Pre-Symmetric w-Cone Distances and Characterization of TVS-Cone Metric Completeness

Abstract

Motivated by two definitions of distance, “pre-symmetric w-distance” and “w-cone distance”, we define the concept of a pre-symmetric w-cone distance in a TVS-CMS and introduce its properties and examples. Also, we discuss the TVS-cone version of the recent results obtained by Romaguera and Tirado. Meanwhile, using Minkowski functionals, we show the equivalency between some consequences concerning a pre-symmetric w-distance in a usual metric space and a pre-symmetric w-cone distance in a TVS-CMS. Then, some types of various w-cone-contractions and the relations among them are investigated. Finally, as an application, a characterization of the completeness of TVS-cone metric regarding pre-symmetric concept is performed, which differentiates our results from former characterizations.

1. Introduction and Preliminaries

Presume that $(X,d)$ is a metric space and $w:X\times X\to [0,+\infty )$ is a function convincing the following properties for all $x,y,z\in X$:

$\left(w_1\right)$

$w(x,z)\le w(x,y)+w(y,z)$;

$\left(w_2\right)$

w is lower semicontinuous (in short, lsc) in its second variable; i.e., if $y_n\to y$ in X, $w(x,y)\le {lim\; inf}_{n}w(x,y_n)$;

$\left(w_3\right)$

For any $ϵ>0$, there is $\delta >0$ provided that $w(z,x)\le \delta$ and $w(z,y)\le \delta$ imply $d(x,y)\le ϵ$.

Then w is called a w-distance.

It is claimed by many mathematicians that the concept of the w-distance, defined by Kada et al. [1] in 1996, is a powerful instrument in metric spaces which has many applications in optimization, especially nonconvex minimization problems. In addition to applications, some authors analyzed the properties and generalizations of a w-distance and discussed several contractions regarding this distance and its extensions in [2,3,4,5,6,7,8,9,10,11,12,13,14] and their references. Moreover, other authors characterized complete metric spaces regarding this distance and its generalizations [15,16,17,18,19,20]. Although each usual metric is a w-distance, the converse is not correct. Also, note that a w-distance has two important differences from a usual metrics. First, $w(x,y)=0$ is not absolutely equivalent to $x=y$. Second, a w-distance is not necessarily symmetric. The distance $w:\mathbb{R}\times \mathbb{R}\to [0,+\infty )$ considered by $w(x,y)=y$ for any $x,y\in \mathbb{R}$ is a good example of these differences. Moreover, we say a w-distance is symmetric whenever $w(x,y)=w(y,x)$. For instance, $w:\mathbb{R}\times \mathbb{R}\to [0,+\infty )$ considered by $w(x,y)=x+y$ for every $x,y\in \mathbb{R}$ is a symmetric w-distance. For more details on w-distance, we refer the reader to Rakočević’s book [21] and Babaei et al.’s survey [22]. On the other hand, note that if $W(x,y)=max\left\{w\right(x,y),w(y,x\left)\right\}$, then the condition $\left(w_2\right)$ is not generally valid for W. But if w is symmetric, then $W=w$ and W satisfies $\left(w_2\right)$. Regarding this discussion, Romaguera and Tirado 2024 [23] defined the concept of pre-symmetric as follows:

Definition 1

([23]). A w-distance is said to be pre-symmetric if

• When $\left(x_n\right)\subset X$ provided that $x_n\to x$ and $w(x_n,x)\to 0$ for a $x\in X$, there exists a subsequence $\left(x_{k\left(n\right)}\right)$ of $\left(x_n\right)$ so that $w(x,x_{k\left(n\right)+1})\le w(x_{k\left(n\right)},x)$ for every positive integer n.

On the other hand, regarding the definition of a topological vector space-cone metric space (in short, TVS-CMS), Ćirić et al. [6] stated the concept of w-cone distance, a generalization of w-distance. To introduce this concept, we first review some basic definitions from [24,25,26,27].

