Fixed Point Theorems for Generalized Kannan-Type Mappings in a New Type of Fuzzy Metric Space

Abstract

In this paper, first, we introduce a new type of fuzzy metric space which is a generalization of fuzzy metric spaces. Second, we study the topological properties of fuzzy metric spaces. Finally, we extend Kannan-type mappings to generalized Kannan-type mappings under gauge functions introduced by Fang in fuzzy metric spaces and prove the existence and uniqueness of fixed point for this kind of mappings. Furthermore, we also obtain the common fixed point theorems for weak compatibility along with property or property. Our results extend and improve very recent theorems in the related literature.

1. Introduction

In 1965, Zadeh [1] introduced the concept of fuzzy sets. Since then, one of the important problems is to obtain an adequate notion of fuzzy metric spaces. In 1975, Kramosil and Michálek [2] reformulated successfully the notion probabilistic metric space, introduced by Menger in 1942, in fuzzy context. After that, George and Veeramani [3] modified the concept of fuzzy metric spaces and defined a Hausdorff topology on this fuzzy space. Another approach for fuzzy metric spaces was proposed by Kaleva and Seikkala [4], by setting the distance between two points to be a nonnegative upper semicontinuous, normal, and convex fuzzy number. In the last several decades, there has been a tremendous development and growth in fuzzy mathematics. One of the branches is to obtain several kinds of generalized fuzzy metric proposed by using related generalized metric spaces: metric spaces [5, 6], metric spaces [7], metric spaces [8], and metric spaces [9], respectively. Very recently, Sedghi et al. [10] introduced an metric space which is a generalization of metric spaces and metric spaces and justified their work by various examples and definitions related to the topology of metric spaces. Now, there arises a natural question: “how the fuzzy metric spaces can be generalized by using the concept of metric spaces?”

In 1968, Kannan [11] introduced the Kannan-type mappings as follows.

Definition 1. (see [11]). Let be a metric space and be a mapping on . We say that is a Kannan-type mapping, if there exists such that.
This kind of mapping is very important in metric fixed point theory. It is well known that Banach’s contraction mappings are continuous while Kannan-type mappings are not necessary continuous. This is a big difference between these two types of mappings. Again, it may also be noted that Banach’s contraction does not characterize metric completeness. In fact, Subrahmanyam [12] proved that every metric space is complete if and only if every Kannan-type mapping has a fixed point. However, it was also pointed out that Kannan’s fixed point result is not an extension of Banach contraction principle. The above shows some of the reasons why the Kannan-type mappings and their generalizations have been considered as constituting an important class of mappings in fixed point theory.
On the other hand, an interesting class of problems in fixed point theory was addressed in recent times by use of gauge (control) functions. There are several gauge (control) functions which have been used to extend Sehgal’s contraction in probabilistic metric spaces. Some examples of such applications are in [13–18]. One of such gauge functions was introduced in the setup of complete Menger probabilistic metric spaces and fuzzy metric spaces by Fang [17], where the gauge function satisfies the condition: for each , there exists such that , which is considered as weaker than the condition presented by Francisco et al. in [19]. It is perceived that the study of fixed points for contractions under some gauge (control) functions is an important category of problems in fixed point theory.
In this reason, in Section 2, we introduce a new type of generalized fuzzy metric spaces called fuzzy metric spaces which is a generalization of fuzzy metric spaces and fuzzy metric spaces. We also study the topological properties and prove some interesting results related its topology and convergent sequences. By using the gauge functions introduced by Fang in fuzzy metric spaces, we introduce in Section 3 the generalized Kannan-type mappings as an extension of the concept of Kannan-type mappings and prove the existence and uniqueness of fixed point for this kind of mapping. Furthermore, we also obtain the common fixed point theorems for weak compatibility along with or property.

2. SFuzzy Metric Spaces

We begin by recalling some basic definitions concerning metric spaces, fuzzy metric spaces, etc.

Definition 2. (see [10]). Let be a nonempty set. An metric on is a function that satisfies the following conditions for : (S1)  (S2) The pair is called an metric space.

