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Coincidence point theorems for generalized cyclic -weak contractions in partially ordered Menger PM-spaces
Fixed Point Theory and Applications volume 2014, Article number: 214 (2014)
Abstract
In this paper, we introduce a new concept of generalized cyclic -weak contraction mappings and establish some coincidence point results for such mappings in complete partially ordered Menger PM-spaces. Our results generalize the main results of Nashine (Nonlinear Anal. 75:6160-6169, 2012) and Gopal et al. (Appl. Math. Comput. 233:955-967, 2014). We also obtain the corresponding coincidence point theorems for generalized cyclic -weak contractions in partially ordered metric spaces.
MSC:47H10, 54H25.
1 Introduction and preliminaries
Fixed point theory is a very useful tool in many fields such as nonlinear operator theory, control theory, game theory, dynamics and economic theory. One of the most fundamental fixed point theorems is the Banach contraction mapping principle. Due to its simplicity and importance, this classical result has been generalized by many authors in different directions (see [1–5]).
In 2004, Ran and Reurings [6] established a fixed point theorem for Banach’s contraction mappings in partially ordered metric spaces. In 2009, Harjani and Sadarangani [7] proved some fixed point theorems for weakly contractive mappings in complete metric spaces endowed with a partial order. In 2012, Nashine [8] presented some fixed point results for cyclic generalized ψ-weakly contractive mappings in complete metric spaces. Other authors also obtained some important results in this area (see [9–14]). On the other hand, in 2012, Samet et al. [15] introduced the concept of α-ψ-contractive and α-admissible mappings in metric spaces. In 2013, Berzig and Karapinar [16] proved some fixed point results for -contractive mappings for a generalized altering distance in complete metric spaces.
In 1942, Menger [17] introduced the concept of a probabilistic metric space, and a large number of authors have done considerable work in such field (see, e.g. [18–22]). Recently, the extension of fixed point theory to generalized structures as partially ordered probabilistic metric spaces has attracted much attention (see, e.g. [23–25]). Gopal et al. [26] established some fixed point results for α-ψ-type contractive mappings and generalized β-type contractive mappings in Menger PM-spaces.
In this paper, we introduce the notion of generalized cyclic -weak contraction mappings to establish some corresponding coincidence point theorems in complete partially ordered Menger PM-spaces. Also, an application is given to show the validity of our results. It is worth pointing out that our results extend and generalize the main results of [8] and [26].
First, we recall some notions, lemmas and examples which will be used in the sequel.
Let R denote the set of reals and the nonnegative reals. A mapping is called a distribution function if it is nondecreasing and left continuous with and . We will denote by the set of all distribution functions and .
Let H denote the specific distribution function defined by
Definition 1.1 ([18])
The mapping is called a triangular norm (for short, a t-norm) if the following conditions are satisfied:
-
(Δ-1) , for all ;
-
(Δ-2) ;
-
(Δ-3) , for , ;
-
(Δ-4) .
Three typical examples of continuous t-norms are , and , for all .
Definition 1.2 ([18])
A triplet is called a Menger probabilistic metric space (shortly, a Menger PM-space), if X is a nonempty set, Δ is a t-norm and ℱ is a mapping from satisfying the following conditions (for , we denote by ):
-
(MS-1) , for all , if and only if ;
-
(MS-2) , for all and ;
-
(MS-3) , for all and .
Definition 1.3 ([19])
is called a non-Archimedean Menger PM-space (shortly, a N.A Menger PM-space), if is a Menger PM-space and Δ satisfies the following condition: for all and ,
Definition 1.4 ([19])
A non-Archimedean Menger PM-space is said to be type of , if there exists a , such that
for all , where . In fact, we obtain , for all and .
Example 1.1 Let be a N.A Menger PM-space, and (or for all ). Then is of -type for defined by (or for and ).
Remark 1.1 Schweizer and Sklar [18] point out that if is a Menger probabilistic metric space and Δ is continuous, then is a Hausdorff topological space in the -topology T, i.e., the family of sets () is a basis of neighborhoods of a point x for T, where .
