and such that for all .(c)For each , there exist and such that for all .(d)For each , there exist and such that for all .(e)For each , there exist and such that for all .(f)For any nonincreasing sequence in , one has .(g) is a function of contractive factor; that is, for any strictly decreasing sequence in , we have .

3. Existence Theorems of Coincidence Points and Fixed Points for Multivalued Non-Self-Maps of Kannan Type and Chatterjea Type

In this section, we prove the existence of coincidence points and fixed points of multivalued non-self-maps of Kannan type and Chatterjea type. For this purpose, we first established a new intersection theorem of and for multivalued non-self-maps in complete metric spaces.

Theorem 8. Let be a complete metric space, a nonempty closed subset of , a multivalued map and a continuous self-map. Suppose that(D1) for all ,(D2) is -invariant (i.e., ) for each ,(D3) there exist a function and such that

Then .

Proof. Since a nonempty closed subset of and is complete, is also a complete metric space. Let . Put and . So . Let be arbitrary. Then . By (D2), we have . Hence (9) implies Inequality (10) shows that Let be given. Take . By (D1), . Choose . If , then and hence from (D2). Hence and the proof is finished. Otherwise, if , then . By (11), we have which implies that there exists such that Next, by (11) again, there exists such that By induction, we can obtain a sequence in satisfying By (16), we have Let , . For , with , we have Since , and hence . This proves that is a Cauchy sequence in . By the completeness of , there exists such that as . By (15) and (D2), we have Since is continuous and , we have Since the function is continuous, by (9), (15), (19), and (20), we get which implies . By the closedness of , we have . From (D2), . Hence we verify . The proof is complete.

Theorem 9. In Theorem 8, if condition (D3) is replaced with one of the following conditions:(K1) there exist a function and such that (K2) there exist a function and such that (K3) there exist a function and such that (K4) there exist a function and such that (K5) there exist a function and such that (K6) there exist a function and such that (K7) there exist a function and such that

Then .

Proof. It is obvious that any of these conditions (K1)–(K7) implies condition (D3) as in Theorem 8. So the desired conclusion follows from Theorem 8 immediately.

The following fixed point theorem for multivalued non-self-maps of generalized Kannan type can be established immediately from Theorem 9 for (the identity mapping).

Theorem 10. Let be a complete metric space, a nonempty closed subset of , and a multivalued map. Suppose that for all and one of the following conditions holds:(P1) there exist a function and such that (P2) there exist a function and such that (P3) there exist a function and such that (P4) there exist a function and such that (P5) there exist a function and such that (P6) there exist a function and such that (P7) there exist a function and such that (P8) there exist a function and such that

Then .

As a consequence of Theorem 10, we obtain the following generalized Kannan type fixed point theorems for multivalued maps.

Corollary 11. Let be a complete metric space, a nonempty closed subset of , and a multivalued map. Suppose that for all and there exists such that Then .

Remark 12. (a) If in Corollary 11, then we can obtain a multivalued version of Kannan’s fixed point theorem [5].
(b) Theorems 810 and Corollary 11 all extend and generalize Kannan’s fixed point theorem.

Theorem 13. Let be a complete metric space, a nonempty closed subset of , a multivalued map, and a continuous self-map. Suppose that conditions (D1) and (D2) as in Theorem 8 hold. If there exist and such that Then .

Proof. Let . Since , by the denseness of , we can find such that . Let be arbitrary. Then . By (D2), we have . Hence (38) has been reduced to Let be given. Take . By (D1), . Choose . If , then and hence from (D2). Hence and the proof is finished. Otherwise, if , then . By (39), we have which implies that there exists such that Let . Then and the last inequality implies Continuing in this way, we can construct inductively a sequence in satisfying for each . Using a similar argument as in the proof of Theorem 8, we have the following: (i); (ii) is a Cauchy sequence in ; (iii)there exists such that as ; (iv) for each ; (v).
By (38), we get which implies . By the closedness of , we have . By (D2), and hence . The proof is complete.

Theorem 14. In Theorem 13, if inequality (38) is replaced with one of the following inequalities: (C1) (C2) (C3) (C4) (C5) (C6) (C7) then .

Applying Theorem 14, we can prove the following fixed point theorems for multivalued maps of generalized Chatterjea type.

Theorem 15. Let a complete metric space, a nonempty closed subset of , and a multivalued map. Suppose that for all and one of the following conditions holds:(Q1) there exist a function and such that (Q2) there exist a function and such that (Q3) there exist a function and such that (Q4) there exist a function and such that (Q5) there exist a function and such that (Q6) there exist a function and such that (Q7) there exist a function and such that (Q8) there exist a function and such that

Then .

