View of Fixed point Results on a Complete Soft Usual Metric Space

Fixed point Results on a Complete Soft Usual Metric Space

Ramakant Bhardwaja,b

a

Department of Mathematics, APS University Rewa(M.P.)

b_{Department of Mathematics, AMITY University Kolkata W.B.}

Email: rkbhardwaj100@gmail.com

Abstract

The present work is related to the theory of fixed point in complete soft usual metric space, which are extension of well-known results on complete metric space as well as complete soft metric space. The obtained results can be used in decision making as well as based on uncertainty problems. Soft weak contractive mapping and soft generalized weak contractive mapping are used to obtained the results.

Keywords: - Soft usual metric space, soft weak contractive mapping, soft fixed point.

Mathematics Subject Classification: - 47H10, 54H25.

1. Introduction and preliminaries

Most of the real-world problems cannot be solved by the theory of crisp sets. Some of them can be solved by theory of probability or by theory of fuzzy sets.In 1999, Molodtsov [4] initiated a novel concept of soft set theory, which is a new mathematical tool for dealing with uncertainties. Soft set is a parameterized general mathematical tool which deals with a collection of approximate descriptions of objects.The detail about soft sets, soft fixed point, soft complete metric can be seen in [1-3, 5-8]

Definition 1.1: Let X be an initial universe set and E be a set of parameters. A pair (F, E) is called a soft set over X if and only if X is a mapping from E into the set of all subsets of the set X, i. e. F: E → P(X), where P(X) is the power set of X.

Definition 1.2: The intersection of two soft sets (F, A) and (G, B) over X is the soft set (H, C), where C = A ∩ B and ∀ε ∈ C, H(ε) = F(ε) ∩ G(ε).This is denoted by (F, A) ∩ (G, B) = (H, C).

Definition 1.3: The union of two soft sets (F, A) and (G, B) over X is the soft set, where C = A ∪ B and ∀ε ∈ C,

H(ε) = {G(ε), if ε ∈ B − AF(ε), if ε ∈ A − B F(ε) ∪ G(ε), ε ∈ A ∩ B

This relationship is denoted by (F, A) ∪ (G, B) = (H, C).

Definition 1.4: The soft set (F, A) over X is said to be a null soft set denoted by Φ if for all ε ∈ A,F(ε) = ϕ (null set).

Definition 1.5: A soft set (F, A) over X is said to be an absolute soft set, if for all ε ∈ A, F(ε) = X.

Definition 1.6: The complement of a soft set (F, A) is denoted by (F, A)c and is defined by (F, A)c _{= (F}c_{, A) where F}c_{: A → P(X) is mapping given by F}c_{(α) = X − F(α), ∀α ∈ A.}

Definition 1.7: Let ℜ be the set of real numbers and B(ℜ) be the collection of all nonempty bounded subsets of ℜ and E taken as a set of parameters. Then a mapping F: E → B(ℜ) is called a soft real set. It is denoted by (F, E). If specifically,(F, E) is a singleton soft set, then identifying (F, E) with the corresponding soft element, it will be called a soft real number and denoted r̃, s̃, t̃ etc.

Definition 1.8: For two soft real numbers

(i) r̃ ≤ s̃, if r̃(e) ≤ s̃(e), for all e ∈ E. (ii) r̃ ≥ s̃, if r̃(e) ≥ s̃(e), for all e ∈ E. (iii) r̃ < s̃, if r̃(e) < s̃(e), for all e ∈ E. (iv) r̃ > s̃, if r̃(e) > s̃(e), for all e ∈ E.

Definition 1.9: A soft set over X is said to be a soft point if there is exactly one e ∈ E, such that P(e) = {x} for some x ∈ X and P(e′_{) = ϕ, ∀e}′_{∈ E\{e}. It will be denoted by x̃}

e.

Definition 1.10: Two soft points x̃_e, ỹ_e are said to be equal if e = e′ and P(e) = P(e′)i.e.x = y. Thus x̃_e ≠ ỹ_e ⟺ x ≠ y or e ≠ e′.

