Fixed-Point Results for Multi-Valued Mappings in Topological Vector Space-Valued Cone Metric Spaces with Applications

Abstract

The objective of this research article is to introduce Kikkawa and Suzuki-type contractions in the setting of topological vector space-valued cone metric space with a solid cone and establish some new fixed point results for multi-valued mappings. The problem of finding fixed points for multi-valued mappings satisfying locally contractive conditions on a closed ball is also addressed. Our findings generalize a number of well-established results in the literature. To highlight the uniqueness of our key finding, we present an example. As a demonstration of the applicability of our principal theorem, we prove a result in homotopy theory.

1. Introduction

Fixed-point (FP) theory is a well-established and prominent area of mathematics, known for its significant implications across various mathematical fields, such as differential equations, geometry, and optimization. Building upon this foundation, Antón-Sancho [1,2] delved into the fixed points of automorphisms on the moduli space of vector bundles over compact Riemann surfaces, as well as fixed points within the space of principal $E_6$-bundles defined over a compact algebraic curve. In this theory, the concept of metric spaces (MSs) is pivotal, and it was originally introduced by M. Fréchet [3] in 1906. This concept provided the groundwork for the development of this theory. Over time, the significance of this idea has inspired many researchers to explore various extensions and generalizations of metric spaces, leading to important advancements in the field in recent years. Stefan Banach [4] is credited with pioneering FP theory, introducing the concept of a contraction mapping and proving the renowned Banach Contraction Principle (BCP). Numerous subsequent studies have focused on extending and generalizing this fundamental result. While BCP is a powerful tool, it falls short in characterizing metric completeness. To address this limitation, Suzuki [5] introduced a generalized contraction principle that characterizes metric completeness. Subsequently, Kikkawa and Suzuki [6] extended this concept to multi-valued mappings, thereby providing a generalization of Nadler’s FP theorem [7].

On the other hand, Huang et al. [8] formally introduced the concept of cone metric spaces (CMSs) with normal cones, seemingly unaware that this idea had already been explored in earlier literature. They worked with a partial order in a real Banach space defined by a cone, and developed fundamental concepts such as convergence, completeness and continuity, subsequently proving BCP in this new framework. Building upon the work of ref. [8], Rezapour et al. [9] demonstrated that the results hold true even for non-normal cones, which is a substantial extension. Later on, Janković et al. [10] provided a comprehensive overview of CMSs, solidifying the foundations of this emerging field. Subsequently, Cho et al. [11] extended the theory of cone metric spaces by defining a generalized Hausdorff distance function and proving an FP theorem for multi-valued mappings. Thereafter, Cho et al. [12] established a result in CMS for the multi-valued mappings satisfying Kannan-type contractions. In subsequent work, various authors [13] generalized many results in CMSs, eliminating the need for the normality assumption of the cone.

Beg et al. [14] extended the theory of CMSs by introducing topological vector space-valued cone metric spaces (tvs-VCMSs). Azam et al. [15,16] strengthened this new notion by establishing new FP results for the mappings satisfying generalized contractive conditions. Shatanawi et al. [17] redefined the concept of the Hausdorff distance function in tvs-VCMSs with non-normal or solid cones and established FP theorems for multi-valued mappings satisfying Mizoguchi–Takahashi-type contractions. Recently, Azam et al. [18] reinforced this thought and produced Kannan-, Chatterjea- and Zamfirescu-type FP results for multi-valued mappings in tvs-VCMSs. For a more in-depth discussion of this subject, please refer to [19,20,21,22,23].

In this research article, we define the notions of Kikkawa and Suzuki-type contractions in the framework of tvs-VCMSs with a solid cone and establish FP results for multi-valued mappings. We also investigate the existence of fixed points for multi-valued mappings that satisfy locally contractive conditions on a closed ball. Our results generalize several well-known findings in the literature, including the main theorems of Kikkawa and Suzuki [6], Nadler [7], Cho et al. [11,12], Mehmood et al. [13] and Azam et al. [18]. Additionally, we apply our primary theorem to prove a result in the homotopy theory.

2. Preliminaries

In this section, we establish the foundational concepts and notation that will be utilized throughout this paper. Let $\mathbb{E}$ be a tvs with its zero vector $\mathbf{0}$. A non-empty subset P of $\mathbb{E}$ is called a convex cone if $a\varsigma +b\varrho \in P$ for all $\varsigma ,\varrho \in P$ and non-negative real numbers $a,b$. A convex cone P is considered to be a pointed cone if it contains no lines, except for the trivial case involving the origin. Formally, a cone P is called pointed if $P\cap (-P)=\left\{\mathbf{0}\right\}.$ A convex cone P is claimed to be normal if $\mathbb{E}$ has a base of neighborhoods of zero consisting of order-convex subsets. For a given cone $P\subseteq E$, we can define a partial ordering ⪯ with respect to P by $\varsigma ⪯\varrho$ if and only if $\varrho -\varsigma \in P$. $\varsigma \prec \varrho$ will stand for $\varsigma ⪯\varrho$ and $\varsigma \ne \varrho$, while $\varsigma \ll \varrho$ will stand for $\varrho -\varsigma \in intP$, where $intP$ denotes the interior of P. The cone P is said to be solid if it has a non-empty interior.

Definition 1

([14]). Let $\mathcal{Z}\ne \varnothing$, and let ($\mathbb{E}$,P) be an ordered tvs. A vector valued function $d:\mathcal{Z}\times \mathcal{Z}\to \mathbb{E}$ is said to be a tvs-valued cone metric if the following conditions hold:

(tvs₁)

$\mathbf{0}⪯d(\varsigma ,\varrho )$ and $d(\varsigma ,\varrho )=0⟺$$\varsigma =\varrho$;

(tvs₂)

$d(\varsigma ,\varrho )=d(\varrho ,\varsigma )$;

(tvs₃)

$d(\varsigma ,\varrho )⪯d(\varsigma ,z)+d(z,\varrho )$,

for all $\varsigma ,\varrho ,z\in \mathcal{Z}$; then, $(\mathcal{Z},d)$ is called a tvs-VCMS.

Example 1

([14]). Let $\mathcal{Z}=[0,1]$, and let $\mathbb{E}$ be the space of all real-valued functions on $\mathcal{Z}$ with continuous derivatives. Under point-wise addition and scalar multiplication, $\mathbb{E}$ forms a vector space over $\mathbb{R}.$ Equip $\mathbb{E}$ with the strongest vector topology $\tau .$ Then, ($\mathcal{Z},\tau$) is a tvs. Now, ($\mathcal{Z},d$) is tvs-VCMS with the metric $d:\mathcal{Z}\times \mathcal{Z}\to \mathbb{E}$ defined by

$$\left(d\left(\varsigma ,\varrho \right)\right)(t)=\left|\varsigma -\varrho \right|e^t,$$

and the cone $P=\left\{\varsigma \in \mathbb{E}:\mathbf{0}⪯\varsigma (t)\mathit{for}\mathit{all}t\in \mathcal{Z}\right\}.$

Remark 1

([14]). The notion of CMS is a special case of the notion of tvs-VCMS.

Definition 2

([14]). Let $(\mathcal{Z},d)$ be a tvs-VCMS, $\varsigma \in \mathcal{Z}$ and ${\left\{{\varsigma }_n\right\}}_{n\ge 1}$ be a sequence in $\mathcal{Z}$. Then,

(i) 

${\left\{{\varsigma }_n\right\}}_{n\ge 1}$ converges to ς whenever for every $c\in \mathbb{E}$ with $0\ll c$ there is a natural number N, such that $d({\varsigma }_n,\varsigma )\ll c$ for all $n\ge N$. We denote this by

$$\underset{n\to \infty }{lim}{\varsigma }_n=\varsigma or{\varsigma }_n\to \varsigma .$$

(ii) 

${\left\{{\varsigma }_n\right\}}_{n\ge 1}$ is a Cauchy sequence whenever for every $c\in \mathbb{E}$ with $0\ll c$ there is a natural number N, such that

$$d({\varsigma }_n,{\varsigma }_m)\ll c,$$

for all $n,m\ge N$.

(iii) 

$(\mathcal{Z},d)$ is a complete tvs-VCMS if every Cauchy sequence is convergent.

Remark 2

([14]). The FP theorems and related results established in CMSs with cones that are non-normal cannot be directly derived from the corresponding theorems in MSs. This is due to the inherent differences between the two types of spaces, as neither of the conditions in Lemmas (i)–(iv) of Huang et al. [8] are satisfied in the context of non-normal solid cones. Moreover, the CMS may not exhibit continuity in general. That is, from ${\varsigma }_n\to \varsigma$ and ${\varrho }_n\to \varrho ,$ it does not necessarily follow that $d({\varsigma }_n,{\varrho }_n)\to d(\varsigma ,\varrho ).$

Lemma 1

([18]). Let $(\mathcal{Z},d)$ be a tvs-VCMS with a solid cone P in the ordered locally convex space $\mathbb{E}$. The following properties will play a pivotal role in our discussion:

(prop₁)

If $\omega \ll \varpi$ and $\varpi ⪯\phi ,$ then $\omega \ll \phi .$

(prop₂)

If $\omega ⪯\varpi$ and $\varpi \ll \phi ,$ then $\omega \ll \phi .$

(prop₃)

If $\omega \ll \varpi$ and $\varpi \ll \phi ,$ then $\omega \ll \phi .$

(prop₄)

If $\mathbf{0}⪯\omega \ll \varpi$ for each $\varpi \in IntP,$ then $\varpi =\mathbf{0}.$

(prop₅)

If $\omega ⪯\varpi +\phi$ for each $\phi \in IntP,$ then $\omega ⪯\varpi .$

(prop₆)

If $\omega ⪯\lambda \omega$ where $0\le \lambda <1$ and $\omega \in P,$ then $\omega =\mathbf{0}.$

(prop₇)

If {${\omega }_n\}\in \mathbb{E}$ and {${\omega }_n\}\to \mathbf{0}$ as $n\to \infty$ and $\varpi \in intP,$ then there exist a natural number aa $n_0$, such that, for all $n>n_0,$ we have ${\omega }_n\ll \varpi .$

Definition 3

([18]). Let $(\mathcal{Z},d)$ be a tvs-VCMS with a solid cone P and $Cl(\mathcal{Z})$ denote a family of non-empty closed subsets of $\mathcal{Z}$. Let $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ be a multi-valued mapping. For $\varsigma \in \mathcal{Z},$ Ξ be a closed subset of $\mathcal{Z}.$ Define

$$W_{\varsigma }(\Xi )=\left\{d(\varsigma ,a):a\in \Xi \right\}.$$

Thus, for any $\varsigma ,\varrho \in \mathcal{Z},$

$$W_{\varsigma }(\Im \varrho )=\left\{d(\varsigma ,\omega ):\omega \in \Im \varrho \right\}.$$

Definition 4

([18]). Let $(\mathcal{Z},d)$ be a tvs-VCMS with a solid cone $P.$ A multi-valued mapping $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ is termed bounded from below if, for each $\varsigma \in \mathcal{Z}$, one can find $z(\varsigma )\in \mathbb{E}$, such that

$$\Im \varsigma -z(\varsigma )\subset P.$$

Definition 5

([18]). Let $(\mathcal{Z},d)$ be a tvs-VCMS with a solid cone $P.$ A multi-valued mapping $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ is said to possess the lower bound property (l.b. property) on $\mathcal{Z}$ if, for any $\varsigma \in \mathcal{Z},$ the multi-valued mapping ${\Im }_{\varsigma }:\mathcal{Z}\to 2^{\mathbb{E}}$ given by

$${\Im }_{\varsigma }(\varrho )=W_{\varsigma }(\Im \varrho ),$$

is bounded from below. In other words, for each $\varsigma ,\varrho \in \mathcal{Z},$ there exists an element $d(\varsigma ,\Im \varrho )\in \mathbb{E}$ satisfying

$$W_{\varsigma }(\Im \varrho )-d(\varsigma ,\Im \varrho )\subset P.$$

This element $d(\varsigma ,\Im \varrho )$ is called lower bound of ℑ associated with $\left(\varsigma ,\varrho \right).$

Definition 6

([18]). Let $(\mathcal{Z},d)$ be a tvs-VCMS with a solid cone $P.$ A multi-valued mapping $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ is said to possess the greatest lower bound property (g.l.b. property) on $\mathcal{Z}$ if the greatest lower bound of $W_{\varsigma }(\Im \varrho )$ exists in $\mathbb{E}$, for all $\varsigma ,\varrho \in \mathcal{Z}.$ We represent the greatest lower bound of $W_{\varsigma }(\Im \varrho )$ by $d(\varsigma ,\Im \varrho )$, i.e.,

$$d(\varsigma ,\Im \varrho )=inf\left\{d(\varsigma ,\omega ):\omega \in \Im \varrho \right\}.$$

Shatanawi et al. [17] defined the concept of the generalized Hausdorff distance function $s:\mathbb{E}\to 2^{\mathbb{E}}$ in the framework of tvs-VCMS as follows: for $u\in \mathbb{E}$, we have

$$s(u)=\left\{v\in \mathbb{E}:u⪯v\right\},$$

and

$$\begin{array}{ccc}\hfill s(u,\Theta )& =& \bigcup _{v\in \Theta }s\left(d(u,v)\right)\hfill \\ & =& \bigcup _{v\in \Theta }\left\{\varpi \in \mathbb{E}:d(u,v)⪯\varpi \right\},\hfill \end{array}$$

and

$$s(\Xi ,\Theta )=\left(\bigcap _{u\in \Xi }s(u,\Theta )\right)\bigcap \left(\bigcap _{v\in \Theta }s(v,\Xi )\right),$$

for all $\Xi ,\Theta \in Cl(\mathcal{Z}).$

The following lemma will be instrumental in proving our main theorem.