Presume that E is a real Hausdorff topological vector space and $\theta$ is a zero vector. A closed $\varnothing \ne P\subset E$ is a cone when $P+P\subset P$, $\alpha P\subset P$ for $\alpha \ge 0$ and $P\cap (-P)=\left\{\theta \right\}$. Also, we define a partial order ⪯ regarding P by $x⪯y$ iff $y-x\in P$. Note that $x\prec y$ when $x⪯y$ and $x\ne y$, and $x\ll y$ iff $y-x\in intP$ in which $intP$ is the interior of P. In this way, P is solid when $intP\ne \varnothing$. Moreover, $(E,P)$ is an ordered TVS. For $x,y\in E$ with $x⪯y$, define $[x,y]=\{z\in E:x⪯z⪯y\}$. $A\subset E$ is an order-convex when $[x,y]\subset A$. Especially, $(E,P)$ is order-convex when it includes a base of neighborhoods of $\theta$ consisting of order-convex subsets. In this manner, P will be a normal cone. When E is a normed space, it means that the unit ball is order-convex, i.e., there exists $K\ge 0$ provided that

$$\begin{array}{c}\hfill \theta ⪯x⪯y\Rightarrow \parallel x\parallel \le K\parallel y\parallel \end{array}$$

for each $x,y\in E$.

Lemma 1

([28]). If a cone in an ordered TVS is normal and solid, TVS will become an ordered normed space.

In 2010, Du [24] first introduced a TVS-CMS as follows:

Presume that $X\ne \varnothing$ and $(E,P)$ is an ordered TVS. A mapping $d:X\times X\to P$ is named a TVS-cone metric when for each $x,y,z\in X$:

$\left(d_1\right)$

$d(x,y)=\theta$ iff $x=y$;

$\left(d_2\right)$

$d(x,y)=d(y,x)$;

$\left(d_3\right)$

$d(x,z)⪯d(x,y)+d(y,z)$.

In this manner, $(X,d)$ will be a TVS-CMS. For the definition of convergence and completeness, see [24].

Then, Du studied the equivalency between Banach contraction principle in a usual metric space and in a TVS-CMS by applying a nonlinear scalarization function (also, for other methods, see [25,29]).

Definition 2

([6]). Presume that $(X,d)$ is a TVS-CMS. A function $q:X\times X\to P$ is named w-cone distance on X when for each $x,y,z\in X$,

$\left(c_1\right)$

$q(x,z)⪯q(x,y)+q(y,z)$;

$\left(c_2\right)$

$q(x,·):X\to P$ is lsc;

$\left(c_3\right)$

for each $c\in intP$, there is $\theta \ll e$ provided that $q(z,x)\ll e$ and $q(z,y)\ll e$ imply $d(x,y)\ll c$.

Remember that a function $f:X\to P$ is lsc at $x\in X$ if there is $n_0\in \mathbb{N}$ provided that $f\left(x\right)⪯f\left(x_n\right)+c$ for any $c\in intP$ and $n\ge n_0$ in which $x_n\to x$ in X.

It should be noted that each w-distance is a w-cone distance but the inverse is not necessarily valid. Also, each TVS-CMS is a w-cone distance but the inverse does not hold. Like w-distances, w-cone distances have two major differences from a TVS-CMS. First, $q(x,y)=\theta$ is not absolutely equivalent to $x=y$; second, a w-cone distance is not necessarily symmetric. For these manners, see Example 1.

Remark 1.

Regarding the definition of c-sequence, $\left(c_2\right)$ can equivalently be as follows:

$\left(c_2^{\prime }\right)$

If $y_n\to y$ in X while $n\to +\infty$ and $g\left(y\right)=q(x,y)$, $g\left(y\right)-g\left(y_n\right)$ will be a c-sequence.

Note that we say that $\left(x_n\right)\subset E$ is a c-sequence when for every $c\in intP$ there is $n_0\in \mathbb{N}$ provided that $x_n\ll c$ for each $n\ge n_0$.

Remark 2.

Notice that the w-cone distance in a TVS-CMS is different from the definition of c-distance introduced by Cho et al. [4] as the following condition is substituted by $\left(c_2\right)$:

• For $x\in X$ and a $y_n\to y$ in X, $q(x,y_n)⪯v$ for a $v=v_x$ and each $n\ge 1$ implies that $q(x,y)⪯v$.

