Fuzzy Mittag–Leffler–Hyers–Ulam–Rassias Stability of Stochastic Differential Equations

Abstract

Stability is the most relevant property of dynamical systems. The stability of stochastic differential equations is a challenging and still open problem. In this article, using a fuzzy Mittag–Leffler function, we introduce a new fuzzy controller function to stabilize the stochastic differential equation (SDE) ${\nu }^{{}^{\prime }}(\gamma ,\mu )=F\left(\gamma ,\mu ,\nu (\gamma ,\mu )\right)$. By adopting the fixed point technique, we are able to prove the fuzzy Mittag–Leffler–Hyers–Ulam–Rassias stability of the SDE.

1. Introduction and Mathematical Preliminaries

Morsi [1] used the concepts of Minkowski functionals of L-fuzzy sets and fuzzy metric space to introduce the notion of fuzzy (pseudo) normed spaces. Subsequently, Jäger and Shi [2], using random normed spaces, introduced the fuzzy normed spaces. In the last years, the fuzzy functional analysis and its applications, especially the Hyers–Ulam–Rassias stability [3,4,5] in fuzzy normed spaces, was widely investigated by several authors [6,7]. Furthermore, several fixed-point ($FP$) results were obtained, with applications to nonlinear functional analysis. To learn more about applications of $FP$ theory, please see references [8,9,10].

Stability is crucial in any dynamical systems. Specifically, the stability of stochastic differential equations is a challenging and still open problem. In this paper, we consider the stochastic differential equation ($SDE$) of the form:

$$\begin{array}{c}\hfill {\nu }^{{}^{\prime }}(\gamma ,\mu )=F\left(\gamma ,\mu ,\nu (\gamma ,\mu )\right).\end{array}$$

(1)

Using a new fuzzy controller function, constructed based on the fuzzy Mittag–Leffler ($FML$) function, we are able to stabilize the pseudo $SDE$ (1). Additionally, by adopting the $FP$ technique [11,12,13] we prove the fuzzy Mittag–Leffler–Hyers–Ulam–Rassias ($MLHUS$) stability of the $SDE$ [14,15]. Our findings extend and improve some existing results [16,17] by using a new fuzzy controller function that allows studying the $MLHUS$ stability of SDEs in fuzzy normed spaces, and by using the alternative of $FP$-theorem [18,19].

In the subsequent analysis, for simplicity, we use the notions: $\Pi =(0,1)$, $J=(0,1]$, $\Omega =[0,\infty ]$ and $\Delta =(0,\infty )$.

Definition 1

([9,20,21]). Consider that S is a linear space and that η represents a fuzzy set from $S\times \Delta$ to J. Then, the ordered pair $(S,\eta )$ is a fuzzy normed ($FN$) space whenever:

(FN1) $\eta (\zeta ,\tau )=1$, $\forall \tau \in \Delta$ iff $\zeta =0$;

(FN2) $\eta (a\zeta ,\tau )=\eta \left(\zeta ,\frac{\tau }{\left|a\right|}\right)$, $\forall \zeta \in S$, $\forall a\in \mathbb{R}\backslash \left\{0\right\}$;

(FN3) $\eta (\xi +\zeta ,\tau +\varsigma )\ge \bigwedge \left(\eta \right(\xi ,\tau ),\eta (\zeta ,\varsigma \left)\right)$, $\forall \xi ,\zeta \in S$, $\forall \tau ,\varsigma \in \Delta$;

(FN4) $\eta (\zeta ,.):\Delta \to J$ is continuous.

A complete $FN$ space is denoted by $FB$ space.

Consider that $(S,\parallel .\parallel )$ is a linear normed space. If for all $\varsigma \in \Delta$

$$\eta (\zeta ,\varsigma )=exp\left(-\frac{\parallel \zeta \parallel }{\varsigma }\right),$$

then $(S,\eta )$ is a $FN$-space.

Consider that $(ϝ,\Sigma ,\xi )$ is a probability measure space. Assume that $(T,{\mathfrak{B}}_T)$ and $(S,{\mathfrak{B}}_S)$ are Borel measurable spaces, in which T and S are $FB$ spaces. A mapping $F:ϝ\times T\to S$ is called a random operator ($RO$) if $\{\gamma :F(\gamma ,\xi )\in B\}\in \Sigma$ for all $\xi$ in T and $B\in {\mathfrak{B}}_S$. In addition, F is $RO$ if $F(\gamma ,\xi )=\zeta \left(\gamma \right)$ is a S-valued random variable for every $\xi$ in T. A $RO$ $F:ϝ\times T\to S$ is called linear if $F(\gamma ,a{\xi }_1+b{\xi }_2)=aF(\gamma ,{\xi }_1)+bF(\gamma ,{\xi }_2)$ almost everywhere for each ${\xi }_1,{\xi }_2$ in T and $a,b$ are scalars, and bounded if there exists a non-negative real-valued random variable $M\left(\gamma \right)$ such that

