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Fixed point results for modified weak and rational α-ψ-contractions in ordered 2-metric spaces
Fixed Point Theory and Applications volume 2014, Article number: 6 (2014)
Abstract
We first introduce the concept of a triangular 2-α-η-admissible mapping which extends the notion of α-admissible mapping with respect to η to 2-metric spaces. Next, we introduce the concepts of modified weak and modified rational α-ψ-contractions and establish the existence and uniqueness of fixed points for such mappings in complete 2-metric spaces. As an application of the obtained results, we prove some fixed point results in partially ordered 2-metric spaces. The presented theorems generalize and improve certain existing results in the literature and provide main results in Dung and Hang (Fixed Point Theory Appl. 2013:161, 2013) as corollaries. Moreover, some examples and an application to integral equations are provided to illustrate the usability of the obtained results.
MSC:46N40, 47H10, 54H25, 46T99.
1 Introduction and preliminaries
There exist many generalizations of the concept of metric spaces in the literature (; , K a cone in an ordered Banach space; 2-metric spaces; probabilistic metric spaces; G-metric spaces etc.; see, for example, [1–9]). The notion of 2-metric was introduced by Gähler in [10]. Note that a 2-metric is not a continuous function of its variables, whereas an ordinary metric is. This led Dhage to introduce the notion of D-metric in [11]. In [12, 13] Mustafa and Sims introduced the notion of G-metric to overcome flaws of a D-metric. After that, many fixed point theorems on G-metric spaces have been proved (see [14] and the references therein). The authors in [15] and [16] noticed that in several situations fixed point results in G-metric spaces can be in fact deduced from fixed point theorems in metric or quasi-metric spaces. It has also been shown by various authors that in several cases the fixed point results in cone metric spaces can be obtained by reducing them to their standard metric counterparts; for example, see [17–19]. It is worth to note that in the above generalizations, a 2-metric space was not known to be topologically equivalent to an ordinary metric.
We recollect some essential notations, required definitions and primary results coherent with the literature.
Definition 1.1 [10]
Let X be a non-empty set and let be a mapping satisfying the following assertions:
(d1) For every pair of distinct points , there exists a point such that ;
(d2) If at least two of three points x, y, z are the same, then ;
(d3) The symmetry:
for all ;
(d4) The rectangle inequality: for all .
Then d is called a 2-metric on X and is called a 2-metric space which will be sometimes denoted by X if there is no confusion. Every member is called a point in X.
Definition 1.2 [10]
Let be a 2-metric space and , . The set
is called a 2-ball centered at a and b with radius r. The topology generated by the collection of all 2-balls as a subbase is called a 2-metric topology on X.
Definition 1.3 [20]
Let be a sequence in a 2-metric space .
-
is said to be convergent to x in , written , if for all , ;
-
is said to be Cauchy in X if for all , , that is, for each , there exists such that for all and ;
-
is said to be complete if every Cauchy sequence is a convergent sequence.
Definition 1.4 [20]
A 2-metric space is said to be compact if every sequence in X has a convergent subsequence.
Lemma 1.1 [20]
Every 2-metric space is a -space.
Lemma 1.2 [20]
in a 2-metric space if and only if in the 2-metric topological space X.
Lemma 1.3 [20]
If is a continuous map from a 2-metric space X to a 2-metric space Y, then in X implies in Y.
It is straightforward from Definitions 1.1-1.3 that every 2-metric is non-negative and every 2-metric space contains at least three distinct points. A 2-metric is sequentially continuous in one argument; moreover, if a 2-metric is sequentially continuous in two arguments, then it is sequentially continuous in all three arguments (see [21]). A convergent sequence in a 2-metric space need not be a Cauchy sequence (see [21]). In a 2-metric space every convergent sequence is a Cauchy sequence if d is continuous (see [21]). There exists a 2-metric space such that every convergent sequence is a Cauchy sequence but d is not continuous (see [21]).
Chatterjea in [22] introduced the notion of C-contraction as follows.
Definition 1.5 Let be a metric space and be a map. Then T is called a C-contraction if there exists such that for all ,
This notion was generalized to a weak C-contraction by Choudhury in [23].