Definition 1.11: A mapping |. |: SP(X̃) × SP(X̃) → ℝ(E)∗, is said to be a usual soft metric on the soft set X̃If |. | satisfies the following conditions:

(M1) |̃(x̃_e1− ỹ_e2)| ≥̃ 0 for all x̃_e1,ỹ_e2 ∈̃ X̃, (M2) |(x̃_e1− ỹ_e2)| = 0 if and only if x̃_e1 = ỹ_e2, (M3) |(x̃_e1−ỹ_e2) = |̃(ỹ_e2 − x̃_e1)| for all x̃_e1, ỹ_e2 ∈̃ X̃,

(M4) |̃(x̃_e1− z̃_e3)| ≤ ̃|̃(x̃_e1 − ỹ_e2)| + |̃(ỹ_e2− z̃_e3)| for all x̃_e1, ỹ_e2,z̃_e3 ∈̃ X̃.

The soft set X̃ with a soft mod (|.| ) on X̃ is called a soft usual metric space and denoted by (X̃, | . |, E).

Definition 1.13 (Cauchy Sequence in usual soft metric): A sequence {x̃_λ,n}_nof soft points in (X̃, | |̃ ,E) is considered as a Cauchy sequence in X̃if corresponding to every ε̃ ≥̃ 0,∃m ∈ N such that |(x̃_λ,i− x̃_λ,j)| ≤̃ ε̃, ∀ i,j ≥ m, i.e.|(x̃_λ,i− x̃_λ,j)| → 0, as i, j → ∞.

Definition 1.14 (Complete soft Usual Metric Space): A soft usual metric space (X̃, | |,E) is called softcomplete usual , if every Cauchy Sequence in X̃ converges to some point of X̃.

2. Main Results: Some soft fixed-point theorems are established for Usual soft metric space

Theorem 2.1: Let (f, φ) ∶ (X̃, | |,E) → (X̃, | |,E) , where(X̃, |, |, E)is ausual soft metric space. if for all x̃_λ, ỹ_μ ∈ SP(X),

|̃ ((f, φ)(x̃λ) − (f, φ)(ỹμ)) | ≤1_{2 [(}x̃λ − (f, φ)(ỹμ)) | + | (ỹμ− (f, φ)(x̃λ)) |]

Where ψ ∶ [0, ∞)2→ [0, ∞) is a continuous mapping such that ψ(x̃_λ, ỹ_μ) = 0 if and only if x̃λ = ỹμ = 0.Then (f, φ) has a unique soft fixed point.

Proof: Let x̃_λ0 be any soft point in SP(X). Set x̃1_λ1 = (f, φ)(x̃_λ0) = (f(x̃_λ0)) φ(λ) − − − − − − − − − − − − − − x̃_λn+1_{n+1= (f, φ)(x̃} λ_n n _{) = (f}n+1_(x̃ λ 0₎₎ φn+1(λ), − −

Ifx̃_λn_n = (f, φ)(x̃_λn−1_n−1) then x̃_λn_n is a fixed point of (f, φ). So we assumex̃_λ

n n _{≠ x̃}

λn+1_n+1

Putting x = x̃_λn−1_n−1 and y = x̃_λn_n in (2.1.1) we have for all n = 0,1,2, … …. |̃(x̃_λn_n − x̃_λn+1_n+1)| = |̃ ((f, φ)x̃λn−1_n−1− (f, φ)x̃λn_n) | ≤ 1₂[|(x̃_λn−1_n−1− (f, φ)x̃n_λn)| + |(x̃_λn_n− (f, φ)x̃_λn−1_n−1)|] −ψ(|(x̃_λn−1_n−1− (f, φ)x̃_λn_n)|, |(x̃_λn_n − (f, φ)x̃_λn−1_n−1)|) ≤ 1₂[|(x̃_λn−1_n−1− x̃_λn+1_n+1)| + |(x̃_λn_n − x̃_λn_n)|] − ψ(|(x̃_λn−1_n−1− x̃_λn+1_n+1)|, |(x̃_λn_n − x̃_λn_n)|) ≤ 1₂[|(x̃_λn−1_n−1− x̃_λn+1_n+1)| + 0] − ψ(|(x̃_λn−1_n−1− x̃_λn+1_n+1)|,0) ψ(|̃(x̃n−1λ_n−1− x̃λn+1_n+1)|,0) = 0, and|̃(x̃λn_n − x̃n+1λ_n+1)| ≤ 1₂[|(x̃λn−1_n−1− x̃λn+1_n+1)|] |(x̃_λn_n − x̃_λn+1_n+1)| ≤ S |̃(x̃_λn−1_n−1, x̃_λn_n)| |̃(x̃_λn_{n − x̃} λ_n+1 n+1_{)| ≤ S}n_|̃(x̃ λ0₀, −x̃λ1₁)|Where S = _1−1/2 1/2 That is |̃(x̃_λn_n − x̃_λn+1_n+1)| ≤ Sn_|̃(x̃ λ₀ 0 _{− x̃} λ₁ 1 _)| For any n > 𝑚, 𝑚, 𝑛 ∈ 𝑁 |̃(x̃_λn_{n− x̃} λ_m m _{)| ≤} Sm 1 − S |(x̃λ0₀ − x̃λ1₁)|

Since 0 ≤ S < 1 and as n → ∞ then

|̃(x̃_λn_{n, x̃}

λm_m)| = 0.

