Abstract and Applied Analysis

VolumeΒ 2012, Article IDΒ 793486, 17 pages

http://dx.doi.org/10.1155/2012/793486

## Generalized - Contractive Type Mappings and Related Fixed Point Theorems with Applications

^{1}Department of Mathematics, Atilim University, Δ°ncek, 06836 Ankara, Turkey^{2}Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia

Received 28 May 2012; Accepted 25 July 2012

Academic Editor: Jean MichelΒ Combes

Copyright Β© 2012 Erdal KarapΔ±nar and Bessem Samet. 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 establish fixed point theorems for a new class of contractive mappings. As consequences of our main results, we obtain fixed point theorems on metric spaces endowed with a partial order and fixed point theorems for cyclic contractive mappings. Various examples are presented to illustrate our obtained results.

#### 1. Introduction and Preliminaries

Let be the family of functions satisfying the following conditions: is nondecreasing; for all , where is the th iterate of . These functions are known in the literature as (c)-comparison functions. It is easily proved that if is a (c)-comparison function, then for any .

Very recently, Samet et al. [1] introduced the following concepts.

*Definition 1.1. *Let be a metric space and be a given mapping. We say that is an - contractive mapping if there exist two functions and such that

Clearly, any contractive mapping, that is, a mapping satisfying Banach contraction, is an - contractive mapping with for all and , .

*Definition 1.2. *Let and . We say that is -admissible if for all , and we have
Various examples of such mappings are presented in [1].

The main results in [1] are the following fixed point theorems.

Theorem 1.3. *Let be a complete metric space and be an - contractive mapping. Suppose that*(i)* is admissible; *(ii)*there exists such that ; *(iii)* is continuous.** Then there exists such that .*

Theorem 1.4. *Let be a complete metric space and be an - contractive mapping. Suppose that *(i)* is admissible; *(ii)*there exists such that ; *(iii)*if is a sequence in such that for all and as , then for all . ** Then there exists such that . *

Theorem 1.5. *Adding to the hypotheses of Theorem 1.3 (resp., Theorem 1.4) the condition, for all , there exists such that and , and one obtains uniqueness of the fixed point. *

In the present work, we introduce the concept of generalized - contractive type mappings, and we study the existence and uniqueness of fixed points for such mappings. Presented theorems in this paper extend and generalize the above results derived by Samet et al. in [1]. Moreover, from our fixed point theorems, we will deduce various fixed point results on metric spaces endowed with a partial order and fixed point results for cyclic contractive mappings.

#### 2. Main Results

We introduce the concept of generalized - contractive type mappings as follows.

*Definition 2.1. *Let be a metric space and be a given mapping. We say that is a generalized - contractive mapping if there exist two functions and such that for all , and we have
where .

*Remark 2.2. *Clearly, since is nondecreasing, every - contractive mapping is a generalized - contractive mapping.

Our first result is the following.

Theorem 2.3. *Let be a complete metric space. Suppose that is a generalized - contractive mapping and satisfies the following conditions: *(i)* is admissible; *(ii)*there exists such that ; *(iii)* is continuous. ** Then there exists such that . *

*Proof. *Let such that (such a point exists from condition (ii)). Define the sequence in by for all . If for some , then is a fixed point of . So, we can assume that for all . Since is admissible, we have
Inductively, we have
From (2.1) and (2.3), it follows that for all , we have
On the other hand, we have
From (2.4) and taking in consideration that is a nondecreasing function, we get that
for all . If for some , we have , from (2.6), we obtain that
a contradiction. Thus, for all , we have
Using (2.6) and (2.8), we get that
for all . By induction, we get
From (2.10) and using the triangular inequality, for all , we have
This implies that is a Cauchy sequence in . Since is complete, there exists such that
Since is continuous, we obtain from (2.12) that
From (2.12), (2.13) and the uniqueness of the limit, we get immediately that is a fixed point of , that is, .

