Common Fixed Points of Intuitionistic Fuzzy Maps for Meir-Keeler Type Contractions

Abstract

The main purpose of this paper is to establish and prove some new common fixed point theorems for intuitionistic fuzzy maps in the context of -cut sets of intuitionistic fuzzy sets on a complete metric space in association with the Hausdorff metric. Furthermore, the technique of Meir-Keeler (shortly, M-K) contraction is applied to obtain common fixed point of intuitionistic fuzzy compatible maps and fixed points of Kannan type intuitionistic fuzzy set-valued contractive mappings. Our results generalize M-K type fixed point theorem along with its various generalizations. Some nontrivial examples have been furnished in the support of the main results.

1. Introduction

The theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena. In particular, fixed point techniques have been applied in such diverse fields as biology, chemistry, economics, engineering, game theory, and physics. The Banach fixed point theorem [1] (also known as a contraction mapping principle) is an important tool in nonlinear analysis. It guarantees the existence and uniqueness of fixed points of self-mappings on complete metric spaces and provides a constructive method to find fixed points. Many extensions of this principle have been done up to now. In , Jungck [2] studied coincidence and common fixed points of commuting mappings and improved the Banach contraction principle. In 1986, Jungck [3] introduced the notion of compatible maps for a pair of self-mappings and existence of common fixed points. In , Meir and Keeler [4] obtained a valuable fixed point theorem for single valued mappings that satisfies the following condition: In , Park and Bae [5] extended it to a pair of commuting single valued mappings. A variety of extensions, generalizations, and applications of this followed; e.g., see [6, 7]. In , Beg and Shahzad [8] derived and proved random fixed point of two random multivalued operators satisfying the M-K [4] condition in Polish spaces. In , Lim [9] wrote on characterization of M-K-contractive maps. In , Abdeljawad et al.'s paper [10] contains a study of M-K type coupled fixed point on ordered partial metric spaces and Chen et al. [11] established and proved common fixed point theorems for the stronger M-K cone-type function in cone ball-metric spaces. In , Karapinar et al. [12] studied the existence and uniqueness of a fixed point of the multidimensional operators in partially ordered metric space which satisfied M-K type contraction condition and improved the results mentioned above and the recent results on these topics in the literature. In the same year, Abdeljawad [13] established and proved M-K-contractive fixed point and common fixed point theorems. Patel et al. [14] formulated and proved a more generalized version of [13]. In , Singh et al. [15] derived a new common fixed point theorem for Suzuki-M-K contractions. In , Redjel et al. [16] proved fixed point theorems for -Meir-Keeler-Khan mappings. Abtahi [17, 18] established and proved fixed point theorems in and common fixed point theorems in for M-K type contractions in metric spaces. Popa and Patriciu [19] derived and proved a general theorem of M-K type for mappings satisfying an implicit relation in partial metric spaces in .

Fuzzy sets were introduced by Zadeh [20] in to represent/manipulate data and information possessing nonstatistical uncertainties. In 1986, the concept of an intuitionistic fuzzy set (IFS) was put forward by Atanassov [21], which can be viewed as an extension of fuzzy set. Intuitionistic fuzzy sets not only define the degree of membership of an element, but also characterize the degree of nonmembership. IFS has much attention due to its significance to remove the vagueness or uncertainty in decision-making. IFS is a tool in modeling real life problems such as psychological investigation and career determination. Abbasizadeh and Davvaz [22] introduced intuitionistic fuzzy topological polygroups. In , Azam et al. [23] proved coincidence and fixed point theorems of intuitionistic fuzzy mappings with applications. In the same year, Azam and Tabassum [24] established and proved fixed point theorems of intuitionistic fuzzy mappings in quasi-pseudometric spaces. Recently, Kumam et al. [25] and Shoaib et al. [26] derived and proved some fuzzy fixed point results for fuzzy mappings in complete b-metric spaces. Humaira et al. [27] established and proved fuzzy fixed point results for contractive mapping and gave some applications. Ertrk and Karakaya [28] stated and proved -tuplet coincidence point theorems in intuitionistic fuzzy normed spaces. Xia Li et al. [29] worked on the intuitionistic fuzzy metric spaces and the intuitionistic fuzzy normed spaces (see also [30–32]).

In this paper, the main focus is to establish the existence of fixed point and common fixed point theorems of M-K type contraction for intuitionistic fuzzy set-valued maps in complete metric spaces. Some nontrivial examples have been furnished in the support of the main results.

2. Preliminaries

We start this section by recalling some pertinent concepts.

