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 [219].
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.
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 ℛpreserving mapping if
In the sequel, let ℕ denote the set of all nonnegative 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.
Definition 1.3 Let . We say that ℛ is Ntransitive on X if
The following remark is a consequence of the previous definition.
(i) If ℛ is transitive, then it is Ntransitive for all .
(ii) If ℛ is Ntransitive, then it is kNtransitive 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;
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
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 ancontractive mapping satisfying the following conditions:
(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
and, similarly, we have
Substituting and in (1), we obtain
So, by (2) and (3) it follows that
Using the monotone property of the ψfunction, we get
It follows that is monotone decreasing, and, consequently, there exists such that
which implies that , then by (a3) we get
On the other hand, by (2) and (i), we obtain
Similarly, by (3) and (i), we get
Now, for some , substituting and in (1), where , we get
So, by (6) and (7), we have
Using the monotone property of the ψfunction, we get
It follows that is monotone decreasing and consequently, there exists such that
which implies that , then by (a3) we get
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
Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfying (10). Then
Then we have
Furthermore, for each , there exist such that . Hence, by (11) we have
On the other hand, we have
Letting in the above inequalities, using (5), (9) and (15), we obtain
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
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 theregularity 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 ψfunction, 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) thatXisdirected, 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
and
Since T is preserving for , from (18) and (19), we get
and
Using (20), (21) and (1), we have
This implies that
Using the monotone property of the ψfunction, we get
It follows that is monotone decreasing and consequently, there exists such that
which implies that , then by (a3) we get
Similarly, we get
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
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
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 ancontractive mapping satisfying the following conditions:
(iv) Fis continuous, orisbiregular.
Then, Fhas a coupled fixed point. Moreover, ifisbidirected, 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
and
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 Ntransitive. 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 Ntransitive 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 Ntransitive binary relation .
Corollary 3.2LetXbe a nonempty set endowed with a binary relation. Suppose that there is a metricdonXsuch thatis complete. Letsatisfy theweaklycontractive conditions, that is,
whereψandφare altering distance functions. Suppose also that the following conditions hold:
(ii) Tis apreserving mapping;
(iv) Tis continuous orisregular.
ThenThas a fixed point. Moreover, ifXisdirected, we have the uniqueness of the fixed point.
Proof In order to link this theorem to the main result, we define the mapping by
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:
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:
(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.
Hence, Definition 2.3 is equivalent to Definition 3.6.
We start by checking that and are Ntransitive. 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 Ntransitive.
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 selfmapping 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 selfmapping 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;
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 1transitive.
Next, we claim that T is contractive mappings. Indeed, in case , we easily get
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 weaklycontractive, that is,
whereψandφare altering distance functions. Suppose also that the following conditions hold:
(i) Fis a mixed monotone mapping;
(iii) Fis continuous orisbiregular.
ThenFhas a coupled fixed point. Moreover, ifisbidirected, 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
Definition 4.4 (Karapınar and Sadarangani [10])
Let be a metric space, let m be a positive integer, let be closed nonempty 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
Corollary 4.5 (Karapınar and Sadarangani [10])
Letbe a complete metric space, letmbe a positive integer, letbe closed nonempty subsets ofXand let. Suppose thatis an altering distance function, andTis a cyclic weakφcontraction, 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 MirceaDan Rus for his remarkable comments, suggestion and ideas that helped to improve this paper.
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