Lemma 2

([17]). Let $(\mathcal{Z},d)$ be a tvs-VCMS with a solid cone P. The following conditions hold:

(i) 

If $u,v\in \mathbb{E}$ and $u⪯v$, then $s(v)\subset s(u)$.

(ii) 

Let $\varsigma \in \mathcal{Z}$ and $\Xi \in Cl(\mathcal{Z})$. If $\mathbf{0}\in s(\varsigma ,\Xi )$, then $\varsigma \in \Xi$.

(iii) 

Let $v\in P$ and let $\Xi ,\Theta \in Cl(\mathcal{Z})$ and $\varsigma \in \Xi$. If $v\in s(\Xi ,\Theta )$, then $v\in s(\varsigma ,\Theta ).$

(iv) 

For all $v\in P$ and $\Xi ,\Theta \in Cl(\mathcal{Z}).$ Then, $v\in s(\Xi ,\Theta )$ if and only if there exist $\varsigma \in \Xi$ and $\varrho \in \Theta$, such that $d(\varsigma ,\varrho )⪯v$.

Remark 3

([17]). Let $(\mathcal{Z},d)$ be a tvs-VCMS with a solid cone P in the ordered locally convex space $\mathbb{E}$. If $\mathbb{E}=\mathbb{R}$ and $P=[0,+\infty )$, then $(\mathcal{Z},d)$ is an MS. Furthermore, for $\Xi ,\Theta \in Cl(\mathcal{Z}),$

$$H(\Xi ,\Theta )=infs(\Xi ,\Theta ),$$

is the Hausdorff distance generated by d. Moreover,

$$s(\{\varsigma \},\{\varrho \})=s(d(\varsigma ,\varrho )),$$

for all $\varsigma ,\varrho \in \mathcal{Z}$.

3. Main Results

In this section, we introduce Kikkawa and Suzuki-type contractions within the framework of tvs-VCMS and obtain novel FP theorems for multi-valued mappings.

Definition 7.

Let $(\mathcal{Z},d)$ be a tvs-VCMS endowed with a solid cone $P.$ A multi-valued mapping $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ is said to be a Kikkawa and Suzuki-type contraction if there exist constants ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that ${\displaystyle \frac{1}{\beta }}d(\varsigma ,\varrho )\in s(\varsigma ,\Im \varsigma )$ implies

$${\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho )\in s(\Im \varsigma ,\Im \varrho ),$$

(1)

for all $\varsigma ,\varrho \in \mathcal{Z}$, where $\beta =max\left\{{\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}},{\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_1}}\right\}.$

Theorem 1.

Let $(\mathcal{Z},d)$ be a complete tvs-VCMS endowed with a solid cone P and $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ be a Kikkawa and Suzuki-type contraction. Then, there exists $\omega \in \mathcal{Z}$, such that $\omega \in \Im \omega$.

Proof. 

Let ${\varsigma }_0\in \mathcal{Z}$ be an arbitrary element. Since $\Im {\varsigma }_0\in Cl(\mathcal{Z}),\Im {\varsigma }_0\ne \varnothing$, such that there is a ${\varsigma }_1\in \Im {\varsigma }_0$. Since $\beta ={\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}}<1,$ so ${\displaystyle \frac{1}{\beta }}\ge 1.$ Hence,

$$d\left({\varsigma }_0,{\varsigma }_1\right)⪯{\displaystyle \frac{1}{\beta }}d\left({\varsigma }_0,{\varsigma }_1\right),$$

by Lemma 2 (iv), we get

$${\displaystyle \frac{1}{\beta }}d\left({\varsigma }_0,{\varsigma }_1\right)\in s\left({\varsigma }_0,\Im {\varsigma }_0\right).$$

By (1), we have

$${\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1)\in s\left(\Im {\varsigma }_0,\Im {\varsigma }_1\right).$$

This implies that

$${\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1)\in \left(\bigcap _{\varsigma \in \Im {\varsigma }_0}s\left(\varsigma ,\Im {\varsigma }_1\right)\right),$$

which further yields

$${\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1)\in s\left(\varsigma ,\Im {\varsigma }_1\right),\mathrm{for}\mathrm{all}\varsigma \in \Im {\varsigma }_0.$$

Since ${\varsigma }_1\in \Im {\varsigma }_0$, we have

$${\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1)\in s\left({\varsigma }_1,\Im {\varsigma }_1\right).$$

By the definition, we have

$${\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1)\in s\left({\varsigma }_1,\Im {\varsigma }_1\right)=\bigcup _{\varsigma \in \Im {\varsigma }_1}s\left(d\left({\varsigma }_1,\varsigma \right)\right).$$

So, there exists some ${\varsigma }_2\in \Im {\varsigma }_1$, such that

$${\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1)\in s\left(d\left({\varsigma }_1,{\varsigma }_2\right)\right).$$

Thus, by definition, we have

$$d\left({\varsigma }_1,{\varsigma }_2\right)⪯{\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1).$$

(2)

Since the multi-valued mapping $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ has the g.l.b. property, we have the following:

$$W_{{\varsigma }_0}-d({\varsigma }_0,\Im {\varsigma }_0)\subset P,$$

and

$$W_{{\varsigma }_1}-d({\varsigma }_1,\Im {\varsigma }_1)\subset P,$$

which yields that

$$d({\varsigma }_0,\Im {\varsigma }_0)⪯d\left({\varsigma }_0,{\varsigma }_1\right),$$

and

$$d({\varsigma }_1,\Im {\varsigma }_1)⪯d\left({\varsigma }_1,{\varsigma }_2\right).$$

Hence, by (2), we have

$$d\left({\varsigma }_1,{\varsigma }_2\right)⪯{\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{3}d\left({\varsigma }_1,{\varsigma }_2\right),$$

which further implies that

$$d\left({\varsigma }_1,{\varsigma }_2\right)⪯\beta d\left({\varsigma }_0,{\varsigma }_1\right),$$

where $\beta ={\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}}<1.$ Now, since ${\displaystyle \frac{1}{\beta }}\ge 1,$ so

$$d\left({\varsigma }_1,{\varsigma }_2\right)⪯{\displaystyle \frac{1}{\beta }}d\left({\varsigma }_1,{\varsigma }_2\right),$$

again by Lemma 2 (iv), we get

$${\displaystyle \frac{1}{\beta }}d\left({\varsigma }_1,{\varsigma }_2\right)\in s\left({\varsigma }_1,\Im {\varsigma }_1\right).$$

Then, by (1), we have

$${\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2)\in s\left(\Im {\varsigma }_1,\Im {\varsigma }_2\right).$$

This implies that

$${\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2)\in \left(\bigcap _{\varsigma \in \Im {\varsigma }_1}s\left(\varsigma ,\Im {\varsigma }_2\right)\right),$$

which further yields

$${\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2)\in s\left(\varsigma ,\Im {\varsigma }_2\right),\mathrm{for}\mathrm{all}\varsigma \in \Im {\varsigma }_1.$$

Since ${\varsigma }_2\in \Im {\varsigma }_1$, we have

$${\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2)\in s\left({\varsigma }_2,\Im {\varsigma }_2\right).$$

By definition, we have

$${\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2)\in s\left({\varsigma }_2,\Im {\varsigma }_2\right)=\bigcup _{\varsigma \in \Im {\varsigma }_2}s\left(d\left({\varsigma }_2,\varsigma \right)\right),$$

so there exists some ${\varsigma }_3\in \Im {\varsigma }_2$, such that

$${\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2)\in s\left(d\left({\varsigma }_2,{\varsigma }_3\right)\right).$$

Then, by definition, we have

$$d\left({\varsigma }_2,{\varsigma }_3\right)⪯{\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2).$$

(3)

Since the multi-valued mapping $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ has the g.l.b. property, we have

$$W_{{\varsigma }_1}-d({\varsigma }_1,\Im {\varsigma }_1)\subset P,$$

and

$$W_{{\varsigma }_2}-d({\varsigma }_2,\Im {\varsigma }_2)\subset P,$$

and since ${\varsigma }_2\in \Im {\varsigma }_1$ and ${\varsigma }_3\in \Im {\varsigma }_2,$ we have

$$d({\varsigma }_1,\Im {\varsigma }_1)⪯d\left({\varsigma }_1,{\varsigma }_2\right),$$

and

$$d({\varsigma }_2,\Im {\varsigma }_2)⪯d\left({\varsigma }_2,{\varsigma }_3\right).$$

Hence, by Inequality (3), we have

$$d\left({\varsigma }_2,{\varsigma }_3\right)⪯{\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{3}d\left({\varsigma }_2,{\varsigma }_3\right),$$

which further implies that

$$d\left({\varsigma }_2,{\varsigma }_3\right)⪯\beta d\left({\varsigma }_1,{\varsigma }_2\right),$$

where $\beta ={\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}}<1.$ By iterating this process, we can construct a sequence $\left\{{\varsigma }_n\right\}$ in $\mathcal{Z}$ with ${\varsigma }_{n+1}\in \Im {\varsigma }_n$, such that

$$d\left({\varsigma }_n,{\varsigma }_{n+1}\right)⪯\beta d\left({\varsigma }_{n-1},{\varsigma }_n\right),$$

which further implies

$$d\left({\varsigma }_n,{\varsigma }_{n+1}\right)⪯\beta d\left({\varsigma }_{n-1},{\varsigma }_n\right)⪯{\beta }^{2}d\left({\varsigma }_{n-2},{\varsigma }_{n-1}\right)⪯...⪯{\beta }^{n}d\left({\varsigma }_0,{\varsigma }_1\right),$$

for all $n\in \mathbb{N}.$ Now, for $n>m,$ we have

$$\begin{array}{ccc}\hfill d\left({\varsigma }_n,{\varsigma }_m\right)& ⪯& d\left({\varsigma }_n,{\varsigma }_{n-1}\right)+d\left({\varsigma }_{n-1},{\varsigma }_{n-2}\right)+...+d\left({\varsigma }_{m+1},{\varsigma }_m\right)\hfill \\ & ⪯& {\beta }^{n-1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\beta }^{n-2}d\left({\varsigma }_0,{\varsigma }_1\right)+...+{\beta }^{m}d\left({\varsigma }_0,{\varsigma }_1\right)\hfill \\ & =& \left({\beta }^{n-1}+{\beta }^{n-2}+...+{\beta }^m\right)d\left({\varsigma }_0,{\varsigma }_1\right)\hfill \\ & ⪯& {\displaystyle \frac{{\beta }^m}{1-\beta }}d\left({\varsigma }_0,{\varsigma }_1\right).\hfill \end{array}$$