That is {x̃_λn_n} is a soft Cauchy sequence, By the soft completeness of X,̃ there is x̃_λ∗ ∈ X̃ such that x̃_λn_n → x̃_λ∗, n → ∞.Let x̃_λn_n → x̃_λ∗ as n → ∞ Then |(x̃_λ∗ _{− (f, φ)x̃} λ ∗_{)| ≤ (x̃} λ ∗ _{− x̃} λ_n+1 n+1_{) + |̃(x̃} λ_n+1 n+1 _{− (f, φ)x̃} λ∗)| ≤ |̃(x̃_λ∗_{− x̃} λ_n+1 n+1_{)| + |̃ ((f, φ)x̃} λ_n n _{− (f, φ)x̃} λ ∗_{) |} ≤ |̃(x̃_λ∗ _{− x̃} λn+1_n+1)| +1₂[|(x̃λn_n − (f, φ)x̃λ∗)| + |(x̃λ∗ − (f, φ)x̃λn_n)|]

−ψ(|(x̃_λn_{n − (f, φ)x̃} λ ∗_{)|, |(x̃} λ ∗ _{− (f, φ)x̃} λn_n)|) ⇒|̃(x̃_λ∗ _{− (f, φ)x̃} λ∗)| ≤1₂|̃(x̃λ∗ − (f, φ)x̃λ∗)|

This contradicts the assumption. So x̃_λ∗ = (f, φ)x̃_λ∗. x̃_λ∗ is a soft fixed point of (f, φ) for usual soft metric space

Uniqueness: If ỹ_μ∗ is anothersoft fixed point of (f, φ) in X̃ such that x̃_λ∗ ≠ ỹ_μ∗, |̃(x̃_λ∗ _{− ỹ}

μ∗) = |̃ ((f, φ)x̃λ∗ − (f, φ)ỹμ∗) |

≤ |̃ ((f, φ)x̃λ∗ − (f, φ)x̃λn_n) | + |̃ ((f, φ)x̃λn_n− (f, φ)ỹμ∗) |

≤ |̃ ((f, φ)x̃∗λ − (f, φ)x̃λn_n) +1₂[|(x̃nλ_n − (f, φ)ỹμ∗)| + |(ỹμ∗ − (f, φ)x̃λn_n)|]

−ψ(|(x̃_λn_n − (f, φ)ỹμ∗)|, |(ỹμ∗ − (f, φ)x̃λn_n)|)

n → ∞ ,|̃(x̃_λ∗ − ỹμ∗)| ≤ 0.This contradict the assumption sox̃λ∗ = ỹμ∗ that is, x̃λ∗ is unique soft fixed

point of (f, φ) in usual soft metric space.

Theorem 2.2: Let (f, φ) ∶ (X̃, |. |̃,E) → (X̃,|. |̃,E) , (where(X̃,| .|,E)is a complete usual soft metric space) be a soft generalized weak contraction that is the following condition is true if for all x̃_λ, ỹ_μ∈ SP(X),

|((f, φ)(x̃λ) − (f, φ)(ỹμ))| ≤ ρ [max {

|(x̃λ, (f, φ)(x̃λ)) − d (ỹμ, (f, φ)(ỹμ)) |,

| (x̃λ, (f, φ)(ỹμ)) | − | (ỹμ, (f, φ)(x̃λ)) − (x̃λ, ỹμ)|

}] − ψ(|(x̃λ− (f, φ)(x̃λ))|, | (ỹμ− (f, φ)(ỹμ)) |, | (x̃λ− (f, φ)(ỹμ)) |,

| (ỹμ− (f, φ)(x̃λ)) |, |(x̃λ, −ỹμ|)

) ( 2.2.1)

Where 𝞺∈ [0, 1), ψ ∶ [0, ∞)5 → [0, ∞) is a continuous mapping :

ψ(x̃1, x̃2, x̃3, x̃4, x̃5) = 0 if and only if one of x̃1, x̃2, x̃3, x̃4, x̃5 is equal to 0.Then (f, φ) has a

uniquesoft fixed point.