Definition 1.6 Let be a metric space and be a map. Then T is called a weak C-contraction if there exists which is continuous, and if and only if such that
for all .
Samet et al. [24] defined the notion of α-admissible mappings as follows.
Definition 1.7 Let T be a self-mapping on X and be a function. We say that T is an α-admissible mapping if
In [24] the authors considered the family Ψ of non-decreasing functions such that for each , where is the n th iterate of ψ, and they gave the following theorem.
Theorem 1.1 Let be a complete metric space and T be an α-admissible mapping. Assume that
for all , where . Also, suppose that the following assertions hold:
-
(i)
there exists such that ;
-
(ii)
either T is continuous or for any sequence in X with for all and as , we have for all .
Then T has a fixed point.
Salimi et al. [25] modified and generalized the notions of α-ψ-contractive and α-admissible mappings as follows.
Definition 1.8 [25]
Let T be a self-mapping on X and be two functions. We say that T is an α-admissible mapping with respect to η if
Note that if we take , then this definition reduces to Definition 1.7. Also, if we take , then we say that T is an η-subadmissible mapping.
The following result properly contains Theorem 1.1 and Theorems 2.3 and 2.4 of [26].
Theorem 1.2 [25]
Let be a complete metric space and T be an α-admissible mapping. Assume that
where and
Also, suppose that the following assertions hold:
-
(i)
there exists such that ;
-
(ii)
either T is continuous or for any sequence in X with for all and as , we have for all .
Then T has a fixed point.
Recently Karapinar et al. [27] introduced the notion of triangular α-admissible mapping as follows.
Definition 1.9 [27]
Let and . We say that T is a triangular α-admissible mapping if
(T1) implies , ,
(T2) implies .
Motivated by the above-mentioned developments, we first introduce the concepts of 2-α-η-admissible mappings and weak and rational α-η-ψ-contractions and establish the existence and uniqueness of fixed points for such mappings in complete 2-metric spaces. As an application of obtained results, we prove some fixed point theorems in partially ordered 2-metric spaces. The presented theorems generalize and improve many existing results in the literature. Moreover, some examples and an application to integral equations are provided to illustrate the usability of the proved results.
2 Fixed point results for weak α-η-C-contraction mappings
Motivated by Karapinar et al. [27] and Salimi et al. [25], we introduce the following notion.
Definition 2.1 Let be a 2-metric space and and be mappings. We say that T is a triangular 2-α-η-admissible mapping if for all ,
(T1) implies , ,
(T2) implies .
If we take , then we say that T is a triangular 2-α-admissible mapping. Also, if we take , then we say that T is a triangular 2-η- subadmissible mapping.
Example 2.1 Let . Define and by ,
Then T is a triangular 2-α-η-admissible mapping.
Lemma 2.1 Let be a 2-metric space and be a triangular 2-α-η-admissible mapping. Assume that there exists such that for all . Define the sequence by . Then
Proof Since there exists such that , then from (T1) we deduce that
By continuing this process, we get
Suppose that . Since
then from (T2) we have .
Again, since
then we deduce .
By continuing this process, we get as required. □
Definition 2.2 Let be a 2-metric space. Let and . We say that T is 2-α-η-continuous on if
If we take , then we say that T is a 2-α-continuous mapping. Also, if we take , then we say that T is a 2-η-continuous mapping.
Example 2.2 Let and . Assume that and are defined by
Clearly, T is not continuous, but T is 2-α-η-continuous on . Indeed, if as and , then and so .
Denote with Ψ the family of continuous functions such that if and only if .
We introduce the following notions as a modification of the approach in [25].
Definition 2.3 Let be a 2-metric space and , be three mappings.
-
We say that T is a weak α-η-C-contraction mapping if
(2.2)
for all , where .
-
We say that T is a modified weak α-C-contraction mapping if
for all , where .
-
We say that T is a modified weak η-C-contraction mapping if
for all , where .
-
We say that T is a weak α-C-contraction mapping of type (I) if
for all , where .