Indeed, every w-cone distance is a c-distance, but the converse is not necessarily correct.

Example 1.

Take $E=C_{\mathbb{R}}[0,1]$ with the maximum norm and cone $P=\{f\in E:f(t)\ge 0\mathit{for}t\in [0,1\left]\right\}$. Also, presume that ${\tau }^{*}$ is the strongest locally convex topology on E. Then, P is a non-normal solid cone. Now, assume that $X\subset [0,+\infty )$ and $d:X\times X\to P\subset (E,{\tau }^{*})$ is defined by $d(x,y)\left(t\right)=|x-y|\phi \left(t\right)$ for a fixed $\phi \in P$. Then, $(X,d)$ will be a TVS-CMS. The following are w-cone distances on this space:

$$\begin{array}{cc}\hfill q_1(x,y)\left(t\right)& =y\phi \left(t\right);\hfill \\ \hfill q_2(x,y)\left(t\right)& =(ax+by)\phi \left(t\right),a,b\in \mathbb{R}.\hfill \end{array}$$

Note that $q_2$ in Example 1 is a symmetric w-cone distance if we take $a=b$. For other properties, results, and examples of these types of distances and c-distances, we refer the reader to [4,5,6,7,8,11,14].

This article is organized into five sections. In Section 1, we gave a historical discussion of the subject of distances and spaces along with some former definitions and results. In Section 2, motivated by two Definitions 1 and 2, we introduce the concept of a pre-symmetric w-cone distance in a TVS-CMS and discuss its properties and examples. Then, we obtain the TVS-cone version of the recent results stated by Romaguera and Tirado, and also, using Minkowski functionals, we establish the equivalency between some consequences with respect to a pre-symmetric w-distance in a usual metric space and a pre-symmetric w-cone distance in a TVS-CMS. In Section 3, some various w-cone-contractions and the relation among them will be investigated. Then, as an application, a characterization of the completeness of TVS-cone metric red regarding the pre-symmetric concept will also be carried out in Section 4. Finally, in Section 5, we will give some suggestions to continue the paper regarding pure and applied issues.

2. Pre-Symmetric $\mathit{w}$-Cone Distance and Suzuki-Type Contraction

It is easily shown by many authors that the TVS-cone metric version of Suzuki’s theorem [16], introduced in the following, is valid.

Theorem 1.

Assume that $(X,d)$ is a complete TVS-CMS and $S:X\to X$ is a mapping. If

$$d(x,Sx)⪯2d(x,y)\Rightarrow d(Sx,Sy)⪯\mu d(x,y)$$

(1)

for each $x,y\in X$ in which $\mu \in (0,1)$, S possesses a unique fixed point (in short, FP).

Note that (1) is known as the cone version of Suzuki-type contraction. Let us start with a new extension of this contraction within a w-cone distance.

Definition 3.

Presume q is a w-cone distance on a TVS-CMS. A self-mapping S on X is named w-cone contraction of Suzuki-type when there is $\mu \in (0,1)$ provided that

$$q(x,Sx)⪯2q(x,y)\Rightarrow q(Sx,Sy)⪯\mu q(x,y)$$

(2)

for any $x,y\in X$.

Note that if we substitute (1) by (2), then the assertion of Theorem 1 may not be correct. For this, we refer to ([23], Example 4) as follows, but we can also consider other cones like the same considered in Examples 1, 3 and 5.

Example 2.

Presume that $X=[0,1]$, $E=\mathbb{R}$, $P=[0,+\infty )$, $d:X\times X\to P$ is defined by $d(x,y)=|x-y|$ and $q(x,y)=y$. It is clear that q is a w-cone distance with the cone P. Also, let $S:X\to X$ be defined by

$$\begin{array}{c}\hfill S\left(x\right)=\left\{\begin{array}{cc}1,\hfill & x=0\hfill \\ \hfill & \\ {\displaystyle \frac{x}{2}},\hfill & x\in (0,1]\hfill \end{array}\right..\end{array}$$

It is clear that S does not have any fixed point, but it satisfies (2) with $\mu =\frac{1}{2}\in (0,1)$. Therefore, the assertion of Theorem 1 is not necessarily valid for w-cone contraction of Suzuki-type.

