Fuzzy Control Problem via Random Multi-Valued Equations in Symmetric F-n-NLS

Abstract

To study an uncertain case of a control problem, we consider the symmetric F-n-NLS which is induced by a dynamic norm inspired by a random norm, distribution functions, and fuzzy sets. In this space, we consider a random multi-valued equation containing a parameter and investigate existence, and unbounded continuity of the solution set of it. As an application of our results, we consider a control problem with multi-point boundary conditions and a second order derivative operator.

1. Introduction

Consider the random operator $Q_1$. A natural generalization of parametric random equations of the form $\alpha =Q_1(\lambda ,\gamma ,\alpha )$, in which $\lambda$ is a element of a probability measure space, is the multi-valued form [1],

$$\alpha \in Q_1(\lambda ,\gamma ,\alpha ).$$

(1)

In regards to solutions, there are many approaches available in the literature, for example the principal eigenvalue-eigenvector method, the monotone minorant method [2,3] and topological degree. The idea in this paper is to use the topological degree for random multi-valued mappings and the method of evaluating solutions. The main idea is presenting an uncertain case of a control problem. To achieve this aim, we use a special space, i.e., symmetric F-n-NLS, that has a dynamic situation and a parameter $\tau$, which can be time, which enable us to consider different cases. We note this kind of space induced by a dynamic norm which is inspired by random norms, probabilistic distances and fuzzy norms was studied; see [4] for details and applications. Our results can be applied in uncertainty problems, risk measures and super-hedging in finance [5].

For the random multi-valued operator $Q_1$, the following sets

$$\mathit{U}=\{(\gamma ,\alpha ):\alpha \in Q_1(\lambda ,\gamma ,\alpha )\},$$

(2)

or

$$\mathit{U}=\{\alpha :\exists \gamma ,\alpha \in Q_1(\lambda ,\gamma ,\alpha )\}.$$

(3)

are solutions of (1). In this paper, we consider a control problem with multi-point boundary conditions and a second order derivative operator as

$$\left\{\begin{array}{c}{\phi }^{″}(\lambda ,\iota )+\nu (\gamma ,\iota )\mu (\phi (\lambda ,\iota ))=0,\iota \in (0,1),\hfill \\ \nu (\gamma ,\iota )\in Q_1(\lambda ,\gamma ,\phi (\lambda ,\iota ))\mathrm{a}.\mathrm{e}.\mathrm{on}[0,1]\hfill \\ \phi (\lambda ,0)=0,\phi (\lambda ,1)={\sum }_{p=1}^n{\omega }_p\phi (\lambda ,{\varsigma }_p).\hfill \end{array}\right.$$

(4)

where ${\varsigma }_p\in (0,1),$ $0\le {\omega }_p$, ${\sum }_{p=1}^n{\omega }_p{\varsigma }_p<1$ and $\lambda \in ϝ$. In Section 2, we introduce our special space, i.e., symmetric F-n-NLS and present some basic results which we need in the main section. In Section 3, we prove some properties of random multi-valued operator. In Section 4, we present an application of our results for a fuzzy control problem.

2. Preliminaries

Here, we let $E_1=[0,1]$, $E_2=(0,1]$, $E_3=[0,\infty )$ and $E_4=[0,\infty ]$.

A mapping $\delta :\mathbb{R}\to E_1$, whose $\mathit{ϵ}$-level set is denoted by

$$\begin{array}{c}\hfill {\left[\delta \right]}_{\mathit{ϵ}}=\{\iota :\delta (\iota )\ge \mathit{ϵ}\},\end{array}$$

is said to be a fuzzy real number if it satisfies the following:

(i)

$\delta$ is normal, i.e., there exists ${\iota }_0\in \mathbb{R}$ such that $\delta ({\iota }_0)=1$;

(ii)

$\delta$ is upper semicontinuous;

(iii)

$\delta$ is fuzzy convex, i.e., $\delta (\iota )\ge min(\delta (\kappa ),\delta (s))$, for each $\iota ,\kappa \in \mathbb{R}$ such that $\kappa \le \iota \le s$ and $\mathit{ϵ}\in E_2$;

(iv)

For each $\mathit{ϵ}\in E_2$, ${\left[\delta \right]}_{\mathit{ϵ}}=[{\delta }_{\mathit{ϵ}}^{-},{\delta }_{\mathit{ϵ}}^{+}]$, where $-\infty <{\delta }_{\mathit{ϵ}}^{-}\le {\delta }_{\mathit{ϵ}}^{+}<+\infty$ and ${\left[\delta \right]}^0=\overline{\{\delta \in \mathbb{R}:\delta (\iota )>0\}}$ is compact.

Let the set $\mathbb{F}$ contain all upper semicontinuous normal convex fuzzy real numbers. ${\mathbb{F}}^{+}$ contains all non-negative fuzzy real numbers of $\mathbb{F}$. For each $\kappa \in \mathbb{R}$, we can define

$$\begin{array}{c}\hfill \overline{\kappa }(\iota )=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\iota =\kappa ,\hfill \\ 0,\hfill & \mathrm{if}\iota \ne \kappa ,\hfill \end{array}\right.\end{array}$$

so $\overline{\kappa }\in \mathbb{F}$ and $\mathbb{R}$ can be embedded in $\mathbb{F}$.

A partial order ⪯ in $\mathbb{F}$ is defined as follows: $\delta ⪯\sigma$ iff for each $\mathit{ϵ}\in E_2$, ${\delta }_{\mathit{ϵ}}^{-}\le {\sigma }_{\mathit{ϵ}}^{-}$ and ${\delta }_{\mathit{ϵ}}^{+}\le {\sigma }_{\mathit{ϵ}}^{+}$ where ${\left[\delta \right]}_{\mathit{ϵ}}=[{\delta }_{\mathit{ϵ}}^{-},{\delta }_{\mathit{ϵ}}^{+}]$ and ${\left[\sigma \right]}_{\mathit{ϵ}}=[{\sigma }_{\mathit{ϵ}}^{-},{\sigma }_{\mathit{ϵ}}^{+}]$. The strict inequality in $\mathbb{F}$ is defined by $\delta \prec \sigma$ iff for each $\mathit{ϵ}\in E_2$, ${\delta }_{\mathit{ϵ}}^{-}<{\sigma }_{\mathit{ϵ}}^{-}$ and ${\delta }_{\mathit{ϵ}}^{+}<{\sigma }_{\mathit{ϵ}}^{+}$ (see [6,7,8]).

