Zeroes of Multifunctions with Noncompact Image Sets

Abstract

In this article we consider zeroes for multifunctions with possibly noncompact image sets. We introduce the notion of multifunction with $\mathbb{K}$-inf-compact support. We also establish three types of applications: the Bayesian approach for analysis of financial operational risk under certain constraints, occupational health and safety measures optimization, and transfer of space innovations and technologies.

1. Introduction and Main Result

Let $(E,\parallel ·\parallel )$ be the finite-dimensional Banach space, $(E,\parallel ·{\parallel }_{*})$ its dual space, and $\langle ·,·\rangle$ the pairing. Let ${\overline{B}}_1\left(\overline{0}\right)=\{y\in E\hspace{0.17em}:\hspace{0.17em}\parallel y\parallel \hspace{0.17em}\le 1\}$ also be the closed unit ball centred at $\overline{0}$, and let $S_1\left(\overline{0}\right)=\{y\in E\hspace{0.17em}:\hspace{0.17em}\parallel y\parallel \hspace{0.17em}=1\}$ be the unit sphere centred at $\overline{0}$. The multifunction $A:{\overline{B}}_1\left(\overline{0}\right)\to 2^E$ is strict if $A\left(y\right)\ne \varnothing$ for each $y\in {\overline{B}}_1\left(\overline{0}\right).$ Further, we use the notation $A:{\overline{B}}_1\left(\overline{0}\right)\rightrightarrows E$ for the strict multifunction A acting from ${\overline{B}}_1\left(\overline{0}\right)$ onto $E.$

For a multifunction $A:{\overline{B}}_1\left(\overline{0}\right)\rightrightarrows E$ we consider the problem: find $\overline{y}$ such that

$$\left\{\begin{array}{c}A\left(\overline{y}\right)\ni \overline{0},\hfill \\ \overline{y}\in {\overline{B}}_1\left(\overline{0}\right).\hfill \end{array}\right.$$

(1)

The investigation of Problem (1) and its various complexities has witnessed substantial advancements in recent years. A considerable portion of contemporary research, as evidenced by works such as those by Hu et al. [1], Gasinski and Papageorgiou [2], Li et al. [3], Zeng et al. [4], and Zgurovsky and Kasyanov [5], has been oriented towards cases where the set A embodies compact values.

Existing methodologies in this realm are predominantly characterized by their focus on the amalgamation of the penalty method with other approximation methods. This structured approach has undoubtedly facilitated significant strides in the quest to resolve zeroes for operator inclusions and variation inequalities, especially for multifunctions with noncompact image sets. However, there exists a latent constraint: the inevitability of an additional limit transition intertwined with the penalty method. This constraint can potentially obfuscate the precision and efficacy of approximating solutions for nonlinear unilateral problems with partial derivatives, operator equations and inclusions, and variation inequalities, especially in infinite-dimensional spaces.

Recognizing this intricacy, our paper aims to bridge this gap. The underlying motivation to delve into Problem 1, especially for multifunctions with potentially noncompact image sets, stems from a pressing desire to eliminate this additional limit transition. By doing so, we can facilitate a direct approximation of solutions of the aforementioned nonlinear problems in infinite-dimensional spaces with operator inclusions in finite-dimensional spaces.

The novelty of our work lies in this generalization. We extend the prevailing methodologies to cater to multifunctions with potentially non-compact values, introducing a fresh perspective that both complements and challenges existing research paradigms.

In essence, our study not only builds upon the foundational work of our predecessors but also carves a new path that underscores both the relevance and the innovation of our approach in addressing Problem (1).

We recall that

$$\mathrm{Gr}A=\{(d,y):y\in {\overline{B}}_1\left(\overline{0}\right),d\in A\left(y\right)\}$$

is graphic of the multifunction A. Consider the upper and lower support functions, respectively:

$$\begin{array}{cc}\hfill & {[A\left(y\right),\omega ]}_{+}=\underset{d\in A\left(y\right)}{sup}\langle d,y\rangle ,\hfill \\ \hfill & {[A\left(y\right),\omega ]}_{-}=\underset{d\in A\left(y\right)}{inf}\langle d,y\rangle ,y\in {\overline{B}}_1\left(\overline{0}\right),\omega \in E.\hfill \end{array}$$

Their properties are considered in [5]. We recall that the multifunction $A:{\overline{B}}_1\left(\overline{0}\right)\rightrightarrows E$ with compact values is upper semi-continuous if for each $\omega \in E$ the function $y\mapsto {[A\left(y\right),\omega ]}_{+}$ is upper semi-continuous [6].