To find a way that solve this problem, we now define a pre-symmetric w-cone distance.

Definition 4.

A w-cone distance q on a TVS-CMS is said to be pre-symmetric if

• When $x_n\to x$ in X and $q(x_n,x)\to \theta$ for a $x\in X$, there exists a ${\left(x_{k\left(n\right)}\right)}_{n\in \mathbb{N}}$ of $\left(x_n\right)$ provided that $q(x,x_{k\left(n\right)+1})⪯q(x_{k\left(n\right)},x)$ for each $n\in \mathbb{N}$.

Let us first review notions in Definition 4.

• $x_n\to x$ signifies that $d(x_n,x)$ is a c-distance in E.
• $q(x_n,x)\to \theta$ means that for each $c\in intP$, there is $n_0\in \mathbb{N}$ so that $q(x_n,x)\ll c$ for each $n\ge n_0$.
• $q(x,x_{k\left(n\right)+1})⪯q(x_{k\left(n\right)},x)$ means that $q(x_{k\left(n\right)},x)-q(x,x_{k\left(n\right)+1})\in P$.

Lemma 2.

Each symmetric w-cone distance on a TVS-CMS is pre-symmetric.

Proof. 

Using Proposition 1 from [23] and the convergence properties in a TVS-CMS, the proof is straightforward. □

Example 3.

Take $E=C_{\mathbb{R}}[0,1]$, maximum norm, P, ${\tau }^{*}$, $a,b$ and d as defined in Example 1. If $X=[0,1]$, then $(X,d)$ is a TVS-CMS. Also, define $q:X\times X\to P$ by $q(x,y)\left(t\right)=(ax+by)\phi \left(t\right)$ in which a is nonnegative, b is positive and $a\ge b$. Evidently, q is a pre-symmetric w-cone distance. If $X=\left\{0\right\}\cup Y$ in which Y is a subset of $[g,+\infty )$ where g is positive, then $(X,d)$ is a discrete TVS-CMS. Again take $q:X\times X\to P$ by $q(x,y)\left(t\right)=(ax+by)\phi \left(t\right)$ in which a is nonnegative and b is positive with $a<b$. Clearly, q is a pre-symmetric w-cone distance which is not symmetric.

Example 4

([23], Example 5). Presume that $X=[0,+\infty )$, $E=\mathbb{R}$, $P=[0,+\infty )$, $d:X\times X\to P$ is given by $d(x,y)=|x-y|$, q is a w-cone distance in which $q(x,y)⪰h$ for a $h\in intP$ and each $x,y\in X$. Obviously, d is a TVS-CMS and q is a pre-symmetric w-cone distance.

Example 5.

Take $E=C_{\mathbb{R}}^1[0,1]$, $\parallel \psi \parallel ={\parallel \psi \parallel }_{\infty }+{\parallel {\psi }^{\prime }\parallel }_{\infty }$ and $P=\{\psi \in E:\psi (t)\ge 0\}$. Also, presume that ${\tau }^{*}$ as defined in the Example 1. Then, $P\subset E$ is a non-normal solid cone. Now, assume that $X\subset [0,+\infty )$ and $d:X\times X\to P\subset (E,{\tau }^{*})$ is given by $d(x,y)\left(t\right)=|x-y|\psi \left(t\right)$ with $\psi \left(t\right)=e^t$. Then, $(X,d)$ is a TVS-CMS. Next, presume that $q:X\times X\to X$ is given by $q(x,y)=x+y$. Evidently, q is a symmetric w-cone distance, and, using Lemma 2, it will be a pre-symmetric w-cone distance, which is not a pre-symmetric w-distance.

The following is the basic theorem of this article that generalizes Theorem 1.

Theorem 2.