Proof: Let x̃_λ0be any soft point in usual soft metric space Set x̃1_λ1 = (f, φ)(x̃_λ0) = (f(x̃_λ0)) φ(λ) − − − − − − − − − − − − − − x̃_λn+1_n+1= (f, φ)(x̃_λn_n) = (fn+1_(x̃ λ 0₎₎ φn+1(λ), − −

If x̃_λn_n = (f, φ)(x̃_λn−1_n−1) then x̃_λn_n is a fixed point of (f, φ). Taking x̃_λn_n ≠ x̃_λn+1_n+1

Putting x = x̃_λn−1_n−1 and y = x̃_λn_n, for all n = 0,1,2, … …. |̃(x̃_λn_{n − x̃}

λn+1_n+1)| = | ((f, φ)x̃λn−1_n−1− (f, φ)x̃λn_n) |

≤ ρ [max {|̃(x̃λn−1n−1− (f, φ)x̃λn−1n−1)|, |̃(x̃λnn − (f, φ)x̃λnn)|, |(x̃λn−1n−1− (f, φ)x̃λnn)|, |̃(x̃n_λn− (f, φ)x̃_λn−1_n−1)|, |̃(x̃n−1_λn−1− x̃_λn_n)| }]

− ψ(|̃(x̃λn−1n−1− (f, φ)x̃λn−1n−1)|,|(x̃λnn − (f, φ)x̃λnn)|, |̃(x̃λn−1n−1− (f, φ)x̃λnn)|, |̃(x̃n_λn− (f, φ)x̃_λn−1_n−1)|, |̃(x̃n−1_λn−1− x̃_λn_n)| ) ≤ ρ[max{|̃(x̃_λn−1_{n−1− x̃} λn_n), |̃(x̃λn_n − x̃λn+1_n+1)|, |̃(x̃n−1λ_n−1− x̃λn+1_n+1)|,0, |̃(x̃λn−1_n−1− x̃λn_n)|}] − ψ(|̃(x̃λn−1_n−1− x̃λn_n)|, |̃(x̃λn_n− x̃λn+1_n+1)|, |(x̃λn−1_n−1− x̃λn+1_n+1)|,0,|̃(x̃λn−1_n−1− x̃λn_n)) ( 2.2.2)

By the condition of the theorem of generalized week contraction ψ(|(x̃_λn−1_{n−1− x̃} λ_n n _{)|, |(x̃} λ_n n _{− x̃} λ_n+1 n+1_{)|, |(x̃} λ_n−1 n−1 _{− x̃} λn+1_n+1)|,0, |̃(x̃λn−1_n−1− x̃λn_n)|) = 0 And |(x̃_λn_{n− x̃} λn+1 n+1_{)| ≤ α[max{|(x̃} λn−1 n−1_{−, x̃} λn n _{)|, |̃(x̃} λn n _{− x̃} λn+1n+1)|, |(x̃λn−1n−1− x̃λn+1n+1)|, |̃(x̃λn−1n−1− x̃λnn)|}] (2.2.3) Case I:If we choose max{|̃(x̃_λn−1_{n−1− x̃} λn_n)|, |̃(x̃λn_n− x̃λn+1_n+1)|, |̃(x̃λn−1_n−1−,x̃n+1λ_n+1)|,0, |(x̃λn−1_n−1− x̃nλ_n)|} = |̃(x̃λn−1_n−1−x̃λn_n)| Now by(2.2.3) |(x̃_λn_n− x̃_λn+1_n+1)| ≤ ρ |(x̃_λn−1_n−1− x̃_λn_n)| Similarly, we can write,|̃(x̃_λn−1_n−1− x̃_λn_n)| ≤ ρ|̃(x̃_λn−2_n−2− x̃_λn−1_n−1)|

|(x̃_λn_{n − x̃} λn+1_n+1)| ≤ ρn|̃(x̃λ0₀ − x̃1λ₁)| For any n > 𝑚, 𝑚, 𝑛 ∈ 𝑁 |̃(x̃_λn_{n − x̃} λm_m)| ≤ |̃(x̃λn_n− x̃λn−1_n−1)| + |̃(x̃λn−1_n−1− x̃λn−2_n−2)| + … … … + |̃(x̃λm+1_m+1− x̃λm_m)| |(x̃_λn_{n − x̃} λm_m)| ≤ ρ m 1 − ρ|̃(x̃λ0₀− x̃λ1₁)| Since 0 ≤ ρ < 1 and as n → ∞, ⇒ |̃(x̃_λn_n− x̃_λm_m)| = 0.