Let $\mathbf{0}\ll c.$ Choose a symmetric neighborhood V of $\mathbf{0}$ such that $c+V$ is contained in the interior of $P.$ Next, select $N_1\in \mathbb{N}$, such that ${\displaystyle \frac{{\beta }^m}{1-\beta }}d\left({\varsigma }_0,{\varsigma }_1\right)\in V,$ for all $m\ge N_1.$ Consequently, ${\displaystyle \frac{{\beta }^m}{1-\beta }}d\left({\varsigma }_0,{\varsigma }_1\right)\ll c,$ for all $m\ge N_1.$ Hence,

$$d\left({\varsigma }_n,{\varsigma }_m\right)⪯{\displaystyle \frac{{\beta }^m}{1-\beta }}d\left({\varsigma }_0,{\varsigma }_1\right)\ll c,$$

for all $m,n>N_1.$ This implies that $\left\{{\varsigma }_n\right\}$ is a Cauchy sequence in $\mathcal{Z}$. Since $\mathcal{Z}$ is complete, so there is $\omega \in \mathcal{Z}$, such that ${\varsigma }_n⟶\omega$. Now, for $c\gg \mathbf{0}$, choose $N(c)\in \mathbb{N}$, such that for $n\ge N(c),$ we have

$$d\left(\omega ,{\varsigma }_{n+1}\right)\ll {\displaystyle \frac{c}{2}}\left({\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_2}}\right),$$

(4)

and

$$d\left({\varsigma }_n,\omega \right)\ll {\displaystyle \frac{c}{2}}\left({\displaystyle \frac{1-{\sigma }_3}{{\sigma }_1+{\sigma }_2}}\right).$$

(5)

We now show that $\omega \in \Im \omega$. Choose ${\varsigma }_{n_1}\in \left\{{\varsigma }_n\right\}$, such that ${\varsigma }_{n_1}\ne \omega$. As ${\varsigma }_n\to \omega$, there exists $N_2\in \mathbb{N}$, such that

$$d\left({\varsigma }_n,\omega \right)⪯{\displaystyle \frac{1}{3}}d\left(\omega ,{\varsigma }_{n_1}\right)\mathrm{for}\mathrm{all}n\ge N_2.$$

Since ${\varsigma }_{n+1}\in \Im {\varsigma }_n$ and

$$\begin{array}{ccc}\hfill \beta d\left({\varsigma }_n,{\varsigma }_{n+1}\right)& ⪯& d\left({\varsigma }_n,{\varsigma }_{n+1}\right)\hfill \\ & ⪯& d\left({\varsigma }_n,\omega \right)+d\left(\omega ,{\varsigma }_{n+1}\right)\hfill \\ & ⪯& {\displaystyle \frac{1}{3}}d\left(\omega ,{\varsigma }_{n_1}\right)+{\displaystyle \frac{1}{3}}d\left(\omega ,{\varsigma }_{n_1}\right)\hfill \\ & =& {\displaystyle \frac{2}{3}}d\left(\omega ,{\varsigma }_{n_1}\right)\hfill \\ & ⪯& d\left(\omega ,{\varsigma }_{n_1}\right)-d\left({\varsigma }_n,\omega \right)\hfill \\ & ⪯& d\left({\varsigma }_n,{\varsigma }_{n_1}\right),\hfill \end{array}$$

which implies

$$d\left({\varsigma }_n,{\varsigma }_{n+1}\right)⪯{\displaystyle \frac{1}{\beta }}d\left({\varsigma }_n,{\varsigma }_{n_1}\right).$$

Hence, by Lemma 2 (iv), we get

$${\displaystyle \frac{1}{\beta }}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)\in s\left({\varsigma }_n,\Im {\varsigma }_n\right).$$

By (1), we have

$${\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\Im {\varsigma }_n\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\Im {\varsigma }_{n_1}\right)\in s\left(\Im {\varsigma }_n,\Im {\varsigma }_{n_1}\right).$$

This implies that

$${\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\Im {\varsigma }_n\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\Im {\varsigma }_{n_1}\right)\in \left(\bigcap _{\varsigma \in \Im {\varsigma }_n}s\left(\varsigma ,\Im {\varsigma }_{n_1}\right)\right),$$

which further yields

$${\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\Im {\varsigma }_n\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\Im {\varsigma }_{n_1}\right)\in s\left(\varsigma ,\Im {\varsigma }_{n_1}\right),\mathrm{for}\mathrm{all}\varsigma \in \Im {\varsigma }_n.$$

Since ${\varsigma }_{n+1}\in \Im {\varsigma }_n$, we have

$${\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\Im {\varsigma }_n\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\Im {\varsigma }_{n_1}\right)\in s\left({\varsigma }_{n+1},\Im {\varsigma }_{n_1}\right).$$

By definition, we have

$${\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\Im {\varsigma }_n\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\Im {\varsigma }_{n_1}\right)\in s\left({\varsigma }_{n+1},\Im {\varsigma }_{n_1}\right)=\bigcup _{\varsigma \in \Im {\varsigma }_{n_1}}s\left(d\left({\varsigma }_{n+1},\varsigma \right)\right),$$

so there exists some ${\varsigma }_{n_2}\in \Im {\varsigma }_{n_1}$ such that

$${\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\Im {\varsigma }_n\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\Im {\varsigma }_{n_1}\right)\in s\left(d\left({\varsigma }_{n+1},{\varsigma }_{n_2}\right)\right),$$

and thus

$$d\left({\varsigma }_{n+1},{\varsigma }_{n_2}\right)⪯{\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\Im {\varsigma }_n\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\Im {\varsigma }_{n_1}\right).$$

(6)

Since the multi-valued mapping $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ has the g.l.b. property, we have

$$W_{{\varsigma }_n}-d\left({\varsigma }_n,\Im {\varsigma }_n\right)\subset P,$$

and

$$W_{{\varsigma }_{n_1}}-d\left({\varsigma }_{n_1},\Im {\varsigma }_{n_1}\right)\subset P,$$

and since ${\varsigma }_{n+1}\in \Im {\varsigma }_{n+1}$ and ${\varsigma }_{n_2}\in \Im {\varsigma }_{n_1},$ we have

$$d\left({\varsigma }_n,\Im {\varsigma }_n\right)⪯d\left({\varsigma }_n,{\varsigma }_{n+1}\right),$$

and

$$d\left({\varsigma }_{n_1},\Im {\varsigma }_{n_1}\right)⪯d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right).$$

Hence, from Inequality (6), we have

$$d\left({\varsigma }_{n+1},{\varsigma }_{n_2}\right)⪯{\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,{\varsigma }_{n+1}\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right).$$

From

$$\begin{array}{ccc}\hfill d\left(\omega ,{\varsigma }_{n_2}\right)& ⪯& d\left(\omega ,{\varsigma }_{n+1}\right)+d\left({\varsigma }_{n+1},{\varsigma }_{n_2}\right)\hfill \\ & ⪯& d\left(\omega ,{\varsigma }_{n+1}\right)+{\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,{\varsigma }_{n+1}\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right).\hfill \end{array}$$

Taking the limit as $n\to +\infty$, we obtain

$$d\left(\omega ,{\varsigma }_{n_2}\right)⪯{\sigma }_{1}d\left(\omega ,{\varsigma }_{n_1}\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right).$$

(7)

Next, we prove that

$${\sigma }_{1}d\left({\varsigma }_{n_1},\omega \right)+{\sigma }_{2}d\left({\varsigma }_{n_1},\Im {\varsigma }_{n_1}\right)+{\sigma }_{3}d\left(\omega ,\Im \omega \right)\in s\left(\Im {\varsigma }_{n_1},\Im \omega \right)\left(\mathrm{recall}{\varsigma }_{n_1}\ne \omega \right).$$

For each $n\in \mathbb{N}$

$$d\left(\omega ,{\varsigma }_{n_2}\right)⪯d\left(\omega ,{\varsigma }_{n_2}\right)+{\displaystyle \frac{1}{n}}d\left({\varsigma }_{n_1},\omega \right).$$

(8)

By the triangle inequality, we have

$$d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right)⪯d\left({\varsigma }_{n_1},\omega \right)+d\left(\omega ,{\varsigma }_{n_2}\right),$$

using (8), we have

$$d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right)⪯d\left({\varsigma }_{n_1},\omega \right)+d\left(\omega ,{\varsigma }_{n_2}\right)+{\displaystyle \frac{1}{n}}d\left({\varsigma }_{n_1},\omega \right).$$

By (7), we have

$$\begin{array}{ccc}\hfill d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right)& ⪯& d\left({\varsigma }_{n_1},\omega \right)+{\sigma }_{1}d\left(\omega ,{\varsigma }_{n_1}\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right)+{\displaystyle \frac{1}{n}}d\left({\varsigma }_{n_1},\omega \right)\hfill \\ & =& ({\displaystyle \frac{1+{\sigma }_1}{1-{\sigma }_3}}+{\displaystyle \frac{1}{n(1-{\sigma }_3)}})d\left(\omega ,{\varsigma }_{n_1}\right),\hfill \end{array}$$

for all $n\in \mathbb{N}.$ Taking the limit as $n\to \infty ,$ we have

$$d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right)⪯\left({\displaystyle \frac{1+{\sigma }_1}{1-{\sigma }_3}}\right)d\left(\omega ,{\varsigma }_{n_1}\right)⪯{\displaystyle \frac{1}{\beta }}d\left(\omega ,{\varsigma }_{n_1}\right),$$

by Lemma 2 (iv), we get

$${\displaystyle \frac{1}{\beta }}d\left({\varsigma }_{n_1},\omega \right)\in s\left({\varsigma }_{n_1},\Im {\varsigma }_{n_1}\right).$$

Then, from the hypothesis, we have

$${\sigma }_{1}d\left({\varsigma }_{n_1},\omega \right)+{\sigma }_{2}d\left({\varsigma }_{n_1},\Im {\varsigma }_{n_1}\right)+{\sigma }_{3}d\left(\omega ,\Im \omega \right)\in s\left(\Im {\varsigma }_{n_1},\Im \omega \right).$$

Similarly, (for ${\varsigma }_{n_2}\in \mathcal{Z}\backslash \left\{\omega \right\}$ ), we have

$${\sigma }_{1}d\left({\varsigma }_{n_2},\omega \right)+{\sigma }_{2}d\left({\varsigma }_{n_2},\Im {\varsigma }_{n_2}\right)+{\sigma }_{3}d\left(\omega ,\Im \omega \right)\in s\left(\Im {\varsigma }_{n_2},\Im \omega \right),$$

and for each $k\in \mathbb{N}$, we obtain

$${\sigma }_{1}d\left({\varsigma }_{n_k},\omega \right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\Im {\varsigma }_{n_k}\right)+{\sigma }_{3}d\left(\omega ,\Im \omega \right)\in s\left(\Im {\varsigma }_{n_k},\Im \omega \right).$$

This implies that

$${\sigma }_{1}d\left({\varsigma }_{n_k},\omega \right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\Im {\varsigma }_{n_k}\right)+{\sigma }_{3}d\left(\omega ,\Im \omega \right)\in \left(\bigcap _{\varsigma \in \Im {\varsigma }_{n_k}}s\left(\varsigma ,\Im \omega \right)\right),$$

which further yields

$${\sigma }_{1}d\left({\varsigma }_{n_k},\omega \right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\Im {\varsigma }_{n_k}\right)+{\sigma }_{3}d\left(\omega ,\Im \omega \right)\in s\left(\varsigma ,\Im \omega \right),\mathrm{for}\mathrm{all}\varsigma \in \Im {\varsigma }_{n_k}.$$

Since ${\varsigma }_{n_{k+1}}\in \Im {\varsigma }_{n_k}$, we have

$${\sigma }_{1}d\left({\varsigma }_{n_k},\omega \right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\Im {\varsigma }_{n_k}\right)+{\sigma }_{3}d\left(\omega ,\Im \omega \right)\in s\left({\varsigma }_{n_{k+1}},\Im \omega \right).$$

By definition, we have

$${\sigma }_{1}d\left({\varsigma }_{n_k},\omega \right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\Im {\varsigma }_{n_k}\right)+{\sigma }_{3}d\left(\omega ,\Im \omega \right)\in s\left({\varsigma }_{n_{k+1}},\Im \omega \right)=\bigcup _{\varsigma \in \Im \omega }s\left(d\left({\varsigma }_{n_{k+1}},\varsigma \right)\right),$$

so there exists some ${\omega }_k\in \Im \omega$, such that

$${\sigma }_{1}d\left({\varsigma }_{n_k},\omega \right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\Im {\varsigma }_{n_k}\right)+{\sigma }_{3}d\left(\omega ,\Im \omega \right)\in s\left(d\left({\varsigma }_{n_{k+1}},{\omega }_k\right)\right).$$

So, by definition, we have

$$d\left({\varsigma }_{n_{k+1}},{\omega }_k\right)⪯{\sigma }_{1}d\left({\varsigma }_{n_k},\omega \right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\Im {\varsigma }_{n_k}\right)+{\sigma }_{3}d\left(\omega ,\Im \omega \right).$$

(9)

Since the multi-valued mapping $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ has the g.l.b. property, we have

$$W_{{\varsigma }_{n_k}}-d\left({\varsigma }_{n_k},\Im {\varsigma }_{n_k}\right)\subset P,$$

and

$$W_{\omega }-d\left(\omega ,\Im \omega \right)\subset P,$$

and since ${\varsigma }_{n_{k+1}}\in \Im {\varsigma }_{n_k}$ and ${\omega }_k\in \Im \omega ,$ we have

$$d\left({\varsigma }_{n_k},\Im {\varsigma }_{n_k}\right)⪯d\left({\varsigma }_{n_k},{\varsigma }_{n_k+1}\right),$$

and

$$d\left(\omega ,\Im \omega \right)⪯d\left(\omega ,{\omega }_k\right).$$

Hence, by Inequality (9), we have

$$d\left({\varsigma }_{n_{k+1}},{\omega }_k\right)⪯{\sigma }_{1}d\left({\varsigma }_{n_k},\omega \right)+{\sigma }_{2}d\left({\varsigma }_{n_k},{\varsigma }_{n_k+1}\right)+{\sigma }_{3}d\left(\omega ,{\omega }_k\right).$$

(10)

To conclude, if for some natural number k, we have ${\varsigma }_{n_k}={\varsigma }_{n_{k+1}}$, then ${\varsigma }_{n_k}$ is an FP of ℑ.