Presume that q is a pre-symmetric w-cone distance on a complete TVS-CMS $(X,d)$ and S is a w-cone contraction of Suzuki-type, i.e., it satisfies (2). Then, S possesses a unique FP $\kappa \in X$. Moreover, $q(\kappa ,\kappa )=\theta$.

To prove Theorem 2, we first need some notions and a preliminary proposition.

Suppose that $U\subset E$ is absolutely convex and absorbing and recall that the Minkowski functional ${\nu }_U\left(y\right)=inf\{\beta >0:y\in \beta U\}$ for $y\in E$ is a semi-norm [26]. Note that $M\subset N$ induces ${\nu }_N\left(y\right)\le {\nu }_M\left(y\right)$ for $y\in E$. Now, let $(E,P)$ be an ordered TVS, P be solid, and $f\in intP$. Then $[-f,f]=(P-f)\cap (f-P)$ is an absolutely convex neighborhood of $\theta$ [27], and ${\nu }_f$ is Minkowski functional ${\nu }_{[-f,f]}$. Also, $int[-f,f]=(intP-f)\cap (f-intP)$. Moreover, ${\nu }_f$ is nondecreasing. Further, note that if P is both solid and normal, then semi-norm ${\nu }_f$ will become a norm. For more details, see [25,26,27,28,29].

Proposition 1.

Presume that P is a solid cone with $f\in intP$, q is a pre-symmetric w-cone distance on a TVS-CMS $(X,d)$, and $(X,d_{\nu })$ is a usual metric space with $d_{\nu }={\nu }_f\circ d$ in which ${\nu }_f$ is a Minkowski functional. Then $w={\nu }_f\circ q$ is a pre-symmetric w-distance.

Proof. 

It was shown by Kadelburg et al. ([25], Theorem $3.1$) that $d_{\nu }$ is a usual metric when d is a TVS-CMS and $d_{\nu }={\nu }_f\circ d$ (also, see [29]). Moreover, it was proved by Kadelburg and Radenović [7] that w is a w-distance when q is a w-cone distance and $w={\nu }_f\circ q$ (also, see [22]). Now, assume that q is a pre-symmetric w-cone distance. Thus, if $x_n\to x$ in X and $q(x_n,x)\to \theta$ for a $x\in X$, there exists a ${\left(x_{k\left(n\right)}\right)}_{n\in \mathbb{N}}$ of $\left(x_n\right)$ provided that $q(x,x_{k\left(n\right)+1})⪯q(x_{k\left(n\right)},x)$ for any positive integer n. That is, if $d(x_n,x)$ is a c-sequence and there is $n_0\in \mathbb{N}$ so that $q(x_n,x)\ll c$ for each $n\ge n_0$ and $c\in intP$, then $q(x,x_{k\left(n\right)+1})⪯q(x_{k\left(n\right)},x)$. Now, applying $d_{\nu }={\nu }_f\circ d$ and $w={\nu }_f\circ q$, it follows from the properties of ${\nu }_f$ and $\left(p_1\right)$-$\left(p_6\right)$ for a TVS-CMS in [7] that when $x_n\to x$ in X and $w(x_n,x)\to 0$ for a $x\in X$, there exists $\left(x_{k\left(n\right)}\right)$ of $\left(x_n\right)$ with $w(x,x_{k\left(n\right)+1})\le w(x_{k\left(n\right)},x)$ for any positive integer n, i.e., w is a pre-symmetric w-distance and the proof ends. □

Proof of the Theorem 2.

Presume that metric $d_{\nu }$ and pre-symmetric distance w are the same defined in Proposition 1; that is, $d_{\nu }={\nu }_f\circ d$ and $w={\nu }_f\circ q$ in which $f\in intP$, ${\nu }_f$ is the Minkowski functional, d is a TVS-CMS, and q is a pre-symmetric x-cone distance. Now, it follows from ${\nu }_f$ and (2) that

$$\begin{array}{c}\hfill w(x,Sx)\le 2w(x,y)\Rightarrow w(Sx,Sy)\le \mu w(x,y)\end{array}$$

for every $x,y\in X$; that is, S satisfies the basic w-contraction of Suzuki-type from Romaguera and Tirado’s point of view. Also, since the TVS-CMS $(X,d)$ is complete, $(X,d_{\nu })$ is complete (Theorem $3.2$ of Kadelburg et al. [25]). Therefore, the result follows from the proof of Theorem 2 of Romaguera and Tirado [23] and, hence, the proof ends. □

Note that the scalarization method of Du [24] also shows that there exists an equivalency between some results in a usual metric space and a TVS-CMS. However, it is not discussed regarding (pre-symmetric) w-cone distance and (pre-symmetric) w-distance.

