On set-valued contractions of Nadler type in tυs-G-cone metric spaces

Abstract

In this article, for a tυs-G-cone metric space (X, G) and for the family of subsets of X, we introduce a new notion of the $t\upsilon s\mathsf{\text{ - }}\mathcal{H}\mathsf{\text{ - cone}}$ metric with respect to G, and we get a fixed result for the stronger Meir-Keeler-G-cone-type function in a complete tυs-G-cone metric space $\left(\mathcal{A},\mathcal{H}\right)$ Our result generalizes some recent results due to Dariusz Wardowski and Radonevic' et al.

MSC: 47H10; 54C60; 54H25; 55M20.

1 Introduction and preliminaries

Recently, Huang and Zhang [1] introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space, and they showed some fixed point theorems of contractive type mappings on cone metric spaces. The category of cone metric spaces is larger than metric spaces. Subsequently many authors like Abbas and Jungck [2] had generalized the results of Huang and Zhang [1] and studied the existence of common fixed points of a pair of self mappings satisfying a contractive type condition in the framework of normal cone metric spaces. However, authors like Jankovic' et al. [3], Rezapour and Hamlbarani [4] studied the existence of common fixed points of a pair of self and nonself mappings satisfying a contractive type condition in the situation in which the cone does not need to be normal. Many authors studied this subject and many results on fixed point theory are proved (see e.g., [4–15]).

Recently, Du [16] introduced the concept of tυs-cone metric and tυs-cone metric space to improve and extend the concept of cone metric space in the sense of Huang and Zhang [1]. Later, in the articles [16–19], the authors tried to generalize this approach by using cones in topological vector spaces tυs instead of Banach spaces. However, it should be noted that an old result shows that if the underlying cone of an ordered tυs is solid and normal, then such tυs must be an ordered normed space. Thus, proper generalizations when passing from norm-valued cone metric spaces to tυs-valued cone metric spaces can be obtained only in the case of nonnormal cones (for details, see [19]).

We recall some definitions and results of the tυs-cone metric spaces that introduced in [19, 20], which will be needed in the sequel.

Let E be a real Hausdorff topological vector space (tυs for short) with the zero vector θ. A nonempty subset P of E is called a convex cone if P + P ⊆ P and λP ⊆ P for λ ≥ 0. A convex cone P is said to be pointed (or proper) if P ∩ (-P) = {θ}; P is normal (or saturated) if E has a base of neighborhoods of zero consisting of order-convex subsets. For a given cone P ⊆ E, we can define a partial ordering ≼ with respect to P by x ≼ y if and only if y - x ∈ P; x ≺ y will stand for x ≼ y and x ≠ y, while x ≪ y will stand for y - x ∈ intP, where intP denotes the interior of P. The cone P is said to be solid if it has a nonempty interior.

In the sequel, E will be a locally convex Hausdorff tυs with its zero vector θ, P a proper, closed, and convex pointed cone in E with int P ≠ ϕ and ≼ a partial ordering with respect to P.

Definition 1 [16, 18, 19] Let X be a nonempty set and (E, P) an ordered tυs. A vector-valued function d: X × X → E is said to be a tυs-cone metric, if the following conditions hold:

(C₁) ∀_{x,y∈X,x≠y}θ ≼ d(x, y);

(C₂) ∀_x,y∈Xd(x, y) = θ ⇔ x = y;

(C₃) ∀_x,y∈Xd(x, y) = d(y, x);

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

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

Definition 2 [16, 18, 19] Let (X, d) be a tυs-cone metric space, x ∈ X and {x _n } a sequence in X.