The next theorem does not require the continuity of .

Theorem 2.4. *Let be a complete metric space. Suppose that is a generalized - contractive mapping and the following conditions hold: *(i)* is admissible; *(ii)*there exists such that ; *(iii)*if is a sequence in such that for all and as , then there exists a subsequence of such that for all . ** Then there exists such that . *

*Proof. *Following the proof of Theorem 2.3, we know that the sequence defined by for all , converges for some . From (2.3) and condition (iii), there exists a subsequence of such that for all . Applying (2.1), for all , we get that
On the other hand, we have
Letting in the above equality, we get that
Suppose that . From (2.16), for large enough, we have , which implies that . Thus, from (2.14), we have
Letting in the above inequality, using (2.16), we obtain that
which is a contradiction. Thus we have , that is, .

With the following example, we will show that hypotheses in Theorems 2.3 and 2.4 do not guarantee uniqueness of the fixed point.

*Example 2.5. *Let be endowed with the Euclidean distance for all . Obviously, is a complete metric space. The mapping is trivially continuous and satisfies for any
for all , where
Thus is a generalized - contractive mapping. On the other hand, for all , we have
Thus is admissible. Moreover, for all , we have . Then the assumptions of Theorem 2.3 are satisfied. Note that the assumptions of Theorem 2.4 are also satisfied; indeed if is a sequence in that converges to some point ββwith for all , then, from the definition of , we have for all , which implies that for all . However, in this case, has two fixed points in .

For the uniqueness of a fixed point of a generalized - contractive mapping, we will consider the following hypothesis. (H) For all , there exists such that and .

Here, denotes the set of fixed points of .

Theorem 2.6. *Adding condition to the hypotheses of Theorem 2.3 (resp., Theorem 2.4), one has obtains that is the unique fixed point of .*

*Proof. *Suppose that is another fixed point of . From (H), there exists such that
Since is admissible, from (2.22), we have
Define the sequence in by for all and . From (2.23), for all , we have
On the other hand, we have
Using the above inequality, (2.24) and the monotone property of , we get that
for all . Without restriction to the generality, we can suppose that for all . If , we get from (2.26) that
which is a contradiction. Thus we have , and
for all . This implies that
Letting in the above inequality, we obtain that
Similarly, one can show that
From (2.30) and (2.31), it follows that . Thus we proved that is the unique fixed point of .

*Example 2.7. *Let be endowed with the standard metric for all . Obviously, is a complete metric space. Define the mapping by
In this case, is not continuous. Define the mapping by
We will prove that (A) is a generalized - contractive mapping, where for all ; (B) is -admissible; (C)there exists such that ; (D)if is a sequence in such that for all and as , then there exists a subsequence of such that for all ; (E)condition (H) is satisfied.

* Proof of (A). *To show (A), we have to prove that (2.1) is satisfied for every . If and , we have
Then (2.1) holds. If and , we have
Then (2.1) holds also in this case. The other cases are trivial. Thus (2.1) is satisfied for every .

* Proof of (B). *Let such that . From the definition of , we have two cases.*Caseββ1* (if ). In this case, we have , which implies that .*Caseββ2* (if ). In this case, we have , which implies that .

So, in all cases, we have . Thus is admissible.

*Proof of (C). *Taking , we have .

*Proof of (D). *Let be a sequence in such that for all and as for some . From the definition of , for all , we have
Since is a closed set with respect to the Euclidean metric, we get that
which implies that . Thus we have for all .

* Proof of (E). *Let . It is easy to show that, for , we have . So, condition (H) is satisfied.

Conclusion. *Now, all the hypotheses of Theorem 2.6 are satisfied; thus has a unique fixed point . In this case, we have .*

#### 3. Consequences

Now, we will show that many existing results in the literature can be deduced easily from our Theorem 2.6.

##### 3.1. Standard Fixed Point Theorems

Taking in Theorem 2.6, for all , we obtain immediately the following fixed point theorem.