The arithmetic operations ⊕, ⊖, ⊙ and ⊘ on $\mathbb{F}\times \mathbb{F}$ are defined by

$$\begin{array}{ccc}\hfill \left(\delta \oplus \sigma \right)(\iota )& =& \underset{\kappa \in \mathbb{R}}{sup}min\left(\delta (\kappa ),\sigma (\iota -\kappa )\right),\iota \in \mathbb{R},\hfill \\ \hfill \left(\delta \ominus \sigma \right)(\iota )& =& \underset{\kappa \in \mathbb{R}}{sup}min\left(\delta (\kappa ),\sigma (\kappa -\iota )\right),\iota \in \mathbb{R},\hfill \\ \hfill \left(\delta \odot \sigma \right)(\iota )& =& \underset{0\ne \kappa \in \mathbb{R}}{sup}min\left(\delta (\kappa ),\sigma (\frac{\iota }{\kappa })\right),\iota \in \mathbb{R},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \left(\delta \oslash \sigma \right)(\iota )& =& \underset{\kappa \in \mathbb{R}}{sup}min\left(\delta (\kappa \iota ),\sigma (\kappa )\right),\iota \in \mathbb{R},\delta ,\sigma (\succ 0)\in \mathbb{F}.\hfill \end{array}$$

Definition 1.

Let ℧ be a real linear space over $\mathbb{R}$ with dim $\mho \ge n$. Suppose $\parallel •,\cdots ,•\parallel :{\mho }^n\to {\mathbb{F}}^{+}$ is a mapping and $L,R:{E_1}^2\to E_1$ are symmetric, nondecreasing mapping satisfying

$$\begin{array}{c}\hfill L(0,0)=0\mathit{and}R(1,1)=1.\end{array}$$

Write

$$\begin{array}{c}\hfill [\parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n{\parallel ]}_{\mathit{ϵ}}=[\parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n{\parallel }_{\mathit{ϵ}}^{-},\parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n{\parallel }_{\mathit{ϵ}}^{+}],\end{array}$$

for ${\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\in \mho$, $\mathit{ϵ}\in E_2$ and suppose that for every linearly independent vectors ${\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n$ $\in \mho$, there exists ${\mathit{ϵ}}_0\in E_2$ independent of ${\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\in \mho$ such that for each $\mathit{ϵ}\le {\mathit{ϵ}}_0$, one has

$$\begin{array}{c}\hfill inf\parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n{\parallel }_{\mathit{ϵ}}^{-}>0,{\parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel }_{\mathit{ϵ}}^{+}<\infty .\end{array}$$

The quadruple $\left({\mho }^n,\parallel •,\cdots ,•\parallel ,L,R\right)$ is said to be a symmetric fuzzy n-normed linear space (F-n-NLS) in the sense of Felbin [8] and $\parallel •,\cdots ,•\parallel$ is a fuzzy n-norm if

(N1)

$\parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel =\overline{0}$ iff ${\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n$ are linearly dependent;

(N2)

$\parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel$ is invariant under any permutation of ${\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\in \mho$;

(N3)

$\parallel c{\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel =\left|c\right|\odot \parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel$ for any $c\in \mathbb{R}$;

(N4)

$\parallel {\vartheta }_0+{\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel ⪯\parallel {\vartheta }_0,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel \oplus \parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel$;

(i)

whenever $\kappa \le \parallel {\vartheta }_0,{\vartheta }_2,\cdots ,{\vartheta }_n{\parallel }_1^{-}$, $\iota \le \parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n{\parallel }_1^{-}$ and $\iota +\kappa \le \parallel {\vartheta }_0+{\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n{\parallel }_1^{-}$,

$$\begin{array}{c}\hfill \parallel {\vartheta }_0+{\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel (\kappa +\iota )\ge L\left(\parallel {\vartheta }_0,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel (\kappa ),\parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel (\iota )\right),\end{array}$$

(ii)

whenever $\kappa \ge \parallel {\vartheta }_0,{\vartheta }_2,\cdots ,{\vartheta }_n{\parallel }_1^{-}$, $\iota \ge \parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n{\parallel }_1^{-}$ and $\iota +\kappa \ge \parallel {\vartheta }_0+{\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n{\parallel }_1^{-}$,

$$\begin{array}{c}\hfill \parallel {\vartheta }_0+{\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel (\kappa +\iota )\le R\left(\parallel {\vartheta }_0,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel (\kappa ),\parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\parallel (\iota )\right).\end{array}$$

Now, we consider a symmetric F-n-NLS in the sense of Narayanan-Vijayabalaji [9] and next we show a relationship between them.

Definition 2

([9]). Assume that ℧ is a linear space and ∗ is a continuous t-norm. Let the fuzzy subset η of ${\mho }^n\times \mathbb{R}$ with dim $\mho \ge n$ satisfy

(FN1) 

For all $\tau \in \mathbb{R}$ with $\tau \le 0$, $\eta ({\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n,\tau )=0$;

(FN2) 

For all $\tau \in \mathbb{R}$ with $\tau >0$, $\eta ({\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n,\tau )=1$ for $\tau \ge 0$ iff ${\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n$ are linearly dependent;

(FN3) 

$\eta ({\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n,\tau )$ is invariant under any permutation of ${\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\in \mho$;

(FN4) 

For all $\tau \in \mathbb{R}$ with $\tau >0$,

$$\begin{array}{c}\hfill \eta \left(c{\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n,\tau \right)=\eta \left({\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n,\frac{\tau }{\left|c\right|}\right)\mathit{if}c\in \mathbb{R}\mathit{with}c\ne 0;\end{array}$$

(FN5) 

For all $\tau \in \mathbb{R}$ with $\tau ,\theta >0$,

$$\begin{array}{c}\hfill \eta ({\vartheta }_0+{\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n,\tau +\theta )\ge \eta \left({\vartheta }_0,{\vartheta }_2,\cdots ,{\vartheta }_n,\tau \right)\ast \eta \left({\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n,\theta \right);\end{array}$$

(FN6) 

$\eta ({\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n,.):\mathring{E_3}\to E_1$ is left continuous;

(FN7) 

${lim}_{\tau \to +\infty }\eta ({\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n,\tau )=1$.

