Soft Fixed Point Theorems for the Soft Comparable Contractions

Abstract

In this article, we introduce the notions of a soft -comparable contraction and soft comparable Meir-Keeler contraction in a soft metric space. Furthermore, we prove two soft fixed point theorems which assure the existence of soft fixed points for these two types of comparable contractions. The obtained results not only generalize but also unify many recent fixed point results in the literature.

1. Introduction and Preliminaries

It is the main feature of mathematical study to produce different methods and tools to perceive the behavior of systems that we have difficulty understanding with known methods. In particular, it may be necessary to deal with systems that contain uncertainties and to use inaccurate data in different situations. With this motivation, one of the mathematical tools used to deal with the necessities of systems established with uncertainty and to analyze the models created by the uncertainties and uncertainties already existing in the data is the Fuzzy Set Theory. Fuzzy sets were introduced by Zadeh [1] for dealing with the uncertainties on its own limits. Another mathematical tool to deal with the uncertainties is the soft set that was introduced by Molodtsov [2]. In this paper, we shall focus on the soft set theory. The topology based on the soft sets was defined by Cagman et al. [3]. They also considered the basic topological notions over soft sets. On the other hand, a soft real set and soft real number were proposed successfully by Das and Samanta [4]. Furthermore, the same authors in considered the notions of a soft metric and its topology, properly. After then, Abbas et al. [5] proved a fixed point theorem by introducing the notion of soft contraction mapping over the soft metric space. Application potential of the soft sets in various distinct research topics is very rich and wide, for example, the smoothness of functions, game theory, operation research, probability theory, and measurement theory. For more details on soft sets and application, we can refer to, e.g., [3, 4, 6–12].

As usual, denotes real numbers and . Furthermore, the letters denote integers and natural numbers, respectively. The symbol denotes the collection of all nonempty bounded subsets of .

We shall denote an initial universe . We set as a set of parameters. As usual, denotes the collection of all subsets of . For a nonempty subset of , we consider a set-valued mapping for all with . We define a pair on as

Here, is called a soft set [2]. The symbol represents the collection of all soft sets on .

A soft set on is called null soft [11] (respectively, absolute soft set [11]) represented by, respectively, if(respectively,) for all. We presume that and are two soft sets on . We define the intersection [11] of the mentioned two sets above as a soft set , denoted by , where , and for each , . As expected, we define the union of and [11] as a soft set , denoted by where and for each ,

We use the notation to indicate the complement [11] of soft set on where is a mapping given by for all .

A mapping is called a soft real set [13]. The symbol is used to denote the set of all nonnegative soft real numbers. If is a singleton soft set, then it is called a soft real number. Regarding the corresponding soft set, soft real numbers will be denoted as , , , etc. In particular, and are the soft real numbers where , for all .

For two soft real numbers, for all , we have the following inequalities [13]: (1) if (2) if (3) if (4) if

Definition 1. (1)The mapping is called soft increasing, if(2)The mapping is called soft continuous at , if for every , there exists such that implies

Moreover, is called soft continuous at every point of , then we call as a continuous mapping.

A soft set on is called a soft point [4, 14], denoted by , if there is a unique such that for some and for all .

Definition 2 (see). Let be an absolute soft set, and let be the collection of all soft points of . A mapping is called a soft metric on if satisfies the following conditions for all :
,
if and only if ,
,
.

The triple is called a soft metric space, in short, s.m.s.

For the sake of simplicity, we set .

Suppose is a s.m.s. and is a nonnegative soft real number. A soft open ball with the center and radius is defined by . Analogously, a soft closed ball with center and radius is . We set that a soft setis soft open inwith respect to if and only if all soft points of are interior points of .

In a soft metric space , a sequence of soft points is called convergent in if there is a soft point such that

Furthermore, a sequence is said to be a Cauchy in if

Moreover, if each Cauchy sequence in converges to some point of , then is called complete soft metric space.

Let be another soft metric space. A soft mapping is soft continuous at a point , if for each of , there exists such that

In other words, for every , there exists such that implies that . Moreover, if is soft continuous for each point of , then it is called soft continuous mapping.

2. Soft Fixed Points for the Soft Inf-Comparable Contraction

In this section, we first introduce the notion of soft -comparable mapping .

Definition 3 (see [15]). Let be a parameter set and . We call a soft -comparable mapping if it satisfies the following two axioms:
for all and ,
for all .

Lemma 4. Let be a soft -comparable mapping. Then, for all , where denotes the -th iteration of .

