Previous Article in Journal
An Efficient Technique to Solve Time-Fractional Kawahara and Modified Kawahara Equations Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

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

1
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 13114-16846, Iran
2
Faculty of Engineering and Natural Sciences, Istinye University, Istanbul 34710, Turkey
3
School of Mathematical and Statistical Sciences, National University of Ireland, Galway, University Road, H91 TK33 Galway, Ireland
4
Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh 11432, Saudi Arabia
*
Author to whom correspondence should be addressed.
Symmetry 2022, 14(9), 1778; https://doi.org/10.3390/sym14091778
Original submission received: 30 July 2022 / Revised: 19 August 2022 / Accepted: 23 August 2022 / Published: 26 August 2022

## 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.
MSC:
47C10

## 1. Introduction

Consider the random operator $Q 1$. A natural generalization of parametric random equations of the form $α = Q 1 ( λ , γ , α )$, in which $λ$ is a element of a probability measure space, is the multi-valued form ,
$α ∈ Q 1 ( λ , γ , α ) .$
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 $τ$, 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  for details and applications. Our results can be applied in uncertainty problems, risk measures and super-hedging in finance .
For the random multi-valued operator $Q 1$, the following sets
$U = { ( γ , α ) : α ∈ Q 1 ( λ , γ , α ) } ,$
or
$U = { α : ∃ γ , α ∈ Q 1 ( λ , γ , α ) } .$
are solutions of (1). In this paper, we consider a control problem with multi-point boundary conditions and a second order derivative operator as
$φ ″ ( λ , ι ) + ν ( γ , ι ) μ ( φ ( λ , ι ) ) = 0 , ι ∈ ( 0 , 1 ) , ν ( γ , ι ) ∈ Q 1 ( λ , γ , φ ( λ , ι ) ) a . e . on [ 0 , 1 ] φ ( λ , 0 ) = 0 , φ ( λ , 1 ) = ∑ p = 1 n ω p φ ( λ , ς p ) .$
where $ς p ∈ ( 0 , 1 ) ,$ $0 ≤ ω p$, $∑ p = 1 n ω p ς p < 1$ and $λ ∈ ϝ$. 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 $E 1 = [ 0 , 1 ]$, $E 2 = ( 0 , 1 ]$, $E 3 = [ 0 , ∞ )$ and $E 4 = [ 0 , ∞ ]$.
A mapping $δ : R → E 1$, whose $ϵ$-level set is denoted by
$[ δ ] ϵ = { ι : δ ( ι ) ≥ ϵ } ,$
is said to be a fuzzy real number if it satisfies the following:
(i)
$δ$ is normal, i.e., there exists $ι 0 ∈ R$ such that $δ ( ι 0 ) = 1$;
(ii)
$δ$ is upper semicontinuous;
(iii)
$δ$ is fuzzy convex, i.e., $δ ( ι ) ≥ min ( δ ( κ ) , δ ( s ) )$, for each $ι , κ ∈ R$ such that $κ ≤ ι ≤ s$ and $ϵ ∈ E 2$;
(iv)
For each $ϵ ∈ E 2$, $[ δ ] ϵ = [ δ ϵ − , δ ϵ + ]$, where $− ∞ < δ ϵ − ≤ δ ϵ + < + ∞$ and $[ δ ] 0 = { δ ∈ R : δ ( ι ) > 0 } ¯$ is compact.
Let the set $F$ contain all upper semicontinuous normal convex fuzzy real numbers. $F +$ contains all non-negative fuzzy real numbers of $F$. For each $κ ∈ R$, we can define
$κ ¯ ( ι ) = 1 , if ι = κ , 0 , if ι ≠ κ ,$
so $κ ¯ ∈ F$ and $R$ can be embedded in $F$.
A partial order ⪯ in $F$ is defined as follows: $δ ⪯ σ$ iff for each $ϵ ∈ E 2$, $δ ϵ − ≤ σ ϵ −$ and $δ ϵ + ≤ σ ϵ +$ where $[ δ ] ϵ = [ δ ϵ − , δ ϵ + ]$ and $[ σ ] ϵ = [ σ ϵ − , σ ϵ + ]$. The strict inequality in $F$ is defined by $δ ≺ σ$ iff for each $ϵ ∈ E 2$, $δ ϵ − < σ ϵ −$ and $δ ϵ + < σ ϵ +$ (see [6,7,8]).
The arithmetic operations ⊕, ⊖, ⊙ and ⊘ on $F × F$ are defined by
$δ ⊕ σ ( ι ) = sup κ ∈ R min δ ( κ ) , σ ( ι − κ ) , ι ∈ R , δ ⊖ σ ( ι ) = sup κ ∈ R min δ ( κ ) , σ ( κ − ι ) , ι ∈ R , δ ⊙ σ ( ι ) = sup 0 ≠ κ ∈ R min δ ( κ ) , σ ( ι κ ) , ι ∈ R ,$
$δ ⊘ σ ( ι ) = sup κ ∈ R min δ ( κ ι ) , σ ( κ ) , ι ∈ R , δ , σ ( ≻ 0 ) ∈ F .$
Definition 1.
Let be a real linear space over $R$ with dim $℧ ≥ n$. Suppose $∥ • , ⋯ , • ∥ : ℧ n → F +$ is a mapping and $L , R : E 1 2 → E 1$ are symmetric, nondecreasing mapping satisfying
