Journal of Applied Mathematics

Volume 2012 (2012), Article ID 196759, 12 pages

doi:10.1155/2012/196759

Research Article

New Nonlinear Conditions and Inequalities for the Existence of Coincidence Points and Fixed Points

Wei-Shih Du and Shao-Xuan Zheng

Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan

Received 22 January 2012; Accepted 3 July 2012

Academic Editor: C. Conca

Copyright © 2012 Wei-Shih Du and Shao-Xuan Zheng. 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.

Some new fixed point theorems for nonlinear maps are established. By using these results, we can obtain some new coincidence point theorems. Our results are quite different in the literature and references therein.

1. Introduction and Preliminaries

Let us begin with some basic definitions and notation that will be needed in this paper. Throughout this paper, we denote by and , the sets of positive integers and real numbers, respectively. Let be a metric space. For each , and , let . Denote by the class of all nonempty subsets of , the family of all nonempty closed 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 .

A point in is a fixed point of a map if (when is a single-valued map) or (when is a multivalued map). The set of fixed points of is denoted by .

Let be a self-map and be a multivalued map. A point in is said to be a coincidence point (see, for instance, [1–4]) of and if . The set of coincidence points of and is denoted by .

The celebrated Banach contraction principle (see, e.g., [5]) plays an important role in various fields of applied mathematical analysis. Since then a number of generalizations in various different directions of the Banach contraction principle have been investigated by several authors in the past; see [6–20] and references therein. In 1969, Nadler [6] first proved a famous generalization of the Banach contraction principle for multivalued maps.

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

Definition 1.1 (see [3, 4, 7–10, 21, 22]). A function is said to be an -function (or -function) if

It is obvious that if is a nondecreasing function or a nonincreasing function, then is an -function. So the set of -functions is a rich class. But it is worth to mention that there exist functions which are not -functions.

Example A (see [4]). Let be defined by

Since ,   is not an -function.

Very recently, Du [4] first proved the following characterizations of -functions.

Theorem 1.2 (see [4]). 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 [10]; that is, for any strictly decreasing sequence in , one has .

In 1989, Mizoguchi and Takahashi [11] proved the following fixed point theorem which is a generalization of Nadler’s fixed point theorem and gave a partial answer of Problem 9 in Reich [12]. It is worth to mention that the primitive proof of Mizoguchi-Takahashi’s fixed point theorem is quite difficult. Recently, Suzuki [13] gave a very simple proof of Mizoguchi-Takahashi’s fixed point theorem.

Theorem MT (Mizoguchi and Takahashi). Let be a complete metric space and be a multivalued map and be a -function. Assume that

then .

In 2007, M. Berinde and V. Berinde [14] proved the following interesting fixed point theorem which generalized Mizoguchi-Takahashi’s fixed point theorem.

Theorem BB (M. Berinde and V. Berinde). Let be a complete metric space, be a multivalued map, be a -function and . Assume that

then .

Let be a metric space. Recall that a function is called a -distance [3, 5, 7, 15–23], if the following are satisfied:

() for any ;()for any , is l.s.c.;()for any , there exists such that and imply .

A function is said to be a -function [3, 7, 16, 18–22], introduced and studied by Lin and Du, if the following conditions hold:

for all ;if and in with such that for some , then ;for any sequence in with , if there exists a sequence in such that , then ;for ,,, and imply .

It is well known that the metric is a -distance and any -distance is a -function, but the converse is not true; see [7, 16].

The following results are crucial in this paper.

Lemma 1.3 (see [7, 20]). Let be a metric space and be a function. Assume that satisfies the condition . If a sequence in with , then is a Cauchy sequence in .

Let be any function. For each and , let .

Lemma 1.4 (see [7, 19, 20]). Let be a closed subset of a metric space and be any function. Suppose that satisfies and there exists such that . Then   if and only if .

Recently, Du [7, 19] first introduced the concepts of -functions and -metrics as follows.

Definition 1.5 (see [7, 19]). Let be a metric space. A function is called a -function if it is a -function on with for all .

Remark 1.6. If is a -function, then, from , if and only if .

Definition 1.7 (see [7, 19]). Let be a metric space and be a -function (resp., -distance). For any , , define a function by

where , then is said to be the - (resp., -) on induced by .

Clearly, any Hausdorff metric is a -metric, but the reverse is not true.

Lemma 1.8 (see [7, 19]). Let be a metric space and be a -metri on induced by a -function . Then every -metric is a metric on .

Recently, Du [7] established the following new fixed point theorems for -metric and -functions to extend Berinde-Berinde’s fixed point theorem.

Theorem D (Du [7, Theorem 2.1]). Let be a complete metric space, be a multivalued map and a -function. Suppose that for each

and further satisfies one of the following conditions:

(D1) is closed;(D2)the map defined by is l.s.c.;(D3)the map defined by is l.s.c.;(D4)for each sequence in with , and , one has ;(D5) for every .

Then .

In [7], Du also gave the generalizations of Kannan’s fixed point theorem, Chatterjea’s fixed point theorem and other new fixed point theorems for nonlinear multivalued contractive maps; see [7] for more detail.

In this paper, we first establish some new types of fixed point theorem. Some applications to the existence for coincidence point and others are also given. Our results are quite different in the literature and references therein.

2. New Inequalities and Nonlinear Conditions for Fixed Point Theorems

In this section, we first establish some new existence theorems for fixed point.