Question: Is it possible to introduce Proposition 1 and Theorem 2 regarding the scalarization method of Du?

Note that this question can also be asked for the results obtained in the following sections.

The next result can be concluded immediately by applying Lemma 2 and Theorem 2.

Corollary 1.

Presume that q is a symmetric w-cone distance on a TVS-CMS $(X,d)$ and S is a w-cone contraction of Suzuki-type, i.e., it satisfies (2). Then, S possesses a unique FP $\kappa \in X$. Moreover, $q(\kappa ,\kappa )=\theta$.

Example 6.

Take $E=C_{\mathbb{R}}[0,1]$, maximum norm, P, ${\tau }^{*}$, $a,b$, and d as defined in the Example 1. Also, take $X=[0,1]$ and the symmetric w-cone distance $q:X\times X\to P$ by $q(x,y)\left(t\right)=(x+y)\phi \left(t\right)$. Moreover, define $S:X\to X$ by

$$\begin{array}{c}\hfill S\left(x\right)=\left\{\begin{array}{cc}0,\hfill & x\in [0,1)\hfill \\ \hfill & \\ {\displaystyle \frac{1}{2}},\hfill & x=1\hfill \end{array}\right..\end{array}$$

Note that Theorem 1 cannot be applied to show the existence of a FP (it is enough to take $x=\frac{1}{2}$ and $y=1$ which shows that (1) is not valid); although $\kappa =0$ is the FP of S. However, (2) is true via these conditions and all hypothesises of Theorem 2 are satisfied. Therefore, we conclude from Theorem 2 that S possesses a unique FP; that is, Theorem 2 guarantees the existence of a unique FP which is $\kappa =0$.

3. Relation between Various Contractions Regarding $\mathit{w}$-Cone Distance

Let us start this section with Theorem 14 of Ćirić et al. [6] in 2012, which a TVS-CMS version of Theorem 2 of Suzuki and Takahashi [15] and it is proved directly.

Theorem 3

([6]). Presume that q is a w-cone distance on a complete TVS-CMS $(X,d)$ and $S:X\to X$ is a w-cone-contractive mapping, i.e., there is $\mu \in (0,1)$ provided that $q(Sx,Sy)⪯\mu q(x,y)$ for each $x,y\in X.$ Then, S possesses a unique FP $\kappa \in X$. Moreover, $q(\kappa ,\kappa )=\theta$.

Proof. 

Using Lemma $2.7$ of [7], the result follows from the proof of Theorem 2 of Suzuki and Takahashi [15]. □

Note that we considered another proof, short and optimal, regarding Minkowski functional which is introduced by Kadelburg et al. [7,25]. It is completely valid for the next consequences on w-cone distances, too.

Although the well-known contractive mapping theorem of Banach is a direct corollary of both Theorems 1 and 3, Theorem 3 is not a straight result of Theorem 2 and it can be considered an indirect conclusion of Theorem 2 by applying the cone version of Shioiji et al.’s results [20] as follows.

Theorem 4.

Assume that q is a w-cone distance on a TVS-CMS $(X,d)$ and S is a w-cone-contractive mapping. Then, there is a symmetric w-cone distance $q^{\prime }$ provided that S is a w-cone-contractive mapping with respect to $q^{\prime }$.

Proof. 

Using Lemma $2.7$ of [7] and Proposition 1, the result can be obtained from the proof of Theorem 1 of Shioiji et al.’s results [20]. □

Using Theorems 3 and 4, we can obtain the result of Theorem 2 regarding Corollary 1.

