A Note on tvs-G-Cone Metric Fixed Point Theory

Definition 1.1 (see [22].)Let (X, d) be a tvs-cone metric space with a solid cone P, and let 𝒜 be a collection of nonempty subsets of X. A map ℋ : 𝒜 × 𝒜 → E is called a tvs-ℋ-cone metric with respect to d if for any A₁, A₂ ∈ 𝒜 the following conditions hold:

•  

  (H₁) ℋ(A₁, A₂) = θ⇒A₁ = A₂,

•  

  (H₂) ℋ(A₁, A₂) = ℋ(A₂, A₁),

•  

  (H₃) $$d(x, y)≼ℋ(A₁, A₂) + ε,

•  

  (H₄) one of the following is satisfied:

  • (i)

    $$ℋ(A₁, A₂)≼d(x, y) + ε,

  • (ii)

    $$ℋ(A₁, A₂)≼d(x, y) + ε.

Theorem 1.2 (see [22].)Let (X, d) be a tvs-cone complete metric space with a solid cone P and let 𝒜 be a collection of nonempty closed subsets of X, 𝒜 ≠ ϕ, and let ℋ : 𝒜 × 𝒜 → E be a tvs-ℋ-cone metric with respect to d. If the mapping T : X → 𝒜 the condition that exists a λ ∈ (0,1) such that for all x, y ∈ X holds

then T has a fixed point in X.

Definition 1.3 (see [16], [18], [19].)Let X be a nonempty set and (E, P) an ordered tvs. A vector-valued function d : X × X → E is said to be a tvs-cone metric, if the following conditions hold:

•  

  (C₁) ∀_{x,y∈X,x≠y} θ≼d(x, y),

•  

  (C₂) ∀_x,y∈X d(x, y) = θ⇔x = y,

•  

  (C₃) ∀_x,y∈X d(x, y) = d(y, x),

•  

  (C₄) ∀_x,y,z∈X d(x, z)≼d(x, y) + d(y, z).

Then the pair (X, d) is called a tvs-cone metric space.

Definition 1.4 (see [16], [18], [19].)Let (X, d) be a tvs-cone metric space, x ∈ X, and {x_n} a sequence in X.

• (1)

  {x_n}  tvs-cone converges to x whenever for every c ∈ E with θ ≪ c, there exists n₀ ∈ ℕ such that d(x_n, x) ≪ c for all n ≥ n₀. We denote this by cone-lim _n→∞ x_n = x;

• (2)

  {x_n} is a tvs-cone Cauchy sequence whenever for every c ∈ E with θ ≪ c, there exists n₀ ∈ ℕ such that d(x_n, x_m) ≪ c for all n, m ≥ n₀;

• (3)

  (X, d) is tvs-cone complete if every tvs-cone Cauchy sequence in X is tvs-cone convergent in X.

Remark 1.5. Clearly, a cone metric space in the sense of Huang and Zhang [1] is a special case of tvs-cone metric spaces when (X, d) is a tvs-cone metric space with respect to a normal cone P.

Remark 1.6 (see [19], [22], [23].)Let (X, d) be a tvs-cone metric space with a solid cone P. The following properties are often used, particularly in the case when the underlying cone is nonnormal.

• (p1)

  If u≼v and v ≪ w, then u ≪ w,

• (p2)

  If u ≪ v and v≼w, then u ≪ w,

• (p3)

  If u ≪ v and v ≪ w, then u ≪ w,

• (p4)

  If θ≼u ≪ c for each c ∈ int P, then u = θ,

• (p5)

  If a≼b + c for each c ∈ int P, then a≼b,

• (p6)

  If E is tvs with a cone P, and if a≼λa where a ∈ P and λ ∈ [0,1), then a = θ,

• (p7)

  If c ∈ int P, a_n ∈ E, and a_n → θ in locally convex tvsE, then there exists n₀ ∈ ℕ such that a_n ≪ c for all n > n₀.

Definition 1.7 (see [25].)Let X be a nonempty set, and let G : X × X × X → [0, ∞) be a function satisfying the following axioms:

• (G1)

  ∀_x,y,z∈X G(x, y, z) = 0⇔x = y = z,

• (G2)

  ∀_{x,y∈X,x≠y} G(x, x, y) > 0,

• (G3)

  ∀_x,y,z∈X G(x, y, z) ≥ G(x, x, y),

• (G4)

  ∀_x,y,z∈X (x, y, z) = G(x, z, y) = G(z, y, x) = ⋯ (symmetric in all three variables),

• (G5)

  ∀_x,y,z,w∈X G(x, y, z) ≤ G(x, w, w) + G(w, y, z).

Then the function G is called a generalized metric, or, more specifically a G-metric on X, and the pair (X, G) is called a G-metric space.

