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.

1. Introduction and Preliminaries

In 2007, 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 studied this subject and many results on fixed point theory are proved (see, e.g., [2–15]). Recently, Du [16] introduced the concept of tvs-cone metric and tvs-cone metric space to improve and extend the concept of cone metric space in the sense of Huang and Zhang [1]. Later, in the papers [16–21], the authors tried to generalize this approach by using cones in topological vector spaces tvs instead of Banach spaces. However, it should be noted that an old result shows that if the underlying cone of an ordered tvs is solid and normal, then such tvs must be an ordered normed space. Thus, proper generalizations when passing from norm-valued cone metric spaces to tvs-valued cone metric spaces can be obtained only in the case of nonnormal cones (for details, see [19]). Further, Radenović et al. [22] introduced the concept of set-valued contraction of Nadler type in the setting of tvs-cone spaces and proved a fixed point theorem in the setting of tvs-cone spaces with respect to a solid cone.

Definition 1.1 (see [<xref ref-type="bibr" rid="B24">22</xref>]).

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:

ℋ(A1,A2)=θ⇒A1=A2,

ℋ(A1,A2)=ℋ(A2,A1),

∀ε∈E,θ≪ε∀x∈A1∃y∈A2d(x,y)≼ℋ(A1,A2)+ε,

one of the following is satisfied:

∀ε∈E,θ≪ε∃x∈A1∀y∈A2ℋ(A1,A2)≼d(x,y)+ε,

∀ε∈E,θ≪ε∃y∈A2∀x∈A1ℋ(A1,A2)≼d(x,y)+ε.

Theorem 1.2 (see [<xref ref-type="bibr" rid="B24">22</xref>]).

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
(1.1)H(Tx,Ty)≼λG(x,y)
then T has a fixed point in X.

We recall some definitions and results of the tvs-cone metric spaces that introduced in [19, 23], which will be needed in the sequel.

Let E be be a real Hausdorff topological vector space (tvs 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 tvs 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.

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:

∀x,y∈X,x≠yθ≼d(x,y),

∀x,y∈Xd(x,y)=θ⇔x=y,

∀x,y∈Xd(x,y)=d(y,x),

∀x,y,z∈Xd(x,z)≼d(x,y)+d(y,z).

Then the pair (X,d) is called a tvs-cone metric space.
Definition 1.4 (see [<xref ref-type="bibr" rid="B16">16</xref>, <xref ref-type="bibr" rid="B18">18</xref>, <xref ref-type="bibr" rid="B19">19</xref>]).

Let (X,d) be a tvs-cone metric space, x∈X, and {xn} a sequence in X.

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

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

(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.

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.

If u≼v and v≪w, then u≪w,

If u≪v and v≼w, then u≪w,

If u≪v and v≪w, then u≪w,

If θ≼u≪c for each c∈intP, then u=θ,

If a≼b+c for each c∈intP, then a≼b,

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

If c∈intP, an∈E, and an→θ in locally convex tvsE, then there exists n0∈ℕ such that an≪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 [24]. In 2003, Mustafa and Sims [25] introduced a more appropriate and robust notion of a generalized metric space as follows.

Definition 1.7 (see [<xref ref-type="bibr" rid="B22">25</xref>]).

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

∀x,y,z∈XG(x,y,z)=0⇔x=y=z,

∀x,y∈X,x≠yG(x,x,y)>0,

∀x,y,z∈XG(x,y,z)≥G(x,x,y),

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

∀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 tvs-cone metrics, we introduce the below notion of tvs-generalized-cone metrics.

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:

∀x,y,z∈XG(x,y,z)=θ if and only if x=y=z,

∀x,y∈X,x≠yθ≪G(x,x,y),

∀x,y,z∈XG(x,x,y)≼G(x,y,z),

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

∀x,y,z,w∈XG(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 {xn} a sequence in X.

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

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

(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 {xn} a sequence in X. The following are equivalent:

{xn}tvs-G-cone converges to x,

G(xn,xn,x)→θ as n→∞,

G(xn,x,x)→ as n→∞,

G(xn,xm,x)→θ as n,m→∞.