Remark 1. (see [10]). Every metric space is a -metric space and every -metric space is an metric space, but in general, the converse of these implications are not true (see [10] for more details).

Definition 3. (see [20]). A mapping is called an order norm if the following conditions are satisfied:(1)(2)(3)  implies  (4)When , we have binary norm, which is well known as norm. Typical examples of norm are Lukasiewicznorm , product norm , and minimum norm .

Definition 4. (see [21]). Let be a norm and let be defined as follows:We say that the norm is of type, if is continuous and the sequence is equicontinuous at ; i.e., for every , there exists , such that if , then , . A trivial example of norm of type is .

Definition 5. (see [3]). A tuple is said to be a fuzzy metric space if is an arbitrary nonempty set, is a continuous norm, and is a fuzzy set on satisfying the following conditions for and : (FM1)  (FM2)  (FM3)  (FM4)  (FM5) is a continuous function.The pair is called a fuzzy metric on . Here, is considered as the degree of nearness of and with respect to .
Next, we define the fuzzy metric space by using the concept of metric.

Definition 6. A triple is called an fuzzy metric space (denoted ) if is an arbitrary nonempty set, is a continuous norm, and is a fuzzy set on satisfying the following conditions for and :        is a continuous function.

Example 1. Let be the real line and be an metric on defined by . Then, is an metric space. Define , . Let be the function on defined by , and .
It is easy to check that is an fuzzy metric space. Moreover, it is neither fuzzy metric space nor fuzzy metric space because is not symmetric.

Example 2. Let and be an metric defined by , . Define , , and let be the function on defined by , and . It is also easy to verify that is an fuzzy metric space.

Example 3. Let be a fuzzy metric space with continuous norm . If we denote by , and , then is an fuzzy metric space.

Proof. (1)It is easy to see that, for every , and is continuous.(2), , .(3)For all and , we haveTherefore, the conditions of Definition 6 are satisfied, and is an fuzzy metric space.

Example 4. Suppose that and are two -fuzzy metric spaces. The fuzzy set is defined as is an fuzzy metric on .

Proof. It is not difficult to verify that the conditions and in Definition 6 hold.
It remains to prove that the condition holds. One has, as required.
Consequently, the fuzzy set is an fuzzy metric on .

Proposition 1. If is an fuzzy metric space, then , and .

Proof. For each , from , we deduceBy taking the limit in above equalities as , we obtain .

Proposition 2. is nondecreasing with respect to , .

Proof. Suppose that for some , with .
Then,By , we haveTherefore, by Proposition 1 and Definition 3, we havewhich is a contradiction. Hence, the conclusion holds.

Proposition 3. Let be an fuzzy metric space. Then, for , it follows that(1) , (2),

Proof. The conclusion easily follows from Definition 6 and Proposition 2.

Theorem 1. Let be an fuzzy metric space. For every , the open ball with center and radius is defined byThen, is a topology on .

Proof. It is obvious that and belong to .
Let and . We will show that .
Choose . There exists such that . As , there exist and such that . Thus, .
Now, let and . We will show that .
If , then , . Thus, one can find and such that , . Putting , one obtains , . for all i in , Indeed, for , we have , . for all i in , Since, , we have , . Thus, , . Hence, , . Therefore, , , so .

Theorem 2. Every fuzzy metric space is Hausdorff.

Proof. Let be an fuzzy metric space, and let be two distinct points of . Then, .
For some , we denote . For every , , we can find such that . Now, we consider the open balls and . We claim thatIndeed, if there exists , thenwhich is a contradiction. Therefore, is Hausdorff.

Definition 7. Let be an fuzzy metric space and . We say that is bounded if, for every and , there exists such that . Furthermore, a sequence in is said to be bounded if, for and all , there exists such that , where is a closed ball with center and radius defined by .

Definition 8. A sequence in is convergent to, if or as for each . That is, for each and , there exists such that, for , or . Furthermore, a subset of an fuzzy metric space is said to be closed if and imply .