Definition 1.5 ([1])
The function is called an altering distance function, if the following properties are satisfied: (a) h is continuous and nondecreasing; (b) if and only if .
Definition 1.6 ([21])
A function is said to be a Ψ-function, if it is nondecreasing and continuous in , as , if and only if .
In the sequel, the class of all Ψ functions will be denoted by Ψ. Any altering distance function h with the additional property generalizes a Ψ-function ψ through and whenever .
Lemma 1.1 ([19])
Let be a sequence in such that for all . If the sequence is not a Cauchy sequence in X, then there exist , and two sequences , of positive integers such that
-
(1)
, and as ;
-
(2)
and , for .
2 Main results
In this section, we first introduce the new notions of generalized κ-admissible mappings, weakly comparable mappings and generalized cyclic -weak contraction mappings in Menger PM-spaces.
Definition 2.1 Let X be a nonempty set, be two self-maps and be a function. S and T are called generalized κ-admissible, if for all , , and , we have . κ is called m-transitive on X, if for all , , , , we have .
Example 2.1 Let , for all ,
In fact, if , for all , , then . Hence, , and so . Thus, S and T are generalized κ-admissible. Also, we can verify that κ is m-transitive.
Let be a partially ordered set, we will write whenever x and y are comparable (that is, or holds).
Definition 2.2 Let be a partially ordered set, be two self-maps and . T is called weakly comparable with respect to S, if such that implies Tx and Ty are comparable (that is, ). ≍ is called m-transitive on X, if such that for all implies .
Example 2.2 Let , , and
Since , we have . Since , we have . Since , we have . Since , we have . Note that and are not comparable. Hence, T is weakly comparable with respect to S.
Definition 2.3 Let X be a nonempty set, m be a positive integer, be subsets of X, and be two self-maps. Then Y is said to be a cyclic representation of Y with respect to S and T, if the following two conditions are satisfied:
-
(i)
, , are nonempty closed sets;
-
(ii)
.
Example 2.3 Let , , , , and . Define by and , for all . Then it is easy to verify that is a cyclic representation of Y with respect to S and T.
Definition 2.4 Let be a partially ordered set and be a N.A Menger PM-space of type . Let be a function and be a lower semi-continuous function. Let m be a positive integer, be subsets of X, , and be two self-maps. T is said to be a generalized cyclic -weak contraction, if Y is a cyclic representation of Y with respect to S and T, , and for and for all , and are comparable such that
for all and , where h is an altering distance function, are two continuous functions such that , if and only if , for all , , and
Now we are ready to state our main results.
Theorem 2.1 Let be a partially ordered set and be a complete N.A Menger PM-space of type . Let m be a positive integer, be subsets of X, , be a generalized cyclic -weak contraction satisfying (2.1). Suppose that the following conditions hold:
-
(i)
S and T are generalized κ-admissible;
-
(ii)
κ and ≍ are m-transitive;
-
(iii)
T is weakly comparable with respect to S;
-
(iv)
there exists such that and for all ;
-
(v)
if a sequence satisfies , for all , and and as , then and for n sufficiently large and for all .
Then S and T have a coincidence point in X, that is, there exists such that .
Proof Since and , there exists an , such that . Since and , there exists an , such that . Continuing this process, we can construct two sequences and defined by , for all , and there exists such that and .
By condition (iv), we get and for all . It follows from (i) and (iii) that and for all . By induction, we obtain
We will complete the proof by the following three steps.
Step 1. We prove that
Without loss of generality, assume that , for all (otherwise, for some , then is the coincidence point of S and T. Hence, the conclusion holds).
Since , , , and are comparable, for , by (2.1) and (2.2), we get
for all , where and
For all , we have . Since is nondecreasing and g is strictly decreasing, we have for all . Hence, . On the other hand, it is obvious that . Thus, .
Suppose that , by (2.4), we have
which implies that . Thus, , that is, for all . Then , which is in contradiction to for all .
Hence, . Let . By (2.4), we get
Since h is nondecreasing, it follows from (2.5) that is a decreasing sequence and bounded from below, for every . Hence, there exists , such that .
