n-Tupled Fixed Points Theorem in Fuzzy Metric Spaces with Application

Abstract

We will introduce the concept of -tupled fixed points (for positive integer ) in fuzzy metric space by mild modification of the concept of -tupled fixed points (for even positive interger ) introduced by Imdad et al. (2013) in metric spaces. As application of the above-mentioned concept, we will establish some -tupled fixed point theorems for contractive type mappings in fuzzy metric space which extends the result of Roldán et al. (2013). Also we have given an application to solve a kind of Lipschitzian systems for variables and an integral system.

1. Introduction

The concept of coupled fixed point was introduced by Bhaskar and Lakshmikantham [1] and it motivated the fixed point theorists to work in the area of multidimensional fixed points; for example, see [2–9].

In recent years some of the fixed point theorists tried to establish the existence of -tupled fixed points and common -tupled fixed points for some contractions in metric spaces, partially ordered metric spaces and asymptotically regular metric spaces. In particular, Imdad et al. [9] introduced the concept of -tupled coincidence points as well as -tupled fixed point (for even positive integer) and utilize these two definitions to obtain -tupled coincidence as well as -tupled common fixed point theorems. Very recently, Soliman et al. [10] proved some -tupled coincident point theorems in partially ordered complete asymptotically regular metric spaces.

The purpose of our results is to introduce -tupled fixed points (for all positive integers) and to prove -tupled fixed points theorems for contractive type mappings in fuzzy metric spaces. Results obtained by us in Section 2 extents the work of Roldán et al. [11]. At the end we have given an application to solve a kind of Lipschitzian systems for variables and an integral system. Results proved in this paper follow the lines of the proof of Roldán et al. [11] for obtaining coincidence and fixed points.

A triangular norm (also called a -norm) is a mapping is associative, commutative, and nondecreasing in both arguments and has as an identity element.

Definition 1 (see [11, 12]). For any , let the sequence be defined by and . Then a -norm is said to be of -type if the sequence is equicontinuous at , that is, for all , there exists such that if , then for all .

The most important and well-known continuous -norm of -type is , which verifies for all . The following result presents a wide range of -norms of -type.

Lemma 2. Let be a real number and let be a -norm. Define as , if and , if . Then is a -norm of -type.

Definition 3 (see [13]). A triplet is said to be a fuzzy metric space (in the sense of Kramosil and Michalek; briefly, a FMS), if is a nonempty set, is a continuous -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 4. is a nondecreasing function on .

Definition 5 (see [14]). Let be a fuzzy metric space; then (1)a sequence in is said to be convergent to if for all ;(2)a sequence in is said to be a Cauchy sequence if, for any , there exists , such that for all and ;(3)a fuzzy metric space is said to be complete if and only if every Cauchy sequence in is convergent.

Given any -norm , it is easy to prove that . Therefore, if is a FMS under , then is a FMS under any (continuous or not) -norm. This is the case in the following examples (in which, obviously, we only define for and ).

Example 6. From a metric space , we can consider a FMS in different ways. For and , define (i);(ii);(iii)
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 FMS under .
Furthermore, is a complete metric space if and only if (or or is a complete FMS. For instance, this is the case of any nonempty and closed subset (or subinterval) of provided with its Euclidean metric.

Definition 7. A function on a fuzzy metric space is said to be continuous at a point if, for any sequence in converging to , the sequence converges to . If is continuous at each , then is said to be continuous on . As usual, if , we will denote .

Remark 8. If and , then implies that . We will use this fact in the following way: implies that .

2. Main Results

Henceforth, will denote a nonempty set and .

Here, we introduce -tupled fixed points (for positive integer ) in fuzzy metric space by slightly modified concept of -tupled fixed points (for natural number ) introduced by Imdad et al. [9] in metric space.

Definition 9. Let and be two mappings. (a)One says that and are commuting if for all .(b)An element is called -tupled coincidence points of the mappings and if

Definition 10. An element is called an -tupled fixed point of map and if

Now, we present our main theorem.

Theorem 11. Let be a -norm of -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 is continuous and commuting with . Suppose that, for all and all , Then there exists a unique such that . In particular, and have, at least, one -tupled coincidence point. Furthermore, is the unique -tupled coincidence point of and if we assume that only in the case that is constant on .
In this result, in order to avoid the indetermination , we assume that for all and all .

