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# Fixed point results for ( α ψ , β φ ) -contractive mappings for a generalized altering distance

Maher Berzig1 and Erdal Karapınar2*

Author Affiliations

1 Tunis College of Sciences and Techniques, Tunis University, 5 Avenue Taha Hussein, Tunis, Tunisia

2 Department of Mathematics, Atilim University, Incek, Ankara, 06836, Turkey

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Fixed Point Theory and Applications 2013, 2013:205  doi:10.1186/1687-1812-2013-205

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/205

 Received: 24 April 2013 Accepted: 23 July 2013 Published: 29 July 2013

© 2013 Berzig and Karapınar; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this manuscript, we extend the concept of altering distance, and we introduce a new notion of -contractive mappings. We prove the existence and uniqueness of a fixed point for such mapping in the context of complete metric space. The presented theorems of this paper generalize, extend and improve some remarkable existing results in the literature. We also present several applications and consequences of our results.

### 1 Introduction and preliminaries

Fixed point theory is one of the core research areas in nonlinear functional analysis since it has a broad range of application potential in various fields such as engineering, economics, computer science, and many others. Banach contraction mapping principle [1] is considered to be the initial and fundamental result in this direction. Fixed point theory and hence the Banach contraction mapping principle have evidently attracted many prominent mathematicians due to their wide application potential. Consequently, the number of publications in this theory increases rapidly; we refer the reader to [2-19].

In this paper, by introducing a new notion of -contractive mappings, we aim to establish a more general result to collect/combine a number of existing results in the literature.

We start by recalling the notion of altering distance function introduced by Khan et al.[12] as follows.

Definition 1.1 A function is called an altering distance function if the following properties are satisfied:

is continuous and nondecreasing.

if and only if .

Now, we present a definition, which will be useful later.

Definition 1.2 Let X be a set, and let ℛ be a binary relation on X. We say that is an ℛ-preserving mapping if

In the sequel, let ℕ denote the set of all non-negative integers, let ℝ denote the set of all real numbers.

Example 1.1 Let and a function defined as .

Define and by

Define the first binary relation by if and only if , and define the second binary relation by if and only if . Then, we easily obtain that T is simultaneously -preserving and -preserving.

Definition 1.3 Let . We say that ℛ is N-transitive on X if

The following remark is a consequence of the previous definition.

Remark 1.1 Let . We have:

(i) If ℛ is transitive, then it is N-transitive for all .

(ii) If ℛ is N-transitive, then it is kN-transitive for all .

Definition 1.4 Let be a metric space and , two binary relations on X. We say that is -regular if for every sequence in X such that as , and

there exists a subsequence such that

Definition 1.5 We say that a subset D of X is -directed if for all , there exists such that

### 2 Main results

Before we start the introduction of the concept -contractive mappings, we introduce the notion of a pair of generalized altering distance as follows:

Definition 2.1 We say that the pair of functions is a pair of generalized altering distance where if the following hypotheses hold:

(a1) ψ is continuous;

(a2) ψ is nondecreasing;

(a3) .

The condition (a3) was introduced by Popescu in [15] and Moradi and Farajzadeh in [14].

Definition 2.2 Let be a metric space, and let be a given mapping. We say that T is -contractive mappings if there exists a pair of generalized distance such that

(1)

where .

In the sequel, the binary relations and are defined as following.

Definition 2.3 Let X be a set and are two mappings. We define two binary relations and on X by

and

Now we are ready to state our first main result.

Theorem 2.1Letbe a complete metric space, , and letbe an-contractive mapping satisfying the following conditions:

(i) isN-transitive for;

(ii) Tis-preserving for;

(iii) there existssuch thatfor;

(iv) Tis continuous.

Then, Thas a fixed point, that is, there existssuch that.

