Abstract

In this paper, we introduce a notion of extended Cone S_b-metric space and prove some fixed point results with various types of contractive conditions. Our results enlarge many results in the literature.

1. Introduction and Preliminaries

In 2007, Huang and Zhang ¹ introduced the idea of cone metric space, which is a generalization of metric space by replacing the real numbers by ordering Banach space. Consequently, several originators consider the development of cone metric space for mappings that satisfying different contractive conditions [2-14].

The concept of S – metric space was initiated by Sedghi et. al. ¹⁵ in 2012, which is distinct from other spaces and established some fixed point results in S - metric space. Many authors enlarged the idea of S - metric space and obtained some fixed point theorems in various contractive conditions ^{16, 17, 18, 19, 20, 21, 22}.

A notion of S_b-metric space was initiated by Souayah and Mlaiki ²³ in 2016. Dhamodharan and krishnakumar ²⁴ expanded the idea of S - metric space to cone S-metric space in 2017 and established various fixed point results. Several authors developed the idea of cone S-metric space in fixed point theory. [1,23-33].

The concept of cone S_b-metric space was initiated by Singh and Singh ³⁴ in 2018 and obtained some fixed point results. Nabil Mlaiki ³¹ introduced the concept of extended S_b-metric space and proved some fixed point theorems for mappings satisfying the different contractive conditions ^{34, 35, 36, 37}.

In this paper, we introduce the notion of extended cone S_b-metric space which is a generalization of cone S_b - metric space and prove some fixed point theorems in extended cone S_b - metric space.

Definition 1.1. ¹⁵ Let Ҳ be a nonempty set and a function Γ: Ҳ³ → [0, ∞) satisfies the following conditions.

1. Γ (v₁, v₂, v₃) ≥ 0.

2. Γ (v₁, v₂, v₃) = 0 if and only if v₁= v =v₃,

3. Γ(v₁,v₂,v₃) ≤ Γ (v₁,v₁,t)+Γ (v₂,v₂,t)+Γ (v₃,v₃,t) for all v₁,v₂,v₃,t ∈ Ҳ.

Then Γ is called S- metric on Ҳ and the pair (Ҳ, Γ) is called an S-metric space.

Example 1.1. ¹⁵ Let Ҳ be a non-empty set and the metric ժ on Ҳ. Then

is an S-metric on Ҳ.

Definition 1.2. ²³ Let Ҳ be a nonempty set and let b≥1 be real number. Define a function Γ_b: Ҳ ₃ → [0,∞) is called an S_b-metric if it is satisfies the following conditions.

1. Γ_b (v₁, v₂, v₃) = 0 iff v₁=v₂= v₃,

2. Γ_b (v₁,v₁,v₂) = Γ_b(v₂, v₂, v₁) for all v₁, v₂ ∈ Ҳ.

3.

Then the pair (Ҳ, Γ_b) is called S_b-metric space.

Definition 1.3. ³¹ Let Ҳ be a nonempty set and ζ: Ҳ³ → [1,∞). A function Γ_ζ : Ҳ³ → [0,∞) satisfies the following conditions.

(i) Γ_ζ(v₁,v₂,v₃) = 0 if and only if v₁=v₂=v₃,

(ii) Γ_ζ(v₁,v₂,v₃) ≤ ζ(v₁,v₂,v₃)(Γ_ζ(v₁,v₁,t) + Γ_ζ(v₂,v₂,t) + Γ_ζ(v₃,v₃,t))

Then the pair (Ҳ, Γ_ζ) is called extended S_b– metric space.

Definition 1.4. ¹ Let E be the real Banach space and M be a subset of E is called a cone if it is satisfies the following conditions.

1. M is closed and non-empty M ≠ 0,

2. pv₁ + qv₂ ∈ M for all v₁, v₂ ∈ M and non-negative real numbers p, q.

3. M ∩ (−M) = 0.

For a given cone M ⊂ E, define a partial ordering ≤ on E with respect to M by v₁ ≤ v₂ if and only if v₂ - v₁ ∈ M, while v₁ ≤ v₂ will stand for v₂ - v₁ ∈ int M (interior of M).