Assume that ${\varsigma }_{n_k}\ne {\varsigma }_{n_{k+1}}$ for all $k\in \mathbb{N}$. This implies that there exists an infinite subset J of $\mathbb{N}$, such that ${\varsigma }_{n_k}\ne \omega ,$ for all $n_k\in J$. As $\left\{{\varsigma }_{n_k}\right\}$ is a subsequence of $\left\{{\varsigma }_n\right\}$, we have

$$\begin{array}{ccc}\hfill d(\omega ,{\omega }_k)& ⪯& d(\omega ,{\varsigma }_{n_k+1})+d({\varsigma }_{n_k+1},{\omega }_k)\hfill \\ & ⪯& d(\omega ,{\varsigma }_{n_k+1})+{\sigma }_{1}d\left({\varsigma }_{n_k},\omega \right)+{\sigma }_{2}d\left({\varsigma }_{n_k},{\varsigma }_{n_k+1}\right)+{\sigma }_{3}d\left(\omega ,{\omega }_k\right)\hfill \\ & ⪯& d(\omega ,{\varsigma }_{n_k+1})+{\sigma }_{1}d\left({\varsigma }_{n_k},\omega \right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\omega \right)+{\sigma }_{2}d\left(\omega ,{\varsigma }_{n_k+1}\right)+{\sigma }_{3}d\left(\omega ,{\omega }_k\right),\hfill \end{array}$$

which implies

$$d(\omega ,{\omega }_k)⪯\left({\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}}\right)d\left({\varsigma }_{n_k},\omega \right)+\left({\displaystyle \frac{1+{\sigma }_2}{1-{\sigma }_3}}\right)d(\omega ,{\varsigma }_{n_k+1}),$$

which implies that, by (4) and (5), we have

$$d(\omega ,{\omega }_k)\ll c,$$

for all $n_k\ge k>N(c)$. Consequently, the sequence ${\omega }_k$ converges to $\omega .$ Given that $\Im \omega$ is a closed set, it follows that $\omega \in \Im \omega$. This completes the proof. □

Note: In the subsequent parts of this section, we will consider $(\mathcal{Z},d)$ to be a complete tvs-VCMS endowed with a solid cone P.

Example 2.

Consider the interval $\mathcal{Z}=[0,1]$ and the set $\mathbb{E}$ comprising all real-valued functions defined on $\mathcal{Z}$ with continuous derivatives. Under standard function addition and scalar multiplication, $\mathbb{E}$ forms a vector space over the field of real numbers $\mathbb{R},$ that is,

$$\left(\varsigma +\varrho \right)(t)=\varsigma (t)+\varrho (t)$$

and

$$(\alpha \varsigma )(t)=\alpha \varsigma (t)$$

for all $\varsigma ,\varrho \in \mathcal{Z}$ and $\alpha \in \mathbb{R}.$ Let τ be the finest locally convex topology on $\mathbb{E}$. Then, $(\mathcal{Z},\tau )$ is a tvs, which is not normable and is not even metrizable. Define a function $d:\mathcal{Z}\times \mathcal{Z}\to \mathbb{E}$ by

$$\left(d(\varsigma ,\varrho )\right)(t)=\left|\varsigma -\varrho \right|e^t,$$

and

$$P=\left\{\varsigma \in \mathbb{E}:\mathbf{0}⪯\varsigma (t),\mathrm{for}\mathrm{all}t\in \mathcal{Z}\right\}.$$

Consequently, $(\mathcal{Z},d)$ constitutes a tvs-VCMS. Now, define the multi-valued mapping $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ by

$$\Im \varsigma =\left[0,{\displaystyle \frac{\varsigma }{20}}\right].$$

Then,

$$\left(d(\varsigma ,\varrho )\right)(t)=\left|\varsigma -\varrho \right|e^t$$

$$d(\varsigma ,\Im \varsigma )(t)=\left({\displaystyle \frac{19\varsigma }{20}}\right)e^t$$

$$d(\varrho ,\Im \varrho )(t)=\left({\displaystyle \frac{19\varrho }{20}}\right)e^t$$

$$s(\Im \varsigma ,\Im \varrho )={\displaystyle \frac{\left|\varsigma -\varrho \right|}{20}}{e}^t.$$

Now, for any ϱ in $\Im \varsigma =[0,{\displaystyle \frac{\varsigma }{20}}],$ we have $0\le \varrho \le {\displaystyle \frac{\varsigma }{20}}.$ Therefore,

$$d(\varsigma ,\varrho )(t)=\left|\varsigma -\varrho \right|e^t\le {\displaystyle \frac{\varsigma }{20}}{e}^t.$$

Thus,

$$s(d(\varsigma ,\varrho ))=\left\{\varpi \in \mathbb{E}:{\displaystyle \frac{\varsigma }{20}}{e}^t⪯\varpi \right\}.$$

Hence,

$$s(\varsigma ,\Im \varsigma )=\left\{\varpi \in \mathbb{E}:{\displaystyle \frac{\varsigma }{20}}{e}^t⪯\varpi \right\}.$$

Then, for ${\sigma }_1={\displaystyle \frac{1}{3}},{\sigma }_2={\displaystyle \frac{1}{4}}$ and ${\sigma }_3={\displaystyle \frac{1}{5}},$ we have $\beta ={\displaystyle \frac{35}{48}}=max\left\{{\displaystyle \frac{35}{48}},{\displaystyle \frac{3}{5}}\right\}<1.$ Now, since ${\displaystyle \frac{1}{\beta }}={\displaystyle \frac{48}{35}}>1$ and

$${\displaystyle \frac{1}{\beta }}d(\varsigma ,\varrho )={\displaystyle \frac{48}{35}}\left|\varsigma -\varrho \right|e^t\ge {\displaystyle \frac{\varsigma }{20}}{e}^t,$$

so,

$${\displaystyle \frac{1}{\beta }}d(\varsigma ,\varrho )\in s(\varsigma ,\Im \varsigma ).$$

Moreover, since

$${\displaystyle \frac{1}{20}}\left|\varsigma -\varrho \right|e^t\le {\displaystyle \frac{1}{3}}\left|\varsigma -\varrho \right|e^t+{\displaystyle \frac{1}{4}}\left({\displaystyle \frac{19\varsigma }{20}}\right)e^t+{\displaystyle \frac{1}{5}}\left({\displaystyle \frac{19\varrho }{20}}\right)e^t,$$

so

$${\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho )\in s(\Im \varsigma ,\Im \varrho ).$$

Thus, all the conditions of Theorem 1 are satisfied, and ℑ has a fixed point 0 in $\mathcal{Z}$.

Corollary 1.

Let $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b. property. Assume that there exists $\sigma \in [0,1)$ such that

$${\displaystyle \frac{1}{\sigma }}d(\varsigma ,\varrho )\in s(\varsigma ,\Im \varsigma )\mathit{implies}\sigma d(\varsigma ,\varrho )\in s(\Im \varsigma ,\Im \varrho ),$$

for all $\varsigma ,\varrho \in \mathcal{Z}.$ Then, there exists $\omega \in \mathcal{Z}$, such that $\omega \in \Im \omega$.

Proof. 

Take ${\sigma }_1=\sigma$ and ${\sigma }_2={\sigma }_3=0$ in Theorem 1. □

We now present a simplified version of Theorem 1, which is obtained by considering the special case where the additional condition ${\displaystyle \frac{1}{\beta }}d(\varsigma ,\varrho )\in s(\varsigma ,\Im \varsigma )$ holds automatically.

Corollary 2.

Let $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b. property. Assume that there exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that

$${\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho )\in s(\Im \varsigma ,\Im \varrho ),$$

for all $\varsigma ,\varrho \in \mathcal{Z}.$ Then, there exists $\omega \in \mathcal{Z}$, such that $\omega \in \Im \omega$.

Proof. 

The proof of this corollary is straightforward. □

Now, we derive a result that follows directly from Corollary 2, which is one of the theorems of Azam et al. [18].

Corollary 3.

Let $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b. property. Assume that there exists $\sigma \in [0,1)$ such that

$$\sigma d(\varsigma ,\varrho )\in s(\Im \varsigma ,\Im \varrho )$$

for all $\varsigma ,\varrho \in \mathcal{Z}.$ Then, there exists $\omega \in \mathcal{Z}$, such that $\omega \in \Im \omega$.

Proof. 

Take ${\sigma }_1=\sigma$ and ${\sigma }_2={\sigma }_3=0$ in Corollary 2. □

Corollary 4.

Let $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b. property. Assume that there exists $\sigma \in [0,{\displaystyle \frac{1}{2}})$, such that

$${\displaystyle \frac{1}{\sigma }}d(\varsigma ,\varrho )\in s(\varsigma ,\Im \varsigma )\mathit{implies}\sigma \left(d(\varsigma ,\Im \varsigma )+d(\varrho ,\Im \varrho )\right)\in s(\Im \varsigma ,\Im \varrho ),$$

for all $\varsigma ,\varrho \in \mathcal{Z}.$ Then, there exists $\omega \in \mathcal{Z}$, such that $\omega \in \Im \omega$.

Proof. 

Take ${\sigma }_1=0$ and ${\sigma }_2={\sigma }_3=\sigma$ in Theorem 1. □

Corollary 5.

Let $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b. property. Assume that there exists $\sigma \in [0,{\displaystyle \frac{1}{2}})$, such that

$$\sigma \left(d(\varsigma ,\Im \varsigma )+d(\varrho ,\Im \varrho )\right)\in s(\Im \varsigma ,\Im \varrho ),$$

for all $\varsigma ,\varrho \in \mathcal{Z}.$ Then, there exists $\omega \in \mathcal{Z}$, such that $\omega \in \Im \omega$.

Proof. 

The proof of it follows directly from the preceding corollary. □

Corollary 6.

Let $\aleph :\mathcal{Z}\to \mathcal{Z}$ be a self-mapping. Assume that there exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that

$$\beta d(\varsigma ,\aleph \varsigma )⪯d(\varsigma ,\varrho )\mathit{implies}d(\aleph \varsigma ,\aleph \varrho )⪯{\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\aleph \varsigma )+{\sigma }_{3}d(\varrho ,\aleph \varrho ),$$

for all $\varsigma ,\varrho \in \mathcal{Z}$, where $\beta =max\left\{{\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}},{\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_1}}\right\}.$ Then, there exists $\omega \in \mathcal{Z}$, such that $\omega =\aleph \omega$.

Proof. 

Define $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ by $\Im (\varsigma )=\left\{\aleph (\varsigma )\right\},$ for every $\varsigma \in \mathcal{Z}$ and for some function $\aleph :\mathcal{Z}\to \mathcal{Z}$ in Theorem 1. □

Remark 4.