$$\eta (F(\gamma ,{\xi }_1)-F(\gamma ,{\xi }_2),M\left(\gamma \right)\tau )\ge \eta ({\xi }_1-{\xi }_2,\tau ),$$

almost everywhere for each ${\xi }_1,{\xi }_2$ in T, $\tau \in \Delta$ and $\gamma \in ϝ$.

In this work, we present the $FP$ technique, which is the second most popular tool for proving the stability of functional equations [22,23].

Theorem 1

([10]). (The alternative of FP). Assume that $(T,\rho )$ is a complete generalized metric space and that $\Lambda :T\to T$ is a strictly contractive function with the Lipschitz constant $\iota <1$. Then, for every $\xi \in T$, either

$$\rho ({\Lambda }^n\xi ,{\Lambda }^{n+1}\xi )=\infty ,$$

for each $n\in \mathbb{N}$, or there is a $n_0\in \mathbb{N}$ for which:

(i) $\rho ({\Lambda }^n\xi ,{\Lambda }^{n+1}\xi )<\infty ,\forall n\ge n_0$;

(ii) the FP ${\xi }^{*}$ of $\Lambda$ is the convergent point of the sequence $\left\{{\Lambda }^n\xi \right\}$;

(iii) in the set $V=\left\{\zeta \in T\mid \rho ({\Lambda }^{n_0}\xi ,\zeta )<\infty \right\}$, ${\xi }^{*}$ is the unique FP of $\Lambda$;

(iv) $(1-\iota )\rho (\zeta ,{\xi }^{*})\le \rho (\zeta ,\Lambda \zeta )$ for every $\zeta \in V$.

Definition 2

([24]). The Mittag–Leffler function is given by the series:

$$\begin{array}{c}\hfill E_q\left(\mu \right)=\sum _{k=0}^{\infty }\frac{{\mu }^k}{\Gamma (qk+1)},\end{array}$$

where $q\in \mathbb{C}$, $Re\left(q\right)>0$ and $\Gamma \left(\mu \right)$ is a gamma function:

$$\begin{array}{c}\hfill \Gamma \left(\mu \right)={\int }_0^{\infty }{e}^{-t}{t}^{\mu -1}dt,\end{array}$$

with $Re\left(\mu \right)>0$. In particular, if $q=1$, we get:

$$\begin{array}{c}\hfill E_1\left(\mu \right)=\sum _{j=0}^{\infty }\frac{{\mu }^j}{\Gamma (j+1)}=\sum _{j=0}^{\infty }\frac{{\mu }^j}{j!}=e^{\mu }.\end{array}$$

Using Definition 2, we introduce the $FML$ function as:

$$\begin{array}{c}\hfill E_q(\mu ,\tau )=\frac{\tau }{\tau +E_q\left(\mu \right)},\forall \tau >0.\end{array}$$

2. Fuzzy MLHUS Stability

The norm $L^1(ϝ\times \Xi ,S)$ is written $\eta {(.,\tau )}_{L^1\left(ϝ\times \Xi \right)}$. We prove the fuzzy $MLHUS$ stability for the $SDE$ ${\nu }^{{}^{\prime }}(\gamma ,\mu )=F\left(\gamma ,\mu ,\nu (\gamma ,\mu )\right)$.

Theorem 2.

Consider that $c\in \mathbb{R}$, $r>0$,

$$\begin{array}{c}\hfill \Xi =\left\{\mu \in \mathbb{R}\mid |\mu -c|\le r\right\},\end{array}$$

and $F:ϝ\times \Xi \times \mathbb{R}\to \mathbb{R}$ is a continuous $RO$ which satisfies a Lipschitz condition:

$$\begin{array}{c}\hfill \eta (F(\gamma ,\mu ,\nu (\gamma ,\mu ))-F(\gamma ,\mu ,\omega (\gamma ,\mu )),\tau )\ge \eta \left(\nu -\omega ,\frac{\tau }{L}\right),\end{array}$$