$L ( 0 , 0 ) = 0 and R ( 1 , 1 ) = 1 .$
Write
$[ ∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ ] ϵ = [ ∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ ϵ − , ∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ ϵ + ] ,$
for $ϑ 1 , ϑ 2 , ⋯ , ϑ n ∈ ℧$, $ϵ ∈ E 2$ and suppose that for every linearly independent vectors $ϑ 1 , ϑ 2 , ⋯ , ϑ n$ $∈ ℧$, there exists $ϵ 0 ∈ E 2$ independent of $ϑ 1 , ϑ 2 , ⋯ , ϑ n ∈ ℧$ such that for each $ϵ ≤ ϵ 0$, one has
$inf ∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ ϵ − > 0 , ∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ ϵ + < ∞ .$
The quadruple $℧ n , ∥ • , ⋯ , • ∥ , L , R$ is said to be a symmetric fuzzy n-normed linear space (F-n-NLS) in the sense of Felbin  and $∥ • , ⋯ , • ∥$ is a fuzzy n-norm if
(N1)
$∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ = 0 ¯$ iff $ϑ 1 , ϑ 2 , ⋯ , ϑ n$ are linearly dependent;
(N2)
$∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥$ is invariant under any permutation of $ϑ 1 , ϑ 2 , ⋯ , ϑ n ∈ ℧$;
(N3)
$∥ c ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ = | c | ⊙ ∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥$ for any $c ∈ R$;
(N4)
$∥ ϑ 0 + ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ ⪯ ∥ ϑ 0 , ϑ 2 , ⋯ , ϑ n ∥ ⊕ ∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥$;
(i)
whenever $κ ≤ ∥ ϑ 0 , ϑ 2 , ⋯ , ϑ n ∥ 1 −$, $ι ≤ ∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ 1 −$ and $ι + κ ≤ ∥ ϑ 0 + ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ 1 −$,
$∥ ϑ 0 + ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ ( κ + ι ) ≥ L ∥ ϑ 0 , ϑ 2 , ⋯ , ϑ n ∥ ( κ ) , ∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ ( ι ) ,$
(ii)
whenever $κ ≥ ∥ ϑ 0 , ϑ 2 , ⋯ , ϑ n ∥ 1 −$, $ι ≥ ∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ 1 −$ and $ι + κ ≥ ∥ ϑ 0 + ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ 1 −$,
$∥ ϑ 0 + ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ ( κ + ι ) ≤ R ∥ ϑ 0 , ϑ 2 , ⋯ , ϑ n ∥ ( κ ) , ∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ ( ι ) .$
Now, we consider a symmetric F-n-NLS in the sense of Narayanan-Vijayabalaji  and next we show a relationship between them.
Definition 2
(). Assume that is a linear space and ∗ is a continuous t-norm. Let the fuzzy subset η of $℧ n × R$ with dim $℧ ≥ n$ satisfy
(FN1)
For all $τ ∈ R$ with $τ ≤ 0$, $η ( ϑ 1 , ϑ 2 , ⋯ , ϑ n , τ ) = 0$;
(FN2)
For all $τ ∈ R$ with $τ > 0$, $η ( ϑ 1 , ϑ 2 , ⋯ , ϑ n , τ ) = 1$ for $τ ≥ 0$ iff $ϑ 1 , ϑ 2 , ⋯ , ϑ n$ are linearly dependent;
(FN3)
$η ( ϑ 1 , ϑ 2 , ⋯ , ϑ n , τ )$ is invariant under any permutation of $ϑ 1 , ϑ 2 , ⋯ , ϑ n ∈ ℧$;
(FN4)
For all $τ ∈ R$ with $τ > 0$,
$η c ϑ 1 , ϑ 2 , ⋯ , ϑ n , τ = η ϑ 1 , ϑ 2 , ⋯ , ϑ n , τ | c | if c ∈ R with c ≠ 0 ;$
(FN5)
For all $τ ∈ R$ with $τ , θ > 0$,
$η ( ϑ 0 + ϑ 1 , ϑ 2 , ⋯ , ϑ n , τ + θ ) ≥ η ϑ 0 , ϑ 2 , ⋯ , ϑ n , τ ∗ η ϑ 1 , ϑ 2 , ⋯ , ϑ n , θ ;$
(FN6)
$η ( ϑ 1 , ϑ 2 , ⋯ , ϑ n , . ) : E 3 ˚ → E 1$ is left continuous;
(FN7)
$lim τ → + ∞ η ( ϑ 1 , ϑ 2 , ⋯ , ϑ n , τ ) = 1$.
Thus, the triple $( ℧ , η , ∗ )$ 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 $( ℧ , η , ∗ )$ be a symmetric F-n-NLS in which $∗ = min$ and
(FN8)
$η ( ϑ 1 , ϑ 2 , ⋯ , ϑ n , τ ) > 0$ for all $τ > 0$ implies $ϑ 1 , ϑ 2 , ⋯ , ϑ n$ are linearly dependent.
Define
$∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ ϵ : = inf [ η ( ϑ 1 , ϑ 2 , ⋯ , ϑ n , τ ) ] ϵ , ϵ ∈ E 1 ˚ .$
Then ${ ∥ • , ⋯ , • ∥ ϵ : ϵ ∈ E 1 ˚ }$ 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