Definition 1.8. Let X be a nonempty set and (E, P) an ordered tvs, and let G : X × X × X → E be a function satisfying the following axioms:

• (G1)

  ∀_x,y,z∈X G(x, y, z) = θ if and only if x = y = z,

• (G2)

  ∀_{x,y∈X,x≠y} θ ≪ G(x, x, y),

• (G3)

  ∀_x,y,z∈X G(x, x, y)≼G(x, y, z),

• (G4)

  ∀_x,y,z∈X G(x, y, z) = G(x, z, y) = G(z, y, x) = ⋯ (symmetric in all three variables),

• (G5)

  ∀_x,y,z,w∈X G(x, y, z)≼G(x, w, w) + G(w, y, z).

Then the function G is called a tvs-generalized-cone metric, or, more specifically, a tvs-G-cone metric on X, and the pair (X, G) is called a tvs-G-cone metric space.

Definition 1.9. Let (X, G) be a tvs-G-cone metric space, x ∈ X, and {x_n} a sequence in X.

• (1)

  {x_n} tvs-G-cone converges to x whenever, for every c ∈ E with θ ≪ c, there exists n₀ ∈ ℕ such that G(x_n, x_m, x) ≪ c for all m, n ≥ n₀. Here x is called the limit of the sequence {x_n} and is denoted by G-cone-lim _n→∞ x_n = x;

• (2)

  {x_n} is a tvs-G-cone Cauchy sequence whenever, for every c ∈ E with θ ≪ c, there exists n₀ ∈ ℕ such that G(x_n, x_m, x_l) ≪ c for all n, m, l ≥ n₀;

• (3)

  (X, G) is tvs-G-cone complete if every tvs-G-cone Cauchy sequence in X is tvs-G-cone convergent in X.

Proposition 1.10. Let (X, G) be a tvs-G-cone metric space, x ∈ X, and {x_n} a sequence in X. The following are equivalent:

• (i)

  {x_n}  tvs-G-cone converges to x,

• (ii)

  G(x_n, x_n, x) → θ as n → ∞,

• (iii)

  G(x_n, x, x)→ as n → ∞,

• (iv)

  G(x_n, x_m, x) → θ as n, m → ∞.

Definition 1.11. One callsφ : int P ∪ {θ} → int P ∪ {θ} a 𝒞ℬ𝒲-tvs-G-cone-type function if the function φ satisfies the following condition

•  

    (φ₁) φ(t) ≪ t for all t ≫ θ and φ(θ) = θ;

•  

  (φ₂) lim _n→∞φⁿ(t) = θ for all t ∈ int P ∪ {θ}.

In this paeper, for a tvs-G-cone metric space (X, G) and for the family 𝒜 of subsets of X, we introduce a new notion of the tvs-ℋ-cone metric ℋ with respect to G, and we get a fixed result for the 𝒞ℬ𝒲-tvs-G-cone-type function in a complete tvs-G-cone metric space (𝒜, ℋ). Our results generalize some recent results in the literature.

Definition 2.1. Let (X, G) be a tvs-G-cone metric space with a solid cone P, and let 𝒜 be a collection of nonempty subsets of X. A map ℋ : 𝒜 × 𝒜 × 𝒜 → E is called a tvs-ℋ-cone metric with respect to G if for any A₁, A₂, A₃ ∈ 𝒜 the following conditions hold:

•  

  H₁ ℋ(A₁, A₂, A₃) = θ⇒A₁ = A₂ = A₃,

•  

  H₂ ℋ(A₁, A₂, A₃) = ℋ(A₂, A₁, A₃) = ℋ(A₁, A₃, A₂) = ⋯(symmetry in all variables),

•  

  (H₃) ℋ(A₁, A₁, A₂)≼ℋ(A₁, A₂, A₃),

•  

  H₄ $$G(x, y, z)≼ℋ(A₁, A₂, A₃) + ε,

•  

  H₅ one of the following is satisfied:

• (i)

  $$,

• (ii)

  $$,

• (iii)

  $$.

Lemma 2.2. Let (X, G) be a tvs-G-cone metric space with a solid cone P, and let 𝒜 be a collection of nonempty subsets of X, 𝒜 ≠ ϕ. If ℋ : 𝒜 × 𝒜 × 𝒜 → E is a tvs-ℋ-cone metric with respect to G, then pair (𝒜, ℋ) is a tvs-G-cone metric space.

Proof. Let {ε_n} ⊂ E be a sequence such that θ ≪ ε_n for all n ∈ ℕ and G-cone-lim _n→∞ε_n = θ. Take any A₁, A₂, A₃ ∈ 𝒜 and x ∈ A₁, y ∈ A₂. From (H₄), for each n ∈ ℕ, there exists z_n ∈ A₃ such that

Therefore, ℋ(A₁, A₂, A₃) + ε_n ∈ P for each n ∈ ℕ. By the closedness of P, we conclude that θ≼ℋ(A₁, A₂, A₃).