Definition 9. A sequence in an fuzzy metric space is said to be Cauchy if, for each and , there exists such that or , . If every Cauchy sequence in is convergent in , then is called a completefuzzy metric space.

Remark 2. (1)Let be an fuzzy metric space. A sequence is convergent to if and only if is convergent to in the topology . That is,(2)It is easy to prove that the fuzzy metric space is complete if and only if the metric space is complete where , and .

Lemma 1. Let be an fuzzy metric space, where is a continuous norm of type. Let be a sequence in with and as . Then, .

Proof. Assume that converges to and . Then, , for each , and , for each :Therefore, .

Lemma 2. Let be an fuzzy metric space, where is a continuous norm of type and be a convergent sequence. Then,(1) is bounded and its limit is unique.(2) is a Cauchy sequence.(3)Every subsequence of converges to the same limit.(4)Every Cauchy sequence is bounded.(5)If every Cauchy sequence in has a subsequence such that , then .

Proof. (1)First, we show that the convergent sequence is bounded. Suppose that . Then, for each and , there exists such that , . Choose and such that and let be such that Then, we can find a number such that So, for , we have Thus, for , , that is, is bounded (the bar means the closure). Now, we show that the limit of is unique. We suppose that the convergent sequence has two distinct limits and . For any and , we can find a number such that . Set . From our assumption, there exist such that , , and , . Taking , we have for , which implies .(2)For each with , one can find such that, for , Therefore, is a Cauchy sequence.(3)Let be a subsequence of . If , then for each and , there exists such that , . If , then and so which implies that converges to .(4)Let be a Cauchy sequence in . Then, for each and , there exists such that , . So, for , we have . Let and choose . Then, , that is, is bounded.(5)Let and . We can find a number such that . Since is a Cauchy sequence in , there exists such that , . From , there exists a positive integer such that and . For , we have so .This completes the proof.

Lemma 3. Let be an fuzzy metric space, where is a continuous norm of type. Let and be two sequences in with , , and . Then, .

Proof. Since , and as , there exists such that for and .
By the nondecreasing property of with respect to , we haveSince is arbitrary chosen, using Definition 8 and the continuity of with respect to , for large enough , we haveTherefore, .

Remark 3. From Lemma 3, we can conclude that is a continuous function on in every fuzzy metric space .

Lemma 4. Let be an fuzzy metric space, where is a continuous norm of type. If there exists such that , , and , then .

Proof. If there exists such that , . Then, we havefor all positive integer . Taking the limit as , we haveHence, we have .

Lemma 5. Let be an fuzzy metric. For each , define a function by , . Then,(1)(2)(3)(4)If , then, for every ,

Proof. It is not difficult to see that (1)–(3) hold.
Now, we prove (4).
Let any and . For , we haveHence, from and , one obtainswhich impliesLetting , we have.

Definition 10. Let be an fuzzy metric space and be a nonempty subset of . We define fuzzy diameter as

Theorem 3. The fuzzy diameter has the following properties:(1) is a singleton.(2)If , then (3)For any , , (4)If , then ,

Proof. The conclusions can be easily obtained from Definition 10.

3. Fixed Point Theorems for Kannan-Type Mappings under Gauge Functions

Definition 11. (see [17]). Let denote the class of all functions satisfying the following condition: for each , there exists such that .

Lemma 6. (see [17]). If , then for , there exists such that .

Definition 12. A map is said to be a function, if(1) is monotonically increasing and continuous.(2)(3)

Definition 13. Let be an fuzzy metric space, where is a continuous norm of type. A mapping is said to be a contraction if there exists a function such that and .

Lemma 7. Let be an fuzzy metric space, where is a continuous norm of type and a self-mapping on be a contraction. Let be the iterative sequence generated by , . If there exists a function such that(1)(2), and (3), Then is a Cauchy sequence in .