From a practical standpoint, the conditions given in Theorem 1 may be overly restrictive, as it is common for a mapping ℑ to satisfy Condition (1) only on a specific subset $\mathcal{Y}$ of $\mathcal{Z}$. Nevertheless, if $\mathcal{Y}$ is complete, ℑ will have an FP within $\mathcal{Y}$. In this way, we present a notable result regarding the existence of FPs for mappings that fulfill the conditions on closed ball $\overline{B_d}\left({\varsigma }_0,r\right)=\{\varrho \in \mathcal{Z}:d(\varsigma ,\varrho )⪯r\}$ of a complete tvs-VCMS $\mathcal{Z}$ with a solid cone $P.$ This result hinges on a carefully chosen initial point ${\varsigma }_0$ ensuring that the iterates ${\varsigma }_n$ remain in $\overline{B_d}\left({\varsigma }_0,r\right).$

Theorem 2.

Let $\Im :\overline{B_d}\left({\varsigma }_0,r\right)\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b property. Suppose that there exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that

$${\displaystyle \frac{1}{\beta }}d(\varsigma ,\varrho )\in s(\varsigma ,\Im \varsigma )$$

implies

$${\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho )\in s(\Im \varsigma ,\Im \varrho ),$$

(11)

for all ${\varsigma }_0,\varsigma ,\varrho \in \overline{B_d}\left({\varsigma }_0,r\right)$, where $\beta =max\left\{{\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}},{\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_1}}\right\}.$ Moreover, there exists $\mathbf{0}\ll r\in P$ such that

$$\left(1-\beta \right)r\in s({\varsigma }_0,\Im {\varsigma }_0).$$

(12)

Then, there exists $\omega \in \overline{B_d}\left({\varsigma }_0,r\right)$, such that $\omega \in \Im \omega$.

Proof. 

By Inclusion (12), we have

$$\left(1-\beta \right)r\in s({\varsigma }_0,\Im {\varsigma }_0)=\bigcup _{\varsigma \in \Im {\varsigma }_0}s\left(d\left({\varsigma }_0,\varsigma \right)\right).$$

Since $\Im {\varsigma }_0$ is a non-empty and closed subset of $\mathcal{Z},$ there exists ${\varsigma }_1\in$$\Im {\varsigma }_0$, such that

$$\left(1-\beta \right)r\in s\left(d\left({\varsigma }_0,{\varsigma }_1\right)\right),$$

which yields

$$d\left({\varsigma }_0,{\varsigma }_1\right)⪯\left(1-\beta \right)r⪯r.$$

(13)

Hence, ${\varsigma }_1\in \overline{B_d}\left({\varsigma }_0,r\right).$ Now, since ${\varsigma }_0,{\varsigma }_1\in \overline{B_d}\left({\varsigma }_0,r\right)$ and ${\varsigma }_1\in$$\Im {\varsigma }_0.$ Moreover,

$$d\left({\varsigma }_0,{\varsigma }_1\right)⪯{\displaystyle \frac{1}{\beta }}d\left({\varsigma }_0,{\varsigma }_1\right).$$

Then, by Lemma 2 (iv), we have

$${\displaystyle \frac{1}{\beta }}d\left({\varsigma }_0,{\varsigma }_1\right)\in s({\varsigma }_0,\Im {\varsigma }_0).$$

Then, by (11), we have

$${\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1)\in s\left(\Im {\varsigma }_0,\Im {\varsigma }_1\right).$$

This implies that

$${\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1)\in \left(\bigcap _{\varsigma \in \Im {\varsigma }_0}s\left(\varsigma ,\Im {\varsigma }_1\right)\right),$$

which further yields

$${\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1)\in s\left(\varsigma ,\Im {\varsigma }_1\right),\mathrm{for}\mathrm{all}\varsigma \in \Im {\varsigma }_0.$$

Since ${\varsigma }_1\in \Im {\varsigma }_0$, we have

$${\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1)\in s\left({\varsigma }_1,\Im {\varsigma }_1\right).$$

By definition, we have

$${\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1)\in s\left({\varsigma }_1,\Im {\varsigma }_1\right)=\bigcup _{\varsigma \in \Im {\varsigma }_1}s\left(d\left({\varsigma }_1,\varsigma \right)\right).$$

Therefore, there exists some ${\varsigma }_2\in \Im {\varsigma }_1$, such that

$${\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1)\in s\left(d\left({\varsigma }_1,{\varsigma }_2\right)\right).$$

Thus, by definition, we have

$$d\left({\varsigma }_1,{\varsigma }_2\right)⪯{\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d({\varsigma }_0,\Im {\varsigma }_0)+{\sigma }_{3}d({\varsigma }_1,\Im {\varsigma }_1).$$

(14)

Since the mapping $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ has the g.l.b. property, we have

$$W_{{\varsigma }_0}-d({\varsigma }_0,\Im {\varsigma }_0)\subset P,$$

and

$$W_{{\varsigma }_1}-d({\varsigma }_1,\Im {\varsigma }_1)\subset P,$$

which yields that

$$d({\varsigma }_0,\Im {\varsigma }_0)⪯d\left({\varsigma }_0,{\varsigma }_1\right),$$

and

$$d({\varsigma }_1,\Im {\varsigma }_1)⪯d\left({\varsigma }_1,{\varsigma }_2\right).$$

Hence, by (14), we have

$$d\left({\varsigma }_1,{\varsigma }_2\right)⪯{\sigma }_{1}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{2}d\left({\varsigma }_0,{\varsigma }_1\right)+{\sigma }_{3}d\left({\varsigma }_1,{\varsigma }_2\right),$$

which further implies that

$$d\left({\varsigma }_1,{\varsigma }_2\right)⪯\beta d\left({\varsigma }_0,{\varsigma }_1\right),$$

(15)

where $\beta ={\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}}<1,$ which implies by (13) that

$$d\left({\varsigma }_1,{\varsigma }_2\right)⪯\beta d\left({\varsigma }_0,{\varsigma }_1\right)⪯\beta \left(1-\beta \right)r.$$

(16)

Now, by the triangle inequality, (15) and (16), we have

$$\begin{array}{ccc}\hfill d\left({\varsigma }_0,{\varsigma }_2\right)& ⪯& d\left({\varsigma }_0,{\varsigma }_1\right)+d\left({\varsigma }_1,{\varsigma }_2\right)\hfill \\ & ⪯& \left(1-\beta \right)r+\beta \left(1-\beta \right)r\hfill \\ & =& \left(1-{\beta }^2\right)r\hfill \\ & ⪯& r,\hfill \end{array}$$

which implies that ${\varsigma }_2\in \overline{B_d}\left({\varsigma }_0,r\right).$ Now, since

$$d\left({\varsigma }_1,{\varsigma }_2\right)⪯{\displaystyle \frac{1}{\beta }}d\left({\varsigma }_1,{\varsigma }_2\right),$$

by Lemma 2 (iv), we get

$${\displaystyle \frac{1}{\beta }}d\left({\varsigma }_1,{\varsigma }_2\right)\in s({\varsigma }_1,\Im {\varsigma }_1).$$

Then, by (11), we have

$${\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2)\in s\left(\Im {\varsigma }_1,\Im {\varsigma }_2\right).$$

This implies that

$${\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2)\in \left(\bigcap _{\varsigma \in \Im {\varsigma }_1}s\left(\varsigma ,\Im {\varsigma }_2\right)\right),$$

which further yields

$${\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2)\in s\left(\varsigma ,\Im {\varsigma }_2\right),\mathrm{for}\mathrm{all}\varsigma \in \Im {\varsigma }_1.$$

Since ${\varsigma }_2\in \Im {\varsigma }_1$, we have

$${\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2)\in s\left({\varsigma }_2,\Im {\varsigma }_2\right).$$

By definition, we have

$${\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2)\in s\left({\varsigma }_2,\Im {\varsigma }_2\right)=\bigcup _{\varsigma \in \Im {\varsigma }_2}s\left(d\left({\varsigma }_2,\varsigma \right)\right),$$

so there exists some ${\varsigma }_3\in \Im {\varsigma }_2$, such that

$${\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2)\in s\left(d\left({\varsigma }_2,{\varsigma }_3\right)\right).$$

Then, by definition, we have

$$d\left({\varsigma }_2,{\varsigma }_3\right)⪯{\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d({\varsigma }_1,\Im {\varsigma }_1)+{\sigma }_{3}d({\varsigma }_2,\Im {\varsigma }_2).$$

(17)

Since the mapping $\Im :\mathcal{Z}\to Cl(\mathcal{Z})$ has the g.l.b. property, we have

$$W_{{\varsigma }_1}-d({\varsigma }_1,\Im {\varsigma }_1)\subset P,$$

and

$$W_{{\varsigma }_2}-d({\varsigma }_2,\Im {\varsigma }_2)\subset P,$$

and since ${\varsigma }_2\in \Im {\varsigma }_1$ and ${\varsigma }_3\in \Im {\varsigma }_2,$ we have

$$d({\varsigma }_1,\Im {\varsigma }_1)⪯d\left({\varsigma }_1,{\varsigma }_2\right),$$

and

$$d({\varsigma }_2,\Im {\varsigma }_2)⪯d\left({\varsigma }_2,{\varsigma }_3\right).$$

Hence, by Inequality (17), we have

$$d\left({\varsigma }_2,{\varsigma }_3\right)⪯{\sigma }_{1}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{2}d\left({\varsigma }_1,{\varsigma }_2\right)+{\sigma }_{3}d\left({\varsigma }_2,{\varsigma }_3\right),$$

which further implies that

$$d\left({\varsigma }_2,{\varsigma }_3\right)⪯\beta d\left({\varsigma }_1,{\varsigma }_2\right),$$

where $\beta ={\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}}<1.$ Then, by (16), we have

$$d\left({\varsigma }_2,{\varsigma }_3\right)⪯{\beta }^2\left(1-\beta \right)r.$$

(18)

Now, by the triangle inequality, (15), (16) and (18), we have

$$\begin{array}{ccc}\hfill d\left({\varsigma }_0,{\varsigma }_3\right)& ⪯& d\left({\varsigma }_0,{\varsigma }_1\right)+d\left({\varsigma }_1,{\varsigma }_2\right)+d\left({\varsigma }_2,{\varsigma }_3\right)\hfill \\ & ⪯& \left(1-\beta \right)r+\beta \left(1-\beta \right)r+{\beta }^2\left(1-\beta \right)r\hfill \\ & =& r-\beta r+\beta r-{\beta }^{2}r+{\beta }^{2}r-{\beta }^{3}r\hfill \\ & =& (1-{\beta }^3)r\hfill \\ & ⪯& r,\hfill \end{array}$$

which implies that ${\varsigma }_3\in \overline{B_d}\left({\varsigma }_0,r\right).$ A sequence {${\varsigma }_n$} in $\overline{B_d}\left({\varsigma }_0,r\right)$ is constructed through successive iterations such that ${\varsigma }_{n+1}\in \Im {\varsigma }_n$ and

$$d\left({\varsigma }_n,{\varsigma }_{n+1}\right)⪯\beta d\left({\varsigma }_{n-1},{\varsigma }_n\right),$$

for all $n\in \mathbb{N}.$ Now, by Inequalities (15) and (18), we have

$$d\left({\varsigma }_n,{\varsigma }_{n+1}\right)⪯{\beta }^{n}d\left({\varsigma }_0,{\varsigma }_1\right).$$

The remaining steps of the proof closely follow the argument presented in Theorem 1; hence, the details are omitted for brevity. To be more specific, the sequence $\left\{{\varsigma }_n\right\}$ satisfies the Cauchy criterion within the closed ball $\overline{B_d}\left({\varsigma }_0,r\right).$ Since $\overline{B_d}\left({\varsigma }_0,r\right)$ is closed, it is complete. Hence, $\left\{{\varsigma }_n\right\}$ converges to some point $\omega \in \overline{B_d}\left({\varsigma }_0,r\right).$ Following the same argument used in Theorem 1, we can conclude that $\omega$ is a fixed point of ℑ. □

Corollary 7.

Let $\Im :\overline{B_d}\left({\varsigma }_0,r\right)\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b property. Suppose that there exists $\sigma \in [0,1)$, such that

$${\displaystyle \frac{1}{\sigma }}d(\varsigma ,\varrho )\in s(\varsigma ,\Im \varsigma )\mathit{implies}\sigma d(\varsigma ,\varrho )\in s(\Im \varsigma ,\Im \varrho ),$$

for all ${\varsigma }_0,\varsigma ,\varrho \in \overline{B_d}\left({\varsigma }_0,r\right)$. Moreover, there exists $\mathbf{0}\ll r\in P$, such that

$$\left(1-\sigma \right)r\in s({\varsigma }_0,\Im {\varsigma }_0).$$

Then, there exists $\omega \in \overline{B_d}\left({\varsigma }_0,r\right)$, such that $\omega \in \Im \omega$.