-
We say that T is a weak η-C-contraction mapping of type (I) if
for all , where .
-
We say that T is a weak α-C-contraction mapping of type (II) if
for all , where and .
-
We say that T is a weak η-C-contraction mapping of type (II) if
for all , and .
Now we are ready to state and prove our first main result of this section.
Theorem 2.1 Let be a complete 2-metric space. Assume that is a weak α-η-C-contraction mapping satisfying the following assertions:
-
(i)
T is a triangular 2-α-η-admissible mapping;
-
(ii)
there exists in X such that for all ;
-
(iii)
T is continuous or 2-α-η-continuous; or
-
(iv)
if is a sequence in X such that for all and as , then for all and all .
Then T has a fixed point.
Proof Let such that for all . Define a sequence by for all . Now, since T is a triangular 2-α-η-admissible mapping, so by Lemma 2.1 we have
From (2.2) we deduce
for all . By taking in (2.4), we get , i.e.,
and so by (2.4) and (2.5) we have
which implies
Hence, the sequence is decreasing in and so it is convergent to , i.e., . Taking limit in (2.6) we get
and then
By taking limit as in (2.4) and applying (2.8), we get
This implies , i.e., . Hence,
If , then by (2.7) we have . Since , we have for all . Since , we have
for all . For , noting that , from (2.10) we have
which implies
Now, since , from (2.11) we get
for all . Hence, from (2.10) and (2.12) we have for all . Now, for all with , we have . Hence,
That is, for all , we have
We now show that is a Cauchy sequence. Suppose to the contrary that is not a Cauchy sequence. Then there are and sequences and such that for all positive integers k,
From (2.13) and (2.14) we deduce
Taking limit as in the above inequality and applying (2.9), we get
Also by (2.13) we get
and
By taking limit as in (2.16) and (2.17) and applying (2.9) and (2.15), we have
Now since , then by (2.3) we have
for all . So by (2.2) we get
Taking limit as in (2.19) and applying (2.15), (2.18) and the continuity of ψ, we deduce
and so . That is, which is a contradiction. Hence, is a Cauchy sequence. Now, since is a complete 2-metric space, then there exists such that . At first we assume that (iii) holds. That is, T is continuous. Then
That is, is a fixed point of T. If T is 2-α-η-continuous on X, as and , then we have
So is a fixed point of T. Next we assume that (iv) holds. That is, for all and all . Then by (2.2) we get
Taking limit as in the above inequality, we get
which implies , i.e., . □
By taking in Theorem 2.1, we have the following corollary.
Corollary 2.1 Let be a complete 2-metric space. Assume that is a modified weak α-C-contraction mapping satisfying the following assertions:
-
(i)
T is a triangular 2-α-admissible mapping;
-
(ii)
there exists in X such that for all ;
-
(iii)
T is continuous or 2-α-continuous; or
-
(iv)
if is a sequence in X such that for all and as , then for all and all .
Then T has a fixed point.
Example 2.3 Let . We define a 2-metric d on X by
Clearly, is a complete 2-metric space. Define , and by
and
Now, we prove that all the hypotheses of Corollary 2.1 (Theorem 2.1) are satisfied and hence T has a fixed point.
Proof Let , if , then . On the other hand, for all , we have . Hence for all . This implies that T is a 2-α-admissible mapping. Clearly, for all . Now, if is a sequence in X such that for all and and as , then and hence . This implies that for all and all .
Let . Then and hence,
That is,
for all . Hence, T is a modified weak α-C-contraction mapping. Then all the hypotheses of Corollary 2.1 (Theorem 2.1) are satisfied and hence T has a fixed point. □
By taking in Theorem 2.1, we have the following corollary.
Corollary 2.2 Let be a complete 2-metric space. Assume that is a modified weak η-C-contraction mapping satisfying the following assertions:
-
(i)
T is a triangular 2-η-subadmissible mapping;
-
(ii)
there exists in X such that for all ;
-
(iii)
T is continuous or 2-η-continuous; or
-
(iv)
if is a sequence in X such that for all and as , then for all and all .