Proof. Let be fixed. If for some , then we have which implies that Thus, we conclude that If for each , then we take , and for all . By the condition () of the soft -comparable mapping , we have that for all , Keeping () in mind and considering that the soft sequenceis bounded from below and also that the soft sequence is strictly decreasing, one can find an such that We assert that . If not, suppose that , then we find a contradiction. So we obtain that

We introduce the notion of soft -comparable contraction, as follows:

Definition 5. Let be a soft metric space and let be a soft -comparable mapping. A mapping is called a soft -comparable contraction if for each soft points ,

Example 6. Set where the soft metric is expressed as with , for .
Let be denoted by and let . Consequently, we find As a result, forms a soft -comparable contraction on .

We say that a soft point is a soft fixed point of a self-soft-mapping if .

Theorem 7. Let be a soft -comparable mapping. Let be a soft -comparable contraction on a complete soft metric space . Then, a soft mapping possesses a soft fixed point.

Proof. Let be given. For each , we put Then, we have for each Since is a soft -comparable mapping, we can conclude that for each , By induction, we obtain that By Lemma 4, we obtained that In what follows, we check whether the sequence is Cauchy: for each , there is such that if , then Suppose, on the contrary, that the statement is false. Then, there exists such that, for any , there are with satisfying that (1) is even and is odd(2)(3) is the smallest even number such that condition holdsBy and , we conclude that Letting , we obtain that On the other hand, Letting , we obtain that By the above arguments, we obtain that Taking , we get . This implies a contradiction. So the sequence is Cauchy.
Since is complete, there exists such that that is, Notice also that Taking , we get that and this is a contradiction unless . Thus, completes the proof.

Example 8. Consider Example 6. All hypotheses of Theorem 7 are fulfilled. Thus, we can conclude that is a fixed soft point of the soft -comparable Meir-Keeler contraction .

3. Observation on the Soft Comparable Meir-Keeler Contractions

We start this section by recalling the Meir-Keeler contraction in the standard setting.

Definition 9. (see). A self-mapping on a metric space is called a Meir-Keeler contraction if the following is fulfilled: for any , there is such that

The mapping is said to be soft comparable, if the following two axioms are fulfilled:

() is a soft increasing and soft continuous function in each coordinate

() for , , and if and only if

Now, we introduce the notion of soft comparable Meir-Keeler contraction.

Definition 10. Let be soft comparable. A self-soft-mapping on a soft metric space is called a soft comparable Meir-Keeler contraction if for each soft real number , there is such that for each soft points .

Example 11. Set where the soft metric is expressed as with , for .
Let be denoted by where Let . Then, So we can conclude that

Consequently, a soft mapping forms a soft comparable Meir-Keeler contraction on .

We establish the following fixed point results for the soft comparable Meir-Keeler contraction.

Theorem 12. Let be a complete soft metric space, and let be a soft comparable. Let be a soft comparable Meir-Keeler contraction on . Then, possesses a soft fixed point.

Proof. Let be given. For each , we put So, for each we have If for some , then by the above inequality and the conditions of the function , we have which implies a contradiction. Hence, for each , we find Thus, the sequence is decreasing and converges to a soft real number, say . In other words, , as .
Notice that . We claim that . Suppose, on the contrary, that . Since is a soft comparable Meir-Keeler contraction, corresponding to , there exists and such that This is a contradiction since . Thus, we obtain that , as .
As a next step, we check whether the sequence is Cauchy in . Suppose, on the contrary, it is not. Thus, there exists a soft real number such that for any , there are with satisfying Further, corresponding to , we can choose in such a way that it is the smallest integer with and . Therefore, So, we derive that for all . As , the inequality above yields that On the other hand, we have Letting in the above inequality, we get Since is a soft comparable Meir-Keeler contraction, we have Moreover, since Taking in the above inequalities, we get that and this is a contradiction. Thus, the sequence is Cauchy.
Keeping the completeness of in mind, one can find such that that is, And, we also have Taking in the inequality above, a contradiction unless . Thus, which completes the proof.

Example 13. Consider Example 11. One can easily check all hypotheses of Theorem 12. Consequently, we conclude that is a fixed soft point of the soft comparable Meir-Keeler contraction .

We next introduce the notion of soft generalized Meir-Keeler contraction, as follows:

Definition 14. Let be a soft metric space. A mapping is called a soft generalized Meir-Keeler contraction if for any soft real number , there exists such that for each soft point ,

It is clear that the soft generalized Meir-Keeler contraction is a comparable soft Meir-Keeler contraction; we can easily conclude the following corollary.

Corollary 15. A soft generalized Meir-Keeler contraction onwhich is a complete soft metric spacepossesses a fixed soft point.

Data Availability

No data is used!

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.

Acknowledgments

This research was supported by the Ministry of Science and Technology of the Republic of China.