(). Let $η E : R × E 3 ˚ → E 2$ be a Euclidean fuzzy norm (Euclidean fuzzy normed spaces were introduced by the authors in ). Then $ϑ 1 , ϑ 2 , ⋯ , ϑ n ∈ ℧$ are linearly independent iff $η ( ϑ 1 , ϑ 2 , ⋯ , ϑ n , τ ) = η E ( 1 , τ )$, for any $τ > 0$.
By the above remark, we have that, $ϑ 1 , ϑ 2 , ⋯ , ϑ n ∈ ℧$ are linearly independent iff
$∥ ϑ 1 , ϑ 2 , ⋯ , ϑ n ∥ ϵ = inf { τ : η ( ϑ 1 , ϑ 2 , ⋯ , ϑ n , τ ) ≥ ϵ , ϵ ∈ E 1 ˚ } = inf { τ : η E ( 1 , τ ) ≥ ϵ , ϵ ∈ E 1 ˚ } = | 1 | ϵ .$
Consider the probability measure space $( ϝ , E 3 ˚ , ξ )$ and let $U , B U$ and $S , B S$ be Borel measurable spaces, where U and S are symmetric F-n-BS. If ${ λ : F ( λ , ξ ) ∈ B } ∈ E 3 ˚$ for every $ξ$ in U and $B ∈ B S$, we say $F : ϝ × U → S$ is a random operator. Let $2 S$ be the family of all subsets of S. The mapping $F : ϝ × U → 2 S$ is said to be random multi-valued operator. A random operator $F : ϝ × U → S$ is said to be linear if $F ( λ , a ξ 1 + b ξ 2 ) = a F ( λ , ξ 1 ) + b F ( λ , ξ 2 )$ almost everywhere for each $ξ 1 , ξ 2$ in U and $a , b$ are scalers, and bounded if there exists a nonnegative real-valued random variable $M ( λ )$ such that
$η F ( λ 1 , ξ 1 , 1 ) − F ( λ 1 , ξ 2 , 1 ) , ⋯ , F ( λ 1 , ξ 1 , n ) − F ( λ 1 , ξ 2 , n ) , M ( λ ) τ ≥ η ξ 1 , 1 − ξ 2 , 1 , ⋯ , ξ 1 , n − ξ 2 , n , τ ,$
almost everywhere for each $ξ 1 , j − ξ 2 , j$ $( j = 1 , 2 , … n )$ in U, $τ ∈ E 3 ˚$ and $λ ∈ ϝ$.
Let $Υ , Ω , ∥ • , ⋯ , • ∥ , L , R$ be a symmetric F-n-BS over $R$ with dim $Υ ≥ n$ and ordering by the cone $Ω$, i.e., $Ω$ is a closed convex subset of $Υ$ such that $γ Ω ⊂ Ω$ for $γ ≥ 0$, $Ω ∩ ( − Ω ) = { 0 }$, and $α p ≤ β p$ iff $β p − α p ∈ Ω$ for $α , β ∈ Υ$ with $α = α 1 , α 2 , ⋯ , α n , β = β 1 , β 2 , ⋯ , β n$ and $1 ≤ p ≤ n$. For nonempty subsets $Δ 1 , Δ 2$ of $Υ$ we write $Δ 1 ≽ 2 Δ 2$ (or, $Δ 2 ≼ 2 Δ 1$) iff for every $α ∈ Δ 1$, we can find a $β ∈ Δ 2$ which $α p ≥ β p$ (or, $β p ≤ α p$) for $1 ≤ p ≤ n$. We say $Ω$ is a normal cone if we can find a constant $K > 0$ where $0 ≤ α p ≤ β p$ for $1 ≤ p ≤ n$ implies $∥ α 1 , α 2 , ⋯ , α n ∥ ϵ ≤ K ∥ β 1 , β 2 , ⋯ , β n ∥ ϵ$. We note in this paper, we consider $Ω$ as a normal cone with $K = 1$. Furthermore,
$c c ( Δ 1 ) = { G ⊂ Δ 1 ⊂ Υ : G is nonempty closed convex }$
Consider the open convex subset $Ξ$ of $Υ$, and let $Ξ Ω = Ω ∩ Ξ$, $∂ Ω Ξ = Ω ∩ ∂ Ξ$ and $Ω ⋅ = Ω ∖ { 0 }$, where $∂ Ξ$ is boundary of $Ξ$ in $Υ$. The mapping $Q 3 : ϝ × Ω ∩ Ξ ¯ → c c ( Ω )$ is said to be compact if $Q 3 ( ϝ × Δ 2 )$ is relatively compact for any bounded subset $Δ 2$ of $Ω ∩ Ξ ¯$, where $Q 3 ( ϝ × Δ 2 ) = ∪ α ∈ Δ 2 Q 3 ( λ , α )$, for any $λ ∈ ϝ$. We say a random multi-valued operator $Q 3$ has the upper semi-continuity property (in short, u.s.c.rmvo) if
${ α = α 1 , α 2 , ⋯ , α n ∈ Ω ∩ Ξ ¯ : Q 3 ( λ , α ) ⊂ W }$
where $Ω ∩ Ξ ¯ = ( Ω ∩ Ξ ¯ ) ∘$ and $λ ∈ ϝ$. Further, if $α ∉ Q 3 ( λ , α )$ for all $α = α 1 , α 2 , ⋯ , α n ∈ ∂ Ω Ξ$ and $λ ∈ ϝ$, the random fixed point index of $Q 3$ in $Ξ$ with respect to $Ω$ is defined which is an integer denoted by $i Ω ( Q 3 , Ξ )$.
Lemma 1.
 Let $Q 3 : ϝ × Ω ∩ Ξ ¯ → c c ( Ω )$ be a compact u.s.c.rmvo. Then
1.
$i Ω ( Q 3 , Ξ ) = 0$ if there exists $φ ∈ Ω ⋅$ such that $α ∉ Q 3 ( λ , α ) + ϱ φ$ for all $α = α 1 , α 2 , ⋯ , α n ∈ ∂ Ω Ξ$, $λ ∈ ϝ$ and $ϱ ≥ 0$.
2.
$i Ω ( Q 3 , Ξ ) = 1$ if $ϱ α ∉ Q 3 ( λ , α )$ for all $λ ∈ ϝ$ and $ϱ ≥ 1$.
The following results are needed later to obtain a generalization of .
Lemma 2.
 Assume that $Q 3 : ϝ × E 3 ⊂ ϝ × Υ → c ( Υ )$ is a u.s.c.rmvo, $α ε , β ε → ( α , β )$ with $β ε ∈ Q 3 ( λ , α ε )$ and $λ ∈ ϝ$. Thus, $β ∈ Q 3 ( λ , α )$.
Lemma 3.
 Let $Ψ : ϝ × E 1 × Ω ∩ Ξ ¯ → c c ( Ω )$ be a compact u.s.c.rmvo with $α ∉ Ψ ( λ , ι , α )$ for all $( ι , α ) ∈ E 1 × ∂ Ω Ξ$ and $λ ∈ ϝ$. Then, $i Ω ( Ψ ( λ , 0 , . ) , Ξ ) = i Ω ( Ψ ( λ , 1 , . ) , Ξ )$.