Then T has a fixed point.
Example 2.4 Let X, d be as in Example 2.3. Define , and by
and
Now, we prove that all the hypotheses of Corollary 2.2 (Theorem 2.1) are satisfied and hence T has a fixed point.
Proof Let , if , then . On the other hand, for all , we have . Hence for all . This implies that T is a 2-η-admissible mapping. Clearly, for all .
Now, if is a sequence in X such that for all and and as , then and hence . This implies that for all and all .
If , then and so
That is,
for all . Hence, T is a modified weak η-C-contraction mapping. Then all the hypotheses of Corollary 2.2 (Theorem 2.1) are satisfied and hence T has a fixed point. □
Corollary 2.3 Let be a complete 2-metric space. Assume that is a weak α-C-contraction mapping of type (I) or a weak α-C-contraction mapping of type (II) satisfying the following assertions:
-
(i)
T is a triangular 2-α-admissible mapping;
-
(ii)
there exists in X such that for all ;
-
(iii)
T is continuous or 2-α-continuous; or
-
(iv)
if is a sequence in X such that for all and as , then for all and all .
Then T has a fixed point.
Corollary 2.4 Let be a complete 2-metric space. Assume that is a weak η-C-contraction mapping of type (I) or weak α-C-contraction mapping of type (II) satisfying the following assertions:
-
(i)
T is a triangular 2-η-admissible mapping;
-
(ii)
there exists in X such that for all ;
-
(iii)
T is continuous or 2-η-continuous; or
-
(iv)
if is a sequence in X such that for all and as , then for all and all .
Then T has a fixed point.
-
(A)
For all , where and for all , there exists such that or and or for all .
Theorem 2.2 Adding condition (A) to the hypotheses of Theorem 2.1 (resp. Corollary 2.1, 2.2, 2.3 and 2.4), we obtain the uniqueness of the fixed point of T.
Proof Assume that and are two fixed points of T. We consider to following cases.
Case 1: Let or for all . Then from (2.2) we have
for all . This implies
That is, for all . So, for all . Hence, .
Case 2: Let and for all . From (A) there exists such that
and
Without loss of generality we can assume
Now, since T is a triangular 2-α-η-admissible mapping, then
for all and all . Then from (2.2) we get
which implies
which implies . Then there exists such that . By taking limit as in (2.20), we get
and so . Therefore, . That is, . Similarly, we can deduce . Then by Lemma 1.1 we get . □
3 Fixed point results for rational contraction in 2-metric spaces
In this section, we prove certain fixed point theorems for a rational contraction mapping via a triangular 2-α-η-admissible mapping.
Denote with the family of continuous functions such that if and only if .
Definition 3.1 Let be a 2-metric space and , be three mappings.
-
We say that T is a modified rational α-η-φ-contraction mapping if
(3.1)
for all , where and
-
We say that T is a modified rational α-φ-contraction mapping if
for all , where .
-
We say that T is a modified rational η-φ-contraction mapping if
for all , where .
-
We say that T is a rational α-φ-contraction mapping if
for all , where .
-
We say that T is a rational η-φ-contraction mapping if
for all , where .
Theorem 3.1 Let be a complete 2-metric space. Assume that is a modified rational α-η-φ-contraction mapping satisfying the following assertions:
-
(i)
T is a triangular 2-α-η-admissible mapping;
-
(ii)
there exists in X such that for all ;
-
(iii)
T is continuous or 2-α-η-continuous; or
-
(iv)
if is a sequence in X such that for all and as , then for all and all .
Then T has a fixed point.
Proof Let such that for all . Define a sequence by for all . Now, since T is a triangular 2-α-η-admissible mapping, so by Lemma 2.1 we have
From (3.1) we deduce
where
and so
By taking in (3.3), we have
and then , i.e., . Hence,
Therefore, .