Thus, the triple $(\mho ,\eta ,\ast )$ is a symmetric F-n-NLS (see [10,11,12]).

A complete symmetric F-n-NLS is called symmetric F-n-BS.

Theorem 1

([9,13,14,15]). Let $(\mho ,\eta ,\ast )$ be a symmetric F-n-NLS in which $\ast =min$ and

(FN8) 

$\eta ({\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n,\tau )>0$ for all $\tau >0$ implies ${\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n$ are linearly dependent.

Define

$$\begin{array}{c}\hfill \parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n{\parallel }_{\mathit{ϵ}}:=inf{\left[\eta ({\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n,\tau )\right]}_{\mathit{ϵ}},\mathit{ϵ}\in \mathring{E_1}.\end{array}$$

Then $\{\parallel •,\cdots ,•{\parallel }_{\mathit{ϵ}}:\mathit{ϵ}\in \mathring{E_1}\}$ is an ascending family of fuzzy n-norms on ℧.

These fuzzy n-norms will be called the ϵ-n-norms on ℧ corresponding to the fuzzy n-norm on ℧.

We note that some applications can be found on [16,17].

Remark 1

([18]). Let ${\eta }_E:\mathbb{R}\times \mathring{E_3}\to E_2$ be a Euclidean fuzzy norm (Euclidean fuzzy normed spaces were introduced by the authors in [18]). Then ${\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\in \mho$ are linearly independent iff $\eta ({\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n,\tau )={\eta }_E(1,\tau )$, for any $\tau >0$.

By the above remark, we have that, ${\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n\in \mho$ are linearly independent iff

$$\begin{array}{ccc}\hfill \parallel {\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n{\parallel }_{\mathit{ϵ}}& =& inf\{\tau :\eta ({\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_n,\tau )\ge \mathit{ϵ},\mathit{ϵ}\in \mathring{E_1}\}\hfill \\ & =& inf\{\tau :{\eta }_E(1,\tau )\ge \mathit{ϵ},\mathit{ϵ}\in \mathring{E_1}\}\hfill \\ & =& {\left|1\right|}_{\mathit{ϵ}}.\hfill \end{array}$$

Consider the probability measure space $(ϝ,\mathring{E_3},\xi )$ and let $\left(U,{\mathit{B}}_U\right)$ and $\left(S,{\mathit{B}}_S\right)$ be Borel measurable spaces, where U and S are symmetric F-n-BS. If $\{\lambda :\mathcal{F}(\lambda ,\xi )\in B\}\in \mathring{E_3}$ for every $\xi$ in U and $B\in {\mathit{B}}_S$, we say $\mathcal{F}:ϝ\times U\to S$ is a random operator. Let $2^S$ be the family of all subsets of S. The mapping $\mathcal{F}:ϝ\times U\to 2^S$ is said to be random multi-valued operator. A random operator $\mathcal{F}:ϝ\times U\to S$ is said to be linear if $\mathcal{F}(\lambda ,\mathbf{a}{\xi }_1+\mathbf{b}{\xi }_2)=\mathbf{a}\mathcal{F}(\lambda ,{\xi }_1)+\mathbf{b}\mathcal{F}(\lambda ,{\xi }_2)$ almost everywhere for each ${\xi }_1,{\xi }_2$ in U and $\mathbf{a},\mathbf{b}$ are scalers, and bounded if there exists a nonnegative real-valued random variable $M(\lambda )$ such that

$$\begin{array}{ccc}& & \eta \left(\mathcal{F}({\lambda }_1,{\xi }_{1,1})-\mathcal{F}({\lambda }_1,{\xi }_{2,1}),\cdots ,\mathcal{F}({\lambda }_1,{\xi }_{1,n})-\mathcal{F}({\lambda }_1,{\xi }_{2,n}),M(\lambda )\tau \right)\hfill \\ & \ge & \eta \left({\xi }_{1,1}-{\xi }_{2,1},\cdots ,{\xi }_{1,n}-{\xi }_{2,n},\tau \right),\hfill \end{array}$$

almost everywhere for each ${\xi }_{1,j}-{\xi }_{2,j}$ $(j=1,2,\dots n)$ in U, $\tau \in \mathring{E_3}$ and $\lambda \in ϝ$.

Let $\left({\rm Y},\Omega ,\parallel •,\cdots ,•\parallel ,L,R\right)$ be a symmetric F-n-BS over $\mathbb{R}$ with dim ${\rm Y}\ge n$ and ordering by the cone $\Omega$, i.e., $\Omega$ is a closed convex subset of ${\rm Y}$ such that $\gamma \Omega \subset \Omega$ for $\gamma \ge 0$, $\Omega \cap (-\Omega )=\{0\}$, and ${\alpha }_p\le {\beta }_p$ iff ${\beta }_p-{\alpha }_p\in \Omega$ for $\alpha ,\beta \in {\rm Y}$ with $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right),\beta =\left({\beta }_1,{\beta }_2,\cdots ,{\beta }_n\right)$ and $1\le p\le n$. For nonempty subsets ${\Delta }_1,{\Delta }_2$ of ${\rm Y}$ we write ${\Delta }_1{\succcurlyeq }_2{\Delta }_2$ (or, ${\Delta }_2{\preccurlyeq }_2{\Delta }_1$) iff for every $\alpha \in {\Delta }_1$, we can find a $\beta \in {\Delta }_2$ which ${\alpha }_p\ge {\beta }_p$ (or, ${\beta }_p\le {\alpha }_p$) for $1\le p\le n$. We say $\Omega$ is a normal cone if we can find a constant $K>0$ where $0\le {\alpha }_p\le {\beta }_p$ for $1\le p\le n$ implies $\parallel {\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n{\parallel }_{\mathit{ϵ}}\le K{\parallel {\beta }_1,{\beta }_2,\cdots ,{\beta }_n\parallel }_{\mathit{ϵ}}$. We note in this paper, we consider $\Omega$ as a normal cone with $K=1$. Furthermore,

$$cc({\Delta }_1)=\{G\subset {\Delta }_1\subset {\rm Y}:G\mathrm{is}\mathrm{nonempty}\mathrm{closed}\mathrm{convex}\}$$

