On Fixed Point Findings for Diverse Contractions in Dislocated-Multiplicative Metric Spaces

Abstract

In our present research study, we present the idea of dislocated-multiplicative metric space (abbrev. -multiplicative metric space) that is generalization of -multiplicative metric space and dislocated-multiplicative metric space. Furthermore, we prove some of the fixed point theorems in -multiplicative metric spaces. Also, we get common fixed point findings for fuzzy mappings in these spaces. Our findings are improved and more generalized form of several findings (see, e.g., [5, 6]).

1. Introduction

In 2008, the idea of multiplicative calculus was defined by Bashirov et al. [1] and then the conception of multiplicative metric spaces (multiplicative distance) was introduced by Çevikel and Özava̧sar [2]. Czerwik [3] presented concepts of -metric space that is the popularization of metric space. Dosenovic et al. in section Future Work in [4] presented the idea of -multiplicative metric spaces. After that, Ali et al. in [5] studied fixed point theorems for single-valued and multivalued mappings on -multiplicative metric spaces.

Furthermore, several authors obtained some fixed point findings for mappings satisfying different contractive conditions (see, e.g., [6–8]). The idea of fuzzy mappings was initially studied by Weiss [9] and Butnariu [10]. Then, the concept of fuzzy mappings was studied by Heilpern [11]. Many of the fixed point theorems for fuzzy contraction mappings in the metric linear space were proved (e.g., [12–15]), which are the fuzzy extension for the Banach contraction principle. The concept of -multiplicative metric spaces, as one of the useful generalizations of multiplicative metric spaces, was first used by Dosenovic et al. in [4],and Ali et al. in [5] study fixed point theorems for single-valued and multivalued mappings on -multiplicative metric spaces.

Our findings in -multiplicative metric space, -multiplicative metric space, and multiplicative metric space can be obtained as corollaries of our findings.

In this part, we list some of the concepts which we will use in our major findings.

The definition of -multiplicative metric space is given as follows.

Definition 1 (see [4, 5]). Suppose that is a nonempty set and is a given real number. A function is considered as a -multiplicative metric if it satisfies the following conditions: ,(i)(ii) iff (iii)(iv)

Example 1 (see [5]). Let . Define a function , , where is any fixed real number. Then, is a -multiplicative metric with .

In [11, 16], an element in any fuzzy set has a degree of belonging, a membership function may be used in order to introduce the value of degree of belonging for any element to a set, and the value of degree of belonging takes real values on the whole closed interval . The membership function is

Suppose is a metric linear space. In , a fuzzy set is a function . Thus, it is an element of , where . If is a fuzzy set and , then the function value is considered as the grade of membership of in .

denotes to the collection of all fuzzy sets in . The -level set of is defined bywhenever is the closure of set (nonfuzzy) .

Definition 2 (see [17]). A fuzzy set in is an approximate quantity if its -level set is a nonempty compact subset (nonfuzzy) of for each .
The set of all an approximate quantities denoted by is a subcollection of .

Ozavsar and Cevikel [2] prove that every multiplicative contraction in a complete multiplicative metric space has a unique fixed point.

Definition 3 (see [2]). Assume that is a multiplicative metric space. A mapping is called multiplicative contraction if

Theorem 1 (see [2]). Assume that is a multiplicative metric space. A mapping is called multiplicative contraction. Then, has a unique fixed point.

Theorem 2 (see [4]). Suppose that is a complete multiplicative metric space and a continuous function , such that

Then, has a unique fixed point.

In 2015, Kang et al. [18] introduced the concept of compatible mappings as follows.

Definition 4. Let be a multiplicative metric space. The mappings ; then, is called compatible if and only if for some in implying .

Many authors studied many fixed point theorems for compatible mappings in multiplicative metric space and employed it to prove a common fixed point theorem (see [4, 18]).

In this paper, we introduce the new notion of -multiplicative metric space. We prove fixed point theorems for single mappings and a common fixed point for fuzzy mappings in -multiplicative metric space. As illustrative application, we state some of our theorems on Cartesian product in these spaces.