If , then from (3.3) we get
Thus, , i.e., for all . Hence by Definition 1.1(d1), for all . Then is a fixed point of T. Now, if , then from (3.3) we get
So, the sequence is decreasing in and so it is convergent to , i.e., . Taking limit in (3.4) we get
which implies . Hence,
From (2.13) in Theorem 2.1 we have for all . We now show that is a Cauchy sequence. Suppose to the contrary that is not a Cauchy sequence. Then there are and sequences and such that for all positive integers k,
As in the proof of Theorem 2.1, we get
Now since , so by (3.2) we have
for all . So by (3.1) we get
where
Taking limit as in (3.9) and applying (3.7) and (3.8), we deduce
where . Then , i.e., , which is a contradiction. Hence, is a Cauchy sequence. Now, since is a complete 2-metric space, then there exists such that . At first we assume that (iii) holds. That is, T is continuous or 2-α-η-continuous. Then
That is, is a fixed point of T. Next we assume that (iv) holds. That is, for all and all . Then by (3.1) we get
for all , where and
Taking limit as in the above inequality, we deduce
Since , then , i.e., for all . Thus, . □
Corollary 3.1 Let be a complete 2-metric space. Assume that is a modified rational α-φ-contraction mapping satisfying the following assertions:
-
(i)
T is a triangular 2-α-admissible mapping;
-
(ii)
there exists in X such that for all ;
-
(iii)
T is continuous or 2-α-continuous; or
-
(iv)
if is a sequence in X such that for all and as , then for all and all .
Then T has a fixed point.
Example 3.1 Let . We define a 2-metric d on X by
where
Clearly, is a complete 2-metric space. Define , and by
and
Now, we prove that all the hypotheses of Corollary 3.1 (Theorem 3.1) are satisfied and hence T has a fixed point.
Proof As in the proof of Example 2.3 we can show that T is a 2-α-admissible mapping, for all and if is a sequence in X such that for all and and as , then for all and all .
Let . Then and hence
That is,
for all . Hence, T is a modified rational α-φ-contraction mapping. Then all the conditions of Corollary 3.1 (Theorem 3.1) are satisfied and hence T has a fixed point. □
By taking in Theorem 3.1, we have the following corollary.
Corollary 3.2 Let be a complete 2-metric space. Assume that is a modified rational η-φ-contraction mapping satisfying the following assertions:
-
(i)
T is a triangular 2-η-admissible mapping;
-
(ii)
there exists in X such that for all ;
-
(iii)
T is continuous or 2-η-continuous; or
-
(iv)
if is a sequence in X such that for all and as , then for all and all .
Then T has a fixed point.
Corollary 3.3 Let be a complete 2-metric space. Assume that is a rational α-φ-contraction mapping satisfying the following assertions:
-
(i)
T is a triangular 2-α-admissible mapping;
-
(ii)
there exists in X such that for all ;
-
(iii)
T is continuous or 2-α-continuous; or
-
(iv)
if is a sequence in X such that for all and as , then for all and all .
Then T has a fixed point.
Corollary 3.4 Let be a complete 2-metric space. Assume that is a rational η-φ-contraction mapping satisfying the following assertions:
-
(i)
T is a triangular 2-η-admissible mapping;
-
(ii)
there exists in X such that for all ;
-
(iii)
T is continuous or 2-η-continuous; or
-
(iv)
if is a sequence in X such that for all and as , then for all and all .
Then T has a fixed point.
4 Fixed point results in partially ordered 2-metric spaces
Recently, there have been so many exciting developments in the field of existence of fixed points in partially ordered sets. This approach was initiated by Ran and Reurings [28] and they also provided some applications to matrix equations. Their results are a hybrid of the two classical theorems: Banach’s fixed point theorem and Tarski’s fixed point theorem. Agarwal et al. [29], Bhaskar and Lakshmikantham [30], Ciric et al. [31] and Hussain et al. [32, 33] presented some new results for nonlinear contractions in partially ordered metric spaces and noted that their theorems can be used to investigate a large class of problems. In this section, as an application of obtained results we prove some fixed point results in partially ordered 2-metric spaces. We also note that the recent fixed point results in [34] can be deduced as simple corollaries.
Recall that if is a partially ordered set and is such that for , implies , then the mapping T is said to be non-decreasing.
