Abstract

In this paper, we study some coincidence point and common fixed point theorems in fuzzy metric spaces by using three-self-mappings. We prove the uniqueness of some coincidence point and common fixed point results by using the weak compatibility of three-self-mappings. In support of our results, we present some illustrative examples for the validation of our work. Our results extend and improve many results given in the literature. In addition, we present an application of fuzzy differential equations to support our work.

1. Introduction

Zadeh [1] introduced the concept of fuzzy sets which is defined as “a set contracted from a function having a domain is a nonempty set and range in is called a fuzzy set, that is if .” In 1975, Kramosil and Michalek [2] introduced the notion of fuzzy metric (FM) space, and they compared the concept of fuzzy metric with the statistical metric space and proved that both the conceptions are equivalent in some cases. Later on, George and Veeramani [3] modify the concept of Kramosil and Michalek [2] and proved that every metric induces a fuzzy metric. They proved some basic properties and Baire’s theorem for fuzzy metric spaces. In 1988, Grabiec [4] used the concept of Kramosil and Michalek [2] and proved fixed point (FP) theorems of “Banach and Edelstein contraction mapping theorems on complete and compact FM spaces, respectively.” Gregori and Sapena [5], Imdad and Ali [6], Mihet [7, 8], Bari and Vetro [9], and Som [10] proved some FP and common fixed point (CFP) theorems in FM spaces. Aliouche et al. [11] and Rao et al. [12] established some related FP in FM spaces.

Hadzic and Pap [13] established some multivalued FP results in probabilistic metric spaces with an application in FM spaces. Later on, Kiany and Amini-Haradi [14] obtained some FP and end-point theorems for set-valued contractive type mappings in FM spaces. In [15], Beg et al. proved some FP results for self-mappings satisfying an implicit relation in a complete FM space. Rolden et al. [16] established some new FP theorems in FM spaces, while in [17], Jeli et al. presented some results by using cyclic ()-contractions in Kaleva-Seikkala’s type FM spaces. Later on, Li et al. [18] proved some strong coupled FP theorems in complete FM spaces with an integral type of application. The concept of rational type fuzzy-contraction is given by Rehman et al. [19]. They proved some unique FP theorems with the application of nonlinear integral in FM spaces. Shamas et al. [20] proved some unique FP results in FM spaces with an application to Fredholm integral equations. Recently, Jabeen et al. [21] presented the concept of weakly compatible self-mappings in fuzzy cone metric spaces, and they proved some coincidence point and CFP theorems in the said space with integral type application. Some more coincidence points, coupled coincidence points, and CFP findings in deferent types of metric spaces can be found in (e.g., see [22–35] the references therein).

In this paper, we establish some unique coincidence points and CFP theorems in FM space by using the concept of Gregori and Sapena [5] and Jabeen et al. [21]. We establish some generalized fuzzy-contraction results for weakly compatible three self-mappings in FM spaces without the assumption that the “fuzzy contractive sequences are Cauchy.” We present some illustrative examples and an application of fuzzy differential equation to support our work. By using this concept, researchers can prove more coincidence points and CFP results for different contractive type mappings in FM spaces with the application of integral operators.

The layout of this paper is as follows: Section 2 consists of preliminary concepts. While Section 3 deals with the main results of this paper in which we shall prove unique coincidence points and CFP theorems by using weakly-compatible three self-mappings in FM spaces with some illustrative examples. In Section 4, we establish an application of fuzzy differential equations to support our main work. While in the last section, that is, Section 5 is the conclusion part of our paper.

2. Preliminaries

In this section, we recall some basic definitions related to our main work such as continuous -norm, FM space, Cauchy sequence, fuzzy-contraction, and weak-compatible mappings. The concept of continuous -norm is given by Schweizer and Sklar [36].