{x̃_λn_{n} is a soft Cauchy sequence, By the completeness of X,̃ there is x̃}

λ∗ ∈ X̃:x̃λn_n → x̃λ∗, n → ∞.

Case – 2:If

max{|̃(x̃n−1_λn−1− x̃_λn_n)|, |̃(x̃_λn_n− x̃_λn+1_n+1)|, |̃(x̃_λn−1_n−1− x̃_λn+1_n+1)|,0,|̃(x̃_λn−1_n−1− x̃n_λn)|} = |̃(x̃_λn_n− x̃_λn+1_n+1)|

⇒ |̃(x̃λn_n − x̃λn+1_n+1)| ≤ ρ |̃(x̃λn_n− x̃λn+1_n+1)|, Since 0 ≤ ρ < 1, this is contradiction.

Case-3:If

max{|̃(x̃n−1_λn−1− x̃_λn_n)|, |̃(x̃n_λn− x̃_λn+1_n+1)|, |̃(x̃_λn−1_n−1− x̃_λn+1_n+1)|,0,|̃(x̃_λn−1_n−1,−x̃_λn_n)|} = |(x̃_λn−1_n−1− x̃_λn+1_n+1)|

Then 2.2.3 can be written as |(x̃_λn_n−,x̃_λn+1_n+1)| ≤ ρ d̃(x̃_λn−1_n−1, x̃_λn+1_n+1) Using the property of Usual soft metric space as triangular inequality

|̃(x̃_λn_n− xλ_n+1 n+1 _{)| ≤ ρ [|(x̃} λn−1_n−1− x̃λn_n)| + |̃(x̃λn_n − x̃λn+1_n+1)|] ⇒|̃(x̃_λn_{n − x̃} λ_n+1 n+1_{|) ≤ δ}n_d̃(x̃ λ₀ 0 _{, x̃} λ₁ 1 _{) Let} ρ 1−ρ = δ

⇒{x̃_λn_{n} is a soft Cauchy sequence⇢x̃} λ ∗ _{∈ X̃.}

It is clear that {x̃_λn_n} is a soft Cauchy sequence and converges to x̃_λ∗ ∈ X̃because X̃is taken to be complete. Let x̃_λn_n → x̃_λ∗ as n → ∞.

|̃(x̃_λ∗ _{− (f, φ)x̃}

≤ |̃(x̃λ∗− x̃λn+1_n+1)| + |̃ ((f, φ)x̃λn_n− (f, φ)x̃λ∗) | ≤ |̃(x̃λ∗− x̃λn+1_n+1)| +ρ [max {|̃(x̃λnn − x̃λn+1n+1)|,|̃(x̃λ∗ − (f, φ)x̃λ∗)|, |(x̃λnn− (f, φ)x̃λ∗)|, |̃(x̃_λ∗_{, −x̃} λ_n+1 n+1_{)|, |̃(x̃} λn_n − x̃λ∗)| }] − ψ (|̃(x̃λn n _{− x̃} λ_n+1 n+1_{)|, |̃(x̃} λ ∗ _{− (f, φ)x̃} λ∗)|, |̃(x̃λn_n − (f, φ)x̃λ∗), |̃(x̃λ∗− x̃λn+1_n+1)|, |̃(x̃λn_n− x̃λ∗)| ) |(x̃_λ∗_{− (f, φ)x̃} λ ∗_{)| ≤ ρ|̃(x̃} λ∗− (f, φ)x̃λ∗)|

This is a contradiction. So x̃_λ∗ = (f, φ)x̃_λ∗.That is x̃_λ∗ is a fixed soft point of(f, φ).

Uniqueness:Ifỹ_μ∗ is another soft fixed point of (f, φ) inX:̃such that x̃_λ∗ ≠ ỹ_μ∗ |(x̃_λ∗ _{− ỹ}

μ∗)| = |̃ ((f, φ)x̃λ∗ − (f, φ)ỹμ∗)|

≤ | ((f, φ)−, (f, φ)x̃λn_n) | + | ((f, φ)x̃λn_n− (f, φ)ỹμ∗)|

Taking n → ∞ ⇒|(x̃_λ∗ − ỹ_μ∗)| ≤ 0.This contradicts the assumption ⇒x̃∗_λ = ỹ_μ∗ that is, x̃_λ∗ is unique soft fixed point of (f, φ).

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