Definition 1 (see [33]). Let be a metric space. The set of all nonempty closed and bounded subsets of is denoted by . The function defined on by for all is a metric on called the Hausdorff metric of ,
where

Definition 2 (see [20]). Let be an arbitrary nonempty set. A fuzzy set in is a function with domain and values in . If is a fuzzy set and , then the function-value is called the grade of membership of in . stands for the collection of all fuzzy sets in unless and until stated otherwise.

Definition 3 (see [21]). Let be a nonempty set. An intuitionistic fuzzy set is defined as where and denote the degree of membership and degree of nonmembership of each element to the set , respectively, such that The collection of all intuitionistic fuzzy sets in is denoted by .

Definition 4 (see [34]). Let A be an intuitionistic fuzzy set and ; then level set of an intuitionistic fuzzy set A is denoted by and is defined as A generalized version of level set of an intuitionistic fuzzy set was investigated in [35, 36].

Definition 5 (see [35, 36]). Let and let be an IFS on ; then cut set of is defined as

Definition 6 (see [23]). Let be an arbitrary set and let be a metric space. A mapping is called an intuitionistic fuzzy mapping.

Definition 7. Mappings and are said to be compatible if whenever there is a sequence satisfying (provided and exist and ), then

Lemma 8 (see [37]). Let be a sequence in and for . If and , then

3. Main Results

Theorem 9. Let be a complete metric space and let be compatible mappings. Suppose for each there exists such that and and the following condition is satisfied:If is continuous, then and have a common fixed point.

Proof. Let , and consider the following sequences and in and , , (which is possible due to the hypothesis ). Then for each there exists such that implies . It follows that . Thus, the sequence is nonincreasing and converges to the greatest lower bound of its range, which we denote by .
Now ; in fact, . Otherwise, if , take so that implies . It implies that which contradicts the fact that Hence . Now it is to prove that is a Cauchy sequence. Suppose that for some . Then for all ; otherwise , a contradiction. Hence, is a Cauchy sequence.
Now assume that for each n. Define and choose (without loss of generality) , , such that (9) is satisfied. Since , there exists an integer such that for . We now let and show that , to prove that is indeed Cauchy. Suppose thatFirst, we show that there exists an integer such thatwhere and are of opposite parity. Let be the smallest integer greater than such that(which is possible due to (11) as ). Moreover,For otherwise, Since , therefore . It implies thatwhich contradicts the fact that is the smallest such that (13) is satisfied. Thus,If and are of opposite parity, we can take in (17) to obtain (12). If and are of the same parity, and are of opposite parity. In this case, Moreover, Thus, Putting , we obtain (12). Hence (12) holds. Now, a contradiction.
Hence is a Cauchy sequence. By completeness of the space, there exists an element such that ; continuity of implies that . Hence, .
Since is a Cauchy sequence in and it follows that is a Cauchy sequence in . By completeness of , there exists such that . Since and , Lemma 8 implies that , that is, . Compatibility of and further implies that Since , therefore , that is, and .
Let ; then, by (9) we have Thus .
Now, Hence, .

Definition 10 (see [38]). Let be a metric space and let be an intuitionistic fuzzy map. A single valued map is said to be a selection of , if there exists such that

Theorem 11. Let be a compact subset of a complete metric space and let be a mapping which satisfies the following conditions:Then, there exists a subset of such that for each Moreover, for each there exists a selection of having as a unique fixed point.

Proof. Let be an arbitrary fixed element of . Two sequences and of elements in and , respectively, will be constructed. is a closed subset of and therefore is compact. There exists a point such that . Similarly, there exists such that . By induction, we prove that sequences and are such that , , . From inequality (28), we haveIf , then (29) implies that , a contradiction. Hence, Thus, is a monotone nonincreasing sequence of nonnegative real numbers. Therefore, converges to . Suppose . Take so that implies that It follows that which is a contradiction to the assumption that . Hence
That is, It follows that By completeness of (see [39]), there exists a set such that . Let ; then If not, let ; then In a limiting case when , we have , a contradiction. Hence, Now, Hence, for all
Next, we will prove that there exists a selection of which has a unique fixed point. For each , is compact. Therefore, for there exists such thatLet defined as be a selection of . Then, for each we have . Let ; then . This implies that Now, It follows that the fixed point of is unique.

The following examples show that our results generalize a number of previous theorems.

Example 12. Let be the set of all nonnegative integers with the Euclidean metric. Let be defined as and let be an intuitionistic fuzzy map defined aswhere For and , For , there exists such that all the hypotheses of Theorem 9 are valid to obtain common fixed point of and . Previously known results are not applicable to this example (even in the case when is single valued, that is, ) since at

Example 13. Let with the Euclidean metric, , and
For , define Define intuitionistic fuzzy map as follows:
when ,When ,For and ,For , there exists , such that satisfies all the assumptions of Theorem 11. In this case, such that for all and corresponding to the mapping defined asis a selection of .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.