Definition 1.

We say that the multifunction $A:{\overline{B}}_1\left(\overline{0}\right)\rightrightarrows E$ has the $\mathbb{K}$-inf-compact support if the function $c_A(y,a):=\langle a,y\rangle ,$ $a\in A\left(y\right),$ $y\in {\overline{B}}_1\left(\overline{0}\right)$ is $\mathbb{K}$-inf-compact on $\mathrm{Gr}A,$ that is, if the following two conditions hold:

(i)

$c_A(·,·)$ is lower semi-continuous on $\mathrm{Gr}A;$

(ii)

if a sequence ${\left\{y_n\right\}}_{n=1,2,\dots }$ with values in ${\overline{B}}_1\left(\overline{0}\right)$ converges and its limit y belongs to ${\overline{B}}_1\left(\overline{0}\right)$, then any sequence ${\left\{a_n\right\}}_{n=1,2,\dots }$ with $a_n\in A\left(y_n\right),$ $n=1,2,\dots ,$ satisfying the condition that the sequence ${\left\{c_A(y_n,a_n)\right\}}_{n=1,2,\dots }$ is bounded above, has a limit point $a\in A\left(y\right).$

The properties of $\mathbb{K}$-inf-compact functions were established in [7,8]; see the papers and references therein.

The main result has the following formulation.

Theorem 1.

Let $A:{\overline{B}}_1\left(\overline{0}\right)\rightrightarrows {\mathbb{R}}^n$ satisfy the following assumptions:

(i)

the set $A\left(y\right)$ is convex for each $y\in {\overline{B}}_1\left(\overline{0}\right);$

(ii)

the multifunction A has a $\mathbb{K}$-inf-compact support;

(iii)

${[A\left(y\right),y]}_{-}\ge 0$ for each $y\in S_1\left(\overline{0}\right);$

(iv)

$\underset{y\in {\overline{B}}_1\left(\overline{0}\right)}{sup}{[A\left(y\right),y]}_{-}<+\infty .$

Then there exists $\overline{y}\in {\overline{B}}_1\left(\overline{0}\right)$ such that $A\left(\overline{y}\right)\ni \overline{0}$.

Proof. 

See Section 3. □

We remark that the unit ball can be replaced by an arbitrary convex body in Theorem 1. The following corollary to Theorem 1 is known as the Acute Angle Lemma for multifunctions (see for example [1,6,9] and references therein).

Corollary 1

(Acute Angle Lemma). Let $A:{\overline{B}}_1\left(\overline{0}\right)\rightrightarrows {\mathbb{R}}^n$ be such that:

(i)

the set $A\left(y\right)$ is convex and compact for each $y\in {\overline{B}}_1\left(\overline{0}\right);$

(ii)

A is upper semi-continuous;

(iii)

${[A\left(y\right),y]}_{-}\ge 0$ for each $y\in S_1\left(\overline{0}\right).$

Then there exists $\overline{y}\in {\overline{B}}_1\left(\overline{0}\right)$ such that $A\left(\overline{y}\right)\ni \overline{0}$.

Proof. 

See Section 3. □

2. Basic Properties of Multifunctions with $\mathbb{K}$-Inf-Compact Support

Multifunctions, especially those with $\mathbb{K}$-inf-compact support, have piqued the interest of researchers given their intricate properties and expansive applications. This section is dedicated to a deep dive into the foundational properties of such multifunctions. Specifically, we engage in a comparative analysis of the prevailing results on the existence of zeroes for multifunctions and how they align with Theorem 1. Moreover, to elucidate our findings, we present examples that offer tangible insights into the generalizations that we have drawn.

The recent surge in research on the zeroes of multifunctions can be evidenced in a plethora of studies, notably those by Zhang et al. [10], Bernhardt [11], Darnell [12], Dong [13], and Hussain [14], among others. A common thread running through these studies is their focus on multifunctions characterized by compact values.

In a departure from this conventional approach, our paper pioneers the exploration of zeroes for multifunctions that are supported by $\mathbb{K}$-inf-compact, which intriguingly permits noncompact image sets. This nuanced shift not only broadens the horizon of multifunction research but also paves the way for newer analytical tools and methodologies.