The following result is a generalized Chatterjea’s type fixed point theorem for multivalued maps in complete metric spaces.

Corollary 16. Let be a complete metric space, a nonempty closed subset of , and a multivalued map. Suppose that for all and there exists such that Then .

Remark 17. (a) If in Corollary 16, then we can obtain a multivalued version of Chatterjea’s fixed point theorem [6].
(b) Theorems 1315 and Corollary 16 all improve and generalize Chatterjea’s fixed point theorem.

4. New Coincidence and Fixed Point Results for Various Multivalued Non-Self-Maps: Mizoguchi-Takahashi Type, Berinde-Berinde Type, and Du Type

In this section, we prove some coincidence and fixed point theorems for multivalued non-self-maps of Mizoguchi-Takahashi type, Berinde-Berinde type, and Du type.

Recall first the following auxiliary result.

Lemma 18 (see [9, Lemma 2.1]). Let be an -function. Suppose that is defined by . Then, is also an -function.

Theorem 19. Let be a complete metric space, a nonempty closed subset of , a multivalued map, and be a continuous self-map. Suppose that conditions (D1) and (D2) as in Theorem 8 hold. If there exist an -function and a function such that then .

Proof. Since is a nonempty closed subset of and is complete, is also a complete metric space. Note first that for each , by (D2), we have for all . So, for each , by (61), we obtain Define by . Then, by Lemma 18, is also an -function. Let be given. Take . Since from (D1), we can choose . If , then and hence from (D2). Thus, and hence we achieved the result. Now, suppose that ; that is, . By (62), we have which implies that there exists such that Next, by (62) again, there exists such that Iteratively, we can obtain a sequences in satisfying for each . Since for all , by (ii), we know that is strictly decreasing in . Since is an -function, by (g) of Theorem 7, we obtain Let . So . By (67), we have Let , . For , with , we have Since , and hence . This proves that is a Cauchy sequence in . By the completeness of , there exists such that as . Thanks to (66) and (D2), we have Since is continuous and , we have Since the function is continuous, by (61), (66), and (72), we get which implies . By the closedness of , we have . By (D2), and hence . The proof is complete.

Theorem 20. In Theorem 19, if inequality (61) is replaced with the following inequality: Then .

Corollary 21. Let be a complete convex metric space, a nonempty closed subset of , a multivalued map, and a continuous self-map. Suppose that(i) for all ,(ii) is -invariant (i.e., ) for each ,(iii) there exist an -function and such that

Then .

Corollary 22. Let be a complete convex metric space, a nonempty closed subset of , a multivalued map, and a continuous self-map. Suppose that(i) for all ,(ii) is -invariant (i.e. ) for each ,(ii)there exist an -function and such that

Then .

As a direct consequence of Theorems 19 and 20, we obtain the following fixed point result for multivalued non-self-maps of Du type in complete metric spaces.

Theorem 23. Let be a complete convex metric space, a nonempty closed subset of , and a multivalued map. Suppose that for all , and one of the following conditions holds:(W1) there exist an -function and a function such that (W2) there exist an -function and a function such that

Then .

Proof. Let be the identity map. It is easy to verify that all the conditions of Theorem 19 (or Theorem 20) are satisfied. Hence the conclusion follows from Theorem 19 (or Theorem 20).

The following fixed point theorems for multivalued non-self-maps of generalized Berinde-Berinde type and generalized Mizoguchi-Takahashi type are established immediately from Theorem 23.

Corollary 24. Let be a complete convex metric space, a nonempty closed subset of , and a multivalued map. Suppose that(i) for all ,(ii) there exist an -function and such that

Then .

Corollary 25. Let be a complete convex metric space, a nonempty closed subset of , and a multivalued map. Suppose that(i) for all ,(ii) there exist an -function and such that

Then .

Corollary 26. Let be a complete convex metric space, a nonempty closed subset of , and a multivalued map. Suppose that(i) for all ,(ii)there exists an -function such that

Then .

Remark 27. (a) If in Theorem 23, then we can obtain Du’s fixed point theorem [12, Theorem 2.6].
(b) Theorems 19, 20 and 23, and Corollaries 2126 all generalize and improve Du’s fixed point theorem, Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, and Banach’s contraction principle.

Acknowledgment

The first author was supported partially by grant no. NSC 101-2115-M-017-001 of the National Science Council of the Republic of China.

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