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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 736063, 9 pages
http://dx.doi.org/10.1155/2011/736063
Research Article

A Suzuki Type Fixed-Point Theorem

Department of Mathematics, Faculty of Science and Arts, Kirikkale University, Yahsihan, 71450 Kirikkale, Turkey

Received 16 December 2010; Accepted 7 February 2011

Academic Editor: Genaro Lopez

Copyright © 2011 Ishak Altun and Ali Erduran. 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 present a fixed-point theorem for a single-valued map in a complete metric space using implicit relation, which is a generalization of several previously stated results including that of Suziki (2008).

1. Introduction

There are a lot of generalizations of Banach fixed-point principle in the literature. See [15]. One of the most interesting generalizations is that given by Suzuki [6]. This interesting fixed-point result is as follows.

Theorem 1.1. Let be a complete metric space, and let be a mapping on . Define a nonincreasing function from into by Assume that there exists , such that for all , then there exists a unique fixed-point of . Moreover, for all .

Like other generalizations mentioned above in this paper, the Banach contraction principle does not characterize the metric completeness of . However, Theorem 1.1 does characterize the metric completeness as follows.

Theorem 1.2. Define a nonincreasing function as in Theorem 1.1, then for a metric space the following are equivalent: (i) is complete,(ii)Every mapping on satisfying (1.2) has a fixed point.

In addition to the above results, Kikkawa and Suzuki [7] provide a Kannan type version of the theorems mentioned before. In [8], it is provided a Chatterjea type version. Popescu [9] gives a Ciric type version. Recently, Kikkawa and Suzuki also provide multivalued versions which can be found in [10, 11]. Some fixed-point theorems related to Theorems 1.1 and 1.2 have also been proven in [12, 13].

The aim of this paper is to generalize the above results using the implicit relation technique in such a way that for , where is a function as given in Section 2.

2. Implicit Relation

Implicit relations on metric spaces have been used in many papers. See [1, 1416].

Let denote the nonnegative real numbers, and let be the set of all continuous functions satisfying the following conditions:: is nonincreasing in variables ,: there exists , such that or or implies ,:, for all .

Example 2.1. , where . It is clear that .

Example 2.2. , where .
Let , then we have . Similarly, let , then we have . Again, let , then . Since , is satisfied with . Also , for all . Therefore, .

Example 2.3. , where .
Let , then we have . Similarly, let , then we have . Again, let , then . Thus, is satisfied with . Also , for all . Therefore, .

Example 2.4. , where .
Let , then we have . Similarly, let , then we have . Again, let , then . Since , is satisfied with . Also , for all . Therefore, .

Example 2.5. , where .
Let , then we have . Similarly, let , then we have . Again, let , then . Thus, is satisfied with . Also , for all . Therefore, .

3. Main Result

Theorem 3.1. Let be a complete metric space, and let be a mapping on . Define a nonincreasing function from into as in Theorem 1.1. Assume that there exists , such that implies for all , then has a unique fixed-point and holds for every .

Proof. Since , holds for every , by hypotheses, we have and so from , By , we have for all . Now fix and define a sequence in by . Then from (3.4), we have This shows that , that is, is Cauchy sequence. Since is complete, converges to some point . Now, we show that For , there exists , such that for all . Then, we have Hence, by hypotheses, we have and so Letting , we have and so By , we have and this shows that (3.6) is true.
Now, we assume that for all , then from (3.6), we have for all .
Case 1. Let . In this case, . Now, we show by induction that for . From (3.4), (3.14) holds for . Assume that (3.14) holds for some with . Since we have and so Therefore, by hypotheses, we have and so then and by , we have Therefore, (3.14) holds.
Now, from (3.6), we have This shows that , which contradicts (3.14).
Case 2. Let . In this case, . Again we want to show that (3.14) is true for . From (3.4), (3.14) holds for . Assume that (3.14) holds for some with . Since we have and so Therefore, as in the previous case, we can prove that (3.14) is true for . Again from (3.6), we have This shows that , which contradicts (3.14).Case 3. Let . In this case, . Note that for , either or holds. Indeed, if then we have which is a contradiction. Therefore, either or holds for every . If holds, then by hypotheses we have and so Letting , we have which contradicts . If holds, then by hypotheses we have and so Letting , we have which contradicts .
Therefore, in all the cases, there exists , such that . Since is Cauchy sequence, we obtain . That is, is a fixed point of . The uniqueness of fixed point follows easily from (3.6).

Remark 3.2. If we combine Theorem 3.1 with Examples 2.1, 2.2, 2.3, and 2.4, we have Theorem 2 of [6], Theorem 2.2 of [7], Theorem 3.1 of [7], and Theorem 4 of [8], respectively.

Using Example 2.5, we obtain the following result.

Corollary 3.3. Let be a complete metric space, and let be a mapping on . Define a nonincreasing function from into as in Theorem 1.1. Assume that implies for all , where , then there exists a unique fixed point of .

Remark 3.4. We obtain some new results, if we combine Theorem 3.1 with some examples of .

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