In this paper, we also introduce the below concept of the 𝒞ℬ𝒲-tvs-G-cone-type function.

Definition 1.11.

One callsφ:intP∪{θ}→intP∪{θ} a 𝒞ℬ𝒲-tvs-G-cone-type function if the function φ satisfies the following condition

φ(t)≪t for all t≫θ and φ(θ)=θ;

limn→∞φn(t)=θ for all t∈intP∪{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.

2. Main Results

Let E be a locally convex Hausdorff tvs 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 tvs-ℋ-cone metric ℋ with respect to tvs-G-cone metric G.

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:

ℋ(A1,A2,A3)=θ⇒A1=A2=A3,

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

ℋ(A1,A1,A2)≼ℋ(A1,A2,A3),

∀ε∈E,θ≪ε∀x∈A1,y∈A2∃z∈A3G(x,y,z)≼ℋ(A1,A2,A3)+ε,

one of the following is satisfied:

∀ε∈E,θ≪ε∃x∈A1∀y∈A2,z∈A3ℋ(A1,A2,A3)≼G(x,y,z)+ε,

∀ε∈E,θ≪ε∃y∈A2∀x∈A1,z∈A3ℋ(A1,A2,A3)≼G(x,y,z)+ε,

∀ε∈E,θ≪ε∃z∈A3∀y∈A2,x∈A1ℋ(A1,A2,A3)≼G(x,y,z)+ε.

We will prove that a tvs-ℋ-cone metric satisfies the conditions of (G1)–(G5).

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 {εn}⊂E be a sequence such that θ≪εn for all n∈ℕ and G-cone-limn→∞εn=θ. Take any A1,A2,A3∈𝒜 and x∈A1, y∈A2. From (H4), for each n∈ℕ, there exists zn∈A3 such that
(2.1)G(x,y,zn)≼H(A1,A2,A3)+εn.
Therefore, ℋ(A1,A2,A3)+εn∈P for each n∈ℕ. By the closedness of P, we conclude that θ≼ℋ(A1,A2,A3).

Assume that A1=A2=A3. From H_{5}, we obtain ℋ(A1,A2,A3)≼εn 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 xn∈A1 such that ℋ(A1,A2,A3)≼G(xn,y,z)+εn for all y∈A2 and z∈A3. From (H4), there exists a sequence {wn}⊂A4 satisfying G(xn,wn,wn)≼ℋ(A1,A4,A4)+εn for every n∈ℕ. Obviously, for any y∈A2 and any z∈A3 and n∈ℕ, we have
(2.2)H(A1,A2,A3)≼G(xn,y,z)+εn≼G(xn,wn,wn)+G(wn,y,z)+εn.
Now for each n∈ℕ, there exists yn∈A2, zn∈A3 such that G(wn,yn,zn)≼ℋ(A4,A2,A3)+εn. Consequently, we obtain that for each n∈ℕ(2.3)H(A1,A2,A3)≼H(A1,A4,A4)+H(A4,A2,A3)+3εn.
Therefore,
(2.4)H(A1,A2,A3)≼H(A1,A4,A4)+H(A4,A2,A3).
In the case when (H5)(ii) or (H5)(iii) holds, we use the analogous method.

In the sequel, we denote by Θ the class of functions φ:intP∪{θ}→intP∪{θ} satisfying the following conditions:

φ is a 𝒞ℬ𝒲-tvs-G-cone-type-function;

φ is subadditive, that is, φ(u1+u2)≼φ(u1)+φ(u2) for all u1,u2∈intP.

Our main result is the following.

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
(2.5)H(Tx,Ty,Tz)≼φ(G(x,y,z)),
then T has a fixed point in X.

Proof.