Consider the open convex subset $\Xi$ of ${\rm Y}$, and let ${\Xi }_{\Omega }=\Omega \cap \Xi$, ${\partial }_{\Omega }\Xi =\Omega \cap \partial \Xi$ and $\stackrel{\cdot }{\Omega }=\Omega \setminus \{0\}$, where $\partial \Xi$ is boundary of $\Xi$ in ${\rm Y}$. The mapping $Q_3:ϝ\times \left(\Omega \cap \overline{\Xi }\right)\to cc(\Omega )$ is said to be compact if $Q_3(ϝ\times {\Delta }_2)$ is relatively compact for any bounded subset ${\Delta }_2$ of $\Omega \cap \overline{\Xi }$, where $Q_3(ϝ\times {\Delta }_2)={\cup }_{\alpha \in {\Delta }_2}{Q}_3(\lambda ,\alpha )$, for any $\lambda \in ϝ$. We say a random multi-valued operator $Q_3$ has the upper semi-continuity property (in short, u.s.c.rmvo) if

$$\begin{array}{c}\hfill \{\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in \Omega \cap \overline{\Xi }:Q_3(\lambda ,\alpha )\subset W\}\end{array}$$

where $\Omega \cap \overline{\Xi }={(\Omega \cap \overline{\Xi })}^{\circ }$ and $\lambda \in ϝ$. Further, if $\alpha \notin Q_3(\lambda ,\alpha )$ for all $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in {\partial }_{\Omega }\Xi$ and $\lambda \in ϝ$, the random fixed point index of $Q_3$ in $\Xi$ with respect to $\Omega$ is defined which is an integer denoted by $i_{\Omega }(Q_3,\Xi )$.

Lemma 1.

[2] Let $Q_3:ϝ\times \left(\Omega \cap \overline{\Xi }\right)\to cc(\Omega )$ be a compact u.s.c.rmvo. Then

1. 

$i_{\Omega }(Q_3,\Xi )=0$ if there exists $\phi \in \stackrel{\cdot }{\Omega }$ such that $\alpha \notin Q_3(\lambda ,\alpha )+\varrho \phi$ for all $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in {\partial }_{\Omega }\Xi$, $\lambda \in ϝ$ and $\varrho \ge 0$.

2. 

$i_{\Omega }(Q_3,\Xi )=1$ if $\varrho \alpha \notin Q_3(\lambda ,\alpha )$ for all $\lambda \in ϝ$ and $\varrho \ge 1$.

The following results are needed later to obtain a generalization of [19].

Lemma 2.

[20] Assume that $Q_3:ϝ\times E_3\subset ϝ\times {\rm Y}\to c({\rm Y})$ is a u.s.c.rmvo, $\left({\alpha }_{\epsilon },{\beta }_{\epsilon }\right)\to (\alpha ,\beta )$ with ${\beta }_{\epsilon }\in Q_3(\lambda ,{\alpha }_{\epsilon })$ and $\lambda \in ϝ$. Thus, $\beta \in Q_3(\lambda ,\alpha )$.

Lemma 3.

[19] Let $\Psi :ϝ\times E_1\times \left(\Omega \cap \overline{\Xi }\right)\to cc(\Omega )$ be a compact u.s.c.rmvo with $\alpha \notin \Psi (\lambda ,\iota ,\alpha )$ for all $(\iota ,\alpha )\in E_1\times {\partial }_{\Omega }\Xi$ and $\lambda \in ϝ$. Then, $i_{\Omega }(\Psi (\lambda ,0,.),\Xi )=i_{\Omega }(\Psi (\lambda ,1,.),\Xi )$.

3. Random Multi-Valued Operator

Lemma 4.

Let $Q_3:ϝ\times E_3\times \Omega \to cc(\Omega )$ be a compact u.s.c.rmvo and $\Xi \subset {\rm Y}$ be open with $0\in \Xi$. Additionally,

1. 

$\iota \alpha \in Q_3(\lambda ,0,\alpha )$, for any $\lambda \in ϝ$, for some $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in \stackrel{\cdot }{\Omega }$ implies $\iota <1$,

2. 

$i_{\varrho }(Q_3(\lambda ,\gamma ,.),\Xi )=0$ if γ is sufficiently large and $\lambda \in ϝ$.

Then $\{\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in {\partial }_{\Omega }\Xi :\exists \gamma >0,\alpha \in Q_3(\lambda ,\gamma ,\alpha )\}\ne \varnothing$.

Proof. 

From the second condition in Lemma 4 we can find ${\gamma }_0>0$ such that $i_{\varrho }(Q_3(\lambda ,\gamma ,.),\Xi )$ $=0$ for all $\lambda \in ϝ$ and $\gamma \ge {\gamma }_0$. Define

$$\begin{array}{ccc}\hfill \omega & =& sup\{\gamma >0:i_{\Omega }(Q_3(\lambda ,\gamma ,.),\Xi )\ne 0\}.\hfill \end{array}$$

We first observe that $\omega >0$. Furthermore,

$$\forall \epsilon >0,\exists ({\iota }_{\epsilon },{\alpha }_{\epsilon })\in E_1\times {\partial }_{\Omega }\Xi :{\alpha }_{\epsilon }\in (1-{\iota }_{\epsilon })Q_3(\lambda ,\epsilon ,{\alpha }_{\epsilon })+{\iota }_{\epsilon }{Q}_3(\lambda ,0,{\alpha }_{\epsilon }).$$

