Fixed Point Theorems for New Type Contractive Mappings

Abstract

In this paper, we introduce some new concepts of contractions called contractions and weak contractions. We prove some fixed point theorems for mappings providing contractions and weak contractions unlike known results in the literature. Also, we present a few examples to illustrate the validity of the results obtained in the paper.

1. Introduction and Preliminaries

The fixed point theory is very important concept in mathematics. In 1922, Banach created a famous result called Banach contraction principle in the concept of the fixed point theory [1]. Later, most of the authors intensively introduced many works regarding the fixed point theory in various of spaces. The concept of a fuzzy metric space was introduced in different ways by some authors (see [2, 3]). Gregori and Sapena [4] introduced the notion of fuzzy contractive mappings and gave some fixed point theorems for complete fuzzy metric spaces in the sense of George and Veeramani, and also for Kramosil and Michalek’s fuzzy metric spaces which are complete in Grabiec’s sense. Mihet [5] developed the class of fuzzy contractive mappings of Gregori and Sapena, considered these mappings in non-Archimedean fuzzy metric spaces in the sense of Kramosil and Michalek, and obtained a fixed point theorem for fuzzy contractive mappings. At the same time, there are lots of different types of fixed point theorems presented by many authors by expanding the Banach’s result in the literature (see [5–12]).

In this work, using a mapping we introduce some new types of contractions called contractions and weak contractions. Later, we prove some fixed point theorems for mappings providing contractions and weak contractions. Some examples are supplied in order to support the usability of our results. Our main results are substantially different and useful compared to some known results in the existing literature.

Definition 1 ([13]). A binary operation is called a continuous triangular norm (in short, continuous norm) if it satisfies the following conditions: (TN-1) is commutative and associative,(TN-2) is continuous,(TN-3) for every ,(TN-4) whenever , and

An arbitrary norm can be extended (by associativity) in a unique way to a nary operator taking for , ; the value is defined, in [14], by ,

Definition 2 ([15]). A fuzzy metric space is an ordered triple such that is a nonempty set, is a continuous t-norm, and is a fuzzy set on , satisfying the following conditions, for all , (FM-1),(FM-2) iff ,(FM-3),(FM-4),(FM-5) is continuous. If, in the above definition, the triangular inequality (FM-4) is replaced by
(NA) for all , , or equivalently,and then the triple is called a non-Archimedean fuzzy metric space [16].

Definition 3. Let be a fuzzy metric space. Then see the following. (i)A sequence in is said to converge to in , denoted by , if and only if for all ; i.e., for each and , there exists such that for all [3, 17].(ii)A sequence is a -Cauchy sequence if and only if for all and , there exists such that for all [15, 17]. A sequence is a -Cauchy sequence if and only if for any and [4, 14, 18].(iii)The fuzzy metric space is called -complete (-complete) if every -Cauchy (-Cauchy) sequence is convergent.

2. Main Results

Definition 4. Let be a strictly increasing, continuous mapping and for each sequence of positive numbers if and only if Let be the family of all functions.
A mapping is said to be a contraction if there exists a such thatfor all and .

Example 5. Let . The different types of the mapping are as follows:,,,.

Remark 6. From and (2) it is easy to conclude that every contraction is a contractive mapping, that is,for all , such that . Thus every -contraction is a continuous mapping.

Theorem 7. Let be a non-Archimedean fuzzy metric space and let be a contraction. Then has a unique fixed point in .

Proof. Let be arbitrary and fixed. Define sequence by If then is a fixed point of ; then the proof is finished. Suppose that for all . Therefore by (2), we get Repeating this process, we getLetting , from (6) we getThen, we haveNow, we want to show that is a Cauchy sequence. Suppose to the contrary, we assume that is not a Cauchy sequence. Then there are and such that for all there exist with andAssume that is the least integer exceeding satisfying inequality (9). Then, we haveand so, for all , we getBy taking in (11) and using (8), we obtainFrom (FM-4), we getTaking the limit as in (13), we obtainBy applying inequality (2) with and Taking the limit as in (15), applying (2), from (12), (14), and continuity of , we obtain which is a contradiction. Thus is a Cauchy sequence in . From the completeness of there exists such thatFinally, the continuity of yields . Now, we show that has a unique fixed point. Suppose that and are two fixed points of . Indeed, if for , , then we getwhich is a contradiction. Thus, has a unique fixed point. Hence, the proof is completed.

Example 8. Let , ,for all Let such that for all and define by for all Clearly, is a non-Archimedean fuzzy metric space.
Case 1. We assume that for all Since and , then So, there exists such that Thereforeand soand, then, we getwhich impliesThat is,Case 2. Let and Since , , then . Hence, we have So, there exists such that That is, Therefore, is a contraction. Then all the conditions of Theorem 7 hold and has a unique fixed point

Definition 9. Let be a non-Archimedean fuzzy metric space. A mapping is said to be a weak contraction if there exists such that for all and

Remark 10. Every contraction is a weak contraction. But the converse is not true.

Example 11. Let where , . and for all Clearly, is a complete non-Archimedean fuzzy metric space. Let such that for all and define by Since is not continuous, is not contraction by Remark 6.
Now, we show that is a weak contraction for all and .
Case 1. Let and , Then we have So, there exists such that Case 2. Let and , Then we have So, there exists such that Case 3. Let and , Then we have So, there exists such that Therefore, is a weak contraction.

Theorem 12. Let be a non-Archimedean fuzzy metric space and let be a weak contraction. Then has a unique fixed point in .

Proof. Let be arbitrary and fixed. Define sequence by If then is a fixed point of ; then the proof is finished. Suppose that for all . Therefore by (29), we getIf there exists such that from (41) becomes which is a contradiction, therefore,for all . That is, from property of , (41), and (44), we getThus, from (41), we have for all . It implies that By taking in (47), we getThen, we haveThe proof that is a Cauchy sequence can be shown as in Theorem 7. From the completeness of there exists such that Now, we show that is a fixed point of Since is continuous, there are two cases.
Case 1. For each , there exists such that and , where Then, we getThis proves that is a fixed point of .
Case 2. There exists such that for all That is, for all It follows from (29), property of ,If , then we haveand there exists such that for all , we getFrom (52), we havefor all Since is continuous, taking the limit as in (55), we obtainwhich is a contradiction. Therefore, ; that is, is a fixed point of
Now, we prove that the fixed point of is unique. Let be two fixed points of Suppose that ; then we have It follows from (29) that we havewhich is a contradiction. Then, , that is, Therefore, the fixed point of is unique.

Example 13. Let be the non-Archimedean fuzzy metric space and let be considered in Example 11. Let such that for all . So, is a weak contraction. Therefore, Theorem 12 can be applicable to and the unique fixed point of is .

Conclusion 14. In this paper, we introduce new contraction types in non-Archimedean fuzzy metric spaces and presented new fixed point results. Our results can be expanded and solutions to new problems can be produced in this way. Also, a new more general contraction can be achieved or common fixed point theorems for a class of mappings can be obtained using contractions and also our result can be extended to other spaces.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares no conflicts of interest.