Let us choose ε∈intP arbitrarily, and let εn∈E be a sequence such that θ≪εn and εn≼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
(2.6)G(x1,x1,x2)≼H(Tx0,Tx0,Tx1)+ε1≼φ(G(x0,x0,x1))+ε1.
Taking into account (2.5), (2.6), and (H4) and since φ∈Θ, there exists x3∈Tx2 such that
(2.7)G(x2,x2,x3)≼H(Tx1,Tx1,Tx2)+ε2≼φ(G(x1,x1,x2))+ε2≼φ(φ(G(x0,x0,x1))+ε1)+ε2≼φ(φ(G(x0,x0,x1)))+φ(ε1)+ε2≪φ2(G(x0,x0,x1))+ε1+ε2≼φ2(G(x0,x0,x1))+ε3+ε32.
We continue in this manner. In general, for xn, n∈ℕ, xn+1 is chosen such that xn+1∈Txn and
(2.8)G(xn,xn,xn+1)≼H(Txn-1,Txn-1,Txn)+εn≪φ(G(xn-1,xn-1,xn))+εn≼φ2(G(xn-2,xn-2,xn-1))+εn-1+εn≼⋯⋯≼φn(G(x0,x0,x1))+∑i=1nεi≼φn(G(x0,x0,x1))+∑i=1nε3i≼φn(G(x0,x0,x1))+12ε.
Since ε is arbitrary, letting ε→θ and by the definition of the 𝒞ℬ𝒲-tvs-G-cone-type function, we obtain that
(2.9)limn→∞G(xn,xn,xn+1)=θ.

Next, we let cm=G(xm,xm+1,xm+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),
(2.10)G(xm,xm+1,xm+1)≪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 mp,np∈ℕ with mp>np≥p satisfying

mp is even and np is odd,

G(xmp,xnp,xnp)≽γ, and

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

Since cm↓θ, by (ii), we have that limp→∞G(xmp,xnp,xnp)=γ and
(2.11)γ≼G(xmp,xnp,xnp)≼G(xmp,xmp+1,xmp+1)+G(xmp+1,xnp+1,xnp+1)+G(xnp+1,xnp,xnp).
It follows from (H4); let us choose ε∈E arbitrarily such that
(2.12)G(xnp+1,xnp+1,xmp+1)≼H(Txnp+1,Txnp+1,Txmp+1)+ε.
Taking into account (2.5), (2.11), and (2.12), we have that
(2.13)γ≼G(xmp,xnp,xnp)≼G(xmp,xmp+1,xmp+1)+H(Txnp+1,Txnp+1,Txmp+1)+ε+G(xnp+1,xnp,xnp)≼G(xmp,xmp+1,xmp+1)+φ(G(xnp,Gxnp,Gxmp))+ε+G(xnp+1,xnp,xnp)≪G(xmp,xmp+1,xmp+1)+G(xnp,Gxnp,Gxmp)+ε+G(xnp+1,xnp,xnp).
Since ε is arbitrarily, letting ε→θ and by letting p→∞, we have
(2.14)γ≪θ+limp→∞G(xmp,xnp,xnp)+θ+θ=γ,
a contradiction. So {xn} is a tvs-G-cone Cauchy sequence. Since (X,G)is a tvs-G-cone complete metric space, {xn} is tvs-G-cone convergent in X and G-cone-limn→∞xn=x. Thus, for every τ∈intP and sufficiently large n, we have that
(2.15)H(Txn,Txn,Tx)≼φ(G(xn,xn,x))≪G(xn,xn,x)≪τ3.
Since for n∈ℕ∪{0}, xn+1∈Txn, by (H4) we obtain that for all n∈ℕ there existsyn∈Tx such that
(2.16)G(xn+1,xn+1,yn+1)≼H(Txn,Txn,Tx)+εn+1≼φ(G(xn,xn,x))+ε3n+1≪G(xn,xn,x)+ε3n+1.
Since ε/3n+1→θ, then for sufficiently large n, we obtain that
(2.17)G(yn+1,x,x)≼G(yn+1,xn+1,xn+1)+G(xn+1,x,x)≪2τ3+τ3=τ,
which implies G-cone-limn→∞yn=x. Since Tx is closed, we obtain that x∈Tx.

For the case φ(t)=kt, k∈(0,1), then φ∈Θ and it is easy to get the following corollary.

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
(2.18)H(Tx,Ty,Tz)≼k⋅G(x,y,z),
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]).

Acknowledgment

The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper.

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