Corollary 3.1. *Let be a complete metric space and be a given mapping. Suppose that there exists a function such that
**
for all . Then has a unique fixed point.*

The following fixed point theorems follow immediately from Corollary 3.1.

Corollary 3.2 (see Berinde [2]). *Let be a complete metric space and be a given mapping. Suppose that there exists a function such that
**
for all . Then has a unique fixed point. *

Corollary 3.3 (see ΔiriΔ [3]). *Let be a complete metric space and be a given mapping. Suppose that there exists a constant such that
**
for all . Then has a unique fixed point. *

Corollary 3.4 (see Hardy and Rogers [4]). *Let be a complete metric space and be a given mapping. Suppose that there exist constants with such that
**
for all . Then has a unique fixed point. *

Corollary 3.5 (see Banach Contraction Principle [5]). *Let be a complete metric space and be a given mapping. Suppose that there exists a constant such that
**
for all . Then has a unique fixed point. *

Corollary 3.6 (see Kannan [6]). *Let be a complete metric space and be a given mapping. Suppose that there exists a constant such that
**
for all . Then has a unique fixed point. *

Corollary 3.7 (see Chatterjea [7]). *
for all . Then has a unique fixed point. *

##### 3.2. Fixed Point Theorems on Metric Spaces Endowed with a Partial Order

Recently there have been so many exciting developments in the field of existence of fixed point on metric spaces endowed with partial orders. This trend was started by Turinici [8] in 1986. Ran and Reurings in [9] extended the Banach contraction principle in partially ordered sets with some applications to matrix equations. The obtained result in [9] was further extended and refined by many authors (see, e.g., [10β15] and the references cited therein). In this section, from our Theorem 2.6, we will deduce very easily various fixed point results on a metric space endowed with a partial order. At first, we need to recall some concepts.

*Definition 3.8. *Let be a partially ordered set and be a given mapping. We say that is nondecreasing with respect to if

*Definition 3.9. *Let be a partially ordered set. A sequence is said to be nondecreasing with respect to if for all .

*Definition 3.10. *Let be a partially ordered set and be a metric on . We say that is regular if for every nondecreasing sequence such that as , there exists a subsequence of such that for all .

We have the following result.

Corollary 3.11. *Let be a partially ordered set and be a metric on such that is complete. Let be a nondecreasing mapping with respect to . Suppose that there exists a function such that
**
for all with . Suppose also that the following conditions hold: *(i)*there exists such that ; *(ii)* is continuous or is regular. ** Then has a fixed point. Moreover, if for all there exists such that and , one has uniqueness of the fixed point.*

*Proof. *Define the mapping by
Clearly, is a generalized - contractive mapping, that is,
for all . From condition (i), we have . Moreover, for all , from the monotone property of , we have
Thus is admissible. Now, if is continuous, the existence of a fixed point follows from Theorem 2.3. Suppose now that is regular. Let be a sequence in such that for all and as . From the regularity hypothesis, there exists a subsequence of such that for all . This implies from the definition of that for all . In this case, the existence of a fixed point follows from Theorem 2.4. To show the uniqueness, and let . By hypothesis, there exists such that and , which implies from the definition of that and . Thus we deduce the uniqueness of the fixed point by Theorem 2.6.

The following results are immediate consequences of Corollary 3.11.