The cone M is called normal if there is a constant K > 0 such that for all v₁, v₂ ∈ E, 0 ≤ v₁ ≤ v₂ implies ||v₁||≤ K ||v₂||.

Then K is called the normal constant of M.

The cone M is called regular if every increasing sequence which is bounded from above is convergent.

Example 1.2. ¹ Let E be the real vector space and K > 1 then,

with supermom norm and the cone M = {pv₁ + q ∈ E: p ≥ 0, q ≥ 0} in E. The cone M is regular and normal.

Definition 1.5. ¹ Let Ҳ be a non-empty set and Γ: Ҳ x Ҳ → E satisfies the following conditions.

1. 0 ≤ Γ (v₁, v₂) for all v₁, v₂ ∈ Ҳ and Γ (v₁, v₂) = 0 if and only if v₁₌v₂.

2. Γ (v₁, v₂) = Γ (v₂, v₁) for all v₁, v₂ ∈ Ҳ.

3. Γ(v₁, v₂) ≤ Γ (v₁, v₃) + Γ (v₃, v₂)

for all v₁, v₂, v₃ ∈ Ҳ. Then Γ is called a cone metric on Ҳ and (Ҳ, Γ) is called a cone metric space.

Definition 1.6. ²⁴ Let M be a cone in E (real Banach space) with int M ≠ 0 and ≤ is a partial ordering with respect to M. Let Ҳ be a non-empty set and define a function Γ: Ҳ ³ → E, if Γ satisfies all the conditions,

1. Γ(v₁, v_2, v₃) ≥ 0

2. Γ(v₁, v₂, v₃) = 0 if and only if v₁=v₂= v₃

3. Γ(v₁,v₂,v₃)≤Γ(v₁,v₁,t)+Γ(v₂,v₂,t)+Γ(v₃,v₃,t) for all v₁,v₂,v₃,t ∈ Ҳ.

Then Γ is called a cone S-metric on Ҳ and (Ҳ, Γ) is called a cone S-metric space.

Example 1.3. ²⁴ Let E=R², M = {(v₁, v₂) ∈ R²: v₁ ≥ 0, v₂ ≥ 0}⊂ R² , Ҳ=R and ժ: Ҳ x Ҳ x Ҳ → E be the metric on Ҳ then Γ: Ҳ³ → E defined by

is a cone S-metric on Ҳ where α > 0 is a constant.

Definition 1.7. ³⁴ Let Ҳ be a nonempty set and M be a cone in E(real Banach space) and define Γ_b : Ҳ³ →E is satisfies the following conditions

1. Γ_b(v₁,v₂,v₃)≥ 0.

2. Γ_b(v₁,v₂,v₃)= 0 if and only if v₁=v₂= v₃.

3. Γ_b(v₁, v₂,v₃)≤r[Γ_b(v₁,v₁,t)+Γ_b(v₂,v₂,t)+Γ_b(v₃,v₃,t)]

for all v₁, v₂, v₃, t ∈ Ҳ, where r ≥1 is a constant then Γ_b is called a cone S_b- metric on Ҳ and (Ҳ, Γ_b) is called an cone S_b-metric space.

2. Main Result

In this section, we introduce an extended cone S_b- metric space and prove some fixed point results in extended cone S_b-metric space.

Definition 2.1. Let Ҳ be a non-empty set and ζ: Ҳ³ → [1,∞) be a function. If Γ_ζ : Ҳ³ → E (Real Banach Space) satisfies the following conditions.

1. Γ_ζ (v₁,v₂,v₃)≥ 0.

2. Γ_ζ(v₁,v₂,v₃)= 0 if and only if v₁₌v_{2 =}v_3.

3.

for all v₁,v₂,v₃,t ∈ Ҳ.

Then (Ҳ, Γ_ζ) is called an extended cone S_b- metric space.

Remark 2.1. If ζ(v₁,v₂,v₃) = 1, then the extended cone S_b- metric space reduces to a cone S- metric space.

Remark 2.2. If ζ(v₁,v₂,v₃) = b ≥ 1 then the extended cone S_b-metric space is said to be cone S_b -metric space.

