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Journal of Applied Mathematics
Volume 2012, Article ID 161470, 11 pages
http://dx.doi.org/10.1155/2012/161470
Research Article

The Existence of Fixed Points for Nonlinear Contractive Maps in Metric Spaces with -Distances

1Department of Mathematics, Payame Noor University, Tehran 19395-4697, Iran
2Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan

Received 20 November 2011; Accepted 4 December 2011

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Hossein Lakzian and Ing-Jer Lin. 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 fixed point theorems for -contractive maps and -contractive maps on a complete metric space are proved. Presented fixed point theorems generalize many results existing in the literature.

1. Introduction and Preliminaries

Branciari [1] established a fixed point result for an integral type inequality, which is a generalization of Banach contraction principle. Kada et al. [2] introduced and studied the concept of -distance on a metric space. They give examples of -distances and improved Caristi’s fixed point theorem, Ekeland’s -variational’s principle, and the nonconvex minimization theorem according to Takahashi (see many useful examples and results on -distance in [25] and in references therein). Kada et al. [2] defined the concept of -distance in a metric space as follows.

Definition 1.1 (see [2]). Let be a metric space endowed with a metric . A function is called a -distance on if it satisfies the following properties:(1) for any ,(2) is lower semicontinuous in its second variable, that is, if and in then ,(3)for each , there exists such that and imply .
We denote by the set of functions satisfying the following hypotheses:(c1) is continuous and nondecreasing,(c2) if and only if .
We denote by the set of functions satisfying the following hypotheses:(h1) is right continuous and nondecreasing,(h2) for all .
Let be a -distance on metric space , and . A map from into itself is a ()-contractive map on if for each , .

The following lemmas are used in the next section.

Lemma 1.2 (see [3]). If , then for each , and if and , then .

Lemma 1.3 (see [2]). Let be a metric space and let be a -distance on .(i)If is a sequence in such that , then . In particular, if , then .(ii)If for any , where and are sequences in converging to 0, then converges to .(iii)Let be a -distance on metric space and a sequence in such that for each there exist such that implies (or ), then is a Cauchy sequence.

Note that if and , then and, by Lemma 1.3, .

In [3], Razani et al. proved a fixed point theorem for -contractive mappings, which is a new version of the main theorem in [1], by considering the concept of the -distance.

The main aim of this paper is to present some generalization fixed point Theorems by Kada et al. [2], Hicks and Rhoades [6] and several other results with respect to ()-contractive maps on a complete metric space.

2. -Contractive Maps

In the next theorem we state one of the main results of this paper generalizing Theorem 4 of [2]. In what follows, we use to denote the composition of with .

Theorem 2.1. Let be a -distance on complete metric space and . Suppose is a map that satisfies for each and that for every with . Then there exists such that . Moreover, if , then .

Proof. Fix . Set with . Then by (2.1) thus and Lemma 1.2 implies and similarly
Now we proof that is a Cauchy sequence. By triangle inequality, continuity of and (2.4), we have as and so which concludes
By induction, for any we have
So, by Lemma 1.3, is a Cauchy sequence, and since is complete, there exists such that in.
Now we prove that is a fixed point of .
From (2.8), for each , there exists such that implies but and is lower semicontinuous, thus Therefore, . Set and we have

Now, assume that . Then by hypothesis, we haveas by (2.4) and (2.10). This is a contradiction. Hence .

If , we have

This is a contradiction. So , and by hypothesis .

Here we give a simple example illustrating Theorem 2.1. In this example, we will show that Theorem  4 in [2] cannot be applied.

Example 2.2. Let , which is a complete metric space with usual metric of reals. Moreover, by defining , is a -distance on . Let be a map as , . Suppose is a continuous and strictly nondecreasing map and , for any . We have and so there is not any such that , and hence Theorem 4 in [2] dose not work. But because for any we have . Also for any we have . So for arbitrary , , hence is satisfied in Theorem 2.1. We note that 0 is a fixed point for .

The next examples show the role of the conditions (2.1) and (2.2).

Example 2.3. Let , , and define by , where . Set and for all . Let us define by and if . We have
If , then and hence (2.1) holds.
Now, we remark that , and Thus, the condition (2.2) is not satisfied, and there is no with . In this case we observe that Theorem 2.1 is invalid without condition (2.2).

Example 2.4. Let , , , and set . Let be as Example 2.3. Let us define by and if . Clearly, has no fixed point in . Now, for each and that for every with , so condition (2.2) is satisfied. But, for , for any . Hence, condition (2.1) dose not hold. We note that Theorem 2.1 dose not work without condition (2.1).

Suppose is Lebesgue-integrable mapping which is summable and , for each . Now, in the next corollary, set and , where . Then, and . Hence we can conclude the following corollary as a special case.

Corollary 2.5. Let be a selfmap of a complete metric space satisfying for all . Suppose that with . Then there exists a such that .

Note that Corollary 2.5 is invalid without condition (2.20). For example, take , which is a complete metric space with usual metric of reals. Define by and for . Set . It is easy to check that , for any ; however, for any and . Clearly, has got no fixed point in .

