Soft Fixed Point Theorems for the Soft Comparable Contractions

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


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 func-tions, 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][7][8][9][10][11][12].
As usual, ℝ denotes real numbers and ℝ + ≔ ½0,∞Þ. Furthermore, the letters ℤ, ℕ denote integers and natural numbers, respectively. The symbol BðℝÞ denotes the collection of all nonempty bounded subsets of ℝ.
We shall denote an initial universe Ω. We set P as a set of parameters. As usual, 2 Ω denotes the collection of all subsets of Ω. For a nonempty subset S of P , we consider a set-valued mapping T : S → 2 Ω for all τ ∉ A with TðτÞ = ϕ. We define a pair ðT, AÞ on Ω as Here, ðT, SÞ is called a soft set [2]. The symbol SðΩÞ represents the collection of all soft sets on Ω.
A mapping T : P → BðℝÞ is called a soft real set [13]. The symbol ℝ + ðP Þ is used to denote the set of all nonnegative soft real numbers. If ðT, P Þ is a singleton soft set, then it is called a soft real number. Regarding the corresponding soft set, soft real numbers will be denoted asγ,η,ξ, etc. In particular, 0 and1 are the soft real numbers where 0ðτÞ = 0,1ðτÞ = 1 for all τ ∈ P .
A soft set ðT, P Þ on Ω is called a soft point [4,14], denoted by e x τ , if there is a unique τ ∈ P such that TðτÞ = fxg for some τ ∈ P and TðωÞ = ϕ for all ω ∈ P \ fτg.
Definition 2 (see). LetX = ðT, P Þ be an absolute soft set, and let SP ðXÞ be the collection of all soft points ofX. A mapping d : SP ðXÞ × SP ðXÞ → ℝ + ðP Þ is called a soft metric onX if d satisfies the following conditions for all f In a soft metric space M, a sequence of soft points fg x λ,n g n is called convergent in M if there is a soft point y ν~∈~X such that Furthermore, a sequence fg x λ,n g n is said to be a Cauchy in Moreover, if each Cauchy sequence inX converges to some point ofX, then M is called complete soft metric space.

Soft Fixed Points for the Soft Inf-Comparable Contraction
In this section, we first introduce the notion of soft inf -comparable mapping ψ : ℝ + ðP Þ → ℝ + ðP Þ.
Proof. Let τ~>~ 0 be fixed. If ψ n 0 ðτÞ = 0 for some n 0 ∈ ℕ, then we have which implies that Thus, we conclude that If ψ n ðτ~Þ>~ 0 for each n ∈ ℕ, then we take e σ n = ψ n ðτÞ, and for all n ∈ ℕ. By the condition (ψ 1 ) of the soft inf -comparable mapping ψ, we have that for all n ∈ ℕ, Keeping (ψ 2 ) in mind and considering that the soft sequencefσ n∈ℕ gis bounded from below and also that the soft sequence is strictly decreasing, one can find an ν~≥~ 0 such that lim n→∞ e σ n =ν: We assert thatν = 0. If not, suppose that ν~>~ 0, then we find a contradiction. So we obtain that lim n→∞ ψ n ðτÞ = 0: We introduce the notion of soft inf -comparable contraction, as follows: Definition 5. Let M be a soft metric space and let ψ : ℝ + ðP Þ → ℝ + ðP Þ be a soft inf -comparable mapping. A mapping ðf , φÞ: M → M is called a soft inf -comparable contraction if for each soft points e x p , e y τ ∈ SP ðXÞ, Example 6. Set R = ð e ℝ,d, P Þ where the soft metric is expressed as with P = ½0,∞Þ, φðtÞ = ð2/3Þt for t ∈ ½0,∞Þ.
We say that a soft point e Proof. Let f x 0 τ 0 ∈ SP ðXÞ be given. For each n ∈ ℕ ∪ f0g, we put Then, we have for each n ∈ ℕ ∪ f0g Since ψ : ℝ + ðP Þ → ℝ + ðP Þ is a soft inf -comparable mapping, we can conclude that for each n ∈ ℕ ∪ f0g, 3 Journal of Function Spaces By induction, we obtain that By Lemma 4, we obtained that In what follows, we check whether the sequence f f x n τ n g is Cauchy: for eachε, there is n 0 ∈ ℕ such that if n, k ≥ n 0 , theñ Suppose, on the contrary, that the statement ð * Þ is false. Then, there exists ε~>~ 0 such that, for any r ∈ ℕ, there are n r , k r ∈ ℕ with n r > k r ≥ r satisfying that (1) n r is even and k r is odd Letting r → ∞, we obtain that On the other hand, Letting r → ∞, we obtain that By the above arguments, we obtain that Taking lim r→∞ inf , we get ε~<~ε. This implies a contradiction. So the sequence f f x n τ n g is Cauchy.

Observation on the Soft Comparable Meir-Keeler Contractions
We start this section by recalling the Meir-Keeler contraction in the standard setting.
We establish the following fixed point results for the soft comparable Meir-Keeler contraction. Proof. Let f x 0 τ 0 ∈ SP ðXÞ be given. For each n ∈ ℕ ∪ f0g, we put So, for each n ∈ ℕ ∪ f0g we havẽ