2. Fixed Point Theorems in -Multiplicative Metric Spaces

In this part, we present the conception of -multiplicative metric space. Also, we introduce some of the fixed point theories and show our main findings with the help of some examples in this space.

Definition 5. Suppose that and is a given real number. A function is called -multiplicative metric space if it satisfies the following conditions: ,(i)(ii) implies (iii)(iv)

Example 2. Let . Define asThen, is -multiplicative metric space with .

Example 3. Let . Define asThen, is -multiplicative metric space with .

Definition 6. Let be an -multiplicative metric space. We say that converges to if and only if

Definition 7. Let be an -multiplicative metric space. We say that is -multiplicative Cauchy if and only if

Definition 8. An -multiplicative metric space is complete if every multiplicative Cauchy sequence in is convergent.

Now, we state the following lemma without proof.

Lemma 1. Suppose that is -multiplicative metric space. Then, any subsequence of convergent sequence in is convergent.

The following theorem is the generalization of Theorem 3.2 in [2].

Theorem 3. Suppose that is a complete -multiplicative metric space and a continuous function ; satisfieswhere and . Then, has a unique fixed point.

Proof. Let be an arbitrary point in ; then by hypothesis, there exists such that . In a similar way, one can obtain a sequence such thatThen,As in (11) and , then is a multiplicative Cauchy sequence.
Since is complete, then is convergent such that . However,Therefore, is a fixed point of . Suppose that , , and .This is a contradiction with assumption; then, . Then, has a unique fixed point.

Example 4. Suppose that , is -multiplicative metric space and , where with . Define such that :Then, (1) holds such that . Therefore, has a unique fixed point .

Corollary 1. Suppose that is a complete multiplicative metric space and a continuous function satisfieswhere and . Then, has a unique fixed point.

Corollary 2. Suppose that is a complete -multiplicative metric space and a continuous function satisfieswhere and . Then, has a unique fixed point.

The following theorem is the generalization of Theorem 2.32 in [4].

Theorem 4. Suppose that is a complete -multiplicative metric space and a continuous function , , such that

Then, has a unique fixed point.

Proof. Let be an arbitrary point in ; then by hypothesis, there exists such that .
In a similar way, one can obtain such that .Otherwise, we have a contradiction, that is, .Continuing in this way, we produce a sequence in such that andfor each . It follows by induction that . However,As , , , and is complete, then is a multiplicative Cauchy sequence in and there exists such that .
From Lemma 1, .
Then, is fixed point of , and . Suppose that has another fixed point such that and .This is a contradiction with assumption; then, .
Then, has a unique fixed point.

Corollary 3. Suppose that is a complete multiplicative metric space and a continuous function satisfieswhere and . Then, has a unique fixed point.

Corollary 4. Suppose that is a complete -multiplicative metric space and a continuous function satisfieswhere and . Then, has a unique fixed point.

3. Common Fixed Point Theorems for Fuzzy Mappings in -Multiplicative Metric Spaces

Definition 9. Suppose that is an arbitrary set and is -multiplicative-metric space. A mapping is stated according to be a fuzzy mapping iff is a function from the set into , i.e., , for each .

Definition 10. Suppose that is a -multiplicative metric space. The functions and . A hybrid pair is called -compatible iff for some implies .

Definition 11. Suppose that is a -multiplicative metric space. Two maps and are said to be occasionally coincidentally idempotent if for some , where refers to the set of all coincidence points of two mappings and , i.e.,

Now, we state the following lemma without proof.

Lemma 2. Suppose that is a -multiplicative metric space and . Then,

Corollary 5. Suppose that is a -multiplicative metric space and and if and only if .

Lemma 3. Suppose that is a -multiplicative metric space, is a fuzzy map, and . Then, there exists such that .

Theorem 5. Suppose that is a complete -multiplicative metric space and two continuous mappings satisfywhere and and two fuzzy mappings , such that(i), (ii)The pairs and are -compatible and occasionally idempotent mappingsThen, there exists such that and .