Remark 2.6. From Theorem 2.1, we can obtain Theorem 4 in [2] as a special case. For this, in the hypotheses of Theorem 2.1, set and for all .

Corollary 2.7. Let be a -distance on complete metric space , and . Suppose is a continuous mapping for into itself such that (2.1), is satisfied. Then there exists such that . Moreover, if , then .

Proof. Assume that there exists with and . Then there exists a sequence such that as . Hence and as . Lemma 1.3 implies that as . Now by assumption and so as . By Lemma 1.2, as . We also have hence as . By Lemma 1.3, we conclude that converges to . Since is continuous, we have This is a contradiction. Therefore, if , then . So, Theorem 2.1 gives desired result.

In Example 2.3, is satisfied in condition (2.1), but it is not continuous. So, the hypotheses in Corollary 2.7are not satisfied. We note that has no fixed point.

It is an obvious fact that, if is a map which has a fixed point , then is also a fixed point of for every natural number . However, the converse is false. If a map satisfies for each , where denotes a set of all fixed points of , then it is said to have property [7, 8]. The following theorem extends and improves Theorem 2 of [7].

Theorem 2.8. Let be a complete metric space with -distance on . Suppose satisfies(i) or(ii)with strict inequality, and for all , . If , then has property .

Proof. We shall always assume that , since the statement for is trivial. Let . Suppose that satisfies . Then, and so . Now from we have . Hence, by Lemma 1.3, we have , and . Suppose that satisfies . If , then there is nothing to prove. Suppose, if possible, that . Then a repetition of the argument for case leads to , that is a contradiction. Therefore, in all cases, and .

The following theorem extends Theorem 2.1 of [6]. A function mapping into the real is -orbitally lower semicontinuous at if is a sequence in and implies that .

Theorem 2.9. Let be a complete metric space with -distance on . Suppose and there exists an such that Then,(i) exists,(ii)(iii) if and only if is -orbitally lower semicontinuous at .

Proof. Observe that and are immediate from the proof of Theorem 2.1. We prove . It is clear that impling is -orbitally lower semicontinuous at .
and is -orbitally lower semicontinuous at implies So, .

The mapping is orbitally lower semicontinuous at if implies that . In the following, we improve Theorem 2 of [9] that it is correct form Theorem 1 of [7].

Theorem 2.10. Let be a -distance on complete metric space and . Suppose is orbitally lower semicontinuous map on that satisfies for each . Then there exists such that . Moreover, if , then .

Proof. Observe that the sequence is a Cauchy sequence immediate from the proof of Theorem 2.1 and so there exists a point in such that as . Since is orbitally lower semicontinuous at , we have . Now, we have and so . Similarly, . Hence, . By Theorem 2.1 we can conclude that if , then .

The following example shows that Theorem  2 in [9] cannot be applicable. So our generalization is useful.

Example 2.11. Let be a metric space with metric defined by ,, which is complete. We define by . Let be as defined before in Corollary 2.5 and ,. Assume that by for any . We have, , and so Theorem  2 in [9] dose not work. But for each . Hence by Theorem 2.10 there exists a fixed point for . We note that 0 is fixed point for .

3. -Contractive Maps

In this section we obtain fixed points for ()-contractive maps (i.e., -contractive maps that for all , where ).

In 1969, Kannan [10] proved the following fixed point theorem. Contractions are always continuous and Kannan maps are not necessarily continuous.

Theorem 3.1 (see [10]). Let be a complete metric space. Let be a Kannan mapping on , that is, there exists such that for all . Then, has a unique fixed point in . For each , the iterative sequence converges to the fixed point.

In the next theorem, we generalize this theorem as follows.

Theorem 3.2. Let be a complete metric space. Let be a -Kannan mapping on , that is, there exists such that for all . Then, has a unique fixed point in . For each , the iterative sequence converges to the fixed point.

Proof. Let and define for any , and set . Then, , and so
Then, from the proof of Theorem 2.1, exists. From (3.4), we have Thus, , and so . Clearly, is unique. This completes the proof.

The set of all subadditive functions in is denoted by . In the following theorems, we generalize Theorems 3.4 and 3.5 due to Suzuki and Takahashi [4].

Theorem 3.3. Let be a -distance on complete metric space and be a selfmap. Suppose there exists such that(i) for each ,(ii) for every with . Then has a fixed point in . Moreover, if is a fixed point of , then .

Proof. Fix . Define and for every . Put . Then, . By hypothesis, since , we have for all . It follows that for all . Using the similar argument as in the proof of Theorem 2.1, we can prove that the sequence is Cauchy and so there exists such that as . Also, we have . Since we have and so . The proof is completed.

Corollary 3.4. Let be a -distance on complete metric space and let be a continuous map. Suppose there exists such that for each .
Then has a fixed point in . Moreover, if is a fixed point of , then .

Proof. It suffices to show that for every with . Assume that there exists with and . Then there exists a sequence in such that . It follows that and as . Hence, . On the other hand, since and (3.9), we have and hence for all . Thus, as . Therefore, . Since is continuous, we have which is a contradiction. Therefore, using Theorem 3.3, . This completes the proof.

Question 1. Can we generalize Theorems 3.2, 3.3, and Corollary 3.4 for ()-contractive maps?

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