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Tripled fixed point theorem in fuzzy metric spaces and applications
Fixed Point Theory and Applicationsvolume 2013, Article number: 29 (2013)
In this paper we prove an existence and uniqueness theorem for contractive type mappings in fuzzy metric spaces. In order to do that, we consider a slight modification of the concept of a tripled fixed point introduced by Berinde et al. (Nonlinear Anal. TMA 74:4889-4897, 2011) for nonlinear mappings. Additionally, we obtain some fixed point theorems for metric spaces. These results generalize, extend and unify several classical and very recent related results in literature. For instance, we obtain an extension of Theorem 4.1 in (Zhu and Xiao in Nonlinear Anal. TMA 74:5475-5479, 2011) and a version in non-partially ordered sets of Theorem 2.2 in (Bhaskar and Lakshmikantham in Nonlinear Anal. TMA 65:1379-1393, 2006). As application, we solve a kind of Lipschitzian systems in three variables and an integral system. Finally, examples to support our results are also given.
In a recent paper, Bhaskar and Lakshmikantham  introduced the concepts of coupled fixed point and mixed monotone property for contractive operators of the form , where X is a partially ordered metric space, and then established some interesting coupled fixed point theorems. They also illustrated these important results by proving the existence and uniqueness of the solution for a periodic boundary value problem. Later, Lakshmikantham and Ćirić  proved coupled coincidence and coupled common fixed point results for nonlinear mappings satisfying certain contractive conditions in partially ordered complete metric spaces. After that many results appeared on coupled fixed point theory (see, e.g., [2–8]).
Fixed point theorems have been studied in many contexts, one of which is the fuzzy setting. The concept of fuzzy sets was initially introduced by Zadeh  in 1965. To use this concept in topology and analysis, many authors have extensively developed the theory of fuzzy sets and its applications. One of the most interesting research topics in fuzzy topology is to find an appropriate definition of fuzzy metric space for its possible applications in several areas. It is well known that a fuzzy metric space is an important generalization of the metric space. Many authors have considered this problem and have introduced it in different ways. For instance, George and Veeramani  modified the concept of a fuzzy metric space introduced by Kramosil and Michalek  and defined the Hausdorff topology of a fuzzy metric space. There exists considerable literature about fixed point properties for mappings defined on fuzzy metric spaces, which have been studied by many authors (see [10, 12–16]). Zhu and Xiao  and Hu  gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Fang  proved some common fixed point theorems under ϕ-contractions for compatible and weakly compatible mappings on Menger probabilistic metric spaces. Moreover, Elagan and Segi Rahmat  studied the existence of a fixed point in locally convex topology generated by fuzzy n-normed spaces.
Very recently, the concept of tripled fixed point has been introduced by Berinde and Borcut . In their manuscript, some new tripled point theorems are obtained using the mixed g-monotone mapping. Their results generalize and extend the Bhaskar and Lakshmikantham’s research for nonlinear mappings. Moreover, these results could be used to study the existence of solutions of a periodic boundary value problem involving . A multidimensional notion of a coincidence point between mappings and some existence and uniqueness fixed points theorems for nonlinear mappings defined on partially ordered metric spaces are studied in .
In this paper, our main aim is to obtain an existence and uniqueness theorem for contractive type mappings in the framework of fuzzy metric spaces. In order to do that, we consider a slight modification of the concept of a tripled fixed point introduced by Berinde and Borcut for nonlinear mappings. The power of this result is two-fold. Firstly, we can particularize it to complete metric spaces, obtaining a Berinde-Borcut type result (in non-fuzzy setting). Moreover, our result, in a unified manner, covers also coupled fixed (see Zhu and Xiao ) and fixed point theorems. Finally, examples to support our results are also given.
Henceforth, X will denote a non-empty set and . Subscripts will be used to indicate the arguments of a function. For instance, will be denoted by and will be denoted by . Furthermore, for brevity, will be denoted by gx.
A metric on X is a mapping satisfying, for all ,
From these properties, we can easily deduce that and for all . The last requirement is called the triangle inequality. If d is a metric on X, we say that is a metric space (briefly, a MS).
Let be a MS. A mapping is said to be Lipschitzian if there exists such that for all . The smallest k (denoted by ) for which this inequality holds is said to be the Lipschitz constant for f. A Lipschitzian mapping is a contraction if .
Theorem 1 (Banach’s contraction principle)
Every contraction from a complete metric space into itself has a unique fixed point.
If provided with the Euclidean metric, examples of Lipschitzian mappings are , , , , and .
