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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

## 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 [1–5]. 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, 14–16].

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 .

#### References

- A. Aliouche and V. Popa, “General common fixed point theorems for occasionally weakly compatible hybrid mappings and applications,”
*Novi Sad Journal of Mathematics*, vol. 39, no. 1, pp. 89–109, 2009. - S. K. Chatterjea, “Fixed-point theorems,”
*Comptes Rendus de l'Académie Bulgare des Sciences*, vol. 25, pp. 727–730, 1972. View at Zentralblatt MATH - Lj. B. Ćirić, “Generalized contractions and fixed-point theorems,”
*Publications de l'Institut Mathématique*, vol. 12(26), pp. 19–26, 1971. View at Zentralblatt MATH - R. Kannan, “Some results on fixed points,”
*Bulletin of the Calcutta Mathematical Society*, vol. 60, pp. 71–76, 1968. View at Zentralblatt MATH - T. Suzuki and M. Kikkawa, “Some remarks on a recent generalization of the Banach contraction principle,” in
*Fixed Point Theory and Its Applications*, pp. 151–161, Yokohama Publ., Yokohama, Japan, 2008. View at Zentralblatt MATH - T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,”
*Proceedings of the American Mathematical Society*, vol. 136, no. 5, pp. 1861–1869, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Kikkawa and T. Suzuki, “Some similarity between contractions and Kannan mappings,”
*Fixed Point Theory and Applications*, vol. 2008, Article ID 649749, 8 pages, 2008. View at Zentralblatt MATH - O. Popescu, “Fixed point theorem in metric spaces,”
*Bulletin of the Transilvania University of Braşov*, vol. 1(50), pp. 479–482, 2008. - O. Popescu, “Two fixed point theorems for generalized contractions with constants in complete metric space,”
*Central European Journal of Mathematics*, vol. 7, no. 3, pp. 529–538, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Kikkawa and T. Suzuki, “Some notes on fixed point theorems with constants,”
*Bulletin of the Kyushu Institute of Technology. Pure and Applied Mathematics*, no. 56, pp. 11–18, 2009. View at Zentralblatt MATH - M. Kikkawa and T. Suzuki, “Three fixed point theorems for generalized contractions with constants in complete metric spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 69, no. 9, pp. 2942–2949, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Enjouji, M. Nakanishi, and T. Suzuki, “A generalization of Kannan's fixed point theorem,”
*Fixed Point Theory and Applications*, vol. 2009, Article ID 192872, 10 pages, 2009. View at Zentralblatt MATH - M. Kikkawa and T. Suzuki, “Some similarity between contractions and Kannan mappings. II,”
*Bulletin of the Kyushu Institute of Technology. Pure and Applied Mathematics*, no. 55, pp. 1–13, 2008. View at Zentralblatt MATH - I. Altun and D. Turkoglu, “Some fixed point theorems for weakly compatible mappings satisfying an implicit relation,”
*Taiwanese Journal of Mathematics*, vol. 13, no. 4, pp. 1291–1304, 2009. View at Zentralblatt MATH - M. Imdad and J. Ali, “Common fixed point theorems in symmetric spaces employing a new implicit function and common property (E.A),”
*Bulletin of the Belgian Mathematical Society. Simon Stevin*, vol. 16, no. 3, pp. 421–433, 2009. View at Zentralblatt MATH - V. Popa, M. Imdad, and J. Ali, “Using implicit relations to prove unified fixed point theorems in metric and 2-metric spaces,”
*Bulletin of the Malaysian Mathematical Sciences Society*, vol. 33, no. 1, pp. 105–120, 2010. View at Zentralblatt MATH