Definition 1 (see [36]). An operation is known as a continuous -norm if it fulfils the following axioms; (1) is associative, commutative, and continuous.(2) and , whenever and , for each .The basic continuous -norms are (see; [36]): the minimum, the product, and the Lukasiewicz -norms are defined respectively as follows;

Definition 2 (see [3]). A 3-tuple is said to be a FM space if is an arbitrary set, is a continuous -norm, and is a fuzzy set on satisfying the following; (i) and (ii).(iii)(iv) is continuousfor all and

Definition 3 (see [3, 5]). Let be a FM space, and be a sequence in . Then (i) is said to converge to if for and where ( represent the set of natural numbers) such that . We represent this by or as .(ii) is said to be a Cauchy sequence, if for and such that .(iii) is complete, if every Cauchy sequence is convergent in .(iv) is known as a fuzzy-contractive, if there is so that

Lemma 4 (see [3]). is nondecreasing .

Lemma 5 (see [3]). Let be a FM space and let a sequence in converges to a point iff as , for .

Definition 6 (see [9]). Let be a FM space. The FM is triangular if for all and .
Note: easily one can prove that a FM is triangular if we define a mapping by

Lemma 7 (see [3]). Let be a FM space. Let and be a sequence in . Then iff for .

Definition 8 (see [5]). Let be a FM space and . Then, is known as a fuzzy-contraction, if there is so that for all and .

Definition 9 (see [37]). Let and be two self-mappings on a nonempty set (i.e., ). If and for some . Then is called a coincidence point of and , and is called a point of coincidence of the mappings and . The mappings and are said to be weakly-compatible if they commute at their coincidence point, i.e., for some , then .

Proposition 10 (see [37]). Let and be weakly-compatible self-mappings on a nonempty set . If and have a unique point of coincidence such that , then is known as the unique CFP of and .

3. Main Results

In this section, we present some unique coincidence points and CFP theorems in FM spaces by using weakly-compatible three self-mappings with illustrative examples.

Theorem 11. Let be a FM space in which a FM is triangular. Let be three self-mappings that satisfy for all , and with . If and is a complete subspace of . Then, , and have a unique point of coincidence. In addition, if and are weakly-compatible, then, , and have a unique CFP.

Proof. Let be arbitrary and by using the hypothesis . Now, we choose a sequence such that Now by the view of (5) and (6), for , we have After simplification, for , we obtain Now three possibilities arise: (i)If is a maximum term in (8) for , then after simplification, we get that (ii)If is a maximum term in (8) for , then after simplification, we get that (iii)If is a maximum term in (8) for , then after simplification, we get that Assume that , then from (9), (10), and (11), we get that Similarly, again from (5) and (6), for , After simplification, for , we obtain Now three possibilities arise, (i)If is a maximum term in (14) for , then after simplification, we get that (ii)If is a maximum term in (14) for , then after simplification, we get that (iii)If is a maximum term in (14) for , then after simplification, we get that Since , as defined in (12), then from (15), (16), and (17), we get that Now, from (12) and (18), and by induction, for , we have Hence, from the above, we get that the sequence is a fuzzy-contractive in , therefore, Next, we have to prove that the sequence is a Cauchy sequence in . Then, from Definition 2 and by using (20), for where for and , we have that This implies that Hence, proved that is a Cauchy sequence. Since, is a complete subspace of , then there exits such that as , therefore, Now, we have to show that . First, we estimate that for . Since, is triangular, therefore, Now by using (5), (20), and (23), for , we have Thus, The above (26) is together with (23) and (24), we get that which is a contradiction, as , therefore, , for . Hence, . Similarly, again by using the triangular property of , Again by using (5), (20), and (23), similar to the above simplification, we get The above (29) is together with (23) and (28), we get that which is a contradiction, as , therefore, , for . Hence, . It follows that is a coincidence point, and is a common point of coincidence point for the self-mappings , and , that is, Next, we prove that the point of coincidence point of the three self-mappings is unique. Let another point such that Now by using (5), for , After simplification, we get that which is a contradiction. Hence, for . Further, by using the weak compatibility of the pairs and and by Proposition 10, it follows that the three self-mappings , and have a unique CFP, that is, .