(2)

for any $\mu \in \Xi$, $\tau \in \Delta$, $\gamma \in ϝ$ and $\nu ,\omega \in \mathbb{R}$, where L is a constant with $rL\in \Pi$. If a continuously differentiable operator $\nu :ϝ\times \Xi \to \mathbb{R}$ satisfies the differential inequality:

$$\begin{array}{c}\hfill \eta \left({\int }_c^{\mu }[F(\gamma ,\xi ,\nu (\gamma ,\xi ))-{\nu }^{{}^{\prime }}(\gamma ,\xi )]d\xi ,\tau \right)\ge E_q(\mu ,\tau ),\end{array}$$

(3)

for any $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$, where $E_q$ is a $FML$ function,

$$\begin{array}{c}\hfill \underset{\xi \in \Xi }{inf}{E}_q(\xi ,\frac{\tau }{2r})\ge E_q\left(\mu ,\frac{\tau }{r}\right),\end{array}$$

(4)

for any $\mu \in \Xi$, $\tau \in \Delta$, $\gamma \in ϝ$, then there exists a unique continuous $RO$ ${\nu }_0:ϝ\times \Xi \to \mathbb{R}$ such that

$$\begin{array}{c}\hfill {\nu }_0(\gamma ,\mu )=\nu (\gamma ,c)+{\int }_c^{\mu }F(\gamma ,\xi ,{\nu }_0(\gamma ,\xi ))d\xi .\end{array}$$

Furthermore, ${\nu }_0$ is a solution of (1) and

$$\begin{array}{c}\hfill \eta (\nu (\gamma ,\mu )-{\nu }_0(\gamma ,\mu ),\tau )\ge E_q(\mu ,(1-rL)\tau ),\end{array}$$

(5)

for any $\mu \in \Xi$, $\tau \in \Delta$, $\gamma \in ϝ$.

Proof. 

Consider the space of continuous ROs

$$\begin{array}{c}\hfill Y=\left\{\alpha :ϝ\times \Xi \to \mathbb{R}\mid \alpha \mathrm{is}\mathrm{a}\mathrm{continuous}RO\right\}.\end{array}$$

(6)

Introduce the below function on $Y^2$ as,

$$\begin{array}{ccc}\hfill & & \rho (\alpha ,\beta )\hfill \\ & & =inf\left\{\lambda \in \Delta \mid \eta \left(\alpha (\gamma ,\mu )-\beta (\gamma ,\mu ),\tau \right)\ge E_q\left(\mu ,\frac{\tau }{\lambda }\right),\forall \mu \in \Xi ,\tau \in \Delta ,\gamma \in ϝ\right\}.\hfill \end{array}$$

(7)

Mihet and Radu [25] proved that $(Y,\rho )$ is a complete generalized metric (see also [26]).

We introduce the $RO$ $\Lambda :Y\to Y$ by:

$$\begin{array}{c}\hfill (\Lambda \alpha )(\gamma ,\mu )=\nu (\gamma ,c)+{\int }_c^{\mu }F\left(\gamma ,\xi ,\alpha (\gamma ,\xi )\right)d\xi ,\end{array}$$

(8)

for every $\alpha \in Y$, $\gamma \in ϝ$ and $\mu \in \Xi$. The continuity of $RO$ $\alpha$ implies the continuity of $\Lambda \alpha$ and well-defined $\Lambda$.

Consider $\alpha ,\beta \in Y$ and $\gamma \in ϝ$. Additionally, consider ${\lambda }_{\alpha ,\beta }\in \Delta$ such that:

$$\begin{array}{c}\hfill \eta \left(\alpha (\gamma ,\mu )-\beta (\gamma ,\mu ),\tau \right)\ge E_q\left(\mu ,\frac{\tau }{{\lambda }_{\alpha \beta }}\right).\end{array}$$

(9)

Assume that $c={\varpi }_1<{\varpi }_2<\cdots <{\varpi }_k=\mu$, $\Delta {\mu }_i={\varpi }_i-{\varpi }_{i-1}=\frac{|\mu -c|}{k}$, $i=1,2,\cdots ,k$, and $\parallel \Delta \mu \parallel ={max}_{1\le i\le k}\left(\Delta {\mu }_i\right)$, for every $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$. Utilizing (2), (4), (8), and (9) we have the following