(5)

for any $\lambda \in ϝ$. Since $Q_3$ is compact, without loss of generality we may assume that ${\iota }_{\epsilon }\to \iota ,{\alpha }_{\epsilon }\to \alpha$ when $\epsilon \to 0$ and $\lambda \in ϝ$. From (5) by Lemma 2 it follows that

$$\alpha \in (1-\iota )Q_3(\lambda ,0,\alpha )+\iota Q_3(\lambda ,0,\alpha )\subset Q_3(\lambda ,0,\alpha ).$$

This contradicts the first condition in Lemma 4. Thus, there exist $\epsilon >0$ such that $(\iota ,\alpha )\notin \Psi (\lambda ,\iota ,\alpha )$ for all $\lambda \in ϝ$ and $(\iota ,\alpha )\in E_1\times {\partial }_{\Omega }\Xi$, where

$$\Psi (\lambda ,\iota ,\alpha )=(1-\iota )Q_3(\lambda ,\epsilon ,\alpha )+\iota Q_3(\lambda ,0,\alpha ).$$

Using Lemma 3 we have

$$i_{\Omega }(Q_3(\lambda ,0,.),\Xi )=i_{\Omega }(Q_3(\lambda ,\epsilon ,.),\Xi ).$$

Using Lemma 1 implies that $i_{\Omega }(Q_3(\lambda ,0,.),\Xi )=1$. Thus, $i_{\Omega }(Q_3(\lambda ,\epsilon ,.),\Xi )=1$, and we deduce $0<\omega <{\gamma }_0$, for each $\lambda \in ϝ$.

Next, for every $\epsilon \in (0,\omega )$ and $\lambda \in ϝ$, there exists ${\gamma }_{\epsilon }\in (\omega -\epsilon ,\omega ]$ with $i_{\Omega }(Q_3(\lambda ,{\gamma }_{\epsilon },.),$ $\Xi )\ne 0$. Consider the random multi-valued operator ${\Psi }_{\epsilon }$ defined by

$${\Psi }_{\epsilon }(\lambda ,\iota ,\alpha )=(1-\iota )Q_3(\lambda ,{\gamma }_{\epsilon },\alpha )+\iota Q_3(\lambda ,\omega +\epsilon ,\alpha ).$$

Now, we prove

$$\{\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in {\partial }_{\Omega }\Xi :\exists \gamma >0,\alpha \in Q_3(\lambda ,\gamma ,\alpha )\}\ne \varnothing .$$

Assume the contrary, that

$$\{\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in {\partial }_{\Omega }\Xi :\exists \gamma >0,\alpha \in Q_3(\lambda ,\gamma ,\alpha )\}=\varnothing .$$

(6)

Then, the random fixed point index of $Q_3(\lambda ,\omega +\epsilon )$ is well defined, for each $\lambda \in ϝ$. If

$$\alpha \notin {\Psi }_{\epsilon }(\lambda ,\iota ,\alpha )\mathrm{for}\mathrm{all}(\iota ,\alpha )\in E_1\times {\partial }_{\Omega }\Xi ,$$

(7)

then, by Lemma 3 we obtain

$$i_{\Omega }(Q_3(\lambda ,{\gamma }_{\epsilon },.),\Xi )=i_{\Omega }(Q_3(\lambda ,\omega +\epsilon ,.),\Xi ),$$

(8)

for each $\lambda \in ϝ$, a contradiction. Then, we can find a $({\iota }_{\epsilon },{\alpha }_{\epsilon })\in E_1\times {\partial }_{\Omega }\Xi$ satisfying

$${\alpha }_{\epsilon }\in (1-{\iota }_{\epsilon })Q_3(\lambda ,{\gamma }_{\epsilon },{\alpha }_{\epsilon })+{\iota }_{\epsilon }{Q}_3(\lambda ,\omega +\epsilon ,{\alpha }_{\epsilon }),$$

(9)

for each $\lambda \in ϝ$. Similarly, there is a $\alpha \in {\partial }_{\Omega }\Xi$ with $\alpha \in Q_3(\lambda ,\omega ,\alpha )$, which shows (6) is not true, and completes the proof. □

Let $\left(\Gamma ,{\Omega }_{\Gamma },{\parallel •,\cdots ,•\parallel }^{\Gamma },L,R\right)$ be asymmetric F-n-BS over $\mathbb{R}$ with dim ${\rm Y}\ge n$ ordered by the normal cone ${\Omega }_{\Gamma }$. Suppose that ${\rm Y}\subset \Gamma ,\Omega \subset {\Omega }_{\Gamma }\cap {\rm Y}$, the embedding $\left({{\rm Y},\parallel •,\cdots ,•\parallel }_{\mathit{ϵ}}\right)\hookrightarrow \left({\Gamma ,\parallel •,\cdots ,•\parallel }_{\mathit{ϵ}}^{\Gamma }\right)$ is continuous, and $Q_1:ϝ\times E_3\times \Omega \to cc({\Omega }_{\Gamma })$ is a compact u.s.c.rmvo. Assume $\Phi :ϝ\times \Gamma \to {\rm Y}$ is a compact random linear operator satisfying $\Phi (\lambda ,{\Omega }_{\Gamma })\subset \Omega$ such that $\Phi =\left({\Phi }_1,{\Phi }_2,\cdots ,{\Phi }_n\right)$, for each $\lambda \in ϝ$.

Theorem 2.

Let

1. 