Proof Let such that for . Define the sequence in X by

If for some , then is a fixed point T. Assume that for all . From (ii) and (iii), we have

Similarly, we have

By induction, from (ii) it follows that

(2)

and, similarly, we have

(3)

Substituting and in (1), we obtain

So, by (2) and (3) it follows that

(4)

Using the monotone property of the ψ-function, we get

It follows that is monotone decreasing, and, consequently, there exists such that

Letting in (4), we obtain

which implies that , then by (a3) we get

(5)

On the other hand, by (2) and (i), we obtain

(6)

Similarly, by (3) and (i), we get

(7)

Now, for some , substituting and in (1), where , we get

So, by (6) and (7), we have

(8)

Using the monotone property of the ψ-function, we get

It follows that is monotone decreasing and consequently, there exists such that

Letting , we obtain

which implies that , then by (a3) we get

(9)

Next, we claim that is a Cauchy sequence. Suppose if we obtain a contradiction, that T is not a Cauchy sequence. Then, there exists , for which we can find subsequences and of with such that

(10)

Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfying (10). Then

(11)

Then we have

(12)

Letting and using (5),

(13)

Furthermore, for each , there exist such that . Hence, by (11) we have

(14)

Again, letting and using (5),

(15)

On the other hand, we have

Letting in the above inequalities, using (5), (9) and (15), we obtain

(16)

By setting and , in (1), we get

that is,

Now, using (6) and (7), we get

Letting , using (15), (16) and the continuity of ψ and φ, we obtain

(17)

which implies by (a3) that , a contradiction with . Hence, our claim holds, that is, is a Cauchy sequence. Since is a complete metric space, then there exists such that

From the continuity of T, it follows that as . Due to the uniqueness of the limit, we derive that , that is, is a fixed point of T. □

Theorem 2.2In Theorem 2.1, if we replace the continuity ofTby the-regularity of, then the conclusion of Theorem 2.1 holds.

Proof Following the lines of the proof of Theorem 2.1, we get that is a Cauchy sequence. Since is a complete metric space, then there exists such that as . Furthermore, the sequence satisfies (2) and (3), that is,

Now, since is -regular, then there exists a subsequence of such that , that is, and , that is, for all k. By setting and , in (1), we obtain

that is,

Using the monotone property of the ψ-function, we get

Letting in the above inequality, we get , that is, . □

Theorem 2.3Adding to the hypotheses of Theorem 2.1 (respectively, Theorem 2.2) thatXis-directed, we obtain uniqueness of the fixed point ofT.

Proof Suppose that and are two fixed points of T. Since X is -directed, there exists such that

(18)

and

(19)

Since T is -preserving for , from (18) and (19), we get

(20)

and

(21)

Using (20), (21) and (1), we have

This implies that

(22)

Using the monotone property of the ψ-function, we get

It follows that is monotone decreasing and consequently, there exists such that

Letting in (22), we obtain

which implies that , then by (a3) we get

(23)

Similarly, we get

(24)

Using (23) and (24), the uniqueness of the limit gives us . □

### 3 Some corollaries

In this section, we derive new results from the previous theorems.

#### 3.1 Coupled fixed point results in complete metric spaces

Let us recall the definition of a coupled fixed point introduced by Guo and Lakshmikantham in [5].

Definition 3.1 (Guo and Lakshmikantham [5])

Let be a given mapping. We say that is a coupled fixed point of F if

Lemma 3.1A pairis a coupled fixed point ofFif and only ifis a fixed point ofTwhereis given by

(25)

Definition 3.2 Let be a metric space and be a given mapping. We say that F is an -contractive mappings if there exists a pair of generalized distance such that

where .

In this section, we define two binary relations and as follows.

Definition 3.3 Let X be a set, and let , be two binary relations on defined by

and

Definition 3.4 Let be a metric space. We say that is -biregular if for all sequences in such that , as , and

there exists a subsequence such that

Definition 3.5 We say that is -bidirected if for all , there exists such that

We have the following result.

Corollary 3.1Letbe a complete metric space andbe an-contractive mapping satisfying the following conditions:

(i) isN-transitive for ();

(ii) For all, we have

(iii) There existssuch that

(iv) Fis continuous, oris-biregular.

Then, Fhas a coupled fixed point. Moreover, ifis-bidirected, then we have the uniqueness of the coupled fixed point.

Proof By Lemma 3.1, a pair is a coupled fixed point of F if and only if is a fixed point of T. Now, consider the complete metric space , where and

From (iv), we have

(26)

and

(27)

Since is nondecreasing, then for all . Hence, for all , we have

where are the functions defined by

and is given by (25). We shall prove that T is -contractive mapping.

Define two binary relations and by

First, we claim that for are N-transitive. Let for all such that

that is,

By definitions of a and b, it follows that

or

Hence, by (i), we have

that is,

or

Then our claim holds.