Theorem 4.1 (Theorems 2.3 and 2.4 of [34])
Let be a complete partially ordered 2-metric space. Assume that is a mapping satisfying the following assertions:
-
(i)
T is non-decreasing;
-
(ii)
there exists in X such that ;
-
(iii)
T is continuous; or
-
(iv)
if is a non-decreasing sequence in X such that as , then for all ;
-
(v)
(4.1)
holds for all with or , where .
Then T has a fixed point.
Proof Define the mapping by
Let , then . From (4.1) we get
Again let such that . This implies that . As the mapping T is non-decreasing, we deduce that and hence for all . Also, let and , then and . So from transitivity we have . That is, for all . Thus T is a triangular 2-α-admissible mapping. The condition (ii) ensures that there exists such that . This implies that for all . Let be a sequence in X such that for all and all and as . So, for all . Then from (iv) we have for all . That is, for all and all . Therefore, all the conditions of Corollary 2.1 are satisfied, so T has a fixed point in X. □
(B) For all which are not comparable, there exists that is comparable to x and y.
Theorem 4.2 (Theorem 2.5 of [34])
Adding condition (B) to the hypotheses of Theorem 4.1, we obtain the uniqueness of the fixed point of T.
Proof Define the mapping as in the proof of Theorem 4.1. Let , where and for all . That is, x and y are not comparable. Hence, by condition (B) there exists that is comparable to x and y, i.e., or and or . That is, or and or for all . Then the conditions of Theorem 2.2 hold and the fixed point of T is unique. □
Theorem 4.3 Let be a complete partially ordered 2-metric space. Assume that is a mapping satisfying the following assertions:
-
(i)
T is non-decreasing;
-
(ii)
there exists in X such that ;
-
(iii)
T is continuous; or
-
(iv)
if is a non-decreasing sequence in X such that as , then for all ;
-
(v)
T is an ordered modified rational φ-contraction mapping, that is,
(4.2)
holds for all with or , where and
Then T has a fixed point.
Proof Define the mapping by
Let , then . From (4.2) we get
Again let such that . This implies that . As the mapping T is non-decreasing, we deduce that and hence for all . Also, let and , then and . So from transitivity we have . That is, for all . Thus T is a triangular 2-α-admissible mapping. The condition (ii) ensures that there exists such that . This implies that for all . Let be a sequence in X such that for all and all and as . So, for all . Then from (iv) we have for all . That is, for all and all . Therefore, all the conditions of Corollary 3.1 are satisfied, so T has a fixed point in X. □
5 Application to existence of solutions of integral equations
Integral equations like (5.1) have been studied in many papers (see, e.g., [32, 35, 36] and the references therein). In this section, we look for a solution to (5.1) in . For the remainder, we gather some definitions from the literature which will be used in the sequel. Let be the set of real continuous functions defined on , and let be defined by
for all . Then is a complete 2-metric space.
Consider the integral equation
and let be defined by
We assume that
-
(A)
is continuous;
-
(B)
is continuous;
-
(C)
is continuous and ;
-
(D)
there exist and two functions such that for all ,
for all and
for all ;
-
(F)
there exists such that ;
-
(G)
if is a sequence in X such that with as , then for all .
Theorem 5.1 Under the assumptions (A)-(G), the integral equation (5.1) has a solution in .
Proof Consider the mapping defined by (5.2). Let with . From (D), we deduce that
Therefore,
That is, implies
Thus F is a weak α-η-C-contraction mapping with .
Hence all the hypotheses of Theorem 2.1 are satisfied and the mapping F has a fixed point which is a solution in of the integral equation (5.1). □
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This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the second and third authors acknowledge with thanks DSR, KAU for financial support. The authors would like to thank the editor and the referees for constructive comments which improved the paper considerably.
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Fathollahi, S., Hussain, N. & Khan, L.A. Fixed point results for modified weak and rational α-ψ-contractions in ordered 2-metric spaces. Fixed Point Theory Appl 2014, 6 (2014). https://doi.org/10.1186/1687-1812-2014-6
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DOI: https://doi.org/10.1186/1687-1812-2014-6