Assume that A₁ = A₂ = A₃. From H₅, we obtain ℋ(A₁, A₂, A₃)≼ε_n for any n ∈ ℕ. So ℋ(A₁, A₂, A₃) = θ.

Let A₁, A₂, A₃, A₄ ∈ 𝒜. Assume that A₁, A₂, A₃ satisfy condition (H₅)(i). Then, for each n ∈ ℕ, there exists x_n ∈ A₁ such that ℋ(A₁, A₂, A₃)≼G(x_n, y, z) + ε_n for all y ∈ A₂ and z ∈ A₃. From (H₄), there exists a sequence {w_n} ⊂ A₄ satisfying G(x_n, w_n, w_n)≼ℋ(A₁, A₄, A₄) + ε_n for every n ∈ ℕ. Obviously, for any y ∈ A₂ and any z ∈ A₃ and n ∈ ℕ, we have

Now for each n ∈ ℕ, there exists y_n ∈ A₂, z_n ∈ A₃ such that G(w_n, y_n, z_n)≼ℋ(A₄, A₂, A₃) + ε_n. Consequently, we obtain that for each n ∈ ℕ Therefore, In the case when (H₅)(ii) or (H₅)(iii) holds, we use the analogous method.

Theorem 2.3. Let (X, G) be a tvs-G-cone complete metric space with a solid cone P, let 𝒜 be a collection of nonempty closed subsets of X, 𝒜 ≠ ϕ, and let ℋ : 𝒜 × 𝒜 × 𝒜 → E be a tvs-ℋ-cone metric with respect to G. If the mapping T : X → 𝒜 satisfies the condition that exists a φ ∈ Θ such that for all x, y, z ∈ X holds

then T has a fixed point in X.

Proof. Let us choose ε ∈ int P arbitrarily, and let ε_n ∈ E be a sequence such that θ ≪ ε_n and ε_n≼ε/3ⁿ. Let us choose x₀ ∈ X arbitrarily and x₁ ∈ Tx₀. If G(x₀, x₀, x₁) = θ, then x₀ = x₁ ∈ T(x₀), and we are done. Assume that G(x₀, x₀, x₁) ≫ θ. Taking into account (2.5) and (H₄), there exists x₂ ∈ Tx₁ such that

Taking into account (2.5), (2.6), and (H₄) and since φ ∈ Θ, there exists x₃ ∈ Tx₂ such that We continue in this manner. In general, for x_n, n ∈ ℕ, x_n+1 is chosen such that x_n+1 ∈ Tx_n and Since ε is arbitrary, letting ε → θ and by the definition of the 𝒞ℬ𝒲-tvs-G-cone-type function, we obtain that

Next, we let c_m = G(x_m, x_m+1, x_m+1), and we claim that the following result holds: for each γ ≫ θ,  there is n₀(ε) ∈ N such that for all m, n ≥ n₀(γ),

We will prove (2.10) by contradiction. Suppose that (2.10) is false. Then there exists some γ ≫ θ such that for all p ∈ ℕ, there are m_p, n_p ∈ ℕ with m_p > n_p ≥ p satisfying

• (i)

  m_p is even and n_p is odd,

• (ii)

  $$, and

• (iii)

  m_p is the smallest even number such that conditions (i), (ii) hold.

Since c_m ↓ θ, by (ii), we have that $$ and It follows from (H₄); let us choose ε ∈ E arbitrarily such that Taking into account (2.5), (2.11), and (2.12), we have that Since ε is arbitrarily, letting ε → θ and by letting p → ∞, we have a contradiction. So {x_n} is a tvs-G-cone Cauchy sequence. Since (X, G)is a tvs-G-cone complete metric space, {x_n} is tvs-G-cone convergent in X and G-cone -lim _n→∞x_n = x. Thus, for every τ ∈ int P and sufficiently large n, we have that Since for n ∈ ℕ ∪ {0}, x_n+1 ∈ Tx_n, by (H₄) we obtain that for all n ∈ ℕ there existsy_n ∈ Tx such that Since ε/3ⁿ⁺¹ → θ, then for sufficiently large n, we obtain that which implies G-cone -lim _n→∞y_n = x. Since Tx is closed, we obtain that x ∈ Tx.

Corollary 2.4. Let (X, G) be a tvs-G-cone complete metric space with a solid cone P, let 𝒜 be a collection of nonempty closed subsets of X, 𝒜 ≠ ϕ, and let ℋ : 𝒜 × 𝒜 × 𝒜 → E be a tvs-ℋ-cone metric with respect to G. If the mapping T : X → 𝒜 satisfies the condition that exists k ∈ (0,1) such that for all x, y, z ∈ X holds

then T has a fixed point in X.

Remark 2.5. Following Corollary 2.4, it is easy to get Theorem 1.2. So our results generalize some recent results in the literature (e.g., [22]).