$$\begin{array}{c}\hfill \begin{array}{cc}& \eta \left((\Lambda \alpha )(\gamma ,\mu )-(\Lambda \beta )(\gamma ,\mu ),\tau \right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(8\right)\hfill \\ & =\eta \left({\int }_c^{\mu }\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right)d\mu ,\tau \right)\hfill \\ & \mathrm{by}\mathrm{integral}\mathrm{definition}\hfill \\ & =\eta \left(\underset{\parallel \Delta \mu \parallel \to 0}{lim}\sum _{i=1}^k\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right)\Delta {\mu }_i,\tau \right)\hfill \\ & \mathrm{by}\mathrm{continuity}\mathrm{property}\mathrm{of}\eta \hfill \\ & =\underset{\parallel \Delta \mu \parallel \to 0}{lim}\eta \left(\sum _{i=1}^k\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right)\Delta {\mu }_i,\tau \right)\hfill \\ & \mathrm{by}\mathrm{triangular}\mathrm{inequality}\hfill \\ & \ge \underset{\parallel \Delta \mu \parallel \to 0}{lim}\bigwedge \eta \left(\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right)\Delta {\mu }_i,\frac{\tau }{k}\right)\hfill \\ & \mathrm{by}\mathrm{property}\mathrm{of}\mathrm{infimum}\hfill \\ & \ge \underset{\xi \in \Xi }{inf}\eta \left(\left(F(\gamma ,\xi ,\alpha (\gamma ,\xi \left)\right)-F(\gamma ,\xi ,\beta (\gamma ,\xi \left)\right)\right),\frac{\tau }{k\Delta {\mu }_i}\right)\hfill \\ & \ge \underset{\xi \in \Xi }{inf}\eta \left(\left(F(\gamma ,\xi ,\alpha (\gamma ,\xi \left)\right)-F(\gamma ,\xi ,\beta (\gamma ,\xi \left)\right)\right),\frac{\tau }{k\parallel \Delta \mu \parallel }\right)\hfill \\ & \ge \underset{\xi \in \Xi }{inf}\eta \left(\left(F(\gamma ,\xi ,\alpha (\gamma ,\xi \left)\right)-F(\gamma ,\xi ,\beta (\gamma ,\xi \left)\right)\right),\frac{k\tau }{k|\mu -c|}\right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(2\right)\hfill \\ & \ge \underset{\xi \in \Xi }{inf}\eta \left(\alpha (\gamma ,\xi )-\beta (\gamma ,\xi ),\frac{\tau }{\left(2rL\right)}\right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(9\right)\hfill \\ & \ge \underset{\xi \in \Xi }{inf}{E}_q\left(\xi ,\frac{\tau }{\left(2rL\right){\lambda }_{\alpha ,\beta }}\right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(4\right)\hfill \\ & \ge E_q\left(\mu ,\frac{\tau }{\left(rL\right){\lambda }_{\alpha ,\beta }}\right),\hfill \end{array}\end{array}$$

for every $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$, that is, $\rho (\Lambda \alpha ,\Lambda \beta )\le \left(rL\right){\lambda }_{\alpha ,\beta }$. Therefore, we can conclude that $\rho (\Lambda \alpha ,\Lambda \beta )\le \left(rL\right)\rho (\alpha ,\beta )$ for any $\alpha ,\beta \in Y$, in which $\left(rL\right)\in \Pi$. Therefore, $\Lambda$ is a strictly contraction mapping.

By (3), (5), (8), and $\nu \in Y$, we obtain:

$$\begin{array}{c}\hfill \begin{array}{cc}& \eta \left((\Lambda \nu )(\gamma ,\mu )-\nu (\gamma ,\mu ),\tau \right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(8\right)\hfill \\ & =\eta \left(\nu (\gamma ,c)+{\int }_c^{\mu }F\left(\gamma ,\xi ,\nu (\gamma ,\xi )\right)d\xi -\nu (\gamma ,\mu ),\tau \right)\hfill \\ & \mathrm{by}\mathrm{property}\mathrm{of}\mathrm{integral}\hfill \\ & =\eta \left({\int }_c^{\mu }[F\left(\gamma ,\xi ,\nu (\gamma ,\xi )\right)-{\nu }^{\prime }(\gamma ,\xi )]d\xi ,\tau \right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(3\right)\hfill \\ & \ge E_q\left(\mu ,\frac{\tau }{1}\right),\hfill \end{array}\end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$. Thus, (7) implies that

$$\begin{array}{c}\hfill \rho (\Lambda \nu ,\nu )<1,\end{array}$$

(10)

and hence,

$$\begin{array}{c}\hfill \rho ({\Lambda }^{n+1}\nu ,{\Lambda }^n\nu )<1<\infty .\end{array}$$

Now, Theorem 1 implies that:

(i) there is a continuous $RO$ ${\nu }_0:ϝ\times \Xi \to \mathbb{R}$ where $\Lambda {\nu }_0={\nu }_0$, that is, ${\nu }_0$ is $FP$ of $\Lambda$, which is uniqu in the set

$$V=\{\alpha \in Y:\rho (\alpha ,\nu )<\infty \}.$$

(ii) ${\Lambda }^n\nu \to {\nu }_0$ in $(Y,\rho )$ as $n\to \infty$.