The following lemma provides that the assumptions of Theorem 1 follow from the assumptions of Corollary 1.

Lemma 1.

Let $A:{\overline{B}}_1\left(\overline{0}\right)\rightrightarrows {\mathbb{R}}^n$ be an upper semi-continuous multifunction with convex and compact values. Then A has $\mathbb{K}$-inf-compact support and $\underset{y\in {\overline{B}}_1\left(\overline{0}\right)}{sup}{[A\left(y\right),y]}_{-}<+\infty$.

Proof. 

See Section 3. □

Let

$$\chi \left(y\right)\hspace{0.17em}:=\left\{\begin{array}{c}0, \mathrm{if} y\in {\overline{B}}_1\left(\overline{0}\right);\hfill \\ +\infty , \mathrm{otherwise},\hfill \end{array}\right.$$

for each $y\in E.$ Let also $\partial \chi$ be its subdifferential, that is,

$$\partial \chi \left(y\right):=\left\{\begin{array}{c}\left\{\overline{0}\right\}, \mathrm{if} y\in B_1\left(\overline{0}\right);\hfill \\ \{p\in E:\langle p,x-y\rangle \le 0 \mathrm{for} \mathrm{each} x\in {\overline{B}}_1\left(\overline{0}\right)\}, \mathrm{if} y\in S_1\left(\overline{0}\right);\hfill \\ \varnothing , \mathrm{otherwise},\hfill \end{array}\right.$$

for each $y\in E.$ We note that ${[\partial \chi \left(y\right),y]}_{-}\equiv 0$ on ${\overline{B}}_1\left(\overline{0}\right).$ Therefore, if A satisfies assumptions of Corollary 1, then the multifunction $A+\partial \chi$ satisfies the assumptions of Theorem 1, in particular, that it has the $\mathbb{K}$-inf-compact support.

In the following example we establish a multifunction satisfying the assumptions of Theorem 1 but which does not satisfy the assumptions of Corollary 1.

Example 1.

Let $n=1,$ and let

$$A\left(y\right)=\left\{\begin{array}{cc}(-\infty ,0],\hfill & y=-1;\hfill \\ \left\{0\right\},\hfill & y=(-1;1);\hfill \\ [0;\infty ),\hfill & y=1,\hfill \end{array}\right.$$

$y\in [-1,1].$ We note that for each $y\in [-1;1]$ the set $A\left(y\right)$ is nonempty and convex by its definition, that is, assumption (i) of Theorem 1 holds.

To prove that the multifunction A has the $\mathbb{K}$-inf-compact support, let us verify that the function $c_A(y,a):=ay,$ $y\in [-1;1],$ $a\in A\left(y\right),$ is $\mathbb{K}$-inf-compact on $\mathrm{Gr}A.$ For this purpose, we firstly note that the function $c_A(·,·)$ is lower semi-continuous on $\mathrm{Gr}A.$ Secondly, let us prove that condition (ii) of Definition 1 holds. Indeed, let ${\left\{y_n\right\}}_{n=1,2,\dots }\subset [-1,1]$ be a sequence that converges and its limit y belongs to $[-1,1],$ and let ${\left\{a_n\right\}}_{n=1,2,\dots }$ be a sequence with $a_n\in A\left(y_n\right),$ $n=1,2,\dots ,$ satisfying the condition that the sequence ${\left\{y_n{a}_n\right\}}_{n=1,2,\dots }$ is bounded above by $\lambda <+\infty .$ To prove that the sequence ${\left\{a_n\right\}}_{n=1,2,\dots }$ has a limit point $a\in A\left(y\right)$, we remark that $\lambda >0$ and $|a_n| \le \lambda$ as soon as $a_n\ne 0,$ $n=1,2,\dots .$ Therefore, assumption (ii) of Theorem 1 holds.

Assumptions (iii) and (iv) of Theorem 1 hold because ${[A\left(y\right),y]}_{-}\equiv 0$ on $[-1,1].$ Therefore, all assumptions of Theorem 1 hold.

On the other hand, since $A(-1)$ is unbounded set, we can observe that assumption (i) of Corollary 1 does not hold.

3. Proofs

In this section we establish the proofs of Theorem 1, Lemma 1, and Corollary 1.