## 3. Random Multi-Valued Operator

Lemma 4.
Let $Q 3 : ϝ × E 3 × Ω → c c ( Ω )$ be a compact u.s.c.rmvo and $Ξ ⊂ Υ$ be open with $0 ∈ Ξ$. Additionally,
1.
$ι α ∈ Q 3 ( λ , 0 , α )$, for any $λ ∈ ϝ$, for some $α = α 1 , α 2 , ⋯ , α n ∈ Ω ⋅$ implies $ι < 1$,
2.
$i ϱ ( Q 3 ( λ , γ , . ) , Ξ ) = 0$ if γ is sufficiently large and $λ ∈ ϝ$.
Then ${ α = α 1 , α 2 , ⋯ , α n ∈ ∂ Ω Ξ : ∃ γ > 0 , α ∈ Q 3 ( λ , γ , α ) } ≠ ∅$.
Proof.
From the second condition in Lemma 4 we can find $γ 0 > 0$ such that $i ϱ ( Q 3 ( λ , γ , . ) , Ξ )$ $= 0$ for all $λ ∈ ϝ$ and $γ ≥ γ 0$. Define
$ω = sup { γ > 0 : i Ω ( Q 3 ( λ , γ , . ) , Ξ ) ≠ 0 } .$
We first observe that $ω > 0$. Furthermore,
$∀ ε > 0 , ∃ ( ι ε , α ε ) ∈ E 1 × ∂ Ω Ξ : α ε ∈ ( 1 − ι ε ) Q 3 ( λ , ε , α ε ) + ι ε Q 3 ( λ , 0 , α ε ) .$
for any $λ ∈ ϝ$. Since $Q 3$ is compact, without loss of generality we may assume that $ι ε → ι , α ε → α$ when $ε → 0$ and $λ ∈ ϝ$. From (5) by Lemma 2 it follows that
$α ∈ ( 1 − ι ) Q 3 ( λ , 0 , α ) + ι Q 3 ( λ , 0 , α ) ⊂ Q 3 ( λ , 0 , α ) .$
This contradicts the first condition in Lemma 4. Thus, there exist $ε > 0$ such that $( ι , α ) ∉ Ψ ( λ , ι , α )$ for all $λ ∈ ϝ$ and $( ι , α ) ∈ E 1 × ∂ Ω Ξ$, where
$Ψ ( λ , ι , α ) = ( 1 − ι ) Q 3 ( λ , ε , α ) + ι Q 3 ( λ , 0 , α ) .$
Using Lemma 3 we have
$i Ω ( Q 3 ( λ , 0 , . ) , Ξ ) = i Ω ( Q 3 ( λ , ε , . ) , Ξ ) .$
Using Lemma 1 implies that $i Ω ( Q 3 ( λ , 0 , . ) , Ξ ) = 1$. Thus, $i Ω ( Q 3 ( λ , ε , . ) , Ξ ) = 1$, and we deduce $0 < ω < γ 0$, for each $λ ∈ ϝ$.
Next, for every $ε ∈ ( 0 , ω )$ and $λ ∈ ϝ$, there exists $γ ε ∈ ( ω − ε , ω ]$ with $i Ω ( Q 3 ( λ , γ ε , . ) ,$ $Ξ ) ≠ 0$. Consider the random multi-valued operator $Ψ ε$ defined by
$Ψ ε ( λ , ι , α ) = ( 1 − ι ) Q 3 ( λ , γ ε , α ) + ι Q 3 ( λ , ω + ε , α ) .$
Now, we prove
${ α = α 1 , α 2 , ⋯ , α n ∈ ∂ Ω Ξ : ∃ γ > 0 , α ∈ Q 3 ( λ , γ , α ) } ≠ ∅ .$
Assume the contrary, that
${ α = α 1 , α 2 , ⋯ , α n ∈ ∂ Ω Ξ : ∃ γ > 0 , α ∈ Q 3 ( λ , γ , α ) } = ∅ .$
Then, the random fixed point index of $Q 3 ( λ , ω + ε )$ is well defined, for each $λ ∈ ϝ$. If
$α ∉ Ψ ε ( λ , ι , α ) for all ( ι , α ) ∈ E 1 × ∂ Ω Ξ ,$
then, by Lemma 3 we obtain
$i Ω ( Q 3 ( λ , γ ε , . ) , Ξ ) = i Ω ( Q 3 ( λ , ω + ε , . ) , Ξ ) ,$
for each $λ ∈ ϝ$, a contradiction. Then, we can find a $( ι ε , α ε ) ∈ E 1 × ∂ Ω Ξ$ satisfying
$α ε ∈ ( 1 − ι ε ) Q 3 ( λ , γ ε , α ε ) + ι ε Q 3 ( λ , ω + ε , α ε ) ,$
for each $λ ∈ ϝ$. Similarly, there is a $α ∈ ∂ Ω Ξ$ with $α ∈ Q 3 ( λ , ω , α )$, which shows (6) is not true, and completes the proof. □