$\varrho \alpha \in \Phi \circ Q_1(\lambda ,0,\alpha )$, for any $\lambda \in ϝ$, for some $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in \stackrel{\cdot }{\Omega }$ implies $\varrho <1$;

2. 

we can find positive numbers $a_1,a_2,a_3$ and a random linear operator $Q_4:ϝ\times \Gamma \to {\mathbb{R}}_{+}$ with $Q_4(\lambda ,\beta )\ne 0$, for any $\lambda \in ϝ$, for some $\beta =\left({\beta }_1,{\beta }_2,\cdots ,{\beta }_n\right)\in \Omega$ such that

(a) 

$Q_4\Phi (\lambda ,\alpha ){\succcurlyeq }_2\{a_1{Q}_4(\lambda ,\alpha )\}$ and

$$Q_4\Phi (\lambda ,\alpha ){\succcurlyeq }_2\{a_1.\parallel \left({\Phi }_1,{\Phi }_2,\cdots ,{\Phi }_n\right)(\lambda ,\alpha ){\parallel }_{\mathit{ϵ}}^{\Gamma }\},$$

for all $\lambda \in ϝ$ and $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in {\Omega }_{\Gamma }$,

(b) 

$Q_4\left(\lambda ,Q_1(\lambda ,\gamma ,\alpha )\right){\succcurlyeq }_2\{a_2\gamma Q_4(\lambda ,\alpha )-a_3\}$ for all $\lambda \in ϝ$, $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in \Omega$, and

(c) 

we can find an increasing map ( on the second part) $\varpi :{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to \mathbb{R}$ such that

$$\underset{\gamma \to \infty }{lim}\varpi (\gamma ,\frac{a_3}{a_1{a}_2\gamma -1})=0$$

(10)

such that $(\varrho ,\gamma ,\alpha )\in E_1\times E_3\times \Omega$ with

$$\alpha \in \varrho \Phi \circ Q_1(\lambda ,\gamma ,\alpha )+(1-\varrho )a_2\gamma \Phi (\lambda ,\alpha )$$

(11)

implies

$$\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}\le \varpi (\gamma ,\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}^{\Gamma }).$$

(12)

Then

$$\mathit{U}=\{\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in \stackrel{\cdot }{\Omega }:\exists \gamma >0,\alpha \in \Phi \circ Q_1(\lambda ,\gamma ,\alpha )\},$$

is an unbounded continuous branch emanating from 0, for each $\lambda \in ϝ$.

Proof. 

Suppose $\Xi \subset {\rm Y}$ is open and bounded where $0\in \Xi$. We use Lemma 4 with $Q_3(\lambda ,\gamma ,\alpha )=\Phi \circ Q_1(\lambda ,\gamma ,\alpha )$ to show $\mathit{U}\cap {\partial }_{\Omega }\Xi \ne \varnothing$, for any $\lambda \in ϝ$. Clearly, condition 1 of Lemma 4 holds. Assume that $(\varrho ,\gamma ,\alpha )\in E_1\times E_3\times \Omega$ satisfies (11), so $\alpha \in \Phi \left(\lambda ,\varrho Q_1(\lambda ,\gamma ,\alpha )+(1-\varrho )a_2\gamma \alpha \right)$, hence, $\alpha =\Phi \left(\lambda ,\varrho {\beta }_{\gamma }+(1-\varrho )a_2\gamma \alpha \right)$, for any $\lambda \in ϝ$, for some ${\beta }_{\gamma }\in Q_1(\lambda ,\gamma ,\alpha )$. By 2(a) and 2(b) we have

$$Q_4(\lambda ,\alpha )\ge a_1{Q}_4(\lambda ,\varrho {\beta }_{\gamma }+(1-\varrho )a_2\gamma \alpha )\ge a_1(a_2\gamma Q_4(\lambda ,\alpha )-a_3),$$

(13)

and

$$\begin{array}{ccc}\hfill Q_4(\lambda ,\alpha )& \ge & a_1{\parallel \left({\Phi }_1,{\Phi }_2,\cdots ,{\Phi }_n\right)\left(\lambda ,\varrho {\beta }_{\gamma }+(1-\varrho )a_2\gamma \alpha \right)\parallel }_{\mathit{ϵ}}^{\Gamma }\hfill \\ \hfill & =& a_1{\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\parallel }_{\mathit{ϵ}}^{\Gamma },\hfill \end{array}$$

(14)

for each $\lambda \in ϝ$. For sufficiently large $\gamma$, (13) and (14) we conclude that

$$\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}^{\Gamma }\le \frac{a_3}{a_1{a}_2\gamma -1},$$

which combining with (12) gives

$$\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}\le \varpi (\gamma ,\frac{a_3}{a_1{a}_2\gamma -1}),$$

(15)

for each $\lambda \in ϝ$. If $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in {\partial }_{\Omega }\Xi$, $0<\epsilon <a_2{\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\parallel }_{\mathit{ϵ}}$ for some $\epsilon$. From (15), (11) it follows that $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\notin \Psi (\lambda ,\varrho ,\alpha )$ for all $\lambda \in ϝ$ and $(\varrho ,\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right))\in E_1\times {\partial }_{\Omega }\Xi$, where $\Psi (\lambda ,\varrho ,\alpha )=\varrho \Phi \circ Q_1(\lambda ,\gamma ,\alpha )+(1-\varrho )a_2\gamma \alpha$. Applying Lemma 1 we obtain $i_{\Omega }(Q_3(\lambda ,\gamma ,\Xi ))=i_{\Omega }(\varphi ,\Xi )$, here $\varphi (\alpha )=a_2\gamma \alpha$, and therefore $i_{\Omega }(Q_3(\lambda ,\gamma ,.),\Xi )=0$, so condition 2 of Lemma 4 holds. The proof is complete. □

4. Applications

In this section, we study an uncertain case of a control problem. For it, we consider the compact u.s.c. rmvo $Q_1:ϝ\times E_3\times {\mathbb{R}}_{+}\to cc({\mathbb{R}}_{+})$ and the continuous map $\mu :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$. We consider the following control problem which contains a parameter:

$$\left\{\begin{array}{c}{\phi }^{\prime \prime }(\lambda ,\iota )+\nu (\gamma ,\iota )\mu (\phi (\lambda ,\iota ))=0,\iota \in \mathring{E_1},\hfill \\ \nu (\gamma ,\iota )\in Q_1(\lambda ,\gamma ,\phi (\lambda ,\iota ))\mathrm{a}.\mathrm{e}.\mathrm{on}{E}_1\hfill \\ \phi (\lambda ,0)=0,\phi (\lambda ,1)={\sum }_{p=1}^n{\omega }_p\phi (\lambda ,{\varsigma }_p)\hfill \end{array}\right.$$

(16)

where ${\varsigma }_p\in \mathring{E_1},$ $0\le {\omega }_p$, ${\sum }_{p=1}^n{\omega }_p{\varsigma }_p<1$ and $\lambda \in ϝ$.