(1) {x _n } tυs-cone converges to x whenever for every c ∈ E with θ ≪ c, there exists $n_0\in \mathbb{N}$ 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 tυs-cone Cauchy sequence whenever for every c ∈ E with θ ≪ c, there exists $n_0\in \mathbb{N}$ such that d(x _n , x _m ) ≪ c for all n, m ≥ n₀;

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

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

Remark 2 [19–21] Let (X, d) be a tυs-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 ≼ υ and υ ≪ w, then u ≪ w;

(p2) If u ≪ υ and υ ≼ w, then u ≪ w;

(p3) If u ≪ υ and υ ≪ w, then u ≪ w;

(p4) If θ ≼ u ≪ c for each c ∈ intP, then u = θ;

(p5) If a ≼ b + c for each c ∈ intP, then a ≼ b;

(p6) If E is tυs with a cone P, and if a ≼ λa where a ∈ P and λ ∈ [0, 1), then a = θ;

(p7) If c ∈ intP, a _n ∈ E and a _n → θ in locally convex tυs E, then there exists $n_0\in \mathbb{N}$ such that a _n ≪ c for all n > n₀.

Metric spaces are playing an important role in mathematics and the applied sciences. To overcome fundamental laws in Dhage's theory of generalized metric spaces [22], flaws that invalidate most of the results claimed for these spaces, Mustafa and Sims [23] introduced a more appropriate and robust notion of a generalized metric space as follows:

Definition 3 [23] Let X be a nonempty set, and let G : X × X × X → [0, ∞) be a function satisfying the following axioms:

(G1) ∀_x,y,z∈XG(x, y, z) = 0 ⇔ x = y = z;

(G2) ∀_{x,y∈X,x≠y}G(x, x, y) > 0;

(G3) ∀_x,y,z∈XG(x, y, z) ≥ G(x, x, y);

(G4) ∀_x,y,z∈XG(x, y, z) = G(x, z, y) = G(z, y, x) = ⋯ (symmetric in all three variables);

(G5) ∀_x,y,z,w∈XG(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.

By using the notions of generalized metrics and tυs-cone metrics, we introduced the below notion of tυs-generalized-cone metrics.

Definition 4 Let X be a nonempty set and (E, P) an ordered tυs, and let G : X × X × X → E be a function satisfying the following axioms:

(G1) ∀_x,y,z∈XG(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∈XG(x, x, y) ≼ G(x, y, z);

(G4) ∀_x,y,z∈XG(x, y, z) = G(x, z, y) = G(z, y, x) = ⋯ (symmetric in all three variables);

(G5) ∀_x,y,z,w∈XG(x, y, z) ≼ G(x, w, w) + G(w, y, z).

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

Definition 5 Let (X, G) be a tυs-G-cone metric space, x ∈ X and {x _n } a sequence in X.

• (1) {x _n } tυs-G-cone converges to x whenever for every c ∈ E with θ ≪ c, there exists $n_0\in \mathbb{N}$ 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 tυs-G-cone Cauchy sequence whenever for every c ∈ E with θ ≪ c, there exists $n_0\in \mathbb{N}$ such that G(x _n , x _m , x _l ) ≪ c for all n, m, l ≥ n₀;

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

Proposition 1 Let (X, G) be a tυs-G-cone metric space, x ∈ X and {x _n } a sequence in X. The following are equivalent

(i) {x _n } tυs-G-cone converges to x;

(ii)G(x _n , x _n , x) → θ as n → ∞;

(iii)G(x _n , x, x) → θ as n → ∞;

(iv)G(x _n , x _m , x) → θ as n, m → ∞.

We also recall the notion of Meir-Keeler type function (see [24]). A function φ : [0, ∞) → [0, ∞) is said to be a Meir-Keeler type function, if φ satisfies the following condition:

$$\forall \eta >0\exists \delta >0\forall t\in \left[0,\infty \right)\left(\eta \le t<\delta +\eta \Rightarrow \phi \left(t\right)<\eta \right).$$

We now define a new notion of stronger Meir-Keeler type function, as follows:

Definition 6 We call φ : [0, ∞) → [0, 1) a stronger Meir-Keeler type function if the function φ satisfies the following condition:

$$\forall \eta >0\exists \delta >0\exists {\gamma }_{\eta }\in \left[0,1\right)\forall t\in \left[0,\infty \right)\left(\eta \le t<\delta +\eta \Rightarrow \phi \left(t\right)<{\gamma }_{\eta }\right).$$

And, we introduce the below concept of the stronger Meir-Keeler tυs-G-cone-type function in a tυs-G-cone metric space.