Before the proof of Theorem 1 we introduce the following notations:

$$\lambda :=\underset{y\in {\overline{B}}_1\left(\overline{0}\right)}{sup}{[A\left(y\right),y]}_{-}<+\infty$$

and

$$\tilde{A}\left(y\right):=\{d\in A\left(y\right):\langle d,y\rangle \le \lambda +1\},y\in {\overline{B}}_1\left(\overline{0}\right).$$

The main idea of the proof of Theorem 1 is to establish that the multifunction $\tilde{A}$ satisfies the assumptions of Corollary 1. The proof of Theorem 1 uses the following auxiliary statement.

Lemma 2.

Let assumptions of Theorem 1 hold. Let ${\{y_n,y\}}_{n=1,2,\dots }\subset {\overline{B}}_1\left(\overline{0}\right)$ be such that $y_n\to y$ as $n\to \infty ,$ and let ${\left\{d_n\right\}}_{n=1,2,\dots }$ be such that $d_n\in \tilde{A}\left(y_n\right)$ for each $n=1,2,\dots .$ Then the sequence ${\left\{d_n\right\}}_{n=1,2,\dots }$ has a limit point. If, additionally, $d_n\to d,$ as $n\to \infty ,$ then $d\in \tilde{A}\left(y\right).$

Proof. 

Let ${\{y_n,y\}}_{n=1,2,\dots }\subset {\overline{B}}_1\left(\overline{0}\right)$ be such that $y_n\to y$ as $n\to \infty ,$ and let ${\left\{d_n\right\}}_{n=1,2,\dots }$ be such that $d_n\in \tilde{A}\left(y_n\right)$ for each $n=1,2,\dots .$ Prove that the sequence ${\left\{d_n\right\}}_{n=1,2,\dots }$ has a limit point. Indeed, since A has $\mathbb{K}$-inf-compact support, then the sequence ${\left\{d_n\right\}}_{n=1,2,\dots }$ has a limit point $d\in A\left(y\right)$, because, according to the definition of $\tilde{A},$ we know that

$$\langle d_n,y_n\rangle \le \lambda +1<+\infty \mathrm{for} \mathrm{each} n=1,2,\dots .$$

(2)

If we additionally assume that $d_n\to d,$ as $n\to \infty ,$ then according to (2), we obtain that $\langle d,y\rangle \le \lambda +1,$ that is, $d\in \overline{A}\left(y\right)$ by the definition of $\tilde{A}.$ □

Proof of Theorem 1.

Fix an arbitrary $y\in {\overline{B}}_1\left(\overline{0}\right).$ The set $\tilde{A}\left(y\right)$ is nonempty because by the definitions of $\lambda$ and the lower support function there exists $\tilde{d}\in A\left(y\right)$ such that

$$\langle \tilde{d},y\rangle \le \underset{d\in A\left(y\right)}{inf}\langle d,y\rangle +1={[A\left(y\right),y]}_{-}\le \lambda +1.$$

Moreover, the set $\tilde{A}\left(y\right)$ is convex by it definition because the set $A\left(y\right)$ is convex. The compactness of $\tilde{A}\left(y\right)$ directly follows from Lemma 2. Therefore, assumption (i) of Corollary 1 holds.

Assumption (ii) of Corollary 1 holds because Lemma 2 and Theorem 2.5 from [15] directly implies that A is upper semi-continuous multifunction.

Assumption (iii) of Corollary 1 follows from the inequalities:

$${[\tilde{A}\left(y\right),y]}_{-}\ge {[A\left(y\right),y]}_{-}\ge 0 \mathrm{for} \mathrm{each} y\in S_1\left(\overline{0}\right),$$

where the first inequality follows from the basic properties of the lower support functions, and the second inequality follows from assumption (iii) of Theorem 1. Therefore, the multifunction $\tilde{A}$ satisfies all the assumptions of Corollary 1. Thus, there exists $\overline{y}\in {\overline{B}}_1\left(\overline{0}\right)$ such that $\tilde{A}\left(\overline{y}\right)\ni \overline{0}$ and, therefore, $A\left(\overline{y}\right)\ni \overline{0}.$ □

Proof of Lemma 1.