Proof. 

Take ${\sigma }_1=\sigma$ and ${\sigma }_2={\sigma }_3=0$ in Theorem 2. □

Corollary 8.

Let $\Im :\overline{B_d}\left({\varsigma }_0,r\right)\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b property. Suppose that there exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that

$${\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho )\in s(\Im \varsigma ,\Im \varrho ),$$

for all ${\varsigma }_0,\varsigma ,\varrho \in \overline{B_d}\left({\varsigma }_0,r\right)$, where $\beta =max\left\{{\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}},{\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_1}}\right\}.$ Moreover, there exists $\mathbf{0}\ll r\in P$, such that

$$\left(1-\beta \right)r\in s({\varsigma }_0,\Im {\varsigma }_0).$$

Then, there exists $\omega \in \overline{B_d}\left({\varsigma }_0,r\right)$, such that $\omega \in \Im \omega$.

Proof. 

The proof of it is obvious. □

Corollary 9.

Let $\Im :\overline{B_d}\left({\varsigma }_0,r\right)\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b property. Suppose that there exists $\sigma \in [0,1)$, such that

$$\sigma d(\varsigma ,\varrho )\in s(\Im \varsigma ,\Im \varrho ),$$

for all ${\varsigma }_0,\varsigma ,\varrho \in \overline{B_d}\left({\varsigma }_0,r\right)$. Moreover, there exists $\mathbf{0}\ll r\in P$, such that

$$\left(1-\sigma \right)r\in s({\varsigma }_0,\Im {\varsigma }_0).$$

Then, there exists $\omega \in \overline{B_d}\left({\varsigma }_0,r\right)$, such that $\omega \in \Im \omega$.

Proof. 

Take ${\sigma }_1=\sigma$ and ${\sigma }_2={\sigma }_3=0$ in the above corollary. □

Corollary 10.

Let $\Im :\overline{B_d}\left({\varsigma }_0,r\right)\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b property. Suppose that there exists $\sigma \in [0,{\displaystyle \frac{1}{2}})$, such that

$${\displaystyle \frac{1}{\sigma }}d(\varsigma ,\varrho )\in s(\varsigma ,\Im \varsigma )\mathit{implies}\sigma \left(d(\varsigma ,\Im \varsigma )+d(\varrho ,\Im \varrho )\right)\in s(\Im \varsigma ,\Im \varrho ),$$

for all ${\varsigma }_0,\varsigma ,\varrho \in \overline{B_d}\left({\varsigma }_0,r\right).$ Moreover, there exists $\mathbf{0}\ll r\in P$, such that

$$\left(1-\sigma \right)r\in s({\varsigma }_0,\Im {\varsigma }_0).$$

Then, there exists $\omega \in \overline{B_d}\left({\varsigma }_0,r\right)$, such that $\omega \in \Im \omega$.

Proof. 

Take ${\sigma }_1=0$ and ${\sigma }_2={\sigma }_3=\sigma$ in Theorem 2. □

Corollary 11.

Let $\Im :\overline{B_d}\left({\varsigma }_0,r\right)\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b property. Assume that there exists $\sigma \in [0,{\displaystyle \frac{1}{2}})$, such that

$$\sigma \left(d(\varsigma ,\Im \varsigma )+d(\varrho ,\Im \varrho )\right)\in s(\Im \varsigma ,\Im \varrho ),$$

for all ${\varsigma }_0,\varsigma ,\varrho \in \overline{B_d}\left({\varsigma }_0,r\right).$ Moreover, there exists $\mathbf{0}\ll r\in P$, such that

$$\left(1-\sigma \right)r\in s({\varsigma }_0,\Im {\varsigma }_0).$$

Then, there exists $\omega \in \overline{B_d}\left({\varsigma }_0,r\right)$, such that $\omega \in \Im \omega$.

Proof. 

The above corollary directly implies this result. □

Corollary 12.

Let $\aleph :\overline{B_d}\left({\varsigma }_0,r\right)\to \mathcal{Z}$ be a single-valued mapping. Suppose that there exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that $\beta d(\varsigma ,\aleph \varsigma )⪯d(\varsigma ,\varrho )$ implies

$$d(\aleph \varsigma ,\aleph \varrho )⪯{\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\aleph \varsigma )+{\sigma }_{3}d(\varrho ,\aleph \varrho ),$$

for all ${\varsigma }_0,\varsigma ,\varrho \in \overline{B_d}\left({\varsigma }_0,r\right)$, where $\beta =max\left\{{\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}},{\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_1}}\right\}.$ Moreover, there exists $\mathbf{0}\ll r\in P$, such that

$$d({\varsigma }_0,\aleph {\varsigma }_0)⪯\left(1-\beta \right)r.$$

Then, there exists $\omega \in \overline{B_d}\left({\varsigma }_0,r\right)$, such that $\omega =\aleph \omega$.

Proof. 

Define $\Im :\overline{B_d}\left({\varsigma }_0,r\right)\to Cl(\mathcal{Z})$ by $\Im (\varsigma )=\left\{\aleph (\varsigma )\right\},$ for every $\varsigma \in \overline{B_d}\left({\varsigma }_0,r\right)$ and for some function $\aleph :\overline{B_d}\left({\varsigma }_0,r\right)\to \mathcal{Z}$ in Theorem 2. □

4. Fixed-Point Results in Cone Metric Spaces

If we take the tvs as a Banach space, then the tvs-VCMS is reduced to a CMS. Throughout this section, we assume that $(\mathcal{Z},d)$ is a complete CMS endowed with a solid cone P. We obtain the following result, which is an extension of the main result of Mehmood et al. [13].

Corollary 13.

Let $\Im :(\mathcal{Z},d)\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b. property. Suppose that there exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that

$${\displaystyle \frac{1}{\beta }}d(\varsigma ,\varrho )\in s(\varsigma ,\Im \varsigma )$$

implies

$${\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho )\in s(\Im \varsigma ,\Im \varrho ),$$

for all $\varsigma ,\varrho \in \mathcal{Z}$, where $\beta =max\left\{{\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}},{\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_1}}\right\}.$ Then, there exists $\omega \in \mathcal{Z}$, such that $\omega \in \Im \omega$.

Proof. 

Take the tvs as a Banach space in Theorem 1. □

Corollary 14.

Let $\Im :(\mathcal{Z},d)\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b. property. Suppose that there exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that

$${\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho )\in s(\Im \varsigma ,\Im \varrho ),$$

for all $\varsigma ,\varrho \in \mathcal{Z}.$ Then, there exists $\omega \in \mathcal{Z}$, such that $\omega \in \Im \omega$.

Proof. 

This result is a straightforward consequence of Corollary 13. □

The above result generalizes the key theorem of Cho et al. [11] in this way.

Corollary 15

([11]). Let $\Im :(\mathcal{Z},d)\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b. property. Suppose that there exists $\sigma \in [0,1)$, such that

$$\sigma d(\varsigma ,\varrho )\in s(\Im \varsigma ,\Im \varrho ),$$

for all $\varsigma ,\varrho \in \mathcal{Z}.$ Then, there exists $\omega \in \mathcal{Z}$, such that $\omega \in \Im \omega$.

Proof. 

Take ${\sigma }_1=\sigma$ and ${\sigma }_1={\sigma }_1=0$ in Corollary 14. □

Now, we derive a result that is one of the results of Cho et al. [12].

Corollary 16

([12]). Let $\Im :(\mathcal{Z},d)\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b. property. Suppose that there exists $\sigma \in [0,{\displaystyle \frac{1}{2}})$, such that

$$\sigma \left(d(\varsigma ,\Im \varsigma )+d(\varrho ,\Im \varrho )\right)\in s(\Im \varsigma ,\Im \varrho ),$$

for all $\varsigma ,\varrho \in \mathcal{Z}.$ Then, there exists $\omega \in \mathcal{Z}$, such that $\omega \in \Im \omega$.

Proof. 

Take ${\sigma }_1=0$ and ${\sigma }_2={\sigma }_3=\sigma$ in Corollary 14. □

Corollary 17.

Let $\Im :\overline{B_d}\left({\varsigma }_0,r\right)\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b. property. Suppose that there exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that

$${\displaystyle \frac{1}{\beta }}d(\varsigma ,\varrho )\in s(\varsigma ,\Im \varsigma )$$

implies

$${\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho )\in s(\Im \varsigma ,\Im \varrho ),$$

for all ${\varsigma }_0,\varsigma ,\varrho \in \overline{B_d}\left({\varsigma }_0,r\right)$, where $\beta =max\left\{{\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}},{\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_1}}\right\}.$ Moreover, there exists $\mathbf{0}\ll r\in P$, such that

$$\left(1-\beta \right)r\in s({\varsigma }_0,\Im {\varsigma }_0),$$

Then, there exists $\omega \in \overline{B_d}\left({\varsigma }_0,r\right)$, such that $\omega \in \Im \omega$.

Proof. 

Take the tvs as a Banach space in Theorem 2. □

Corollary 18.

Let $\Im :\overline{B_d}\left({\varsigma }_0,r\right)\to Cl(\mathcal{Z})$ be a multi-valued mapping having the g.l.b. property. Suppose that there exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that

$${\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho )\in s(\Im \varsigma ,\Im \varrho ),$$

for all ${\varsigma }_0,\varsigma ,\varrho \in \overline{B_d}\left({\varsigma }_0,r\right)$, where $\beta =max\left\{{\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}},{\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_1}}\right\}.$ Moreover, there exists $\mathbf{0}\ll r\in P$, such that

$$\left(1-\beta \right)r\in s({\varsigma }_0,\Im {\varsigma }_0).$$

Then, there exists $\omega \in \overline{B_d}\left({\varsigma }_0,r\right)$, such that $\omega \in \Im \omega$.

Proof. 

It follows directly from the above corollary. □

5. Fixed-Point Results in Metric Spaces

Throughout this section, we work with a complete MS $(\mathcal{Z},d).$ By specializing our main result. We recover the primary theorem of Kikkawa and Suzuki [6] in this way.

Corollary 19

([6]). Let $\Im :(\mathcal{Z},d)\to Cl(\mathcal{Z})$. Suppose that there exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that $\beta d(\varsigma ,\Im \varsigma )\le d(\varsigma ,\varrho )$ implies

$$H(\Im \varsigma ,\Im \varrho )\le {\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho )$$

for all $\varsigma ,\varrho \in \mathcal{Z}$, where $\beta =max\left\{{\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}},{\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_1}}\right\}.$ Then, there exists $\omega \in \mathcal{Z}$, such that $\omega \in \Im \omega$.

Proof. 

Take $\mathbb{E}=\mathbb{R}$ and take the infimum of $s(\Xi ,\Theta )$ in Theorem 1. □

Corollary 20.

Let $\Im :(\mathcal{Z},d)\to Cl(\mathcal{Z})$. Suppose that there exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that

$$H(\Im \varsigma ,\Im \varrho )\le {\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho ),$$

for all $\varsigma ,\varrho \in \mathcal{Z}.$ Then, there exists $\omega \in \mathcal{Z}$, such that $\omega \in \Im \omega$.

Proof. 

It is obvious from the above corollary. □

Remark 5.

By setting ${\sigma }_2={\sigma }_2=0,$ in the above corollary, we obtain Nadler’s main theorem [7].

Corollary 21.

Let $\Im :\overline{B_d}\left({\varsigma }_0,r\right)\to Cl(\mathcal{Z})$. Suppose that there exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that $\beta d(\varsigma ,\Im \varsigma )\le d(\varsigma ,\varrho )$ implies

$$H(\Im \varsigma ,\Im \varrho )\le {\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho ),$$

for all ${\varsigma }_0,\varsigma ,\varrho \in \overline{B_d}\left({\varsigma }_0,r\right)$, where $\beta =max\left\{{\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}},{\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_1}}\right\}.$ Moreover, there exists $r>0$, such that

$$d({\varsigma }_0,\Im {\varsigma }_0)\le \left(1-\beta \right)r.$$

Then, there exists $\omega \in \overline{B_d}\left({\varsigma }_0,r\right)$, such that $\omega \in \Im \omega$.

Proof. 

Take $\mathbb{E}=\mathbb{R}$ and take the infimum of $s(\Xi ,\Theta )$ in Theorem 2. □

Corollary 22.