Definition 2 A triangular norm (also called a t-norm) is a map that is associative, commutative, nondecreasing in both arguments and has 1 as identity. For each , the sequence is defined inductively by and . A t-norm ∗ is said to be of H-type (see ) if the sequence is equicontinuous at , i.e., for all , there exists such that if , then for all .
The most important and well-known continuous t-norm of H-type is , that verifies for all . The following result presents a wide range of t-norms of H-type.
Lemma 3 Let be a real number and let ∗ be a t-norm. Define as , if , and , if . Then is a t-norm of H-type.
Definition 4 
A triple is called a fuzzy metric space (in the sense of Kramosil and Michalek; briefly, a FMS) if X is an arbitrary non-empty set, ∗ is a continuous t-norm and is a fuzzy set satisfying the following conditions, for each , and :
if and only if ;
is left continuous;
In this case, we also say that is a FMS under ∗. In the sequel, we will only consider FMS verifying:
for all .
Lemma 5 is a non-decreasing function on .
Definition 6 Let be a FMS under some t-norm. A sequence is Cauchy if, for any and , there exists such that for all . A sequence is convergent to , denoted by if, for any and , there exists such that for all . A FMS in which every Cauchy sequence is convergent is called complete.
Given any t-norm ∗, it is easy to prove that . Therefore, if is a FMS under min, then is a FMS under any (continuous or not) t-norm. This is the case in the following examples (in which, obviously, we only define for and ).
Example 7 From a metric space , we can consider a FMS in different ways. For and , define:
It is well known that is a FMS under the product ∗ = ⋅, called the standard FMS on , since it is the standard way of viewing the metric space as a FMS. However, it is also true (though lesser-known) that , and are FMSs under min.
Furthermore, is a complete metric space if and only if (or or ) is a complete FMS. For instance, this is the case of any non-empty and closed subset (or subinterval) of ℝ provided with its Euclidean metric.
Definition 8 A function on a FMS is said to be continuous at a point if, for any sequence in X converging to , the sequence converges to . If g is continuous at each , then g is said to be continuous on X. As usual, if , we will denote .
Remark 9 If and , then implies that . We will use this fact in the following way: implies that .
The main result
Definition 10 Let and be two mappings.
We say that F and g are commuting if for all .
A point is called a tripled coincidence point of the mappings F and g if , and .
Theorem 11 Let ∗ be a t-norm of H-type such that for all . Let and be real numbers such that , let be a complete FMS and let and be two mappings such that and g is continuous and commuting with F. Suppose that for all and all ,
Then there exists a unique such that . In particular, F and g have, at least, one tripled coincidence point. Furthermore, is the unique tripled coincidence point of F and g if we assume that only in the case that is constant on .
In this result, in order to avoid the indetermination 00, we assume that for all and all .
Proof Suppose that F is constant in , i.e., there exists such that for all . As F and g are commuting, we deduce that . Therefore, and is a tripled coincidence point of F and g. Now, suppose that and is another tripled coincidence point of F and g. Then , so . Similarly, and is the unique tripled coincidence point of F and g.
Next, suppose that F is not constant in . In this case, and the proof is divided into five steps. Throughout this proof, n and p will denote non-negative integers and .
Step 1. Definition of the sequences , and . Let be three arbitrary points of X. Since , we can choose such that , and . Again, from , we can choose such that , and . Continuing this process, we can construct sequences , and such that, for , , and .
Step 2. , and are Cauchy sequences. Define, for and all , . Since is a non-decreasing function and , we have that
From inequality (1) we deduce, for all and all ,
According to (3), (4), (5) and Remark 9, we have that
This proves that, for all and all ,
Swapping t by , we deduce, for all and , that
Taking into account that ∗ is commutative and ∗ ≥ ⋅, and (3), (4), (5), we observe that
If we join this property to (2),
Repeatedly applying the first inequality, we deduce that for all and . This means that for all ,
Properties (6) and (8) imply that
Next, we claim that
We prove it by induction methodology in . If , (11) is true for all and all by (10). Suppose that (11) is true for all and all for some p, and we are going to prove it for . Applying (1), the induction hypothesis and that ∗ ≥ ⋅,
Arguing in the same way, we come to . Applying the axiom (v) of a FMS, (7) and the induction hypothesis,
The same reasoning is also valid for and . Therefore, (11) is true. This permits us to show that is Cauchy. Suppose that and are given. By the hypothesis, as ∗ is a t-norm of H-type, there exists such that for all and for all . By (9), , so there exists such that for all . Hence (11), we get for all and . Therefore, is a Cauchy sequence. Similarly, and are also Cauchy sequences.