Definition 7 Let (X, G) be a tυs-G-cone metric space with a solid cone P. We call φ : P → [0, 1) a stronger Meir-Keeler tυs-G-cone-type function in X if the function φ satisfies the following condition:

$$\forall \eta \gg \theta \exists \delta \gg \theta \exists {\gamma }_{\eta }\in \left[0,1\right)\forall x,y,z\in X\left(\eta \preccurlyeq G\left(x,y,z\right)\ll \delta +\eta \Rightarrow \phi \left(G\left(x,y,z\right)\right)<{\gamma }_{\eta }\right).$$

The Nadler's results [25] concerning set-valued contractive mappings in metric spaces became the inspiration for many authors in the metric fixed point theory (see for example [26–28]). Particularly Wardowski [29] established a new cone metric $\mathcal{H}:\mathcal{A}\times \mathcal{A}\to E$ for a cone metric space (X, d) and for the family of subsets of X, and introduced the concept of set-valued contraction of Nadler type and prove a fixed point theorem. Later, in [21], the concept of set-valued contraction of Nadler type in the setting of tυs-cone spaces was introduced and a fixed point theorem in the setting of tυs-cone spaces with respect to a solid cone was proved.

In this article, for a tυs-G-cone metric space (X, G) and for the family of subsets of X, we introduce a new notion of the $t\upsilon s\mathsf{\text{ - }}\mathcal{H}\mathsf{\text{ - cone}}$ metric with respect to G, and we get a fixed result for the stronger Meir-Keeler type function in a complete tυs-generalized-cone metric space $\left(\mathcal{A},\mathcal{H}\right)$. Our result generalizes some recent results due to Radonevic' et al. [21] and Dariusz Wardowski [29].

2 Main results

Let E be a locally convex Hausdorff tυs with its zero vector θ, P a proper, closed, and convex pointed cone in E with intP ≠ ϕ and ≼ a partial ordering with respect to P. We introduce the below notion of the $t\upsilon s\mathsf{\text{ - }}\mathcal{H}\mathsf{\text{ - cone}}$ metric with respect to tυs-G-cone metric G.

Definition 8 Let (X, G) be a tυs-G-cone metric space with a solid cone P and let be a collection of nonempty subsets of X. A map $\mathcal{H}:\mathcal{A}\times \mathcal{A}\times \mathcal{A}\to E$ is called a $t\upsilon s-\mathcal{H}-cone$ metric with respect to G if for any $A_1,A_2,A_3\in \mathcal{A}$ the following conditions hold:

(H₁) $\mathcal{H}\left(A_1,A_2,A_3\right)=\theta \Rightarrow A_1=A_2=A_3$;

(H₂) $\mathcal{H}\left(A_1,A_2,A_3\right)=\mathcal{H}\left(A_1,A_2,A_3\right)=\mathcal{H}\left(A_1,A_2,A_3\right)=\cdots \left(symmetryinallvariables\right)$;

(H₃) $\mathcal{H}\left(A_1,A_2,A_3\right)\preccurlyeq \mathcal{H}\left(A_1,A_2,A_3\right)$;

(H₄) ${\forall }_{\epsilon \in E,\theta \ll \epsilon }{\forall }_{x\in A_1,y\in A_2}{\exists }_{z_{\in }{A}_3}G\left(x,y,z\right)\preccurlyeq \mathcal{H}\left(A_1,A_2,A_3\right)+\epsilon$;

(H₅) one of the following is satisfied:

(i) ${\forall }_{\epsilon \in E,\theta \ll \epsilon }{\exists }_{x_{\in }{A}_1}{\forall }_{y\in A_2,z\in A_3}\mathcal{H}\left(A_1,A_2,A_3\right)\preccurlyeq G\left(x,y,z\right)+\epsilon$;

(ii) ${\forall }_{\epsilon \in E,\theta \ll \epsilon }{\exists }_{x_{\in }{A}_2}{\forall }_{x\in A_1,z\in A_3}\mathcal{H}\left(A_{1,}{A}_{2,}{A}_3\right)\preccurlyeq G\left(x,y,z\right)+\epsilon$;

(iii) ${\forall }_{\epsilon \in E,\theta \ll \epsilon }{\exists }_{z_{\in }{A}_3}{\forall }_{y\in A_2,z\in A_1}\mathcal{H}\left(A_{1,}{A}_{2,}{A}_3\right)\preccurlyeq G\left(x,y,z\right)+\epsilon$.

Lemma 1 Let (X, G) be a tυs-G-cone metric space with a solid cone P and let be a collection of nonempty subsets of X. $\mathcal{A}\ne \varphi$. If $\mathcal{H}:\mathcal{A}\times \mathcal{A}\times \mathcal{A}\to E$ is a $t\upsilon s-\mathcal{H}-cone$ metric with respect to G, then pair $\left(\mathcal{A},\mathcal{H}\right)$ is a tυs-G-cone metric space.

Proof Let {ε _n } ⊂ E be a sequence such that θ ≪ ε _n for all $n\in \mathbb{N}$ and G-cone-lim_n→∞ε _n = θ. Take any $A_1,A_2,A_3\in \mathcal{A}$ and x ∈ A₁, y ∈ A₂. From (H₄), for each $n\in \mathbb{N}$, there exists z _n ∈ A₃ such that

$$G\left(x,y,z_n\right)\preccurlyeq \mathcal{H}\left(A_{1,}{A}_{2,}{A}_3\right)+{\epsilon }_n.$$

Therefore, $\mathcal{H}\left(A_1,A_2,A_3\right)+{\epsilon }_n\in P$ for each $n\in \mathbb{N}$. By the closedness of P, we conclude that θ ≼ $\mathcal{H}\left(A_1,A_2,A_3\right)$.

Assume that A₁ = A₂ = A₃. From H₅, we obtain $\mathcal{H}\left(A_1,A_2,A_3\right)\preccurlyeq {\epsilon }_n$ for any $n\in \mathbb{N}$. So $\mathcal{H}\left(A_1,A_2,A_3\right)=\theta$.

Let $A_1,A_2,A_3,A_4\in \mathcal{A}$. Assume that A₁, A ₂ , A ₃ satisfy the condition (H₅)(i). Then for each $n\in \mathbb{N}$, there exists x _n ∈ A₁ such that $\mathcal{H}\left(A_1,A_2,A_3\right)\preccurlyeq G\left(x_n,y,z\right)+{\epsilon }_n$ for all y ∈ A₂ and z ∈ A₃. From (H₄), there exists a sequence {w _n } ⊂ A₄ satisfying $G\left(x_n,w_n,w_n\right)\preccurlyeq \mathcal{H}\left(A_1,A_4,A_4\right)+{\epsilon }_n$ for every $n\in \mathbb{N}$. Obviously for any y ∈ A₂ and any z ∈ A₃ and $n\in \mathbb{N}$, we have

$$\begin{array}{ll}\hfill \mathcal{H}\left(A_{1,}{A}_{2,}{A}_3\right)& \preccurlyeq G\left(x_{n,}y,z\right)+{\epsilon }_n\\ \preccurlyeq G\left(x_{n,}{w}_{n,}{w}_n\right)+G\left(w_{n,}y,z\right)+{\epsilon }_n.\end{array}$$