Let $Γ , Ω Γ , ∥ • , ⋯ , • ∥ Γ , L , R$ be asymmetric F-n-BS over $R$ with dim $Υ ≥ n$ ordered by the normal cone $Ω Γ$. Suppose that $Υ ⊂ Γ , Ω ⊂ Ω Γ ∩ Υ$, the embedding $Υ , ∥ • , ⋯ , • ∥ ϵ ↪ Γ , ∥ • , ⋯ , • ∥ ϵ Γ$ is continuous, and $Q 1 : ϝ × E 3 × Ω → c c ( Ω Γ )$ is a compact u.s.c.rmvo. Assume $Φ : ϝ × Γ → Υ$ is a compact random linear operator satisfying $Φ ( λ , Ω Γ ) ⊂ Ω$ such that $Φ = Φ 1 , Φ 2 , ⋯ , Φ n$, for each $λ ∈ ϝ$.
Theorem 2.
Let
1.
$ϱ α ∈ Φ ∘ Q 1 ( λ , 0 , α )$, for any $λ ∈ ϝ$, for some $α = α 1 , α 2 , ⋯ , α n ∈ Ω ⋅$ implies $ϱ < 1$;
2.
we can find positive numbers $a 1 , a 2 , a 3$ and a random linear operator $Q 4 : ϝ × Γ → R +$ with $Q 4 ( λ , β ) ≠ 0$, for any $λ ∈ ϝ$, for some $β = β 1 , β 2 , ⋯ , β n ∈ Ω$ such that
(a)
$Q 4 Φ ( λ , α ) ≽ 2 { a 1 Q 4 ( λ , α ) }$ and
$Q 4 Φ ( λ , α ) ≽ 2 { a 1 . ∥ Φ 1 , Φ 2 , ⋯ , Φ n ( λ , α ) ∥ ϵ Γ } ,$
for all $λ ∈ ϝ$ and $α = α 1 , α 2 , ⋯ , α n ∈ Ω Γ$,
(b)
$Q 4 λ , Q 1 ( λ , γ , α ) ≽ 2 { a 2 γ Q 4 ( λ , α ) − a 3 }$ for all $λ ∈ ϝ$, $α = α 1 , α 2 , ⋯ , α n ∈ Ω$, and
(c)
we can find an increasing map ( on the second part) $ϖ : R + × R + → R$ such that
$lim γ → ∞ ϖ ( γ , a 3 a 1 a 2 γ − 1 ) = 0$
such that $( ϱ , γ , α ) ∈ E 1 × E 3 × Ω$ with
$α ∈ ϱ Φ ∘ Q 1 ( λ , γ , α ) + ( 1 − ϱ ) a 2 γ Φ ( λ , α )$
implies
$∥ α 1 , α 2 , ⋯ , α n ∥ ϵ ≤ ϖ ( γ , ∥ α 1 , α 2 , ⋯ , α n ∥ ϵ Γ ) .$
Then
$U = { α = α 1 , α 2 , ⋯ , α n ∈ Ω ⋅ : ∃ γ > 0 , α ∈ Φ ∘ Q 1 ( λ , γ , α ) } ,$
is an unbounded continuous branch emanating from 0, for each $λ ∈ ϝ$.
Proof.
Suppose $Ξ ⊂ Υ$ is open and bounded where $0 ∈ Ξ$. We use Lemma 4 with $Q 3 ( λ , γ , α ) = Φ ∘ Q 1 ( λ , γ , α )$ to show $U ∩ ∂ Ω Ξ ≠ ∅$, for any $λ ∈ ϝ$. Clearly, condition 1 of Lemma 4 holds. Assume that $( ϱ , γ , α ) ∈ E 1 × E 3 × Ω$ satisfies (11), so $α ∈ Φ λ , ϱ Q 1 ( λ , γ , α ) + ( 1 − ϱ ) a 2 γ α$, hence, $α = Φ λ , ϱ β γ + ( 1 − ϱ ) a 2 γ α$, for any $λ ∈ ϝ$, for some $β γ ∈ Q 1 ( λ , γ , α )$. By 2(a) and 2(b) we have
$Q 4 ( λ , α ) ≥ a 1 Q 4 ( λ , ϱ β γ + ( 1 − ϱ ) a 2 γ α ) ≥ a 1 ( a 2 γ Q 4 ( λ , α ) − a 3 ) ,$
and
$Q 4 ( λ , α ) ≥ a 1 ∥ Φ 1 , Φ 2 , ⋯ , Φ n λ , ϱ β γ + ( 1 − ϱ ) a 2 γ α ∥ ϵ Γ = a 1 ∥ α 1 , α 2 , ⋯ , α n ∥ ϵ Γ ,$
for each $λ ∈ ϝ$. For sufficiently large $γ$, (13) and (14) we conclude that
$∥ α 1 , α 2 , ⋯ , α n ∥ ϵ Γ ≤ a 3 a 1 a 2 γ − 1 ,$
which combining with (12) gives
$∥ α 1 , α 2 , ⋯ , α n ∥ ϵ ≤ ϖ ( γ , a 3 a 1 a 2 γ − 1 ) ,$