(iii) using (10) we obtain:

$$\begin{array}{c}\hfill \rho (\nu ,{\nu }_0)\le \frac{1}{1-rL}\rho (\Lambda \nu ,\nu )\le \frac{1}{1-rL},\end{array}$$

which implies the validity of (5) for each $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$. □

Consider that $(Y,\eta )$ is a $FN$ space. We introduce the fuzzy set ${\eta }_B$ as:

$$\begin{array}{c}\hfill {\eta }_B\left(\alpha (\gamma ,\xi ),\tau \right):=\underset{\xi \in \Xi }{inf}\left\{\eta \left(\alpha (\gamma ,\xi ),\frac{\tau }{e^{\theta \xi }}\right):\theta \in \Delta ,\Xi \subset {\mathbb{R}}_{+}\right\},\end{array}$$

for every $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$. Then, $(Y,{\eta }_B)$ is a $FN$ space (Bielecki $FN$ space). In fact, (FN1), (FN2), and (FN4) are obvious. Now, we prove only (FN3). Observe that:

$$\begin{array}{ccc}& & \bigwedge \left({\eta }_B(\alpha (\gamma ,\xi ),\tau ),{\eta }_B(\beta (\gamma ,\xi ),\varsigma )\right)\hfill \\ & =& \bigwedge \left(\underset{\xi \in \Xi }{inf}\left\{\eta \left(\alpha (\gamma ,\xi ),\frac{\tau }{e^{\theta \xi }}\right)\right\},\underset{\xi \in \Xi }{inf}\left\{\eta \left(\beta (\gamma ,\xi ),\frac{\varsigma }{e^{\theta \xi }}\right)\right\}\right)\hfill \\ & \le & \bigwedge \left(\underset{\xi \in \Xi }{inf}\left\{\eta \left(\alpha (\gamma ,\xi ),\frac{\tau }{e^{\theta \xi }}\right),\eta \left(\beta (\gamma ,\xi ),\frac{\varsigma }{e^{\theta \xi }}\right)\right\}\right)\hfill \\ & \le & \underset{\xi \in \Xi }{inf}\left\{\eta \left((\alpha +\beta )(\gamma ,\xi ),\frac{(\tau +\varsigma )}{e^{\theta \xi }}\right)\right\}\hfill \\ & =& \underset{\xi \in \Xi }{inf}\left\{\eta \left((\alpha +\beta )(\gamma ,\xi ),\frac{(\tau +\varsigma )}{e^{\theta \xi }}\right)\right\}\hfill \\ & =& {\eta }_B((\alpha +\beta )(\gamma ,\xi ),(\tau +\varsigma )),\hfill \end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$, which proves the triangle inequality (FN3).

Now, we prove the fuzzy $MLHUS$ stability of the random Equation (1) via the Bielecki fuzzy norm.

Theorem 3.

Assume that $c\in \mathbb{R}$, $r>0$ and $\Xi =\left\{\mu \in \mathbb{R}\mid |\mu -c|\le r\right\}$. Consider that $F:ϝ\times \Xi \times \mathbb{R}\to \mathbb{R}$ is a continuous $RO$ which satisfies in the Lipschitz condition:

$$\begin{array}{c}\hfill \eta \left(F(\gamma ,\mu ,\nu (\gamma ,\mu \left)\right)-F(\gamma ,\mu ,\omega (\gamma ,\mu \left)\right),\tau \right)\ge \eta \left(\nu -\omega ,\frac{\tau }{L}\right),\end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$, $\gamma \in ϝ$ and where L is a constant with $rL\in \Pi$. If a continuously differentiable function $\nu :ϝ\times \Xi \to \mathbb{R}$ satisfies the differential inequality:

$$\begin{array}{c}\hfill \eta \left({\int }_c^{\mu }[F(\gamma ,\xi ,\nu (\gamma ,\xi ))-{\nu }^{{}^{\prime }}(\gamma ,\xi )]d\xi ,\tau \right)\ge E_q(\mu ,\tau ),\end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$, where $E_q$ is a $FML$ function; then, with the Bielecki fuzzy norm, the fuzzy $MLHUS$ stability is verified for the Equation (1).

Proof. 