Now for each $n\in \mathbb{N}$, there exists y _n ∈ A₂, z _n ∈ A₃ such that $G\left(w_n,y_n,z_n\right)\preccurlyeq \mathcal{H}\left(A_4,A_2,A_3\right)+{\epsilon }_n$. Consequently, we obtain that for each $n\in \mathbb{N}$

$$\mathcal{H}\left(A_{1,}{A}_{2,}{A}_3\right)\preccurlyeq \mathcal{H}\left(A_{1,}{A}_{4,}{A}_{4,}\right)+\mathcal{H}\left(A_{4,}{A}_{2,}{A}_{3,}\right)+3{\epsilon }_n.$$

Therefore,

$$\mathcal{H}\left(A_{1,}{A}_{2,}{A}_3\right)\preccurlyeq \mathcal{H}\left(A_{1,}{A}_{4,}{A}_{4,}\right)+\mathcal{H}\left(A_{4,}{A}_{2,}{A}_{3,}\right).$$

In the case when (H₅)(ii) or (H ₅)(iii) hold, we use the analog method. □

Our main result is the following.

Theorem 1 Let (X, G) be a tυs-G-cone complete metric space with a solid cone P and let be a collection of nonempty closed subsets of X, $\mathcal{A}\ne \varphi$, and let $\mathcal{H}:\mathcal{A}\times \mathcal{A}\times \mathcal{A}\to E$ be a $t\upsilon s-\mathcal{H}-cone$ metric with respect to G. If the mapping $T:X\to \mathcal{A}$ satisfies the condition that exists a stronger Meir-Keeler tυs-G-cone-type function φ : P → [0, 1) such that for all x, y, z ∈ X holds

$$\mathcal{H}\left(Tx,Ty,Tz\right)\preccurlyeq \phi \left(G\left(x,y,z\right)\right)\cdot G\left(x,y,z\right),$$

(1)

then T has a fixed point in X.

Proof. 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. Put G(x₀, x₀, x₁) = η₀, η₀ ≫ θ. By the definition of the stronger Meir-Keeler tυs-G-cone-type function φ : P → [0, 1), corresponding to η₀ use, there exist δ₀ ≫ θ and ${\gamma }_{{\eta }_0}\in \left(0,1\right)$ with η₀ ≼ G(x₀, x₀, x 1) ≺ η₀ + δ₀ such that $\phi \left(G\left(x_0,x_0,x_1\right)\right)<{\gamma }_{{\eta }_0}$. Let ε ∈ intP and ε₁ ∈ E such that θ ≪ ε₁ and ${\epsilon }_1\preccurlyeq {\gamma }_{{\eta }_0}\cdot \epsilon$. Taking into account (1) and (H₄), there exists x₂ ∈ Tx₁ such that

$$\begin{array}{ll}\hfill G\left(x_1,x_1,x_2\right)& \preccurlyeq \mathcal{H}\left(T{x}_0,T{x}_0,T{x}_1\right)+{\epsilon }_1\\ \preccurlyeq \phi \left(G\left(x_0,x_0,x_1\right)\right)\cdot G\left(x_0,x_0,x_1\right)+{\epsilon }_1\\ \preccurlyeq {\gamma }_{{\eta }_0}\cdot G\left(x_0,x_0,x_1\right)+{\epsilon }_1.\end{array}$$

(2)

Now, put G(x₁, x₁, x₂) = η₁, η₁ ≫ θ. By the definition of the stronger Meir-Keeler tυs-G-cone-type function φ : P - [0, 1), corresponding to η₁ use, there exist δ ₁ ≫ θ and ${\gamma }_{{\eta }_1}\in \left(0,1\right)$ with η ₁ ≼ G(x₁, x₁, x₂) ≺ η ₁ + δ ₁ such that $\phi \left(G\left(x_1,x_1,x_2\right)\right)<{\gamma }_{{\eta }_1}$. Put ${\alpha }_0={\gamma }_{{\eta }_0}$ and ${\alpha }_1=\mathsf{\text{max}}\left\{{\gamma }_{{\eta }_0},{\gamma }_{{\eta }_1}\right\}$. Then α₀, α ₁ ∈ (0, 1) and

$$\phi \left(G\left(x_0,x_0,x_1\right)\right)<{\gamma }_{{\eta }_0}\le \alpha <1\mathsf{\text{and}}\phi \left(G\left(x_{\mathsf{\text{1}}},x_1,x_2\right)\right)<{\gamma }_{{\eta }_1}\le {\alpha }_1<1.$$

