Research Article | Open Access
Some Theorems about -Contraction in Fuzzy Metric Spaces
We previously proved a number of fixed point theorems for some kinds of contractions like -contraction and contraction in fuzzy metric spaces. In this paper, we discuss the problem of existence of fixed point for -contraction in fuzzy metric spaces in sense of George and Veeramani.
Probabilistic metric spaces are generalizations of metric spaces which have been introduced by Menger . George and Veeramani  modified the concept of the fuzzy metric spaces, which were introduced by Kramosil and Michalek . Fixed point theory for contraction type mappings in fuzzy metric spaces is closely related to the fixed point theory for the same type of mappings in probabilistic metric spaces. Hicks introduced -contraction . Radu in  extended -contraction to the generalized -contraction. Mihet in  presented the notion of a -contraction of -type. He also introduced the class of -contraction in fuzzy metric spaces  which is a generalization of the -contraction . Ciric also proved some new results for Banach contractions and Edelstin contractive mappings on fuzzy metric spaces . We obtained the fixed point for -contraction in probabilistic metric spaces and we introduced the generalized -contraction too . The outline of the paper is as follows. In Section 2, we briefly recall some basic concepts. In Section 3, two fixed point theorems about -contractions are proved.
2. Preliminary Notes
Since the definitions of the notion of fuzzy metric space are closely related to the definition of generalized Menger spaces, we review a number of definitions from probabilistic metric space theory used in this paper as an example and Schweizer and Sklar’s definition can be considered. For more details, we refer the reader to [11, 12].
A mapping is called a distance distribution function if it is nondecreasing and left continuous with . The class of all distance distribution functions is denoted by . A probabilistic metric space is an ordered pair where is a nonempty set and is a mapping from to . The value of at is denoted by satisfying the following conditions:(1).(2).(3)
A binary operation is called a triangular norm (abbreviated -norm) if the following conditions are satisfied:(1).(2).(3).(4)
Definition 1. A Menger space is a triple where is a probabilistic metric space; is a -norm and the following inequality holds:If is a Menger space with satisfying , then the family , where , is a base for a uniformity on and is called the -uniformity on . A topology on is determined by this -uniformity, -topology.
Definition 2. A sequence is called an -convergent sequence to if, for all and , there exists such that .
Definition 3. A sequence is called an -Cauchy sequence if, for all and , there exists such that , for all , for all .
A probabilistic metric space is called -sequentially complete if every -Cauchy sequence is -convergent.
It is helpful to mention that George and Veeramani  have extended fuzzy metric spaces (GV-fuzzy metrics spaces).
Definition 4. The 3-tuple is said to be a fuzzy metric space if is a continuous -norm and is a fuzzy set on satisfying the following conditions:(1).(2).(3).(4).(5) is continuous.Gregori and Romaguera introduced the next definition .
Definition 5. Let be a fuzzy metric space. A sequence in is said to be point convergent to (shown as ) if there exists such thatNow, we recall the definition of the generalized -contraction ; let be the family of all the mappings such that the following conditions are satisfied:(1).(2).(3) is continuous.
Definition 6. Let be a probabilistic metric space and . A mapping is a generalized -contraction if there exist a continuous, decreasing function such that , , and such that the following implication holds for every and for every :If and for every , we obtain Hicks’s definition.
Mihet in  showed that if is a -contraction and is a complete fuzzy metric space, then has a unique fixed point, and Ćirić et al. presented the theorem of fixed and periodic points for nonexpansive mappings in fuzzy metric spaces .
The comparison functions from the class of all mappings have the following properties:(1) is an increasing bijection.(2)Since every comparison mapping of this type is continuous, if , then, for every
Definition 7. Let be a probabilistic space, , and let be a map from to . A mapping is called a -contraction on if it satisfies the following condition:
In the rest of the paper, we suppose that is an increasing bijection.
Example 8. Let and (for )Suppose that , and the Łukasiewicz -norm defined by . Then is a fuzzy metric space .
Assume are such that :(i)If , then . This indicates that ; that is,(ii)If , then ; hence again . Thus, the mapping is a -contraction on with and .