Now, we will introduce the cone version of some former definitions and find cone relation between our result.

Definition 5.

Presume that q is a w-cone distance and $S:X\to X$ is a self-mapping. S is a $q^{*}$-contraction when

$$\begin{array}{c}\hfill q(Sx,Sy)⪯\mu q(y,x)\end{array}$$

(3)

for each $x,y\in X$ in which $\mu \in (0,1)$.

If we take $E=\mathbb{R}$ and $P=[0,+\infty )$, Definition 5 is the same defined in Section 2 of ([20], Definition $W{C}_2\left(X\right)$).

Regarding Definition 5, we can obtain the next theorem easily by applying Lemma $2.7$ of [7] and Theorem 5 of [23].

Theorem 5.

Presume that q is a w-cone distance on a TVS-CMS and $S:X\to X$ is a w-cone-contraction. Then, there is a w-cone distance $q^{*}$ provided that S is $q^{*}$-contraction.

The first main result of this part is next theorem.

Theorem 6.

Assume that q is a w-cone distance on a complete TVS-CMS and S is an arbitrary $q^{*}$-contractive, i.e., it satisfies (3). Then S possesses a unique FP $\kappa \in X$ and $q(\kappa ,\kappa )=\theta .$

Proof. 

It is enough to apply Lemma $2.7$ of [7] and follow the proof of Theorem 6 of [23]. □

Now, let us explain what the role of Theorem 5 is. Presume that the hypothesises of Theorem 3 holds, i.e., q is a w-cone distance on a complete TVS-CMS and $S:X\to X$ is an arbitrary w-cone-contraction. Then, using Theorem 5, S is $q^{*}$-contraction in which $q^{*}$ is a w-cone distance. Now, applying Theorem 6, S has a unique FP and $q(\kappa ,\kappa )=\theta$ which are the same assertions of Theorem 3, too. Hence, Theorem 6 is an actual extension of Theorem 3.

Up until now, we have proved that both Theorems 2 and 6 are a genuine generalization of Theorem 3. Now, we ask another question and answer it. Is there a similar extension of Theorem 6? Our response to this question shows the importance of the basic result of this article, Theorem 2. To this end, we first define $q^{*}$-contraction of Suzuki-type.

Definition 6.

Presume that q is a w-cone distance on a TVS-CMS $(X,d)$. $S:X\to X$ is called a $q^{*}$-contractive mapping of Suzuki-type whenever there is $\mu \in (0,1)$, provided that

$$\begin{array}{c}\hfill q(x,Sx)⪯2q(x,y)\Rightarrow q(Sx,Sy)⪯\mu q(y,x)\end{array}$$

(4)

for each $x,y\in X$.

As an analogue to Theorem 6, we have the next result for a pre-symmetric w-cone distance, which is the second basic result of this part.

Theorem 7.

Assume that q is a w-cone distance on a complete TVS-CMS and S is an arbitrary $q^{*}$-contraction of Suzuki-type, i.e., it satisfies (4). Then S possesses a unique FP $\kappa \in X$ and $q(\kappa ,\kappa )=\theta .$

Proof. 

It is enough to apply Proposition 1 and follow the proof of Theorem 7 of [23]. □

4. Characterization of the Completeness of a TVS-CMS

In 1996, Suzuki and Takahashi ([15], Theorem 4) characterized complete metric spaces as follows:

Theorem 8.

Regarding a metric space $(X,d)$, the following sentences are equivalent:

(1) 

$(X,d)$ is complete;

(2) 

For each w-distance, any w-contractive self-mapping possesses a unique FP.

Note that the cone version of this paper is valid due to Theorem 3.

Theorem 9.

The following sentences are equivalent:

(1) 

A TVS-CMS $(X,d)$ is complete;

(2) 

For each w-cone distance q, any w-cone contractive self-mapping possesses a unique FP.

Proof. 

To prove, it is enough to apply Lemma $2.7$ of [7], Theorems $3.1$ and $3.2$ of [25], and Theorems 3 and 8. □

Fusing Theorems 5, 6 and 9, we have an alternative to Theorem 9.