Let ε₂ ∈ E such that θ ≪ ε₂ and ${\epsilon }_2\preccurlyeq {\gamma }_{{\eta }_1}^2\cdot \epsilon$. Then

$${\epsilon }_1\preccurlyeq {\alpha }_1\cdot \epsilon \mathsf{\text{and}}{\epsilon }_2\preccurlyeq {\alpha }_1^2\cdot \epsilon .$$

Taking into account (1), (2), and (H₄), there exists x₃ ∈ Tx₂ such that

$$\begin{array}{ll}\hfill G\left(x_2,x_2,x_3\right)& \preccurlyeq \mathcal{H}\left(T{x}_1,T{x}_1,T{x}_2\right)+{\epsilon }_2\\ \preccurlyeq \phi \left(G\left(x_1,x_1,x_2\right)\right)\cdot G\left(x_1,x_1,x_2\right)+{\epsilon }_2\\ \preccurlyeq {\alpha }_1\cdot G\left(x_1,x_1,x_2\right)+{\epsilon }_2\\ \preccurlyeq {\alpha }_1\left({\alpha }_1\cdot G\left(x_0,x_0,x_1\right)+{\epsilon }_1\right)+{\epsilon }_2\\ \preccurlyeq {\alpha }_1^2\cdot G\left(x_0,x_0,x_1\right)+{\alpha }_1\cdot {\epsilon }_1+{\epsilon }_2\\ \preccurlyeq {\alpha }_1^2\cdot G\left(x_0,x_0,x_1\right)+2{\alpha }_1^2\cdot \epsilon .\end{array}$$

(3)

We continue in this manner. In general, for x _n , $n\in \mathbb{N}$, x _{n +1} is chosen such that x _{n +1} ∈ Tx _n . Put G(x _n , x _n , x _{n +1}) = η _n , η _n ≫ θ. By the definition of the stronger Meir-Keeler tυs-G-cone-type function φ : P → [0, 1), corresponding to η _n use, there exist δ _n ≫ θ and ${\gamma }_{{\eta }_n}\in \left(0,1\right)$ with η _n ≼ G(x _n , x _n , x _{n +1}) ≺ η _n + δ _n such that $\phi \left(G\left(x_n,x_n,x_{n+1}\right)\right)<{\gamma }_{{\eta }_n}$. Put α_n = max $\left\{{\gamma }_{{\eta }_0},{\gamma }_{{\eta }_1},\dots ,{\gamma }_{{\eta }_n}\right\},n\in \mathbb{N}$. Then α _n ∈ (0, 1) and

$$\phi \left(G\left(x_i,x_i,x_{i+1}\right)\right)<{\gamma }_{{\eta }_i}\le {\alpha }_n<1,\mathsf{\text{for}}\mathsf{\text{all}}i\in \left\{0,1,2,\dots ,n\right\}.$$

(4)

On the other hand, for each $n\in \mathbb{N}$, corresponding to ${\gamma }_{{\eta }_n}$ use, we choose ε _{n +1} ∈ E such that θ ≪ ε _{n +1} and ${\epsilon }_{n+1}\preccurlyeq {\gamma }_{{\eta }_n}^{n+1}\cdot \epsilon$. Then

$${\epsilon }_{n+1}\preccurlyeq {\alpha }_n^{n+1}\cdot \epsilon .$$

(5)

From above argument, we can construct a sequence {x _n } in X, a non-decreasing sequence {α _n } and a sequence {ε _n } recursively as follow:

$$\begin{array}{c}{x}_{n+1}\in T{x}_n,\\ {\alpha }_n=\mathsf{\text{max}}\left\{{\gamma }_{{\eta }_0},{\gamma }_{{\eta }_1},\dots ,{\gamma }_{{\eta }_n}\right\}<1,\\ {\epsilon }_{n+1}\preccurlyeq {\gamma }_{{\eta }_n}^{n+1}\cdot \epsilon \preccurlyeq {\alpha }_n^{n+1}\cdot \epsilon ,\end{array}$$

for all $n\in \mathbb{N}\cup \left\{0\right\}$.