Proof. Suppose is an arbitrary point in . Then, there is there exists , whereFrom (11) in Theorem 3,As , that implies is a multiplicative Cauchy sequence.
Since is complete, then is convergent such thatNext, we prove that .From Lemma 1, is a convergent sequence, i.e., as ,i.e.,Then, we have and Corollary 5 illustrates that .
Since , there exists .
Similar to the previous steps, we can prove that .
As two pairs and are -compatible,Moreover,Now, we show that and . Since and are convergent sequences, from Lemma 1,Then, and .

Example 5. Suppose that , is a -multiplicative metric space defined by , and . Define maps as , . Also, define two fuzzy mappings asandNow, for , and for , ; i.e., and are -compatible. Finally, and ; i.e., and are occasionally coincidentally idempotent. Then, is a common fixed point.

Example 6. Suppose that , is a -multiplicative metric space defined by . Define maps as , . Also, define two fuzzy mappings asandNow, for ,and for ,Hence, and are -compatible. Finally,andTherefore, and are occasionally coincidentally idempotent. Furthermore,Then, is a common fixed point.

We concluded the following corollary when we set in Theorem 5.

Corollary 6. Let , and suppose continuous mapping satisfies , where , and fuzzy mapping such that(i)(ii)The pair is -compatible and occasionally idempotent mappingsThen, there exists such that .

Theorem 6. Let and suppose two continuous mappings , satisfywhere , , and such that ,(i) and (ii)The pairs and are -compatible and occasionally idempotent mappingsThen, there exists such that and .

Proof. The proof of this theorem is completed, when putting and in Theorem 5.

Remark 1. If and , then Theorem 6 implies Theorem 5.

Theorem 7. Suppose that is a complete -multiplicative metric space and two continuous mappings satisfywhere , , and two fuzzy mappings , such that(i), (ii)The pairs and are -compatible and occasionally idempotent mappingsThen, there exists : and .

Proof. Let be an arbitrary point in ; then, there exists and from Lemma 3, there exists such that, and from Theorem 4,As , that implies is a multiplicative Cauchy sequence. Since is complete, then is convergent such thatAs is a Cauchy sequence in and is joint orbitally complete, then there exists such thatWe prove that andAs , then . However, is a convergent sequence; i.e.,Then, we have and Corollary 5 illustrates that .
Since , there exists .
Similar to the previous steps, we can prove that .
As two pairs and are -compatible,and thereforeNow, we show that and . Since and are convergent sequences, thenThen, have a common fixed point.

Theorem 8. Suppose that is a complete -multiplicative metric space, , and two continuous mappings satisfywhere , , and , such that(i), , , , and (ii)The pairs and are -compatible and occasionally idempotent mappingsThen, there exist such that and .

4. Applications

In this section, we give some applications on our main results. We state some of our theorems on Cartesian product without proof.

Theorem 9. Suppose that is a complete -multiplicative metric space. The map , satisfieswhere and . Then, has a unique fixed point.

The next example illustrates the previous theory.

Example 7. Suppose that and . Define as. Then, is a complete -multiplicative metric space with .
Let be a function defined by . Then, condition (3) holds. However,with . It is obvious that is a unique fixed point of a map .

Theorem 10. Suppose that is a complete -multiplicative metric space and , , such that

Then, has a unique fixed point.

Theorem 12. Suppose that is a complete -multiplicative metric space and two continuous mappings satisfywhere , , and two fuzzy mappings , such that(i) and (ii)The pairs and are -compatible and occasionally idempotent mappingsThen have a common fixed point.

Theorem 13. Suppose that is a complete -multiplicative metric space, , and two mappings satisfywhere , , and such that(i), , , , and (ii)The pairs and are -compatible and occasionally idempotent mappingsThen, there exists such that and .

Theorem 14. Suppose that is a complete -multiplicative metric space and two mappings , , satisfywhere , , , such that(i), , , , and (ii)The pairs and are -compatible and occasionally idempotent mappingsThen, there exists such that and .

5. Conclusion

In this paper, we introduced the concept of -multiplicative metric spaces. We studied some of the fixed point theorems in these spaces. Also, we obtain common fixed point theorems for fuzzy mappings in complete -multiplicative metric spaces. Finally, we get some of applications on our main findings. We hope that our presented idea herein will be a source of motivation for other researchers to extend and improve these findings for their applications.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.