for each $λ ∈ ϝ$. If $α = α 1 , α 2 , ⋯ , α n ∈ ∂ Ω Ξ$, $0 < ε < a 2 ∥ α 1 , α 2 , ⋯ , α n ∥ ϵ$ for some $ε$. From (15), (11) it follows that $α = α 1 , α 2 , ⋯ , α n ∉ Ψ ( λ , ϱ , α )$ for all $λ ∈ ϝ$ and $( ϱ , α = α 1 , α 2 , ⋯ , α n ) ∈ E 1 × ∂ Ω Ξ$, where $Ψ ( λ , ϱ , α ) = ϱ Φ ∘ Q 1 ( λ , γ , α ) + ( 1 − ϱ ) a 2 γ α$. Applying Lemma 1 we obtain $i Ω ( Q 3 ( λ , γ , Ξ ) ) = i Ω ( ϕ , Ξ )$, here $ϕ ( α ) = a 2 γ α$, and therefore $i Ω ( Q 3 ( λ , γ , . ) , Ξ ) = 0$, 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 $Q 1 : ϝ × E 3 × R + → c c ( R + )$ and the continuous map $μ : R + → R +$. We consider the following control problem which contains a parameter:
$φ ′ ′ ( λ , ι ) + ν ( γ , ι ) μ ( φ ( λ , ι ) ) = 0 , ι ∈ E 1 ˚ , ν ( γ , ι ) ∈ Q 1 ( λ , γ , φ ( λ , ι ) ) a . e . on E 1 φ ( λ , 0 ) = 0 , φ ( λ , 1 ) = ∑ p = 1 n ω p φ ( λ , ς p )$
where $ς p ∈ E 1 ˚ ,$ $0 ≤ ω p$, $∑ p = 1 n ω p ς p < 1$ and $λ ∈ ϝ$.
Denote $Θ = ∑ p = 1 n ω p ς p$ for every $( ι , κ ) ∈ E 1 × E 1$, and let
$ϖ ( ι , κ ) = κ ( 1 − ι ) , κ ≤ ι , ι ( 1 − κ ) , κ > ι$
$Q 5 ( ι , κ ) = ι 1 − Θ ∑ p = 1 n ω p ϖ ( ς p , κ ) + ϖ ( ι , κ ) ;$
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
$Υ = α = α 1 , α 2 , ⋯ , α n ∈ C 1 ( E 1 ) : α ( 0 ) = 0 ,$
and
$Γ = α = α 1 , α 2 , ⋯ , α n ∈ C ( E 1 ) : α ( 0 ) = 0 .$
Let $Φ : ϝ × Γ → Υ$ be a random linear operator ($Φ = Φ 1 , Φ 2 , ⋯ , Φ n$) defined by
$Φ ( λ , φ ) ( ι ) = ∫ 0 1 Q 5 ( ι , κ ) φ ( λ , κ ) d κ ,$
for each $ι ∈ E 1$ and $λ ∈ ϝ$. We solve
$α = α 1 , α 2 , ⋯ , α n ∈ Φ ∘ Q 3 ( λ , γ , α ) ,$
where the random multi-valued operator $Q 3$ is defined by
$Q 3 ( λ , γ , α ) ( ι ) = Q 1 λ , γ , α ( ι ) μ α ( ι ) ,$
for each $ι ∈ E 1$ and $λ ∈ ϝ$, since it is equivalent (17).
Theorem 3.
Let $b 1 = sup ι ∈ E 1 ∫ 0 1 Q 5 ( ι , κ ) d κ − 1$. Suppose we can find $b 2 > 0 , b 3 > 0 , b 4 ∈ ( 0 , b 1 )$, $λ ∈ ϝ$ and $s ∈ ( 0 , 2 )$ such that
1.
$Q 1 ( λ , 0 , α ) μ ( α ) ≼ 2 γ α , α = α 1 , α 2 , ⋯ , α n > 0$,
2.
$b 2 γ α − b 3 ≼ 2 Q 1 ( λ , γ , α ) μ ( α )$,
3.
$Q 1 ( λ , γ , α ) ≼ 2 1 + γ s 2 | α | s for all ( γ , α ) ∈ E 3 ˚ × R +$.
Thus, the positive fuzzy solution set $U$ for (18) is unbounded continuous in $C 1 ( E 1 )$, originating from 0.
Proof.
Use Theorem 2 and cones
$Ω = { α = α 1 , α 2 , ⋯ , α n ∈ Υ : α ( ι ) ≥ 0 , ι ∈ E 1 } ,$
and
$Ω Γ = { α = α 1 , α 2 , ⋯ , α n ∈ Γ : α ( ι ) ≥ 0 , ι ∈ E 1 } .$
Then, $Γ$ and $Υ$, resp., are symmetric F-n-BSs with the norms
$∥ α 1 , α 2 , ⋯ , α n ∥ ϵ Γ = sup ι ∈ E 1 ∥ α 1 , α 2 , ⋯ , α n ( ι ) ∥ ϵ ,$
and
$∥ α 1 , α 2 , ⋯ , α n ∥ ϵ = ∥ α 1 , α 2 , ⋯ , α n ′ ∥ ϵ Γ .$
Suppose $α = α 1 , α 2 , ⋯ , α n ∈ Ω ⋅$, $λ ∈ ϝ$ and $ϱ$ satisfies $ϱ α ∈ Φ ∘ Q 3 ( λ , 0 , α )$, so we can find $φ ( λ , κ ) ∈ Q 1 ( λ , 0 , α ( κ ) )$ such that
$ϱ α ( ι ) = ∫ 0 1 Q 5 ( ι , κ ) φ ( λ , κ ) μ ( α ( κ ) ) d κ ≤ b 4 ∥ α 1 , α 2 , ⋯ , α n ∥ ϵ Γ ∫ 0 1 Q 5 ( ι , κ ) d κ ≤ ∥ α 1 , α 2 , ⋯ , α n ∥ ϵ Γ ,$