Theorem 2.1. Let be a complete metric space, be a -function and be a multivalued map. Suppose that

() there exist two functions such that is an -function and for each , it holds

() further satisfies one of the following conditions:(H1) is closed, that is, , the graph of is closed in ;(H2)the map defined by is l.s.c.;(H3)the map defined by is l.s.c.;(H4)for any sequence in with , and , one has ;(H5) for every .

Then .

Proof. Let . If , then we are done. If , then by Lemma 1.4. Choose . If , then is a fixed point of . Otherwise, if , then, by , we have

Since , there exists such that

which implies

Since and is a -function, . Since , we have and hence

Continuing in this way, we can construct inductively a sequence in satisfying , is strictly decreasing in and

for each . Since is an -function, applying Theorem 1.2, we get

Put and . Then , and

By (2.6) and (2.8), we have

We claim that . Let , . For , with , we have

Since , and, from (2.10), we get

Applying Lemma 1.3, is a Cauchy sequence in . By the completeness of , there exists such that as . From and (2.10), we have

Now, we verify that . If (H1) holds, since is closed, and as , we have .

If (H2) holds, by the lower semicontinuity of , as and (2.11), we obtain

which implies . By Lemma 1.4, we get .

Suppose that (H3) holds. Since is convergent in , . Since

we have and hence .

If (H4) holds, by (2.11), there exists with and such that . By , . Since , it follows that as . By the closedness of , we get or .

Finally, assume that (H5) holds. On the contrary, suppose that . Then, by (2.10) and (2.12), we obtain

a contradiction. Therefore . The proof is completed.

If we put in Theorem 2.1, then we have the following result.

Corollary 2.2. Let be a complete metric space and be a multivalued map. Suppose that

() there exist two functions such that is an -function and for each , it holds

() further satisfies one of the following conditions:

    (h1) is closed;    (h2) the map defined by is l.s.c.;    (h3) for any sequence in with , and , one has ;    (h4) for every .

Then .

The following result is immediate from Theorem 2.1.

Theorem 2.3. Let be a complete metric space, be a -function, and be a multivalued map. Suppose that the condition holds and further assume that

() there exists an -function such that for each ,

then .

Corollary 2.4. Let be a complete metric space and be a multivalued map. Suppose that the condition holds and further assume that

there exists an -function such that for each ,

then .

Theorem 2.5. Let be a complete metric space, be a -function, and be a multivalued map. Suppose that the condition holds and further assume that

there exist such that for each , for all ,

then .

Proof. Let be defined by and for all . Then implies and the conclusion follows from Theorem 2.1.

Corollary 2.6. Let be a complete metric space and be a multivalued map. Suppose that the condition holds and further assume that

there exist such that for each , for all ,

then .

Theorem 2.7. Let be a complete metric space, be a -function, and be a multivalued map. Suppose that the condition holds and further assume that

there exists such that for each , for all ,

then .

Proof. Let . Then implies and the conclusion follows from Theorem 2.5.

Remark 2.8. and are equivalent. Indeed, in the proof of Theorem 2.7, we have shown that implies . If holds, then put . So and holds. Hence and are equivalent. Therefore Theorem 2.5 can also be proved by using Theorem 2.7 and we know that Theorems 2.5 and 2.7 are indeed equivalent.

Corollary 2.9. Let be a complete metric space and be a multivalued map. Suppose that the condition holds and further assume that

there exists such that for each , for all ,

then .

Remark 2.10. Corollaries 2.6 and 2.9 are equivalent.

3. Applications of Theorem 2.1 to the Existence of Coincidence Points

By applying Theorem 2.1, we can prove easily the following new coincidence point theorem.

Theorem 3.1. Let be a complete metric space, be a -function, be a self-map, be a multivalued map, and . Suppose that the condition holds and further assume that

is -invariant (i.e., ) for each ; there exist two functions such that is an -function and it holds

then .

Proof. For each , if , from , we have . So . Hence implies . Applying Theorem 2.1, . So there exists such that . By , . Therefore, and the proof is complete.

The following result is immediate from Theorem 3.1.

Corollary 3.2. Let be a complete metric space, be a self-map, be a multivalued map, and . Suppose that the condition holds and further assume

is -invariant (i.e., ) for each ; there exist two functions such that is an -function and it holds

then .

As an application of Theorem 3.1, one has the following fixed point theorem.

Theorem 3.3. Let be a complete metric space, be a -function, be a multivalued map, and . Suppose that the condition holds and further assume that

there exist two functions such that is an -function and it holds

then .

Corollary 3.4. Let be a complete metric space, be a multivalued map and . Suppose that the condition holds and further assume that

there exist two functions such that is an -function and it holds

then .

Theorem 3.5. Let be a complete metric space, be a -function, be a -metric on , be a multivalued map, be a self-map, and . Suppose that the conditions and hold and further assume that

there exist two functions such that is an -function and it holds

then .

Corollary 3.6. Let be a complete metric space, be a multivalued map, be a self-map, and . Suppose that the conditions and hold and further assume that

there exist two functions such that is an -function and it holds

then .

Theorem 3.7. Let be a complete metric space, be a -function, be a -metric on , be a multivalued map, and . Suppose that the condition holds and further assume that

there exist two functions such that is an -function and it holds

then .

Corollary 3.8. Let be a complete metric space, be a multivalued map and . Suppose that the condition holds and further assume that

there exist two functions such that is an -function and it holds

then .

Acknowledgments

This research was supported partially by Grant no. NSC 100-2115-M-017-001 of the National Science Council of the Republic of China.

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