Proof. From Assumption 1, it follows that , .
From Assumption 2, we obtain, by induction, and .
Now, we will prove that, for ,Since as , , there exists such that . Since , there exists such that . Thus, for , there exists such that , . Using the inequalities , , (31), and the monotonicity property of with respect to , we have. Thus, (32) holds.
Since , by Lemma 6, we deduce that, for any , there exists such that .
Let be given. Now, we show that for ,From (34), we obtainThus, (34) holds for . Assume that (34) holds for some . Since is monotone, from in Definition 6 and, then, by conditions (31) and (34), it follows thatwhich completes the claim.
Next, we will show that is a Cauchy sequence in , that is, , .
Fix and . Since is equicontinuous at and , there exists such that for and .
We first prove that , . Since , from (32), there exists such that , .
Hence, by (34), we have , .
Thus, we have shown that, for ,By with in Definition 6, we have that, for ,From (37), it follows that, for ,Thus, by using the continuity of , we conclude thatTherefore, we have proved that, for ,This shows that is a Cauchy sequence in .

Lemma 8. Let be an fuzzy metric space , where is a continuous norm of type. If there exists a function such that , , and , then .

Proof. Since is monotone, it is obvious that , . By induction, we have , with and also have, from the assumption, that.
To prove that , we need to verify that for . On the contrary, suppose that there exists such that . Since , there exists such that.
Since , there exists such that . Therefore, we can choose large enough such that . By the monotonicity of , using (42) and (43), we havewhich is a contradiction. Therefore, , . Consequently, .

Theorem 4. Let be a complete fuzzy metric space, where is a continuous norm of type. If the mapping is a contraction, then has a unique fixed point and, for , .

Proof. Let be an arbitrary point in and let us define the sequence by , .
Since is a contraction, by (30), we have that, for ,By Lemma 7, we conclude that is a Cauchy sequence in . Since is complete, there exists such that , that is, for , .
Now, we will prove that is a fixed point of .
Since , by Lemma 6, we have that for , there exists such that . Putting in of Definition 6, we haveSince and is continuous, taking limit as in above inequality, one hasSo, , that is, is a fixed point of .
Next, suppose that with is another fixed point of . Then, for , we havewhich implies, according to Lemma 8, that . Therefore, has a unique fixed point.

Example 5. Let and , . Define the functions by , , and . Then, is an fuzzy metric space.
Let be a mapping defined by and be defined byNotice that and , .
Now, we verify that satisfies the contraction condition:This shows that is a contraction; then, has a fixed point in . Indeed, 0 is the fixed point of .

Definition 14. Let be an fuzzy metric space, where is a continuous norm of type and assume that is a function. A mapping is called generalized Kannan-type mapping under a gauge function if, for ,where with , with , .

Lemma 9. Let be a complete fuzzy metric space, where is a continuous norm of type and be a generalized Kannan-type mapping under a gauge function on . Assume that , . If is the iterative sequence generating by , then , .

Proof. Let , , , and , , , , and be positive real numbers with . From (51), for , we haveFor , putting and in the above inequality, we haveNow, we claim thatSuppose, to the contrary, there exists such thatBy property of function and (53), we havewhich is a contradiction. So (53) and (54) imply the following inequality:If we apply induction to the above inequality, we see thatOur additional assumption on fuzzy metric spaces implies that .
Thus, by taking the limit as , we have that, for all ,

Theorem 5. Let be an fuzzy metric space, where is a continuous norm of type and assume that , , . Let be a generalized Kannan-type mapping under a gauge function on . Then, has a unique fixed point and, for every , .

Proof. Choose and let be the iterative sequence generated by . We will show that is a Cauchy sequence. Indeed, if not so, by definition, there exists for which we can find and subsequences and of with for all positive such thatSo, for with and with , we haveTherefore,By Lemma 9, for , we getSo we can choose large enough such thatTherefore, from (62) and (64) and the definition of function, it follows thatwhich is a contradiction. So is a Cauchy sequence. The completeness of fuzzy metric space implies that for some . Now, we claim that . If possible, let , for some .
Since , we have chosen such thatThen, we haveBy Lemma 9 and the convergence of to , there exists a positive integer such that, for ,Thus, it follows thatwhich is a contradiction. Hence, , ; therefore, is a fixed point of .
In order to prove the uniqueness, suppose that and are two fixed points of . Therefore, with all of the above assumptions on , for , we haveThus, , completing the proof.
Next, we prove the common fixed point theorems for weakly compatible mappings along with property and property.