Denote $\Theta ={\sum }_{p=1}^n{\omega }_p{\varsigma }_p$ for every $(\iota ,\kappa )\in E_1\times E_1$, and let

$$\varpi (\iota ,\kappa )=\left\{\begin{array}{c}\kappa (1-\iota ),\kappa \le \iota ,\\ \iota (1-\kappa ),\kappa >\iota \end{array}\right.$$

$$Q_5(\iota ,\kappa )=\frac{\iota }{1-\Theta }\sum _{p=1}^n{\omega }_p\varpi ({\varsigma }_p,\kappa )+\varpi (\iota ,\kappa );$$

Let $C(E_1)$, resp., $C^1(E_1)$, be the symmetric F-n-BS of all continuous, resp., continuously differentiable, functions on $E_1$. Denote

$${\rm Y}=\left\{\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in C^1(E_1):\alpha (0)=0\right\},$$

and

$$\Gamma =\left\{\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in C(E_1):\alpha (0)=0\right\}.$$

Let $\Phi :ϝ\times \Gamma \to {\rm Y}$ be a random linear operator ($\Phi =\left({\Phi }_1,{\Phi }_2,\cdots ,{\Phi }_n\right)$) defined by

$$\Phi (\lambda ,\phi )(\iota )={\int }_0^1{Q}_5(\iota ,\kappa )\phi (\lambda ,\kappa )d\kappa ,$$

(17)

for each $\iota \in E_1$ and $\lambda \in ϝ$. We solve

$$\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in \Phi \circ Q_3(\lambda ,\gamma ,\alpha ),$$

(18)

where the random multi-valued operator $Q_3$ is defined by

$$Q_3(\lambda ,\gamma ,\alpha )(\iota )=Q_1\left(\lambda ,\gamma ,\alpha (\iota )\right)\mu \left(\alpha (\iota )\right),$$

for each $\iota \in E_1$ and $\lambda \in ϝ$, since it is equivalent (17).

Theorem 3.

Let $b_1={\left\{{sup}_{\iota \in E_1}{\int }_0^1{Q}_5(\iota ,\kappa )d\kappa \right\}}^{-1}$. Suppose we can find $b_2>0,b_3>0,b_4\in (0,b_1)$, $\lambda \in ϝ$ and $s\in (0,2)$ such that

1. 

$Q_1(\lambda ,0,\alpha )\mu (\alpha ){\preccurlyeq }_2\gamma \alpha ,\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)>0$,

2. 

$b_2\gamma \alpha -b_3{\preccurlyeq }_2{Q}_1(\lambda ,\gamma ,\alpha )\mu (\alpha )$,

3. 

$Q_1(\lambda ,\gamma ,\alpha ){\preccurlyeq }_{2}1+{\gamma }^{\frac{s}{2}}{\left|\alpha \right|}^s\mathit{for}\mathit{all}(\gamma ,\alpha )\in \mathring{E_3}\times {\mathbb{R}}_{+}$.

Thus, the positive fuzzy solution set $\mathit{U}$ for (18) is unbounded continuous in $C^1(E_1)$, originating from 0.

Proof. 

Use Theorem 2 and cones

$$\Omega =\{\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in {\rm Y}:\alpha (\iota )\ge 0,\iota \in E_1\},$$

and

$${\Omega }_{\Gamma }=\{\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in \Gamma :\alpha (\iota )\ge 0,\iota \in E_1\}.$$

Then, $\Gamma$ and ${\rm Y}$, resp., are symmetric F-n-BSs with the norms

$$\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}^{\Gamma }=\underset{\iota \in E_1}{sup}{\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)(\iota )\parallel }_{\mathit{ϵ}},$$

and

$$\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}={\parallel {\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)}^{\prime }\parallel }_{\mathit{ϵ}}^{\Gamma }.$$

Suppose $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in \stackrel{\cdot }{\Omega }$, $\lambda \in ϝ$ and $\varrho$ satisfies $\varrho \alpha \in \Phi \circ Q_3(\lambda ,0,\alpha )$, so we can find $\phi (\lambda ,\kappa )\in Q_1(\lambda ,0,\alpha (\kappa ))$ such that

$$\begin{array}{ccc}\hfill \left|\varrho \alpha (\iota )\right|& =& \left|{\int }_0^1{Q}_5(\iota ,\kappa )\phi (\lambda ,\kappa )\mu (\alpha (\kappa ))d\kappa \right|\hfill \\ & \le & b_4{\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\parallel }_{\mathit{ϵ}}^{\Gamma }\left|{\int }_0^1{Q}_5(\iota ,\kappa )d\kappa \right|\hfill \\ & \le & \parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}^{\Gamma },\hfill \end{array}$$

for each $\iota \in E_1$ and $\lambda \in ϝ$. Then $\varrho <1$. By [21], we can conclude that the compact random linear operator $\Phi$ have an eigen-value ${\varrho }_0>0$ and a positive eigen-map ${\phi }_0$. Define the random linear operator $Q_4$ on $\Gamma$, by $Q_4(\lambda ,\alpha )={\int }_0^1\alpha (\kappa ){\phi }_0(\kappa )d\kappa$. From condition 2. we have

$$\begin{array}{ccc}\hfill Q_4\left(\lambda ,Q_3(\lambda ,\gamma ,\alpha )\right)& {\succcurlyeq }_2& {\int }_0^1(b_2\gamma \alpha (\kappa )-b_3){\phi }_0(\kappa )d\kappa \hfill \\ & \ge & b_2\gamma Q_4(\lambda ,\alpha )-a_3,\hfill \end{array}$$