for each $ι ∈ E 1$ and $λ ∈ ϝ$. Then $ϱ < 1$. By , we can conclude that the compact random linear operator $Φ$ have an eigen-value $ϱ 0 > 0$ and a positive eigen-map $φ 0$. Define the random linear operator $Q 4$ on $Γ$, by $Q 4 ( λ , α ) = ∫ 0 1 α ( κ ) φ 0 ( κ ) d κ$. From condition 2. we have
$Q 4 λ , Q 3 ( λ , γ , α ) ≽ 2 ∫ 0 1 ( b 2 γ α ( κ ) − b 3 ) φ 0 ( κ ) d κ ≥ b 2 γ Q 4 ( λ , α ) − a 3 ,$
for each $λ ∈ ϝ$; here $a 3 = b 3 ∫ 0 1 φ 0 ( κ ) d κ$. When $β = β 1 , β 2 , ⋯ , β n$ in which $β ( 0 ) = 0$ and $β ( 1 ) ≥ 0$, then we can find a number $b 5 > 0$ such that $β ( ι ) ≥ b 5 ∥ β 1 , β 2 , ⋯ , β n ∥ ϵ Γ φ 0 ( ι )$ on $E 1$. For $α = α 1 , α 2 , ⋯ , α n ∈ Ω Γ$, $Φ α$ is a concave function with $Φ α ( λ , 0 ) = 0$ and $Φ α ( λ , 1 ) ≥ 0$, and we have $Φ α ( λ , ι ) ≥ b 5 ∥ Φ 1 , Φ 2 , ⋯ , Φ n ( λ , α ) ∥ ϵ Γ φ 0 ( ι )$, for each $λ ∈ ϝ$. From Fubini’s Theorem it follows that
$Q 4 λ , Φ ( λ , α ) = ∫ 0 1 ∫ 0 1 Q 5 ( ι , κ ) α ( κ ) d κ φ 0 ( ι ) d ι = ∫ ∫ E 1 × E 1 Q 5 ( ι , κ ) α ( κ ) φ 0 ( ι ) d κ d ι = ∫ 0 1 ∫ 0 1 Q 5 ( ι , κ ) φ 0 ( ι ) d ι α ( κ ) d κ = ∫ 0 1 Φ φ 0 ( λ , κ ) α ( κ ) d κ = ϱ 0 ∫ 0 1 φ 0 ( κ ) α ( κ ) d κ = ϱ 0 Q 4 ( λ , α ) ,$
for each $λ ∈ ϝ$. Consequently, there is constant $a 1 > 0$ satisfying
$Q 4 λ , Φ ( λ , α ) ≥ a 1 Q 4 ( λ , α ) and Q 4 λ , Φ ( λ , α ) ≥ a 1 ∥ Φ 1 , Φ 2 , ⋯ , Φ n ( λ , b 2 ) ∥ ϵ Γ ,$
for each $λ ∈ ϝ$. Now, assume $( ϱ , γ , α ) ∈ E 1 × E 3 × Ω$ with
$α ∈ ϱ Φ ∘ Q 3 ( λ , γ , α ) + ( 1 − ϱ ) b 2 γ Φ ( λ , α ) .$
This implies
$− α ″ ∈ ϱ Q 3 ( λ , γ , α ) + ( 1 − ϱ ) b 2 γ α ,$
for each $λ ∈ ϝ$. Now, $n q , q = 0 , 1 , 2 , ⋯ , 6$ and n are constant, not depending on $γ , α$ and $ι ∈ E 1$. Using Theorem 2 implies that
$∥ α 1 , α 2 , ⋯ , α n ∥ ϵ Γ ≤ a 3 a 1 b 2 γ − 1 .$
Therefore we can choose $n 1$ such that
$γ ∥ α 1 , α 2 , ⋯ , α n ∥ ϵ Γ ≤ n 1 .$
From (22), the well-known inequality
$∥ α 1 , α 2 , ⋯ , α n ′ ∥ ϵ Γ 2 ≤ n 2 ∥ α 1 , α 2 , ⋯ , α n ∥ ϵ Γ . ∥ α 1 , α 2 , ⋯ , α n ′ ′ ∥ ϵ Γ$
and (21) we obtain
$∥ α 1 , α 2 , ⋯ , α n ″ ∥ ϵ Γ ≤ n 3 1 + γ s 2 ∥ α 1 , α 2 , ⋯ , α n ∥ ϵ Γ s + b 2 n 1 ≤ n 4 1 + γ s 2 ∥ α 1 , α 2 , ⋯ , α n ∥ ϵ Γ s .$
Furthermore, for $α = α 1 , α 2 , ⋯ , α n ∈ Ω$, we have
$∥ α 1 , α 2 , ⋯ , α n ∥ ϵ Γ ≤ n 0 ∥ α 1 , α 2 , ⋯ , α n ′ ∥ ϵ Γ .$
Combining the inequalities, (22), (23), (24) and (25) we get
$∥ α 1 , α 2 , ⋯ , α n ″ ∥ ϵ Γ ≤ n 5 ( 1 + ∥ α 1 , α 2 , ⋯ , α n ″ ∥ ϵ Γ s 2 ) ≤ n 6 .$
From (23) we can choose n such that
$∥ α 1 , α 2 , ⋯ , α n ′ ∥ ϵ Γ ≤ n ∥ α 1 , α 2 , ⋯ , α n ∥ ϵ Γ 1 2 .$
Since $∥ α 1 , α 2 , ⋯ , α n ∥ ϵ = ∥ α 1 , α 2 , ⋯ , α n ′ ∥ ϵ Γ$ we have condition (2c) of Theorem 2 satisfied with the function $ϖ ( γ , ι ) = n ι 1 2$. □