Corollary 3.12. *Let be a partially ordered set and be a metric on such that is complete. Let be a nondecreasing mapping with respect to . Suppose that there exists a function such that
**
for all ββwith . Suppose also that the following conditions hold: *(i)*there exists such that ; *(ii)* is continuous or is regular. ** Then has a fixed point. Moreover, if for all there exists such that and , one has uniqueness of the fixed point.*

Corollary 3.13. *Let be a partially ordered set and be a metric on such that is complete. Let be a nondecreasing mapping with respect to . Suppose that there exists a constant such that
**
for all ββwith . Suppose also that the following conditions hold: *(i)*there exists such that ; *(ii)* is continuous or is regular. **
Then has a fixed point. Moreover, if for all there exists such that and , one has uniqueness of the fixed point. *

Corollary 3.14. *Let be a partially ordered set and be a metric on such that is complete. Let be a nondecreasing mapping with respect to . Suppose that there exist constants with such that
**
for all with . Suppose also that the following conditions hold: *(i)*there exists such that ; *(ii)* is continuous or is regular. *

Corollary 3.15 (see Ran and Reurings [9], Nieto and LΓ³pez [16]). *Let be a partially ordered set and be a metric on such that is complete. Let be a nondecreasing mapping with respect to . Suppose that there exists a constant such that
**
for all with . Suppose also that the following conditions hold: *(i)*there exists such that ; *(ii)* is continuous or is regular. *

Corollary 3.16. *Let be a partially ordered set and be a metric on such that is complete. Let be a nondecreasing mapping with respect to . Suppose that there exists a constant such that
**
for all with . Suppose also that the following conditions hold: *(i)*there exists such that ; *(ii)* is continuous or is regular. *

Corollary 3.17. *
for all with . Suppose also that the following conditions hold: *(i)*there exists such that ; *(ii)* is continuous or is regular. **
Then has a fixed point. Moreover, if for allββ there exists such that and , one has uniqueness of the fixed point. *

##### 3.3. Fixed Point Theorems for Cyclic Contractive Mappings

One of the remarkable generalizations of the Banach Contraction Mapping Principle was reported by Kirk et al. [17] via cyclic contraction. Following the paper [17], many fixed point theorems for cyclic contractive mappings have appeared (see, e.g., [18β23]). In this section, we will show that, from our Theorem 2.6, we can deduce some fixed point theorems for cyclic contractive mappings.

We have the following result.

Corollary 3.18. *Let be nonempty closed subsets of a complete metric space and be a given mapping, where . Suppose that the following conditions hold: *(I)* and ; *(II)*there exists a function such that
**Then has a unique fixed point that belongs to . *

*Proof. *Since and are closed subsets of the complete metric space , then is complete. Define the mapping by
From (II) and the definition of , we can write
for all . Thus is a generalized - contractive mapping.

Let such that . If , from (I), , which implies that . If , from (I), , which implies that . Thus in all cases, we have . This implies that is -admissible.

Also, from (I), for any , we have , which implies that .

Now, let be a sequence in such that for all and as . This implies from the definition of that
Since is a closed set with respect to the Euclidean metric, we get that
which implies that . Thus we get immediately from the definition of that for all .

Finally, let . From (I), this implies that . So, for any , we have and . Thus condition (H) is satisfied.

Now, all the hypotheses of Theorem 2.6 are satisfied, and we deduce that has a unique fixed point that belongs to (from (I)).

The following results are immediate consequences of Corollary 3.18.

Corollary 3.19 (see Pacurar and Rus [21]). *Let be nonempty closed subsets of a complete metric space and be a given mapping, where . Suppose that the following conditions hold: *(I)* and ; *(II)*there exists a function such that
**Then has a unique fixed point that belongs to . *

Corollary 3.20. *Let be nonempty closed subsets of a complete metric space and be a given mapping, where . Suppose that the following conditions hold: *(I)* and ; *(II)*there exists a constant such that
**Then has a unique fixed point that belongs to . *

Corollary 3.21. * and ; *(II)*there exist constants with such that
**Then has a unique fixed point that belongs to . *

Corollary 3.22 (see Kirk et al. [17]). * and ; *(II)*there exists a constant such that
**Then has a unique fixed point that belongs to . *

Corollary 3.23. * and ; *(II)*there exists a constant such that
**Then has a unique fixed point that belongs to . *

Corollary 3.24. * and ; *(II)*there exists a constant such that
**Then has a unique fixed point that belongs to .*

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