Theorem 10.

The following sentences are equivalent:

(1) 

A TVS-CMS $(X,d)$ is complete;

(2) 

For each w-cone distance q, any $q^{*}$-contractive self-mapping on X possesses a unique FP.

Proof. 

Using Lemma $2.7$ of [7] and Theorems $3.1$ and $3.2$ of [25], the result follows from Theorem 9 of [23]. □

Considering Theorem 1, we obtain the next characterization.

Theorem 11.

The following statements are equivalent:

(1) 

A TVS-CMS $(X,d)$ is complete;

(2) 

Each cone version of Suzuki-type contractive self-mapping possesses a unique FP.

Gathering all the results that have been introduced up until now, we can introduce a complete theorem about characterization.

Theorem 12.

For a TVS-CMS $(X,d)$, the following are equivalent:

(1) 

$(X,d)$ is complete.

(2) 

For each pre-symmetric w-cone distance q, any w-cone contractive self-mapping of Suzuki-type possesses a unique FP.

(3) 

For each pre-symmetric w-cone distance q, any $q^{*}$-contractive self-mapping of Suzuki-type possesses a unique FP.

(4) 

For each symmetric w-cone distance q, any w-cone contractive mapping of Suzuki-type possesses a unique FP.

Proof. 

Using Theorems 2, 7 and 11 and Lemma 2, the proof is clear and left to the reader. □

Remark 3.

Using Proposition 1, Lemma $2.7$ of [7], Theorems $3.1$ and $3.2$ of [25], and Theorem 11 of [23], one can prove Theorem 12 indirectly.

5. Conclusions

In this paper, we defined a new concept in a TVS-CMS, named pre-symmetric w-cone distance, and introduced its properties and examples. Also, we discussed the TVS-cone version of the recent results obtained by Romaguera and Tirado; meanwhile, using Minkowski functionals, we showed the equivalency between some consequences with respect to a pre-symmetric w-distance in a usual metric space and a pre-symmetric w-cone distance in a TVS-CMS. Then, some types of various w-cone-contractions and the relation among them was investigated. Finally, a characterization of the completeness of TVS-cones metric regarding the pre-symmetric concept was performed, which differentiated our results from former characterizations. To follow the result of this article, many questions can be stated as follows:

(Q₁)

Is it possible to define a pre-symmetric generalized distance in Menger probabilistic metric spaces and characterize Menger probabilistic metric completeness?

(Q₂)

Can one obtain the result of this paper for a pre-symmetric c-distance in cone metric spaces (scalar weighted cone metric spaces (Tomar and Joshi, 2021 [30])) without direct proof?

(Q₃)

Is it possible to define a pre-symmetric $wt$-distance (generalized c-distance) in b-metric (cone b-metric) spaces and characterize b-metric (cone b-metric) completeness?

(Q₄)

Is it possible to define a pre-symmetric $\omega$-distance in G-metric spaces and characterize G-metric completeness?

(Q₅)

As we know, there exist fuzzy versions of various metric spaces. For example, fuzzy metric spaces are discussed by Kramosil and Michalek, 1975 [31] and, similarly, fuzzy cone metric spaces are introduced by Bag, 2003 [32] and Öner et al. (2015, 2022) [33,34]. On the other hand, as we mentioned, there exists a distance in several metric spaces. It is clear that one can define fuzzy versions of distances, as it is performed by several authors. For example, a fuzzy w-distance in metric spaces is introduced by [35] and a distance in fuzzy cone metric spaces is defined by Bag, 2015 [36]. In 2020, characterization of the completeness of fuzzy metric spaces is also discussed by Romaguera [37]. Regarding these works, one can also think about the following questions:

(Q₅-1)

Is it possible to define fuzzy pre-symmetric w-distance and characterize fuzzy metric completeness?

(Q₅-2)

Is it possible to define fuzzy pre-symmetric w-cone distance and characterize fuzzy cone metric completeness? If yes, can topological concepts like this paper be used to reduce the direct proof to an indirect one?