3. Main Results
In this section, we recall some contraction through the several definitions and then we prove the existence of fixed point for these contractions. It would be interesting to compare different types of contraction mapping in fuzzy metric spaces. It is useful to note that the concept of -contraction has been introduced by Mihet .
As we mentioned, existence of convergent sequence is sometimes a difficult condition. Gregori and Romaguera presented another type of convergence that is called -convergence . A GV-fuzzy metric space with the point convergence is a space with convergence in sense of Fréchet too.
First, we introduce contraction; then we review the fixed point theorem by -convergent subsequence.
Let be a fuzzy metric space and . A mapping is called a contraction on if the following condition holds:Consider the mapping ; we say that the -norm is -convergent if, , , ;
A theorem for contraction on a GV-fuzzy metric space is as follows .
Theorem 9. Let be a GV-fuzzy metric space and and let be a contraction. Suppose that, for some and and the sequence has a convergent subsequence. If is -convergent and is convergent, then will have a fixed point.
In this paper, due to the next theorem, the existence of fixed point for -contraction is proved under the new conditions.
Theorem 10. Let be a GV-fuzzy metric space and and let be a -contraction. Suppose that, for some and and and the sequence has a convergent subsequence. If is -convergent and is convergent, then will have a fixed point.
Proof. Let, for every and
By the assumption , soand by induction for every We show that the sequence is a Cauchy sequence; that is, for every and there exists an integer such that, for all .
Let and be given; since the series converges, there exists such that Then, for every ,Let be such that Since is -convergent, such a number exists. By using (10), we obtain for every and Suppose is a convergent subsequence of which converges to . Then, for every .
Let be given. Since , there is a such that . Since and then are Cauchy sequence, we can take large enough such that and ; then which implies that . By -contraction condition from , we haveIt means . Since the convergence is -convergence in a GV-fuzzy metric space, we get which means is a fixed point.
We mention an example for Theorem 13.
Example 11. Let be a complete fuzzy metric space where , and is defined as is given by and . If we take , then is a -contraction if we take , and as . Therefore,On the other hand, , and . So is a convergent sequence and hence has a convergent subsequence to . It is clear that is -convergent and . It means is a unique fixed point for .
For more information, reader can refer to .
Definition 12. Let be a GV-fuzzy metric space and . The mapping is a generalized -contraction if there exist a continuous, decreasing function such that , and such that the following implication holds for every and for every :If , and for every , we obtain the Mihet definition.
In the next theorem, we will prove a theorem for existing fixed point in the generalized -contraction.
Now, we prove the new theorem for this kind of contraction.
Theorem 13. Let be a GV-fuzzy metric space with t-norm such that and let be a generalized -contraction such that is continuous on and for every . Suppose that there exists such that and satisfied . Suppose that for some the sequence has a convergent subsequence. Then has a fixed point.
Proof. satisfies the following condition:for all and for all , and are given as in Definition 6. Let and First, we show that the sequence is a Cauchy sequence. We prove that for every and there exists an integer such that, for every and ,
By the assumption, there is a such that
From it follows thatwhich implies that , and by continuing in this way we obtain that for every Suppose is a natural number such that for every . Then implies thatNow let ; then we obtainwhich means that is a Cauchy sequence.
Suppose has a convergent subsequence which is convergent to . Then, for every ,Let be given since ; there is a such that . Since and then are Cauchy sequences, we can take large enough such that , and , ; then which implies thatLet be such that and such that . Since and are continuous at zero and , such numbers, and , exist.
If we take in relation (22), then we have soand this implies thatThusthen , and since convergence is -convergence in a GV-fuzzy metric space, then which means is a fixed point.
By an example, we describe Theorem 13 more.
Example 14. Let and the mappings , and be the same as in Example 11. Set for every and It is obvious that is continuous on . The mapping is generalized -contraction and for every . On the other hand, there exists such that and . By the previous example, has a convergent subsequence. So is the unique fixed point for .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author thanks the reviewers for their useful comments.
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