By the same method used in the proof of Theorem 2, we assume that $c={\varpi }_1<{\varpi }_2<\cdots <{\varpi }_k=\mu$, $\Delta {\mu }_i={\varpi }_i-{\varpi }_{i-1}=\frac{|\mu -c|}{k}$, $i=1,2,\cdots ,k$ and $\parallel \Delta \mu \parallel ={max}_{1\le i\le k}\left(\Delta {\mu }_i\right)$. Now, we show the contraction of $\Lambda$ on $Y$ with respect to the Bielecki fuzzy norm introduced in (6):

$$\begin{array}{c}\hfill \begin{array}{cc}& \eta \left((\Lambda \alpha )(\gamma ,\mu )-(\Lambda \beta )(\gamma ,\mu ),\tau \right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(8\right)\hfill \\ & =\eta \left({\int }_c^{\mu }\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right)d\mu ,\tau \right)\hfill \\ & \mathrm{by}\mathrm{integral}\mathrm{definition}\hfill \\ & =\eta \left(\underset{\parallel \Delta \mu \parallel \to 0}{lim}\sum _{i=1}^k\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right)\Delta {\mu }_i,\tau \right)\hfill \\ & \mathrm{by}\mathrm{continuity}\mathrm{property}\mathrm{of}\eta \hfill \\ & =\underset{\parallel \Delta \mu \parallel \to 0}{lim}\eta \left(\sum _{i=1}^k\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right)\Delta {\mu }_i,\tau \right)\hfill \\ & \mathrm{by}\mathrm{triangular}\mathrm{inequality}\hfill \\ & \ge \underset{\parallel \Delta \mu \parallel \to 0}{lim}\bigwedge \eta \left(\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right)\Delta {\mu }_i,\frac{\tau }{k}\right)\hfill \\ & \mathrm{by}\mathrm{property}\mathrm{of}\mathrm{infimum}\hfill \\ & \ge \underset{\xi \in \Xi }{inf}\eta \left(\left(F(\gamma ,\xi ,\alpha (\gamma ,\xi \left)\right)-F(\gamma ,\xi ,\beta (\gamma ,\xi \left)\right)\right),\frac{\tau }{k\Delta {\mu }_i}\right)\hfill \\ & \ge \underset{\xi \in \Xi }{inf}\eta \left(\left(F(\gamma ,\xi ,\alpha (\gamma ,\xi \left)\right)-F(\gamma ,\xi ,\beta (\gamma ,\xi \left)\right)\right),\frac{\tau }{k\parallel \Delta \mu \parallel }\right)\hfill \\ & \ge \underset{\xi \in \Xi }{inf}\eta \left(\left(F(\gamma ,\xi ,\alpha (\gamma ,\xi \left)\right)-F(\gamma ,\xi ,\beta (\gamma ,\xi \left)\right)\right),\frac{k\tau }{k|\mu -c|}\right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(2\right)\hfill \\ & \ge \underset{\xi \in \Xi }{inf}\eta \left(\alpha (\gamma ,\xi )-\beta (\gamma ,\xi ),\frac{\tau }{\left(rL\right)}\right)\hfill \\ & \ge \underset{\xi \in \Xi }{inf}\eta \left(\alpha (\gamma ,\xi )-\beta (\gamma ,\xi ),\frac{\tau }{\left(rL\right)e^{\theta \xi }}\right)\hfill \\ & \mathrm{by}\mathrm{definition}{\eta }_B\hfill \\ & \ge {\eta }_B\left(\alpha -\beta ,\frac{\tau }{\left(rL\right)}\right).\hfill \end{array}\end{array}$$

then,

$$\begin{array}{c}\hfill \eta \left((\Lambda \alpha )(\gamma ,\mu )-(\Lambda \beta )(\gamma ,\mu ),\tau \right)\ge {\eta }_B\left(\alpha -\beta ,\frac{\tau }{\left(rL\right)}\right),\end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$, $\gamma \in ϝ$, that is, $\rho (\Lambda \alpha ,\Lambda \beta )\le {\eta }_B\left(\alpha -\beta ,\frac{\tau }{\left(rL\right)}\right)$. Hence, we can conclude that $\rho (\Lambda \alpha ,\Lambda \beta )\le \left(rL\right)\rho (\alpha ,\beta )$ for any $\alpha ,\beta \in Y$. By letting $\left(rL\right)\in \Pi$, we obtain the strict continuity. Furthermore, by Theorem 1, we obtain:

$$\begin{array}{c}\hfill \rho (\nu ,{\nu }_0)\le \frac{1}{1-rL}\rho (\Lambda \nu ,\nu )\le \frac{1}{1-rL},\end{array}$$

so, the fuzzy $MLHUS$ stability of Equation (1) is verified. □

Theorem 4.