Firstly, let us prove that

$$\underset{y\in {\overline{B}}_1\left(\overline{0}\right)}{sup}{[A\left(y\right),y]}_{-}<+\infty .$$

(3)

Since $A:{\overline{B}}_1\left(\overline{0}\right)\rightrightarrows {\mathbb{R}}^n$ is an upper semi-continuous multifunction with compact values, Theorem 2.5 from [15] implies that $A\left({\overline{B}}_1\left(\overline{0}\right)\right)$ is a compact set. Therefore,

$$\underset{y\in {\overline{B}}_1\left(\overline{0}\right)}{sup}{[A\left(y\right),y]}_{-}\le \underset{y\in {\overline{B}}_1\left(\overline{0}\right)}{sup}\underset{d\in A\left(y\right)}{inf}{\parallel d\parallel }_{E^{*}}<+\infty ,$$

that is, inequality (3) holds.

Let us prove that A has $\mathbb{K}$-inf-compact support. Indeed, lower semi-continuity of the function $c_A(y,a):=\langle a,y\rangle ,$ $a\in A\left(y\right),$ $y\in {\overline{B}}_1\left(\overline{0}\right),$ on $\mathrm{Gr}A$ holds, because the bilinear form is continuous on the product of the finite-dimensional vector spaces and $\mathrm{Gr}A$ is a closed set as a graphic of a strict upper semi-continuous multifunction with compact values. So, to finish the proof, it is sufficient to establish property (ii) of Definition 1. Let ${\left\{y_n\right\}}_{n=1,2,\dots }$ be a sequence with values in ${\overline{B}}_1\left(\overline{0}\right)$ that converges and its limit y belongs to ${\overline{B}}_1\left(\overline{0}\right).$ Let ${\left\{a_n\right\}}_{n=1,2,\dots }$ also be a sequence with $a_n\in A\left(y_n\right),$ $n=1,2,\dots ,$ satisfying the condition that the sequence ${\left\{c_A(y_n,a_n)\right\}}_{n=1,2,\dots }$ is bounded above by $\mu <+\infty .$ The sequence ${\left\{a_n\right\}}_{n=1,2,\dots }$ has a limit point $a\in A\left(y\right)$ because the set

$$\begin{array}{cc}\hfill \left\{\right(x,a)& :x\in {\overline{B}}_1\left(\overline{0}\right),a\in A\left(x\right),c_A(x,a)\le \mu \}\hfill \\ & =\mathrm{Gr}A\cap \left\{\right(x,a)\in E\times E:\langle a,x\rangle \le \mu \}\hfill \end{array}$$

is compact as an intersection of compact and closed sets, respectively. □

Proof of Corollary 1.

The statement of Corollary 1 follows from [1,6,9]. Alternatively, according to Lemma 1, assumptions of Corollary 1 imply the assumptions of Theorem 1. We note that the statement of Corollary 1 is the classical Acute Angle Lemma stated in [1,6,9]. □

4. Examples of Applications

In this section we establish three types of applications: the Bayesian approach for analysis of financial operational risk under certain constraints (see Section 4.1), occupational health and safety measures optimization (see Section 4.2), and transfer of space innovations and technologies (see Section 4.3). More potential applications can be found in such papers [16,17,18,19,20].

4.1. Remarks on Bayesian Approach for Analysis of Financial Operational Risk under Certain Constraints

Let us establish the sketch of methodology of using the Bayesian approach for analysis of financial operational risk under certain constraints that are represented by an operator in finite-dimensional spaces. This could involve, for example, updating the probabilities of different operational risks given new data, while also satisfying certain financial or regulatory constraints.

The solution to such a problem would depend on the specific nature of on operator and constraints [21]. However, some general solutions might involve the use of mathematical programming or optimization techniques. For example, constraint optimization methods can be used to find the optimal solution that satisfies all constraints. This could involve techniques such as linear programming, quadratic programming, or other advanced methods such as semidefinite programming or convex optimization.

Additionally, when considering the Bayesian approach, one might employ Bayesian inference techniques to update the probabilities of various risks based on new data. This could involve the use of computational methods such as Markov chain Monte Carlo (MCMC) methods or variational inference.

In all cases, it is crucial to correctly formulate the problem and accurately specify the operator and constraints, as well as to ensure that the Bayesian updating process accurately reflects the new data and the prior knowledge. Theorem 1 allows us to consider the question of the consistency of constraints in the greatest generality.

4.2. Application of Operator Inclusion in Finite-Dimensional Spaces to Optimization of Occupational Health and Safety Measures

In this subsection, we describe the use of operator inclusion in finite-dimensional spaces to analyse the complex system of rules and procedures in place for occupational safety. Our method involves treating these rules and procedures as operators acting within a finite-dimensional space, which allows a sophisticated mathematical analysis [22].