Let such that and . Using condition (ii), we obtain immediately that and . Then T is -preserving for . Moreover, from condition (iii), we know that there exists such that for . If F is continuous, then T also is continuous. Then all the hypotheses of Theorem 2.1 are satisfied. If is -biregular, then we easily have that is -regular. Hence, Theorem 2.2 yields the result. We deduce the existence of a fixed point of T that gives us from (25) the existence of a coupled fixed point of F. Now, since is -bidirected, one can easily derive that is -directed by regarding Lemma 3.1 and Definition 3.5. Finally, by using Theorem 2.3, we obtain the uniqueness of the fixed point of T, that is, the uniqueness of the coupled fixed point of F. □

#### 3.2 Fixed point results on metric spaces endowed with N-transitive binary relation

In [18], Samet and Turinci established fixed point results for contractive mappings on metric spaces, endowed with an amorphous arbitrary binary relation. Very recently, this work has been extended by Berzig in [2] to study the coincidence and common fixed points.

In this section, we establish a fixed point theorem on metric space endowed with N-transitive binary relation .

Corollary 3.2LetXbe a non-empty set endowed with a binary relation. Suppose that there is a metricdonXsuch thatis complete. Letsatisfy the-weakly-contractive conditions, that is,

whereψandφare altering distance functions. Suppose also that the following conditions hold:

(i) isN-transitive ();

(ii) Tis a-preserving mapping;

(iii) there existssuch that;

(iv) Tis continuous oris-regular.

ThenThas a fixed point. Moreover, ifXis-directed, we have the uniqueness of the fixed point.

Proof In order to link this theorem to the main result, we define the mapping by

(28)

and we define the mapping by

(29)

Next, by using (28), (29) and Definition 2.3, the conclusion follows directly from Theorems 2.1, 2.2 and 2.3. □

#### 3.3 Fixed point results for cyclic contractive mappings

In [13], Kirk et al. have generalized the Banach contraction principle. They obtained a new fixed point results for cyclic contractive mappings.

Theorem 3.1 (Kirk et al.[13])

For, letbe a nonempty closed subsets of a complete metric space, and letbe a given mapping. Suppose that the following conditions hold:

(i) for allwith;

(ii) there existssuch that

ThenThas a unique fixed point in.

Let us define the binary relations and .

Definition 3.6 Let X be a nonempty set and let , be nonempty closed subsets of X. We define two binary relations for by

Now, based on Theorem 2.2, we will derive a more general result for cyclic mappings.

Corollary 3.3For, letbe nonempty closed subsets of a complete metric space, and letbe a given mapping. Suppose that the following conditions hold:

(i) for allwith;

(ii) there exist two altering distance functionsψandφsuch that

ThenThas a unique fixed point in.

Proof Let . For all , we have by assumption that each is nonempty closed subset of the complete metric space X, which implies that is complete.

Define the mapping by

and define the mapping by

Hence, Definition 2.3 is equivalent to Definition 3.6.

We start by checking that and are N-transitive. Indeed, let such that and for all , that is, and for all such that

which implies that . Hence, we obtain and , that is, and , which implies that and are N-transitive.

Next, from (ii) and the definition of α and β, we can write

for all . Thus, T is -contractive mapping.

We claim next that T is -preserving and -preserving. Indeed, let such that and , that is, and ; hence, there exists such that , . Thus, , then and , that is, and . Hence, our claim holds.

Also, from (i), for any for all , we have , which implies that and , that is, and .

Now, we claim that Y is -regular. Let be a sequence in Y such that as , and

that is,

It follows that there exist such that

so

By letting

we conclude that the subsequence satisfies

hence and for all k, that is, and , which proves our claim.

Hence, all the hypotheses of Theorem 2.2 are satisfied on , and we deduce that T has a fixed point in Y. Since for some and for all , then .

Moreover, it is easy to check that X is -directed. Indeed, let with , , . For , we have and . Thus, X is -directed.

Finally, the uniqueness follows by Theorem 2.3. □

### 4 Related fixed point theorems

In this section, we show that many existing results in the literature can be deduced from our results.

#### 4.1 Classical fixed point results

Corollary 4.1 (Dutta and Choudhury [4])

Letbe a complete metric space, and letbe a self-mapping satisfying the inequality

whereψandφare altering distance functions. ThenThas a unique fixed point.