Suppose that a and b are real numbers such that $a<b$. Let $\Xi =[a,b]$ and $c\in \Xi$. Assume that K and L are positive constants such that $LK\in \Pi$. Consider that $F:ϝ\times \Xi \times \mathbb{R}\to \mathbb{R}$ is a continuous $RO$ which satisfies a Lipschitz condition:

$$\begin{array}{c}\hfill \eta \left(F(\gamma ,\mu ,\nu )-F(\gamma ,\mu ,\omega ),\tau \right)\ge \eta \left(\nu -\omega ,\frac{\tau }{L}\right),\end{array}$$

(11)

for any $\mu \in \Xi$, $\tau \in \Delta$, $\gamma \in ϝ$ and $\nu ,\omega \in \mathbb{R}$. If a continuously differentiable operator $\nu :ϝ\times \Xi \to \mathbb{R}$ satisfies the differential inequality:

$$\begin{array}{c}\hfill \eta \left({\int }_0^{\mu }[F(\gamma ,\xi ,\nu (\gamma ,\xi ))-{\nu }^{{}^{\prime }}(\gamma ,\xi )]d\xi ,\tau \right)\ge E_q(\mu ,\tau ),\end{array}$$

(12)

for any $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$, where $E_q$ is a $FML$ function,

$$\begin{array}{c}\hfill \underset{\xi \in \Xi }{inf}{E}_q(\xi ,\frac{\tau }{(b-a)})\ge E_q\left(\mu ,\frac{\tau }{K}\right),\end{array}$$

(13)

for any $\mu \in \Xi$, $\tau \in \Delta$, $\gamma \in ϝ$, then there exists a unique continuous $RO$ ${\nu }_0:ϝ\times \Xi \to \mathbb{R}$ such that:

$$\begin{array}{c}\hfill {\nu }_0(\gamma ,\mu )=\nu (\gamma ,c)+{\int }_0^{\mu }F\left(\gamma ,\xi ,{\nu }_0(\gamma ,\xi )\right)d\xi .\end{array}$$

Furthermore, ${\nu }_0$ is a solution of (1) and

$$\begin{array}{c}\hfill \eta \left(\nu (\gamma ,\mu )-{\nu }_0(\gamma ,\mu ),\tau \right)\ge E_q(\mu ,(1-rL)\tau ),\end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$, $\gamma \in ϝ$.

Proof. 

Consider the space of continuous ROs:

$$\begin{array}{c}\hfill Y=\left\{\alpha :ϝ\times \Xi \to \mathbb{R}\mid \alpha \mathrm{is}\mathrm{a}\mathrm{continuous}RO\right\}.\end{array}$$

Introduce the below function on $Y^2$ as,

$$\begin{array}{c}\hfill \rho (\alpha ,\beta )=inf\left\{\lambda \in \Delta \mid \eta \left(\alpha (\gamma ,\mu )-\beta (\gamma ,\mu ),\tau \right)\ge E_q\left(\mu ,\frac{\tau }{\lambda }\right),\forall \mu \in \Xi ,\tau \in \Delta ,\gamma \in ϝ\right\}.\end{array}$$

(14)

Further, introduce the $RO$ $\Lambda :Y\to Y$ by:

$$\begin{array}{c}\hfill (\Lambda \alpha )(\gamma ,\mu )=\nu (\gamma ,c)+{\int }_0^{\mu }F\left(\mu ,\xi ,\alpha (\gamma ,\xi )\right)d\xi ,\end{array}$$

(15)

for every $\alpha \in Y$, $\gamma \in ϝ$ and $\mu \in \Xi$. The continuity of $RO$ $\alpha$ implies the continuity of $\Lambda \alpha$ and well defining $\Lambda$.

Consider $\alpha ,\beta \in Y$ and $\gamma \in ϝ$. Let ${\lambda }_{\alpha ,\beta }\in \Delta$ such that:

$$\begin{array}{c}\hfill \eta \left(\alpha (\gamma ,\mu )-\beta (\gamma ,\mu ),\tau \right)\ge E_q\left(\mu ,\frac{\tau }{{\lambda }_{\alpha ,\beta }}\right).\end{array}$$

(16)