And, we have that for each $n\in \mathbb{N}\cup \left\{0\right\}$

$$G\left(x_{n+1},x_{n+1},x_{n+2}\right)\preccurlyeq \mathcal{H}\left(T{x}_n,T{x}_n,T{x}_{n+1}\right)+{\epsilon }_{n+1}.$$

Taking into account (4), (5), and (H₄), there exists x _{n +2} ∈ Tx _{n +1} such that

$$\begin{array}{ll}\hfill G\left(x_{n+1,}{x}_{n+1},x_{n+2}\right)& \preccurlyeq \mathcal{H}\left(T{x}_n,T{x}_n,T{x}_{n+1}\right)+{\epsilon }_{n+1}\\ \preccurlyeq \phi \left(G\left(x_n,x_n,x_{n+1}\right)\right)\cdot G\left(x_n,x_n,x_{n+1}\right)+{\epsilon }_{n+1}\\ \preccurlyeq {\alpha }_n\cdot G\left(x_n,x_n,x_{n+1}\right)+{\alpha }_n^{n+1}\cdot \epsilon \\ \preccurlyeq {\alpha }_n\left[\mathcal{H}\left(T{x}_{n-1},T{x}_{n-1},T{x}_n\right)+{\epsilon }_n\right]+{\alpha }_n^{n+1}\cdot \epsilon \\ \preccurlyeq {\alpha }_n\left[\phi \left(G\left(x_{n-1},x_{n-1},x_n\right)\right)\cdot G\left(x_n,x_n,x_{n+1}\right)+{\epsilon }_n\right]+{\alpha }_n^{n+1}\cdot \epsilon \\ \preccurlyeq {\alpha }_n\left[{\alpha }_n\cdot G\left(x_n,x_n,x_{n+1}\right)+{\epsilon }_n\right]+{\alpha }_n^{n+1}\cdot \epsilon \\ \preccurlyeq {\alpha }_n^2\cdot G\left(x_n,x_n,x_{n+1}\right)+{\alpha }_n\cdot {\epsilon }_n+{\alpha }_n^{n+1}\cdot \epsilon \\ \preccurlyeq {\alpha }_n^2\cdot G\left(x_n,x_n,x_{n+1}\right)+2{\alpha }_n^{n+1}\cdot \epsilon \\ \preccurlyeq \cdots \cdots \\ \preccurlyeq {\alpha }_n^{n+1}\cdot G\left(x_0,x_0,x_1\right)+\left(n+1\right){\alpha }_n^{n+1}\cdot \epsilon .\end{array}$$

(6)

Let m, $n\in \mathbb{N}$ be such that m > n. From (6) we conclude that

$$\begin{array}{ll}\hfill G\left(x_{n,}{x}_{n,}{x}_m\right)& \preccurlyeq \sum _{j=n}^{m-1}G\left(x_j,x_j,x_{j+1}\right)\\ \preccurlyeq \sum _{j=n}^{m-1}\left[{\alpha }_{j-1}^j\cdot G\left(x_0,x_0,x_1\right)+j{\alpha }_{j-1}^j\cdot \epsilon \right].\end{array}$$

(7)