Lemma 2.1. Let (Ҳ, Γ_ζ) be an extended cone S_b-metric space. Then we have Γ_ζ(v₁, v₁, v₂) = Γ_ζ (v₂, v₂ ,v₁).

Definition 2.2. Let (Ҳ, Γ_ζ) be an extended cone S_b- metric space and M be a normal cone.

1) A sequence {v_n}∈ Ҳ converges to w if and only if w ∈ Ҳ such that Γ_ζ (v_n, v_n, w) → 0 as n →∞. we can write this

2) A sequence {v_n} is said to be Cauchy sequence if and only if Γ_ζ (v_n,v_n,v_m) → 0 as n, m →∞.

3) If every Cauchy sequence {v_n} converges to w ∈ Ҳ, then (Ҳ, Γ) is said to be a complete extended cone S_b- metric space.

Example 2.1. Let E = R² and M be a cone in E. Let Ҳ = [0,∞) define a function Γ_ζ : Ҳ³ →E such that

where α > 0 is a constant and a function ζ : Ҳ ³ → [1,∞) by ζ(v₁,v₂,v₃) = max{v₁,v₂}+ v₃ + 1 then (Ҳ, Γ_ζ) is a complete extended cone S_b - metric space

Theorem 2.1. Let (Ҳ, Γ_ζ) be a complete extended cone S_b- metric space and T be a self-mapping on Ҳ satisfying the following condition

(1)

for all v₁, v₂, v₃ ∈ Ҳ where 0 ≤ c₁ + c₂ + c₃ + c₄ < 1 and for 0 ≤ b < , then T has a unique fixed point.

Proof. Let v₀ ∈ Ҳ, define a sequence {v_n} by from (1)

where 0 ≤ b < 1/2 continue this process to obtain

for all m, n ∈ N and n < m. Hence by triangle inequality

by the hypothesis of the theorem

by Ratio test series

converges.

Let A = and An = for m > n, we have

Taking limit as n, m → ∞, the sequence {v_n} is a Cauchy sequence. Since Ҳ is complete. {v_n} converges to υ ∈ Ҳ.

By (1) and the triangle inequality,

Taking limit as n → ∞,

that implies Tυ = υ. Hence υ is a fixed point of T. To prove that uniqueness, assume that there exists υ ≠ w ∈ Ҳ such that T υ = υ and T w = w.

Thus,

which is a contradiction. Therefore, T has a unique fixed point.

If c₁ = c and c₂ = c₃ = c₄ = 0 in Theorem 2.1, then the following corollary is obtained.

Corollary 2.1. Let (Ҳ, Γ_ζ) be a complete extended cone S_b- metric space and T be a self-mapping on Ҳ satisfying the following condition

(2)

For all v₁, v₂, v₃ ∈ Ҳ where 0 ≤ c < 1/ 2 and < 1 /2c, then T has a unique fixed point.

If c₁ = 0 and c₂ = c₃ = c₄ = c in the Theorem 2.1, then the following corollary is obtained.

Corollary 2.2. Let (Ҳ, Γ_ζ) be a complete extended cone S_b-metric space and T: Ҳ→Ҳ satisfy the following conditions

for all v₁, v₂, v₃ ∈ Ҳ where 0 ≤ c < 1/ 2 and < 1 /2c, then T has a unique fixed point.

Example 2.2. Let E = R² and M be a cone in E. Let Ҳ = [0, ∞) define a function Γ_ζ : Ҳ³ → E such that

where α > 0, is a constant and a function ζ: Ҳ³→ [1,∞) defined by

Then (Ҳ, Γ_ζ) is a complete extended cone S_b-metric space. Consider the mapping T: Ҳ → Ҳ defined by

Then

where c ∈ thus T satisfies all the conditions of Corollary 2.1 and hence T has a unique fixed point.

3. Conclusion

Fixed point theory plays an essential role in all branches of Mathematics. In this paper, we introduced an extended cone S_b-metric space and proved some fixed results in various contractive conditions. Our results extends several results in existing literature.

Acknowledgements

The authors would like to express their thanks to the editors and reviewers for valuable advice in helping to improve the manuscript.

Conflict of Interest

There is no conflict of interest.

References