1

Mathematical Representation of Occupational Safety Measures:

Each rule or procedure is represented as an operator acting within this space. This operator is capable of transforming the state of the space, analogous to how an actual rule or procedure can alter the state of a workplace in terms of safety. The entire system of safety measures is thus represented as a set of these operators.

2

Defining Objective Function:

The effectiveness of a set of rules and procedures is evaluated using an objective function, which maps the state of the finite-dimensional space (i.e., the state of the workplace safety) to a scalar value. This function might take into account factors such as the number of accidents, severity of injuries, and compliance costs, among others.

3

Constraint Optimization:

Next, we use constraint optimization techniques to find the set of operators (rules and procedures) that minimize (or maximize) our objective function. Regulatory requirements, budget limits, or any other practical limitations could act as the aforementioned constraints.

4

Operator Inclusion:

We then perform operator inclusion to account for the interactions between different rules and procedures. This step recognizes that the overall effect of multiple operators acting together might be different from the sum of their individual effects.

5

Data Analysis:

The model’s predictions are then compared against actual data. These comparisons are used to refine the model, for example, by adjusting the weights assigned to different factors in the objective function.

6

Optimized Set of Rules and Procedures:

Finally, the output of this analysis is an optimized set of rules and procedures. This set of operators (rules and procedures) represents a solution that satisfies the given constraints and optimizes workplace safety as measured by our objective function.

This methodology represents a novel way of using operator inclusion in finite-dimensional spaces to address the complex, multifaceted challenge of occupational safety. Its advantage lies in its ability to handle multiple interacting factors and constraints, offering a sophisticated tool for data-driven decision making in the realm of occupational safety.

4.3. Operator Inclusions for Increasing the Transfer of Space Innovations and Technologies by Bringing Together the Scientific Community, Industry, and Startups in the Space Industry

Operator inclusions in finite-dimensional spaces are a complex mathematical tool that can be used to model various systems and processes. These theoretical models can be adapted to develop effective algorithms or tools that can assist startups and other organizations working in the space industry. Transforming these theoretical concepts into practical tools requires several steps:

1

Understanding the mathematical concept:

Deep understanding of the mathematical model is the first requirement. This may involve collaborating with mathematicians who specialize in this field.

2

Defining the problem:

Specific challenges or problems in space technology need to be identified that could potentially benefit from the application of theoretical models.

3

Creating the algorithm:

With a specific problem in mind, work on transforming the mathematical model into an algorithm that can be implemented in software.

4

Testing and verification:

After developing an initial version of the algorithm, it should be thoroughly tested to verify its productivity and reliability.

5

Integration and implementation:

If the testing is successful, the next step would be to integrate the algorithm into existing systems or tools used by startups or other organizations within the EuroSpaceHub project [23].

6

Documentation and training:

Users would then be provided with comprehensive documentation and training for the algorithm users.

7

Iterative improvement:

Lastly, feedback and performance data would be collected for improvement.

With this process, the theoretical concepts established in Section 1 can be transformed into practical tools for the space industry [24,25].

5. Conclusions

Throughout this study, we embarked on a detailed exploration of zeroes for multifunctions, particularly focusing on those with potentially noncompact image sets. A notable introduction in our research was the concept of multifunctions with $\mathbb{K}$-inf-compact support. This conceptual framework laid the groundwork for examining its practical implications in diverse fields.

Our findings yielded three distinct types of applications:

1

Financial Operational Risk: Utilizing the Bayesian approach, we undertook an analysis of financial operational risk, factoring in specific constraints. This model offers a fresh perspective on risk assessment and potentially revolutionizes how organizations approach this critical domain.

2

Occupational Health and Safety: The multifunctions with $\mathbb{K}$-inf-compact support provided invaluable insights into the optimization of occupational health and safety measures. By ensuring the workplace is both efficient and safe, our findings may contribute to more robust safety standards across industries.

3

Space Innovations and Technologies Transfer: Last but certainly not least, our research has implications for the realm of space innovations. The potential for transferring cutting-edge space technologies can be viewed through the lens of our introduced multifunction concept, offering a promising avenue for future technological advancements.

In conclusion, our exploration into the zeroes for multifunctions with potentially noncompact image sets has opened up a plethora of application areas, each with its own set of challenges and opportunities. As we venture further into this domain, the potential for groundbreaking advancements in these sectors becomes increasingly evident.