By using the continuities of h and φ, letting in (2.5), we get , which implies that . Thus , that is, for all . Hence, (2.3) holds.
Step 2. We prove that is a Cauchy sequence. To prove this fact, we first prove the following claim.
Claim: for every and , there exists , such that with then , that is, .
In fact, suppose this is not true, then there exist and , such that for any , we can find with satisfying , that is, .
Now, take . Then corresponding to , we can choose in such a way that it is the smallest integer with satisfying and . Therefore, . Using the non-Archimedean Menger triangular inequality and Definition 1.4, we have
Since for all , letting in (2.6), we obtain
By , we know that and lie in different adjacently labeled sets and , for . Since ≍ and κ are m-transitive, we obtain and for all . Using the fact that T is a generalized cyclic -weak contraction and letting , we have
where and
By (2.3) and (2.7), we have and . According to the continuities of h and φ, letting in (2.8), we get
Thus, , that is, . Then , which is in contradiction to .
Therefore, our claim is proved. Next, we will prove that is a Cauchy sequence.
By the continuity of g and , we have , for any given . Since g is strictly decreasing, there exists , such that .
For any given and , there exists , such that . By the claim, we can find , such that if with , then
Since , we can also find , such that
for any .
Suppose that and . Then there exists such that . Therefore, , for , . Thus we have
By (2.9), (2.10), and (2.11), we get
Since g is strictly decreasing, by (2.12), we have . This proves that is a Cauchy sequence.
Step 3. We show that S and T have a coincidence point in X.
Since is a Cauchy sequence and is a complete Menger PM-space, there exists , such that . Since is closed and , we know that . Hence, there exists , such that . As is a cyclic representation of Y with respect to S and T, the sequence has infinite terms in each for .
First, suppose that , then , and we can choose a subsequence of with (the existence of this subsequence is guaranteed by the above discussion). Since and , by (v), we obtain the result that and are comparable, for all and k sufficiently large. Letting , by (2.1), we have
where and
Since ϕ is lower semi-continuous and , by (2.3), we have
Let be the set of all discontinuous points of . Since g, h, φ, and L are continuous, we find that is also the set of all discontinuous points of , , , and . Moreover, we know that is a countable set. Let . If (t is a continuous point of ), by (2.3), we have and
Letting in (2.13), we get
which implies that . Hence, and . Then
If with , by the density of real numbers, there exist , such that . Since the distribution is nondecreasing, we have
Hence, from (2.14) and (2.15), we have for any . Thus, , that is, z is the coincidence point of S and T. □
Corollary 2.1 Let be a partially ordered set and be a complete N.A Menger PM-space of type , be a function, be two self-maps and . Suppose that for , Sx and Sy are comparable, we have
for all , where is a continuous function, for and , L is the same as the one in Theorem 2.1,
and
Also, assume that the conditions (i)-(v) of Theorem 2.1 are satisfied, where .
Then S and T have a coincidence point in X, that is, there exists , such that .
Proof Letting , , , and in Theorem 2.1, the conclusion follows immediately. □
Definition 2.5 Let X be a nonempty set, be two self-maps and be a function. S and T are called generalized α-admissible, if for all , , implies . α is called 1-transitive on X, if for all , , , implies .
Theorem 2.2 Let be a partially ordered set and be a complete N.A Menger PM-space, Δ be a continuous t-norm and , be a function, be two self-maps and . Suppose that for , Sx and Sy are comparable, we have
for all , such that , where , , L and Φ are the same as the ones in Corollary 2.1,
and
Also assume that the following conditions hold:
-
(i)
S and T are generalized α-admissible;
-
(ii)
α and ≍ are 1-transitive;
-
(iii)
T is weakly comparable with respect to S;
-
(iv)
there exists such that and for all ;
-
(v)
if a sequence satisfies , for all and , and as , then and for n sufficiently large and for all .
Then S and T have a coincidence point in X.
Proof Let be a function defined by for and . Then . Since , we have for all . Hence, is a N.A Menger PM-space of . Let be a function and for all and .