From above argument and the inequality (7), we put α = max{α _{n -1}, α _n , α _{n+ 1}, ..., α _{m -2}}. Then, we get α = α _{m -2} < 1 and

$$\begin{array}{ll}\hfill G\left(x_n,x_n,x_m\right)& \preccurlyeq \sum _{j=n}^{m-1}\left[{\alpha }^j\cdot G\left(x_0,x_0,x_1\right)+j{\alpha }^j\cdot \epsilon \right]\\ \preccurlyeq \frac{{\alpha }^n}{1-\alpha }G\left(x_0,x_0,x_1\right)+\sum _{j=n}^{m-1}j{\alpha }^j\cdot \epsilon \\ \preccurlyeq \frac{{\alpha }^n}{1-\alpha }G\left(x_0,x_0,x_1\right)+{\alpha }^n\frac{n+\alpha }{{\left(1-\alpha \right)}^2}\cdot \epsilon .\end{array}$$

Since $\underset{n\to \infty }{\mathsf{\text{lim}}}\frac{{\alpha }^n}{1-\alpha }=0$ and $\underset{n\to \infty }{\mathsf{\text{lim}}}{\alpha }^n\frac{n+\alpha }{{\left(1-\alpha \right)}^2}=0$ we obtain that

$$\frac{\alpha }{1-\alpha }G\left(x_{0,}{x}_{0,}{x}_1\right)+{\alpha }^n\frac{n+\alpha }{{\left(1-\alpha \right)}^2}\cdot \epsilon \to \theta$$

in locally convex space E as → ∞.

Apply Remark 2, we conclude that for every τ ∈ E with θ ≪ τ there exists $n_0\in \mathbb{N}$ such that G(x _n , x _n , x _m ) ≪ τ for all m, n ≥ n₀. So {x _n } is a tυs-G-cone Cauchy sequence. Since (X, G) is a tυs-G-cone complete metric space, {x _n } is tυs-G-cone convergent in X and G-cone-lim_n→∞x _n = x. Thus, for every τ ∈ intP and sufficiently large n, we have

$$\mathcal{H}\left(T{x}_{n,}T{x}_{n,}Tx\right)\preccurlyeq \alpha \cdot G\left(x_{n,}{x}_{n,}x\right)\cdot \alpha \frac{\tau }{3\alpha }=\frac{\tau }{3}.$$

Since for $n\in \mathbb{N}\cup \left\{0\right\}$, x _{n +1} ∈ Tx _n , by (H₄), we obtain that for all $n\in \mathbb{N}$ there exist y _n ∈ T _x such that

$$G\left(x_{n+1,}{x}_{n+1,}{y}_{n+1}\right)\preccurlyeq \mathcal{H}\left(T{x}_{n,}T{x}_{n,}Tx\right)+{\epsilon }_{n+1}\preccurlyeq \alpha \cdot G\left(x_{n,}{x}_{n,}x\right)+{\alpha }^{n+1}\epsilon .$$

Then for sufficiently large n, we obtain that

$$G\left(y_{n+1,}x,x\right)\preccurlyeq G\left(y_{n+1,}{x}_{n+1,}{x}_{n+1}\right)+G\left(x_{n+1,}x,x\right)\ll \frac{2\tau }{3}+\frac{\tau }{3}=\tau ,$$

which implies G-cone-lim_n→∞y _n = x. Since Tx is closed, we obtain that x ∈ Tx. □

Follows Theorem 1, we immediate get the following corollary.

Corollary 1 Let (X, G) be a tυs-G-cone complete metric space with a solid cone P and let be a collection of nonempty closed subsets of X, $\mathcal{A}\ne \varphi$, and let $\mathcal{H}:\mathcal{A}\times \mathcal{A}\times \mathcal{A}\to E$ be a $t\upsilon s-\mathcal{H}-cone$ metric with respect to G. If the mapping $T:X\to \mathcal{A}$ satisfies the condition that exists α ∈ (0, 1) such that for all x, y, z ∈ X holds

$$\mathcal{H}\left(Tx,Ty,Tz\right)\preccurlyeq \alpha \cdot G\left(x,y,z\right)$$

then T has a fixed point in X.