Proof Let be the mapping defined by for all . Then T is -contractive mappings. It is easy to show that all the hypotheses of Theorems 2.1 and 2.2 are satisfied. Consequently, T has a unique fixed point. □

Corollary 4.2 (Rhoades [17])

Letbe a complete metric space, and letbe a self-mapping satisfying the inequality

whereφis an altering distance functions. ThenThas a unique fixed point.

Proof Following the lines of the proof of Corollary 4.1, by taking , we get the desired result. □

#### 4.2 Fixed point results in partially ordered metric spaces

We start by defining the binary relations for and the concept of ≤-directed.

Definition 4.1 Let be a partially ordered set.

1. We define two binary relations and on X by

2. We say that X is ≤-directed if for all there exists a such that and .

Corollary 4.3 (Harjani and Sadarangani [8])

Letbe a partially ordered set anddbe a metric onXsuch thatis complete. Suppose that the mappingis weakly contractive, that is,

whereψandφare altering distance functions. Suppose also that the following conditions hold:

(i) Tis a nondecreasing mapping;

(ii) there existswith;

If either:

(iii) Tis continuous or,

(iii′) for every sequenceinXsuch that, andis a nondecreasing sequence, there exists a subsequencesuch thatfor all.

ThenThas a fixed point. Moreover, ifXis ≤-directed, we have the uniqueness of the fixed point.

Proof Using Definition 2.3, we can define the binary relations and by the mappings :

and

In case , the functions α and β are well defined, because the altering functions and are null only, and only if , that is, which is not the case.

We can verify easily that and are 1-transitive.

Next, we claim that T is -contractive mappings. Indeed, in case , we easily get

and in case , we have

hence, our claim holds.

Moreover, from the monotone property of T, we get

and similarly, we have

Thus T is -preserving for . Now, if condition (iii) is satisfied, that is, T is continuous, the existence of a fixed point follows from Theorem 2.1. Suppose now, that the condition (iii′) is satisfied, and let be a nondecreasing sequence in X, that is, and for all n. Suppose also that as . From (iii′), there exists a subsequence such that for all k. This implies from the definition of α and β that and for all k, which implies for and for all k. In this case, the existence of a fixed point follows from Theorem 2.2.

To show the uniqueness, suppose that X is ≤-directed, that is, for all there exists a such that and , which implies from the definition of α and β that and . Hence, Theorem 2.3 gives us the uniqueness of this fixed point. □

#### 4.3 Coupled fixed point theorems

Next, in order to prove a coupled fixed point results in partially ordered set, we need to define an order relation on the set .

Let be a partially ordered set endowed with a metric d, and let be a given mapping. We endow the product set with the partial order:

Definition 4.2 Let X be a set and binary relations for on defined by

Corollary 4.4 (Harjani et al.[6])

Letbe a partially ordered set and suppose that there exists a metricdinXsuch thatis a complete metric space. Suppose that the mappingis weakly-contractive, that is,

whereψandφare altering distance functions. Suppose also that the following conditions hold:

(i) Fis a mixed monotone mapping;

(ii) there existswith;

(iii) Fis continuous oris-biregular.

ThenFhas a coupled fixed point. Moreover, ifis-bidirected, we have the uniqueness of the fixed point.

Proof The conclusions then follows directly from Corollary 3.1. □

#### 4.4 Fixed point results for cyclic contractive mappings

In this section, we will derive from our results the fixed point theorem of Karapınar and Sadarangani [11] for cyclic weak -contractive mappings.

Definition 4.3 (Păcurar and Rus [16])

Let X be a nonempty set, m a positive integer and an operator. By definition, is a cyclic representation of X with respect to T if

1. , are nonempty sets;

2. , .

Definition 4.4 (Karapınar and Sadarangani [10])

Let be a metric space, let m be a positive integer, let be closed non-empty subsets of X, and let . An operator is called a cyclic weak -contraction if

1. is a cyclic representation of Y with respect to T, and

2. is an altering distance function such that

(30)

for any , , where .

Corollary 4.5 (Karapınar and Sadarangani [10])

Letbe a complete metric space, letmbe a positive integer, letbe closed non-empty subsets ofXand let. Suppose thatis an altering distance function, andTis a cyclic weakφ-contraction, whereis a cyclic representation ofYwith respect toT. Then, Thas a unique fixed point.

Proof The proof follows immediately from Corollary 3.3. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

### Acknowledgements

The authors thank Professor Mircea-Dan Rus for his remarkable comments, suggestion and ideas that helped to improve this paper.

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