for each $\lambda \in ϝ$; here $a_3=b_3{\int }_0^1{\phi }_0(\kappa )d\kappa$. When $\beta =\left({\beta }_1,{\beta }_2,\cdots ,{\beta }_n\right)$ in which $\beta (0)=0$ and $\beta (1)\ge 0$, then we can find a number $b_5>0$ such that $\beta (\iota )\ge b_5{\parallel \left({\beta }_1,{\beta }_2,\cdots ,{\beta }_n\right)\parallel }_{\mathit{ϵ}}^{\Gamma }{\phi }_0(\iota )$ on $E_1$. For $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in {\Omega }_{\Gamma }$, $\Phi \alpha$ is a concave function with $\Phi \alpha (\lambda ,0)=0$ and $\Phi \alpha (\lambda ,1)\ge 0$, and we have $\Phi \alpha (\lambda ,\iota )\ge b_5{\parallel \left({\Phi }_1,{\Phi }_2,\cdots ,{\Phi }_n\right)(\lambda ,\alpha )\parallel }_{\mathit{ϵ}}^{\Gamma }{\phi }_0(\iota )$, for each $\lambda \in ϝ$. From Fubini’s Theorem it follows that

$$\begin{array}{ccc}\hfill Q_4\left(\lambda ,\Phi (\lambda ,\alpha )\right)& =& {\int }_0^1\left({\int }_0^1{Q}_5(\iota ,\kappa )\alpha (\kappa )d\kappa \right){\phi }_0(\iota )d\iota \hfill \\ & =& \int {\int }_{E_1\times E_1}{Q}_5(\iota ,\kappa )\alpha (\kappa ){\phi }_0(\iota )d\kappa d\iota \hfill \\ & =& {\int }_0^1\left({\int }_0^1{Q}_5(\iota ,\kappa ){\phi }_0(\iota )d\iota \right)\alpha (\kappa )d\kappa \hfill \\ & =& {\int }_0^1\Phi {\phi }_0(\lambda ,\kappa )\alpha (\kappa )d\kappa \hfill \\ & =& {\varrho }_0{\int }_0^1{\phi }_0(\kappa )\alpha (\kappa )d\kappa \hfill \\ & =& {\varrho }_0{Q}_4(\lambda ,\alpha ),\hfill \end{array}$$

for each $\lambda \in ϝ$. Consequently, there is constant $a_1>0$ satisfying

$$Q_4\left(\lambda ,\Phi (\lambda ,\alpha )\right)\ge a_1{Q}_4(\lambda ,\alpha )\mathrm{and}{Q}_4\left(\lambda ,\Phi (\lambda ,\alpha )\right)\ge a_1{\parallel \left({\Phi }_1,{\Phi }_2,\cdots ,{\Phi }_n\right)(\lambda ,b_2)\parallel }_{\mathit{ϵ}}^{\Gamma },$$

(19)

for each $\lambda \in ϝ$. Now, assume $(\varrho ,\gamma ,\alpha )\in E_1\times E_3\times \Omega$ with

$$\alpha \in \varrho \Phi \circ Q_3(\lambda ,\gamma ,\alpha )+(1-\varrho )b_2\gamma \Phi (\lambda ,\alpha ).$$

(20)

This implies

$$-{\alpha }^{″}\in \varrho Q_3(\lambda ,\gamma ,\alpha )+(1-\varrho )b_2\gamma \alpha ,$$

(21)

for each $\lambda \in ϝ$. Now, $n_q,q=0,1,2,\cdots ,6$ and n are constant, not depending on $\gamma ,\alpha$ and $\iota \in E_1$. Using Theorem 2 implies that

$$\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}^{\Gamma }\le \frac{a_3}{a_1{b}_2\gamma -1}.$$

Therefore we can choose $n_1$ such that

$$\gamma \parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}^{\Gamma }\le n_1.$$

(22)

From (22), the well-known inequality

$${\left(\parallel {\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)}^{\prime }{\parallel }_{\mathit{ϵ}}^{\Gamma }\right)}^2\le n_2\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}^{\Gamma }.{\parallel {\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)}^{\prime \prime }\parallel }_{\mathit{ϵ}}^{\Gamma }$$

(23)

and (21) we obtain

$$\begin{array}{ccc}\hfill \parallel {\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)}^{″}{\parallel }_{\mathit{ϵ}}^{\Gamma }& \le & n_3\left(1+{\gamma }^{\frac{s}{2}}{\left(\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}^{\Gamma }\right)}^s\right)+b_2{n}_1\hfill \\ \hfill & \le & n_4\left(1+{\gamma }^{\frac{s}{2}}{\left(\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}^{\Gamma }\right)}^s\right).\hfill \end{array}$$

(24)

Furthermore, for $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)\in \Omega$, we have

$$\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}^{\Gamma }\le n_0{\parallel {\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)}^{\prime }\parallel }_{\mathit{ϵ}}^{\Gamma }.$$

(25)

Combining the inequalities, (22), (23), (24) and (25) we get

$$\parallel {\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)}^{″}{\parallel }_{\mathit{ϵ}}^{\Gamma }\le n_5(1+{\left(\parallel {\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)}^{″}{\parallel }_{\mathit{ϵ}}^{\Gamma }\right)}^{\frac{s}{2}})\le n_6.$$

(26)

From (23) we can choose n such that

$$\parallel {\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)}^{\prime }{\parallel }_{\mathit{ϵ}}^{\Gamma }\le n{\left(\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}^{\Gamma }\right)}^{\frac{1}{2}}.$$

Since $\parallel \left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right){\parallel }_{\mathit{ϵ}}={\parallel {\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)}^{\prime }\parallel }_{\mathit{ϵ}}^{\Gamma }$ we have condition (2c) of Theorem 2 satisfied with the function $\varpi (\gamma ,\iota )=n{\iota }^{\frac{1}{2}}$. □

5. Conclusions

In this paper, using a generalized norm which has a dynamic case and is inspired by a random norm and fuzzy sets, we introduced a symmetric F-n-NLS to study the existence, and unbounded continuity of the solution set of random multi-valued equation containing a parameter. These results allow us to consider an uncertain control problem. The applied procedure can hopefully be useful in the future to consider other types of fuzzy control problems.