Let $\Im :\overline{B_d}\left({\varsigma }_0,r\right)\to Cl(\mathcal{Z})$. Suppose that there exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that

$$H(\Im \varsigma ,\Im \varrho )\le {\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho ),$$

for all ${\varsigma }_0,\varsigma ,\varrho \in \overline{B_d}\left({\varsigma }_0,r\right)$, where $\beta =max\left\{{\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}},{\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_1}}\right\}.$ Moreover, there exists $r>0$, such that

$$d({\varsigma }_0,\Im {\varsigma }_0)\le \left(1-\beta \right)r.$$

Then, there exists $\omega \in \overline{B_d}\left({\varsigma }_0,r\right)$, such that $\omega \in \Im \omega$.

Proof. 

It is obvious from above corollary. □

6. Homotopy Theory

The study of homotopy in the fixed-point theory for multi-valued mappings has garnered significant attention, with important results achieved through generalizations of classical theorems, such as the Brouwer fixed-point theorem and the Kakutani fixed-point theorem. These results allow the extension of single-valued mappings to multi-valued ones, broadening the applicability to problems where uncertainty, randomness, or multiple possible outcomes must be considered. In addition, homotopy techniques have been successfully employed in various applied contexts, such as the study of differential inclusions, optimization problems, and equilibrium theory in economics. By leveraging these topological tools, researchers can gain a deeper understanding of the qualitative behavior of solutions in complex systems, especially where classical methods fall short. This section explores recent advances in homotopy-based fixed-point results for multi-valued mappings, highlighting their theoretical significance and practical applications.

Theorem 3.

Let $(\mathcal{Z},d)$ be a complete tvs-VCMS with a solid cone P, V be an open subset of $\mathcal{Z}$, and $\partial V$ be the boundary of the open subset V in $\mathcal{Z}.$ Assume that that the following conditions hold:

(i) 

There exists a multi-valued mapping $\mathcal{W}:[0,1]\times \overline{V}\to Cl(\mathcal{Z})$ having the g.l.b property and $\varsigma \notin \mathcal{W}(\mathit{\ell },\varsigma ),$ for every $\mathit{\ell }\in [0,1]$ and every $\varsigma \in \partial V$;

(ii) 

There exists a continuous increasing function $\psi :(0,1]\to P$, such that

$$\psi (\hslash )-\psi (\mathit{\ell })\in s(\mathcal{W}(\mathit{\ell },\varsigma ),\mathcal{W}(\hslash ,\varrho )),$$

(19)

also

$$\psi (\hslash )\in \psi (\mathit{\ell })+P,$$

(20)

for all $\hslash ,\mathit{\ell }\in [0,1]$ and each $\varsigma ,\varrho \in \overline{V}$;

(iii) 

There exist ${\sigma }_1,{\sigma }_2,{\sigma }_3\in [0,1)$ with ${\sigma }_1+{\sigma }_2+{\sigma }_3<1$, such that

$${\displaystyle \frac{1}{\beta }}d(\varsigma ,\varrho )\in s(\varsigma ,\mathcal{W}(\mathit{\ell },\varsigma ))$$

implies

$${\sigma }_{1}d(\varsigma ,\varrho )+{\sigma }_{2}d(\varsigma ,\Im \varsigma )+{\sigma }_{3}d(\varrho ,\Im \varrho )\in s(\mathcal{W}(\mathit{\ell },\varsigma ),\mathcal{W}(\hslash ,\varrho )),$$

(21)

for each $\varsigma ,\varrho ,\stackrel{\circ }{\varsigma }\in \overline{V}$, where $\beta =max\left\{{\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}},{\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_1}}\right\}$ and there exists

$$\mathbf{0}\ll r={\displaystyle \frac{2}{1-\beta }}(\psi (\hslash )-\psi (\mathit{\ell }))$$

such that

$$(1-\beta )r\in s(\stackrel{\circ }{\varsigma },\mathcal{W}(\mathit{\ell },\stackrel{\circ }{\varsigma })).$$

(22)

Then, $\mathcal{W}(0,·)$ has an FP if and only if $\mathcal{W}(1,·)$ has an FP.

Proof. 

Define the set

$$\aleph :=\{\mathit{\ell }\in [0,1]:\varsigma \in \mathcal{W}(\mathit{\ell },\varsigma )\mathrm{for}\mathrm{some}\varsigma \in V\}.$$

Assume that $\mathcal{W}(0,·)$ has a fixed-point $\varsigma$, so $\varsigma \in \mathcal{W}(0,\varsigma )$. Then, by Assumption (i), it follows that $\varsigma \in V$.

Then, it is evident that ℵ is non-empty. We introduce the following partial-order ≾ on ℵ in the following way:

$$(\mathit{\ell },\varsigma )\precsim (\hslash ,\varrho )\iff \mathit{\ell }\le \hslash \mathrm{and}d(\varsigma ,\varrho )⪯{\displaystyle \frac{2}{1-\beta }}(\psi (\hslash )-\psi (\mathit{\ell })).$$

To demonstrate the closedness of ℵ, it suffices to show that any totally ordered subset O of ℵ has a least upper bound in ℵ. Let

$$\stackrel{\circ }{\mathit{\ell }}=sup\{\mathit{\ell }:(\mathit{\ell },\varsigma )\in O\}.$$

Let ${\left\{\left({\mathit{\ell }}_n,{\varsigma }_n\right)\right\}}_{n\ge 0}$ be a sequence in O satisfying $\left({\mathit{\ell }}_n,{\varsigma }_n\right)\precsim \left({\mathit{\ell }}_{n+1},{\varsigma }_{n+1}\right)$ and ${\mathit{\ell }}_n\to \stackrel{\circ }{\mathit{\ell }}$ as $n\to \infty$. Then, for $m>n$, and using the continuity and monotonicity of $\psi ,$ we have

$$d\left({\varsigma }_m,{\varsigma }_n\right)⪯{\displaystyle \frac{2}{1-\beta }}\left(\psi \left({\mathit{\ell }}_m\right)-\psi \left({\mathit{\ell }}_n\right)\right)\to \mathbf{0},$$

as $n,m\to \infty .$ This implies that $\left\{{\varsigma }_n\right\}$ is a Cauchy sequence. Since $(\mathcal{Z},d)$ is a complete tvs-VCMS, so there is an element $\stackrel{\circ }{\varsigma }\in \mathcal{Z}$, such that ${\varsigma }_n\to \stackrel{\circ }{\varsigma }$. Choose $n_0(c)\in \mathbb{N}$, such that for $c\gg \mathbf{0}$ we have $d\left(\stackrel{\circ }{\varsigma },{\varsigma }_n\right)\ll \left({\displaystyle \frac{1-{\sigma }_3}{{\sigma }_1+{\sigma }_2}}\right){\displaystyle \frac{c}{2}}$ and $d\left(\stackrel{\circ }{\varsigma },{\varsigma }_n\right)\ll \left({\displaystyle \frac{1-{\sigma }_3}{1+{\sigma }_2}}\right){\displaystyle \frac{c}{2}}$ for all $n\ge n_0(c)$. Choose an arbitrary ${\varsigma }_{n_1}\in \mathcal{Z}\backslash \left\{\stackrel{\circ }{\varsigma }\right\}$ from the sequence $\left\{{\varsigma }_n\right\}$. As ${\varsigma }_n⟶\stackrel{\circ }{\varsigma }$, there exists some $n_1\in \mathbb{N}$, such that

$$d\left({\varsigma }_n,\stackrel{\circ }{\varsigma }\right)⪯{\displaystyle \frac{1}{3}}d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_1}\right),$$

for all $n\in \mathbb{N}$ with $n\ge n_1$. As ${\varsigma }_{n+1}\in \mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right)$ and

$$\begin{array}{ccc}\hfill \beta d\left({\varsigma }_n,{\varsigma }_{n+1}\right)& ⪯& d\left({\varsigma }_n,{\varsigma }_{n+1}\right)\hfill \\ & ⪯& d\left({\varsigma }_n,\stackrel{\circ }{\varsigma }\right)+d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n+1}\right)\hfill \\ & ⪯& {\displaystyle \frac{2}{3}}d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_1}\right)\hfill \\ & ⪯& d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_1}\right)-d\left({\varsigma }_n,\stackrel{\circ }{\varsigma }\right)\hfill \\ & ⪯& d\left({\varsigma }_n,{\varsigma }_{n_1}\right),\hfill \end{array}$$

that is,

$$d\left({\varsigma }_n,{\varsigma }_{n+1}\right)⪯{\displaystyle \frac{1}{\beta }}d\left({\varsigma }_n,{\varsigma }_{n_1}\right),$$

by Lemma 2 (iv), we have

$${\displaystyle \frac{1}{\beta }}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)\in s\left({\varsigma }_n,\mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right)\right).$$

Then, by assumption, we have

$${\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right)\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right)\in s\left(\mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right),\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right).$$

This implies that

$${\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right)\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right)\in \left(\bigcap _{\varsigma \in \mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right)}s\left(\varsigma ,\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right)\right),$$

which further yields

$${\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right)\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right)\in s\left(\varsigma ,\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right),$$

for all $\varsigma \in \mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right).$ Since ${\varsigma }_{n+1}\in \mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right),$ we have

$${\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right)\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right)\in s\left({\varsigma }_{n+1},\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right).$$

By definition, we have

$${\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right)\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right)\in \bigcup _{\varsigma \in \mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)}s\left(d\left({\varsigma }_{n+1},\varsigma \right)\right).$$

This implies that there is ${\varsigma }_{n_2}\in \mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)$, such that

$${\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right)\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right)\in s\left(d\left({\varsigma }_{n+1},{\varsigma }_{n_2}\right)\right)$$

that is,

$$d\left({\varsigma }_{n+1},{\varsigma }_{n_2}\right)⪯{\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,\mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right)\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right).$$

(23)

Since the mapping $\mathcal{W}:[0,1]\times \overline{V}\to Cl(\mathcal{Z})$ has the g.l.b property, we have

$$W_{{\varsigma }_n}-d\left({\varsigma }_n,\mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right)\right)\subset P,$$

and

$$W_{{\varsigma }_{n_1}}-d\left({\varsigma }_{n_1},\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right)\subset P,$$

and since ${\varsigma }_{n+1}\in \mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right)$ and ${\varsigma }_{n_2}\in \mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right),$ we have

$$d\left({\varsigma }_n,\mathcal{W}\left({\mathit{\ell }}_n,{\varsigma }_n\right)\right)⪯d\left({\varsigma }_n,{\varsigma }_{n+1}\right)$$

and

$$d\left({\varsigma }_{n_1},\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right)⪯d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right).$$

Hence, by Inequality (23), we have

$$d\left({\varsigma }_{n+1},{\varsigma }_{n_2}\right)⪯{\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,{\varsigma }_{n+1}\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right)$$

Since

$$\begin{array}{ccc}\hfill d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_2}\right)& ⪯& d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n+1}\right)+d\left({\varsigma }_{n+1},{\varsigma }_{n_2}\right)\hfill \\ & ⪯& d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n+1}\right)+{\sigma }_{1}d\left({\varsigma }_n,{\varsigma }_{n_1}\right)+{\sigma }_{2}d\left({\varsigma }_n,{\varsigma }_{n+1}\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right),\hfill \end{array}$$

letting $n\to +\infty ,$ we obtain

$$d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_2}\right)⪯{\sigma }_{1}d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_1}\right)++{\sigma }_{3}d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right).$$

Now, we prove that

$$\sigma d\left({\varsigma }_{n_1},\stackrel{\circ }{\varsigma }\right)\in s\left(\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right),\mathcal{W}(\mathring{\mathit{\ell }},\varsigma )\right).$$

For each $n\in \mathbb{N}$, it holds that

$$d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_2}\right)⪯d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_2}\right)+{\displaystyle \frac{1}{n}}d\left({\varsigma }_{n_1},\stackrel{\circ }{\varsigma }\right).$$