## 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.

## Author Contributions

R.S., methodology and project administration. T.A., writing—original draft preparation. D.O., methodology, writing—original draft preparation and project administration. F.S.A., writing—original draft preparation and project administration. All authors have read and agreed to the published version of the manuscript.

## Funding

The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) for funding and supporting this work through Research Partnership Program no RP-21-09-08.

Not applicable.

Not applicable.

Not applicable.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

1. Feng, B.; Zhou, H.C.; Yang, X.G. Uniform boundness of global solutions for a n-dimensional spherically symmetric combustion model. Appl. Anal. 2019, 98, 2688–2722. [Google Scholar] [CrossRef]
2. Huy, N.B.; Binh, T.T.; Tri, V.V. The monotone minorant method and eigenvalue problem for multivalued operators in cones. Fixed Point Theory 2019, 19, 275–286. [Google Scholar] [CrossRef]
3. Huy, N.B. Global Continua of Positive Solutions for Equations with nondifferentiable operators. J. Math. Anal. Appl. 1999, 239, 449–456. [Google Scholar] [CrossRef]
4. Cho, Y.J.; Rassias, T.M.; Saadati, R. Fuzzy Operator Theory in Mathematical Analysis; Springer: Berlin/Heidelberg, Germany, 2018. [Google Scholar]
5. Ren, Y.; Wang, J.; Hu, L. Multi-valued stochastic differential equations driven by G-Brownian motion and related stochastic control problems. Int. J. Control 2017, 90, 1132–1154. [Google Scholar] [CrossRef]
6. Kaleva, O.; Seikkala, S. On fuzzy metric spaces. Fuzzy Sets Syst. 1984, 12, 215–229. [Google Scholar] [CrossRef]
7. Morsi, N.N. On fuzzy pseudo-normed vector spaces. Fuzzy Sets Syst. 1988, 27, 351–372. [Google Scholar] [CrossRef]
8. Felbin, C. Finite-dimensional fuzzy normed linear space. Fuzzy Sets Syst. 1992, 48, 239–248. [Google Scholar] [CrossRef]
9. Narayanan, A.; Vijayabalaji, S. Fuzzy n-normed linear space. Int. J. Math. Math. Sci. 2005, 2005, 3963–3977. [Google Scholar] [CrossRef]
10. Saadati, R.; Cho, Y.J.; Vaezpour, S.M. A note to paper “On the stability of cubic mappings and quartic mappings in random normed spaces”. J. Inequal. Appl. 2009, 2008, 1–6. [Google Scholar] [CrossRef]
11. Mao, W.; Hu, L.; You, S.; Mao, X. The averaging method for multivalued SDEs with jumps and non-Lipschitz coefficients. Discret. Contin. Dyn. Syst.-B 2019, 24, 4937–4954. [Google Scholar] [CrossRef]
12. O’Regan, D. A coincidence continuation theory between multi-valued maps with continuous selections and compact admissible maps. J. Nonlinear Sci. Appl. 2021, 14, 118–123. [Google Scholar] [CrossRef]
13. Mardones-Perez, I.; de Prada Vicente, M.A. An application of a representation theorem for fuzzy metrics to domain theory. Fuzzy Sets Syst. 2016, 300, 72–83. [Google Scholar] [CrossRef]
14. Mardones-Perez, I.; de Prada Vicente, M.A. Fuzzy pseudometric spaces vs. fuzzifying structures. Fuzzy Sets Syst. 2015, 267, 117–132. [Google Scholar] [CrossRef]
15. Lee, K.Y. Approximation properties in fuzzy normed spaces. Fuzzy Sets Syst. 2016, 282, 115–130. [Google Scholar] [CrossRef]
16. Jang, J.S.R.; Sun, C.T.; Mizutani, E. Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence; Pearson India: Chennai, India, 2015. [Google Scholar]
17. Debnath, P.; Mohiuddine, S.A. Soft Computing Techniques in Engineering, Health, Mathematical and Social Sciences; CRC Press: Boca Raton, FL, USA, 2021. [Google Scholar]
18. Deschrijver, G.; O’Regan, D.; Saadati, R.; Vaezpour, S.M. $ℒ$–fuzzy Euclidean normed spaces and compactness. Chaos Solitons Fractals 2009, 42, 40–45. [Google Scholar] [CrossRef]
19. Phong, T.T.; Tri, V.V. The continuity of solution set of a multivalued equation and applications in control problem. Adv. Theory Nonlinear Anal. Appl. 2021, 5, 330–336. [Google Scholar]
20. Hu, S.; Papageorgiou, N.S. Handbook of Multivalued Analysis; Kluwer: Dordrecht, The Netherlands, 1997; Volume I. [Google Scholar]
21. Webb, J.R.L.; Lan, K.Q. Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type. Topol. Methods Nonlinear Anal. 2006, 27, 91–115. [Google Scholar]
 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Share and Cite

MDPI and ACS Style

Saadati, R.; Allahviranloo, T.; O’Regan, D.; Alshammari, F.S. Fuzzy Control Problem via Random Multi-Valued Equations in Symmetric F-n-NLS. Symmetry 2022, 14, 1778. https://doi.org/10.3390/sym14091778

AMA Style

Saadati R, Allahviranloo T, O’Regan D, Alshammari FS. Fuzzy Control Problem via Random Multi-Valued Equations in Symmetric F-n-NLS. Symmetry. 2022; 14(9):1778. https://doi.org/10.3390/sym14091778

Chicago/Turabian Style

Saadati, Reza, Tofigh Allahviranloo, Donal O’Regan, and Fehaid Salem Alshammari. 2022. "Fuzzy Control Problem via Random Multi-Valued Equations in Symmetric F-n-NLS" Symmetry 14, no. 9: 1778. https://doi.org/10.3390/sym14091778

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.