Definition 15. Let and be two self-mappings of an fuzzy metric space . The mappings are said to be weakly compatible, if they commute at their coincidence points; that is, implies .

Definition 16. Let and be two self-mappings of an fuzzy metric space . We say that and satisfy the property if there exists a sequence such that , for some .

Remark 4. Note that the weakly compatible and property are independent to each other as is follows from the following examples.

Example 6. Let be an fuzzy metric space, where and is defined by , and . is defined by , .
Define byThen, for the sequence , we haveThus, and satisfy property.
Furthermore, and are weakly compatible, since is their coincidence point and .

Example 7. Let be an fuzzy metric space, where , and , and .
Define byConsider the sequence , . Then, we haveAt the same time, for the sequence , , one hasThus, and satisfy the property. However, and are not weakly compatible since each and are coincidence points of and while they do not commute. Moreover, they commute at but none of them are coincidence points of and . Hence, property does not imply weak compatibility.

Definition 17. Let and are two self-mappings of an fuzzy metric space . We say that and satisfy the common limit in the range of property, if there exists a sequence such that , for some .

Definition 18. Let and be two self-mappings of an fuzzy metric space . The pair of mappings is said to be generalized Kannan-type mappings under a gauge function if, for ,where with , , .
In what follows, we prove a result concerning weakly compatible mappings along with property.

Theorem 6. Let and be two self-mappings of an fuzzy metric space with a continuous norm of type. Suppose that the following conditions are satisfied:(1) and satisfy property.(2)The pair is generalized Kannan-type mappings under a gauge function (3) is a closed subspace of (4) and are weakly compatible on , provided , Then, and have a unique common fixed point.

Proof. Since and satisfy property, there exists a sequence in such thatSince is a closed subspace of , every convergent sequence of points of has a limit in . Therefore, there is such thatWe now show that .
If possible, let , for some . Since , we can choose such thatThen, we getBy (78), there exists a positive integer such that, for ,Then, we havewhich is a contradiction. Hence, , , which implies .
Since and are weakly compatible, it follows that ; i.e., .
Now, we will prove that is a common fixed point of and .
From condition (2), we deducewhere . Taking the limit as , we haveHence, ; that is, is a common fixed point of and .
In order to prove the uniqueness, let be two common fixed points of and . Therefore, for with , , one hasHence, ; that is, is a unique common fixed point of and .
Finally, we present a theorem for weakly compatible mappings along with property as follows.

Theorem 7. Let and be two self-mappings of an fuzzy metric space with a continuous norm of type. Suppose that the following conditions hold:(1) and satisfy property.(2)The pair is generalized Kannan-type mappings under a gauge function (3) and are weakly compatible on , provided , Then, and have a unique common fixed point.

Proof. Since and satisfy the property, there exists a sequence in such thatWe will show that .
If possible, let , for some . Since , we can choose such thatThen, we haveBy (86), there exists a positive integer such that, for ,Then, we havewhich is a contradiction. Therefore, , , which implies .
Using the similar argument to that in the proof of Theorem 6, we can obtain that is a unique common fixed point of and .

Data Availability

No data sets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Mi Zhou was partially supported by the Hainan Provincial Natural Science Foundation of China (Grant no. 118MS081) and Science and Technology Cooperation Project of Sanya City (Grant no. 2018YD13 and 2018YD08). Xiao-lan Liu was partially supported by the National Natural Science Foundation of China (Grant nos. 61573010 and 11872043), Sichuan Science and Technology Program (Grant no. 2019YJ0541), Scientific Research Project of Sichuan University of Science and Engineering (Grant nos. 2017RCL54 and 2019RC42), and Opening Project of Sichuan Province University Key Laboratory of Bridge Non-Destruction Detecting and Engineering Computing (Grant no. 2019QZJ03).