Moreover,

$$\begin{array}{ccc}\hfill d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right)& ⪯& d\left({\varsigma }_{n_1},\stackrel{\circ }{\varsigma }\right)+d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_2}\right)\hfill \\ & ⪯& d\left({\varsigma }_{n_1},\stackrel{\circ }{\varsigma }\right)+d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_2}\right)+{\displaystyle \frac{1}{n}}d\left({\varsigma }_{n_1},\stackrel{\circ }{\varsigma }\right)\hfill \\ & ⪯& d\left({\varsigma }_{n_1},\stackrel{\circ }{\varsigma }\right)+{\sigma }_{1}d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_1}\right)+{\sigma }_{3}d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right)+{\displaystyle \frac{1}{n}}d\left({\varsigma }_{n_1},\stackrel{\circ }{\varsigma }\right),\hfill \end{array}$$

which implies that

$$d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right)⪯\left({\displaystyle \frac{1+{\sigma }_1}{1-{\sigma }_3}}+{\displaystyle \frac{1}{n(1-{\sigma }_3)}}\right)d\left({\varsigma }_{n_1},\stackrel{\circ }{\varsigma }\right),$$

for all $n\in \mathbb{N}$, we have

$$d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right)⪯\left({\displaystyle \frac{1+{\sigma }_1}{1-{\sigma }_3}}\right)d\left({\varsigma }_{n_1},\stackrel{\circ }{\varsigma }\right)⪯{\displaystyle \frac{1}{\beta }}d\left({\varsigma }_{n_1},\stackrel{\circ }{\varsigma }\right).$$

This gives

$$d\left({\varsigma }_{n_1},{\varsigma }_{n_2}\right)⪯{\displaystyle \frac{1}{\beta }}d\left({\varsigma }_{n_1},\stackrel{\circ }{\varsigma }\right).$$

By Lemma 2 (iv), we have

$${\displaystyle \frac{1}{\beta }}d\left({\varsigma }_{n_1},\stackrel{\circ }{\varsigma }\right)\in s\left({\varsigma }_{n_1},\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right).$$

From the assumption of the theorem, we have

$${\sigma }_{1}d\left({\varsigma }_{n_1},\stackrel{\circ }{\varsigma }\right)+{\sigma }_{2}d\left({\varsigma }_{n_1},\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right)\right)+{\sigma }_{3}d\left(\stackrel{\circ }{\varsigma },\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right)\in s\left(\mathcal{W}\left({\mathit{\ell }}_{n_1},{\varsigma }_{n_1}\right),\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right).$$

Similarly, we conclude

$${\sigma }_{1}d\left({\varsigma }_{n_k},\stackrel{\circ }{\varsigma }\right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right)\right)+{\sigma }_{3}d\left(\stackrel{\circ }{\varsigma },\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right)\in s\left(\mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right),\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right),$$

for each $k\in \mathbb{N}$. This implies that

$${\sigma }_{1}d\left({\varsigma }_{n_k},\stackrel{\circ }{\varsigma }\right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right)\right)+{\sigma }_{3}d\left(\stackrel{\circ }{\varsigma },\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right)\in \left(\bigcap _{\varsigma \in \mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right)}s\left(\varsigma ,\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right)\right),$$

which further yields

$${\sigma }_{1}d\left({\varsigma }_{n_k},\stackrel{\circ }{\varsigma }\right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right)\right)+{\sigma }_{3}d\left(\stackrel{\circ }{\varsigma },\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right)\in s\left(\varsigma ,\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right),$$

for all $\varsigma \in \mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right).$ Since ${\varsigma }_{n_{k+1}}\in \mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right)$, we have

$${\sigma }_{1}d\left({\varsigma }_{n_k},\stackrel{\circ }{\varsigma }\right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right)\right)+{\sigma }_{3}d\left(\stackrel{\circ }{\varsigma },\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right)\in s\left({\varsigma }_{n_{k+1}},\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right).$$

By definition, we have

$$\begin{array}{ccc}\hfill {\sigma }_{1}d\left({\varsigma }_{n_k},\stackrel{\circ }{\varsigma }\right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right)\right)+{\sigma }_{3}d\left(\stackrel{\circ }{\varsigma },\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right)& \in & s\left({\varsigma }_{n_{k+1}},\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right)\hfill \\ & =& \bigcup _{\varsigma \in \mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })}s\left(d\left({\varsigma }_{n_{k+1}},\varsigma \right)\right).\hfill \end{array}$$

Therefore, there exists some ${\stackrel{\circ }{\varsigma }}_n\in \mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })$, such that

$${\sigma }_{1}d\left({\varsigma }_{n_k},\stackrel{\circ }{\varsigma }\right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right)\right)+{\sigma }_{3}d\left(\stackrel{\circ }{\varsigma },\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right)\in s\left(d\left({\varsigma }_{n_{k+1}},{\stackrel{\circ }{\varsigma }}_n\right)\right).$$

Thus, by definition, we have

$$d\left({\varsigma }_{n_{k+1}},{\stackrel{\circ }{\varsigma }}_n\right)⪯{\sigma }_{1}d\left({\varsigma }_{n_k},\stackrel{\circ }{\varsigma }\right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right)\right)+{\sigma }_{3}d\left(\stackrel{\circ }{\varsigma },\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right).$$

(24)

Since the mapping $\mathcal{W}:[0,1]\times \overline{V}\to Cl(\mathcal{Z})$ has the g.l.b property, we have

$$W_{{\varsigma }_{n_k}}-d\left({\varsigma }_{n_k},\mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right)\right)\subset P,$$

and

$$W_{\stackrel{\circ }{\varsigma }}-d\left(\stackrel{\circ }{\varsigma },\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right)\subset P,$$

and since ${\varsigma }_{n_k+1}\in \mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right)$ and ${\stackrel{\circ }{\varsigma }}_n\in \mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma }),$ we have

$$d\left({\varsigma }_{n_k},\mathcal{W}\left({\mathit{\ell }}_{n_k},{\varsigma }_{n_k}\right)\right)⪯d\left({\varsigma }_{n_k},{\varsigma }_{n_k+1}\right),$$

and

$$d\left(\stackrel{\circ }{\varsigma },\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right)⪯d\left(\stackrel{\circ }{\varsigma },{\stackrel{\circ }{\varsigma }}_n\right).$$

Now, by the inequality, we have

$$d\left({\varsigma }_{n_{k+1}},{\stackrel{\circ }{\varsigma }}_n\right)⪯{\sigma }_{1}d\left({\varsigma }_{n_k},\stackrel{\circ }{\varsigma }\right)+{\sigma }_{2}d\left({\varsigma }_{n_k},{\varsigma }_{n_k+1}\right)+{\sigma }_{3}d\left(\stackrel{\circ }{\varsigma },{\stackrel{\circ }{\varsigma }}_n\right).$$

Further,

$$\begin{array}{ccc}\hfill d\left(\stackrel{\circ }{\varsigma },{\stackrel{\circ }{\varsigma }}_n\right)& ⪯& d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_{k+1}}\right)+d\left({\varsigma }_{n_{k+1}},{\stackrel{\circ }{\varsigma }}_n\right)\hfill \\ & ⪯& d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_{k+1}}\right)+{\sigma }_{1}d\left({\varsigma }_{n_k},\stackrel{\circ }{\varsigma }\right)+{\sigma }_{2}d\left({\varsigma }_{n_k},{\varsigma }_{n_k+1}\right)+{\sigma }_{3}d\left(\stackrel{\circ }{\varsigma },{\stackrel{\circ }{\varsigma }}_n\right)\hfill \\ & ⪯& d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_{k+1}}\right)+{\sigma }_{1}d\left({\varsigma }_{n_k},\stackrel{\circ }{\varsigma }\right)+{\sigma }_{2}d\left({\varsigma }_{n_k},\stackrel{\circ }{\varsigma }\right)+{\sigma }_{2}d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_k+1}\right)+{\sigma }_{3}d\left(\stackrel{\circ }{\varsigma },{\stackrel{\circ }{\varsigma }}_n\right),\hfill \end{array}$$

which implies

$$\begin{array}{ccc}\hfill d\left(\stackrel{\circ }{\varsigma },{\stackrel{\circ }{\varsigma }}_n\right)& ⪯& \left({\displaystyle \frac{{\sigma }_1+{\sigma }_2}{1-{\sigma }_3}}\right)d\left({\varsigma }_{n_k},\stackrel{\circ }{\varsigma }\right)+\left({\displaystyle \frac{1+{\sigma }_2}{1-{\sigma }_3}}\right)d\left(\stackrel{\circ }{\varsigma },{\varsigma }_{n_k+1}\right)\hfill \\ & \ll & {\displaystyle \frac{c}{2}}+{\displaystyle \frac{c}{2}}=c.\hfill \end{array}$$

Thus, ${\stackrel{\circ }{\varsigma }}_n\to \stackrel{\circ }{\varsigma }\in \mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })$ and since $\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\in Cl(\mathcal{Z}),$ $\stackrel{\circ }{\varsigma }\in V$. From here, we get $(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\in \aleph$. Thus, $(\mathit{\ell },\varsigma )\precsim (\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })$ for all $(\mathit{\ell },\varsigma )\in O$, which shows that $(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })$ is an upper bound of O. Thus, according to Zorn’s Lemma, ℵ contains a maximal element $(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })$. We assert that $\mathring{\mathit{\ell }}=1$. Suppose that $\mathring{\mathit{\ell }}<1$ and choose $r\gg \mathbf{0}$ and $\mathit{\ell }\ge \mathring{\mathit{\ell }}$, such that

$$B_d(\stackrel{\circ }{\varsigma },r)\subset V,$$

where $r={\displaystyle \frac{2}{1-\beta }}\left(\psi (\mathit{\ell })-\psi (\mathring{\mathit{\ell }})\right).$ Now, using Assumption (ii), we have

$$\psi (\mathit{\ell })-\psi (\mathring{\mathit{\ell }})\in s\left(\mathcal{W}\left(\mathit{\ell },\varsigma \right),\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right)$$

$$\psi (\mathit{\ell })-\psi (\mathring{\mathit{\ell }})\in s\left(\stackrel{\circ }{\varsigma },\mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\right),$$

for $\stackrel{\circ }{\varsigma }\in \mathcal{W}(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })$. Therefore, there exists some $\varsigma \in \mathcal{W}(\mathit{\ell },\varsigma )$, such that

$$\psi (\mathit{\ell })-\psi (\mathring{\mathit{\ell }})\in s(d(\stackrel{\circ }{\varsigma },\varsigma )).$$

Thus, we have

$$d(\stackrel{\circ }{\varsigma },\varsigma )⪯\psi (\mathit{\ell })-\psi (\mathring{\mathit{\ell }})⪯{\displaystyle \frac{(1-\beta )r}{2}}\prec (1-\sigma )r.$$

Thus, we deduce that $\mathcal{W}(\mathit{\ell },·):B(\stackrel{\circ }{\varsigma },r)\to Cl(\mathcal{Z})$ satisfies all assumptions of Theorem (2) for all $\mathit{\ell }\in [0,1]$. Therefore, for all $\mathit{\ell }\in [0,1]$, there exists $\varsigma \in B_d(\stackrel{\circ }{\varsigma },r)$, such that $\varsigma \in \mathcal{W}(\mathit{\ell },\varsigma )$. Thus, $(\varsigma ,\mathit{\ell })\in \aleph$. From

$$d(\varsigma ,\stackrel{\circ }{\varsigma })\prec r={\displaystyle \frac{2}{1-\beta }}(\psi (\mathit{\ell })-\psi (\mathring{\mathit{\ell }})),$$

we have $(\mathring{\mathit{\ell }},\stackrel{\circ }{\varsigma })\precsim (\mathit{\ell },\varsigma )$, which contradicts our assumption. Therefore, $\mathring{\mathit{\ell }}$ must equal 1. Consequently, $\mathcal{W}(·,1)$ has an FP. Conversely, if $\mathcal{W}(1,·)$ possesses an FP, a similar argument can be employed to demonstrate the existence of an FP for $\mathcal{W}(0,·)$. □

7. Conclusions

In this research article, we introduced the Kikkawa and Suzuki-type contractions within the framework of complete tvs-VCMS with a solid cone. By establishing new fixed-point results under these generalized contractive conditions, we extended and refined existing results in the literature. Specifically, our theorems generalized the seminal contributions of Kikkawa and Suzuki [6], Nadler [7], Cho et al. [11,12], Mehmood et al. [13], and Azam et al. [18]. We provided a concrete example of the innovative nature of our leading result. As a demonstration of the applicability of our principal theorem, we proved a result in the Homotopy theory. It is anticipated that the findings presented in this article will contribute to future advancements in the field of complete tvs-VCMS.

8. Future Direction

This work can be further extended to tvs-cone b-metric and tvs-G-cone metric spaces for multi-valued and fuzzy mappings, opening up new avenues of research within functional analysis. Future studies could investigate the application of different iterative procedures, beyond the Picard iteration, to potentially enhance the efficiency of the proposed approach. To demonstrate the practical significance of our findings, future research could explore the application of these results to solve differential and integral inclusions, as outlined in [24,25].