Step 3. We claim that g and F have a tripled coincidence point. Since X is complete, there exist such that , and . As g is continuous, we have that , and . The commutativity of F with g implies that . By (1),
Letting , we deduce that . Hence, . In a similar way, we can show that and , so is a tripled coincidence point of the mappings F and g.
Step 4. We claim that , and . We note that by condition (1),
Let for all and . By (13), (14) and (15),
This proves that for all and all . Repeating this process,
Now, by (16), (13), (14) and (15),
Therefore, for all and . Since for all , we have, taking limit in (17), (18) and (19), that , and . This shows, using (12), that
Step 5. We will prove that . Let for all . Then, by condition (1),
If we use these three inequalities at the same time,
We find that implies that for all and . By (20), (21) and (22),
Letting , we have for all , and this means that for all , i.e., . The unicity of x follows from (1). □
Remark 12 The unicity of the coincidence point of F and g is not always true. For instance, if is constant and is also constant, then every is a coincidence point of F and g.
Remark 13 In the previous theorem, we have only used the continuity of ∗ at , that is, if are sequences such that and , then . And this is true because .
Example 14 Consider as in Example 7. Let and be positive real numbers such that (in particular, ). Define and as and for all . Clearly, g is continuous, F and g are commuting and . We also note that verifies
Therefore, applying Theorem 11, we deduce that F and g have a tripled coincidence point.
In the proof of the next result, the view of as the crisp FMS is used (see Example 7). This approach allows us to deduce results for metric spaces from the corresponding result in the fuzzy setting. Moreover, Theorem 15 is just a tripled coincidence point result, similar to Berinde-Borcut one, see [, Theorem 7] and [, Theorem 4], in a not necessarily partially ordered set.
Theorem 15 Let be a complete metric space and let and be two mappings such that and g is continuous and commuting with F. Suppose that F and g verify some of the following conditions for all :
for some .
for some and some .
for some such that .
Then there exists a unique such that .
Proof (a) Consider defined as in Example 7. As is complete, then is a complete FMS. Fix and , and we are going to prove (1) using and . If or or , then (1) is obvious. Suppose that , and . This means that , and . Therefore, and . Hence, and (1) is also true.
(b) In this case,
(c) If ,
Example 16 If , for all and are such that , the mappings and , defined as and for all , verify the hypothesis of Theorem 15(c). It is easy to check that , where , is the unique tripled coincidence point of F and g and verifies .
Now, we prove the existence of a coupled coincided point for and g that generalizes Theorem 4.1 in , taking . That is, the main result of the paper also covers the main theoretical results of Zhu and Xiao .
Corollary 17 Let ∗ be a t-norm of H-type such that for all . Let and be real numbers such that , let be a complete FMS and let and be two mappings such that and g is continuous and commuting with F. Suppose that
for all and all . Then there exists a unique such that .
Proof Define and as for all . Then and is commuting with g (). Furthermore,
Then there exists a unique such that . If verifies , then , so . □
Corollary 18 ([, Theorem 2.2])
Let be a complete metric space and let and be two mappings such that and g is continuous and commuting with F. Suppose that F and g verify some of the following conditions for all :
for some .
for some and some .
for some such that .
Then there exists a unique such that .
Proof Similar to the proof of Theorem 15. □
Remark 19 In fact, the previous result is proved for X, a partially ordered set in .
Moreover, from a similar procedure, we can deduce the celebrated Banach contraction principle (Theorem 1).
Let be Lipschitzian mappings and let be real numbers. Define as for all . Then h is another Lipschitzian mapping and . Obviously, if , then h is a contraction, so there exists a unique such that .
Next, define as for all . It is clear that for all . Furthermore,
If , then F verifies (1) with for all .
Corollary 20 Let be Lipschitzian mappings on ℝ (provided with the Euclidean metric) and let such that . Then the system
has a unique solution, which is , where is the only real solution of .
Consider the system
If we choose , and , then , and are Lipschitzian mappings, and and . Let , and . Then . As system () is equal to (S), then () has a unique solution, which is of the form , where is the only solution of
Finding, for example, the root by the bisection method, we get, approximately, .
An integral system
Let with and let . Consider with the distance , where ∫ represents the Lebesgue integral. It is well known that is a complete MS. Let be real numbers and let be a mapping verifying and
If , we want to find functions such that
holds for all , .
For all and all , define
On the one hand, it is not difficult to prove that , hence is well defined. On the other hand,
If we suppose that , then F verifies (1) with for all . Then the system (23) has a unique solution, which is of the form , where is the only solution of the equation
(this exists as a simple application of the Banach contraction principle).
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The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.