Fixed point results for -contractive mappings for a generalized altering distance
© Berzig and Karapınar; licensee Springer 2013
Received: 24 April 2013
Accepted: 23 July 2013
Published: 29 July 2013
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  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.  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.
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 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.
The following remark is a consequence of the previous definition.
If ℛ is transitive, then it is N-transitive for all .
If ℛ is N-transitive, then it is kN-transitive for all .
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:
ψ is continuous;
ψ is nondecreasing;
In the sequel, the binary relations and are defined as following.
Now we are ready to state our first main result.
is N-transitive for ;
T is -preserving for ;
there exists such that for ;
T is continuous.
Then, T has a fixed point, that is, 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 2.2 In Theorem 2.1, if we replace the continuity of T by the -regularity of , then the conclusion of Theorem 2.1 holds.
Letting in the above inequality, we get , that is, . □
Theorem 2.3 Adding to the hypotheses of Theorem 2.1 (respectively, Theorem 2.2) that X is -directed, we obtain uniqueness of the fixed point of T.
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 .
Definition 3.1 (Guo and Lakshmikantham )
In this section, we define two binary relations and as follows.
We have the following result.
is N-transitive for ();
- (ii)For all , we have
- (iii)There exists such that
F is continuous, or is -biregular.
Then, F has a coupled fixed point . Moreover, if is -bidirected, then we have the uniqueness of the coupled fixed point.
and is given by (25). We shall prove that T is -contractive mapping.
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 , 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  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 .
is N-transitive ();
T is a -preserving mapping;
there exists such that ;
T is continuous or is -regular.
Then T has a fixed point. Moreover, if X is -directed, we have the uniqueness of the fixed point.
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 , 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. )
for all with ;
- (ii)there exists such that
Then T has a unique fixed point in .
Let us define the binary relations and .
Now, based on Theorem 2.2, we will derive a more general result for cyclic mappings.
for all with ;
- (ii)there exist two altering distance functions ψ and φ such that
Then T has 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.
Hence, Definition 2.3 is equivalent to Definition 3.6.
which implies that . Hence, we obtain and , that is, and , which implies that and are N-transitive.
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 .
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 )
where ψ and φ are altering distance functions. Then T has 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 )
where φ is an altering distance functions. Then T has 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.
- 1.We define two binary relations and on X by
We say that X is ≤-directed if for all there exists a such that and .
Corollary 4.3 (Harjani and Sadarangani )
T is a nondecreasing mapping;
there exists with ;
T is continuous or,
(iii′) for every sequence in X such that , and is a nondecreasing sequence, there exists a subsequence such that for all .
Then T has a fixed point. Moreover, if X is ≤-directed, we have the uniqueness of the fixed point.
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.
hence, our claim holds.
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 .
Corollary 4.4 (Harjani et al. )
F is a mixed monotone mapping;
there exists with ;
F is continuous or is -biregular.
Then F has a coupled fixed point . Moreover, if is -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  for cyclic weak -contractive mappings.
Definition 4.3 (Păcurar and Rus )
, are nonempty sets;
Definition 4.4 (Karapınar and Sadarangani )
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 )
Let be a complete metric space, let m be a positive integer, let be closed non-empty subsets of X and let . Suppose that is an altering distance function, and T is a cyclic weak φ-contraction, where is a cyclic representation of Y with respect to T. Then, T has a unique fixed point .
Proof The proof follows immediately from Corollary 3.3. □
The authors thank Professor Mircea-Dan Rus for his remarkable comments, suggestion and ideas that helped to improve this paper.
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