Since ψ and are both nondecreasing, we have . Hence, . By the definition of g, we have
For , Sx and Sy are comparable, by (2.17) and (2.18), we get
Since ψ is continuous and as , it follows from (2.19) that (2.16) holds. Also, S and T also satisfy (i)-(v) of Theorem 2.1.
Thus, all the conditions of Corollary 2.1 are satisfied. Therefore, S and T have a coincidence point in X. □
3 Coincidence point results in partially ordered metric spaces
In this section, we will apply the results obtained in Section 2 to establish the corresponding coincidence point theorems for generalized cyclic -weak contractions in partially ordered metric spaces. We first introduce a new notion in metric spaces that we will use in Theorem 3.1.
Definition 3.1 Let S and T be two self-maps of a metric space , be a function. S and T are called generalized κ-admissible, if for all , implies . κ is called m-transitive on X, if , implies .
Theorem 3.1 Let be an ordered complete metric space and be a function. Let m be a positive integer, be subsets of X, , S and be two self-maps, Y be a cyclic representation of Y with respect to S and T. Suppose that , and for and for all , and are comparable, we have
for all , where h is a continuous and nondecreasing linear function, if and only if , are two continuous functions such that , if and only if , and for all , for all ,
and
Also, assume that the following conditions hold:
-
(i)
S and T are generalized κ-admissible;
-
(ii)
κ and ≍ are m-transitive;
-
(iii)
T is weakly comparable with respect to S;
-
(iv)
there exists such that and ;
-
(v)
if a sequence satisfies , for all , and as , then and for n sufficiently large.
Then S and T have a coincidence point in X, that is, there exists such that .
Proof Let be the induced PM-space, where ℱ is defined by , for all and .
In fact, for , and , by , we have
which implies that (1.1) holds and . Hence, by Example 1.1, we know that is a N.A Menger PM-space of -type for defined by for and . It is not difficult to prove that a sequence in converges to a point if and if only in τ-converges to . Since is a complete metric space, is a τ-complete N.A Menger PM-space of type .
For , and are comparable, by (3.1) and the properties of h, φ, L, for , we have
Since for , we have for . It follows from (3.2) that (2.1) holds. In fact, S and T also satisfy (i)-(v) of Theorem 2.1.
Thus, all the conditions of Theorem 2.1 are satisfied when . Therefore, the conclusion holds. □
4 An illustration
In this section, we give an example to demonstrate Theorem 2.1.
Example 4.1 Let , for all , for all and . Define by
for all . Then is a complete N.A Menger PM-space of -type. Suppose that , , , and . Define and by
Then S and T are generalized κ-admissible and Y is a cyclic representation of Y with respect to S and T, and T is weakly comparable with respect to S, and κ and ≍ are 3-transitive.
Let , for all , , , , , for all . Now, we verify inequality (2.1) in Theorem 2.1. By the definitions of F, g, ϕ, h, φ, and L, we only need to prove that
for all , that is,
where if or , and if otherwise.
We consider the following cases:
Case 1. If , then . By the definition of T, we have
which implies that (4.1) holds.
Case 2. If and , then and . By the definition of T, we have
which implies that (4.1) holds. Similarly, if and , we also have (4.1) holds.
Case 3. If , then and . By the definition of T, we have
which implies that (4.1) holds.
Thus, all the conditions of Theorem 2.1 are satisfied. Therefore, S and T have a coincidence point in X, indeed, and are coincidence points of S and T.
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Acknowledgements
The authors thank the editor and the referees for their valuable comments and suggestions. The research was supported by the National Natural Science Foundation of China (11361042, 11071108, 11326099, 11461045) and the Provincial Natural Science Foundation of Jiangxi, China (20132BAB201001, 2010GZS0147, 20142BAB211016).
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Zhu, C., Xu, W. & Wu, Z. Coincidence point theorems for generalized cyclic -weak contractions in partially ordered Menger PM-spaces. Fixed Point Theory Appl 2014, 214 (2014). https://doi.org/10.1186/1687-1812-2014-214
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DOI: https://doi.org/10.1186/1687-1812-2014-214