Proof. Suppose that is constant in ; that is, there exists such that As and are commuting, we deduce that Therefore, and is -tupled coincidence point of and .
Now, suppose that and is another -tupled coincidence point of and . Then, Similarly, and is the unique -tupled coincidence points of and .
Next, suppose that is not constant in .
In this case, and the proof is divided into five steps. In the entire proof, and stand to be nonnegative integers and .
Step  1. First we define of the sequences .
Let be arbitrary points in .
Since . we can choose such that Again from . we can choose such that Continuing in this way, we can construct sequences in such that for .
Step  2. Now, we will show that the sequences are Cauchy sequences. Define, for and all , Since is a nondecreasing function and , we have that From inequality (6) we deduce, for all and all , Using (15) and Remark 8, we have This proves that, for all and all , Swapping by , we deduce, for all and , that Taking into account that is commutative and and (15), we observe that If we join this property to (14), Repeatedly applying the first inequality, we deduce that for all and . This means that, for all , Properties (17) and (20) imply that Next, we claim that for all and . We prove it by using mathematical induction on . If , (24) is true for all and all by (23). Suppose that (24) is true for all and all for some , and we are going to prove it is also true for . Applying (6), the induction hypothesis, and that , Arguing in the same way, we arrive at the following: Applying the axiom of a FMS, (18), and the induction hypothesis, Arguing in the same manner, we have Therefore, (24) is true.
This permits us to show that is Cauchy. Suppose that for a given .
By the hypothesis, as is a -norm of -type, there exists such that for all and for all .
By (22), , so there exists such that for all .
Hence (24), we get for all and .
Therefore, is a Cauchy sequence. Similarly, we can show that are also Cauchy sequences.
Step  3. Now, we claim that and have a tripled coincidence point. Since is complete, there exists such that As is continuous, we have that The commutativity of with implies that By (6) and Remark 8, we get Letting , we deduce that Hence, .
Similarly, we can show that So is -tupled coincidence point of the mappings and :
Step  4. Now, we claim that We note that, by condition (6), Let for all and .
By (38), This proves that for all and . Repeating this process, for all and .
Now, by (40) and (38), Therefore, for all and .
Since for all , we have, taking limit in (41), that This shows, using (36), that
Step  5. Now, we will prove that .
Let for all .
Using condition (6), we have Combining all the inequalities, we will have the following: We conclude that implies that for all and . From (46), Letting , we have for all , and it means that for all ; that is, The uniqueness of follows directly from (6).

Remark 12. The uniqueness of the coincidence point of and is not always true. For instance, if is constant and is also constant, then every is a coincidence point of and .

Consequences. In the proof of the next result, the view of as the crisp FMS is used (see Example 6). Here we deduce results for -tupled coincidence point in metric spaces which are generalization of the results of Roldán et al. [11] which are not in a partially ordered set.

Theorem 13. Let be a complete metric space and let and be two mappings such that and is continuous and commuting with . Suppose that and satisfy some of the following conditions for all ;(1)  for some ,(2)  for some and some ,(3)  for some , such that .
Then there exists a unique such that .

Proof. Consider defined as in Example 6. As is complete, then is a complete FMS. Fix and .
Now we are ready to prove (6).
Put and . If or or or , then (6) is obvious.
Suppose that This means that Therefore, Hence, and (6) is also true.
In this case, If .

Remark 14. We can obtain Theorems 11 and 15 due to Roldán et al. [11] as corollaries, if we establish our main result for the mappings and .

Applications. Now we are ready to establish application.

Lipschitzian Systems. Let be Lipschitzian mappings and let be real numbers. Define as for all . Then is another Lipschitzian mapping and . Obviously, if , then is a contraction, so there exists a unique such that .

Next, define as for all . It is clear that for all . Furthermore, If , then verifies (6) with for all .

Corollary 15. Let be Lipschitzian mappings on (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 .

An Integral System. Let with and let . Consider with the distance , where represents the Lebesgue integral. It is well known that is a complete metric space. Let , be real numbers and let be a mapping verifying , and for all .

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 verifies (6) with for all . Then system (62) has a unique solution, which is of the form , where is the only solution of the equation for all .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author (P. P. Murthy) is thankful to University Grants Commission, New Delhi, India, for financial assistance through Major Research Project File no. 42-32/2013 (SR). Also the authors are grateful to the referees for their valuable comments and suggestions to improve this paper.