In addition, let $a={\varpi }_1<{\varpi }_2<\cdots <{\varpi }_k=b$, $\Delta {\mu }_i={\varpi }_i-{\varpi }_{i-1}=\frac{b-a}{k}$, $i=1,2,\cdots ,k$ and $\parallel \Delta \mu \parallel ={max}_{1\le i\le k}\left(\Delta {\mu }_i\right)$, for every $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$. Utilizing (11), (13), (14) and (15) we have the following

$$\begin{array}{c}\hfill \begin{array}{cc}& \eta \left((\Lambda \alpha )(\gamma ,\mu )-(\Lambda \beta )(\gamma ,\mu ),\tau \right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(15\right)\hfill \\ & =\eta \left({\int }_0^{\mu }\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right)d\mu ,\tau \right)\hfill \\ & \mathrm{by}\mathrm{integral}\mathrm{definition}\hfill \\ & =\eta \left(\underset{\parallel \Delta \mu \parallel \to 0}{lim}\sum _{i=1}^k\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right)\Delta {\mu }_i,\tau \right)\hfill \\ & \mathrm{by}\mathrm{continuity}\mathrm{property}\mathrm{of}\eta \hfill \\ & =\underset{\parallel \Delta \mu \parallel \to 0}{lim}\eta \left(\sum _{i=1}^k\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right)\Delta {\mu }_i,\tau \right)\hfill \\ & \mathrm{by}\mathrm{triangular}\mathrm{inequality}\hfill \\ & \ge \underset{\parallel \Delta \mu \parallel \to 0}{lim}\bigwedge \eta \left(\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right)\Delta {\mu }_i,\frac{\tau }{k}\right)\hfill \\ & \mathrm{by}\mathrm{property}\mathrm{of}\mathrm{infimum}\hfill \\ & \ge \underset{\xi \in \Xi }{inf}\eta \left(\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right),\frac{\tau }{k\Delta {\mu }_i}\right)\hfill \\ & \ge \underset{\xi \in \Xi }{inf}\eta \left(\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right),\frac{\tau }{k\parallel \Delta \mu \parallel }\right)\hfill \\ & \ge \underset{\xi \in \Xi }{inf}\eta \left(\left(F(\gamma ,{\varpi }_i,\alpha (\gamma ,{\varpi }_i))-F(\gamma ,{\varpi }_i,\beta (\gamma ,{\varpi }_i))\right),\frac{k\tau }{k(b-a)}\right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(11\right)\hfill \\ & \ge \underset{\xi \in \Xi }{inf}\eta \left(\alpha (\gamma ,\xi )-\beta (\gamma ,\xi ),\frac{\tau }{L(b-a)}\right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(16\right)\hfill \\ & \ge \underset{\xi \in \Xi }{inf}{E}_q\left(\xi ,\frac{\tau }{L(b-a){\lambda }_{\alpha ,\beta }}\right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(13\right)\hfill \\ & \ge E_q\left(\mu ,\frac{\tau }{LK{\lambda }_{\alpha ,\beta }}\right),\hfill \end{array}\end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$, that is, $\rho (\Lambda \alpha ,\Lambda \beta )\le \left(LK\right){\lambda }_{\alpha ,\beta }$. Hence, we can conclude that $\rho (\Lambda \alpha ,\Lambda \beta )\le \left(LK\right)\rho (\alpha ,\beta )$ for all $\alpha ,\beta \in Y$, in which $\left(LK\right)\in \Pi$. Therefore, $\Lambda$ is a strict contraction mapping.

By (3), (5), (7) and $\nu \in Y$, we obtain:

$$\begin{array}{c}\hfill \begin{array}{cc}& \eta \left((\Lambda \nu )(\gamma ,\mu )-\nu (\gamma ,\mu ),\tau \right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(15\right)\hfill \\ & =\eta \left(\nu (\gamma ,c)+{\int }_c^{\mu }F\left(\gamma ,\xi ,\nu (\gamma ,\xi )\right)d\xi -\nu (\gamma ,\mu ),\tau \right)\hfill \\ & \mathrm{by}\mathrm{property}\mathrm{of}\mathrm{integral}\hfill \\ & =\eta \left({\int }_c^{\mu }[F\left(\gamma ,\xi ,\nu (\gamma ,\xi )\right)-{\nu }^{\prime }(\gamma ,\xi )]d\xi ,\tau \right)\hfill \\ & \mathrm{by}\mathrm{equality}\left(12\right)\hfill \\ & \ge E_q\left(\mu ,\frac{\tau }{1}\right),\hfill \end{array}\end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$. Thus, (7) implies that:

$$\begin{array}{c}\hfill \rho (\Lambda \nu ,\nu )<1,\end{array}$$

(17)

and hence,

$$\begin{array}{c}\hfill \rho ({\Lambda }^{n+1}\nu ,{\Lambda }^n\nu )<1<\infty .\end{array}$$

Now, Theorem 1 implies that:

(i) there is a continuous $RO$ ${\nu }_0:ϝ\times \Xi \to \mathbb{R}$ where $\Lambda {\nu }_0={\nu }_0$, that is, ${\nu }_0$ is $FP$ of $\Lambda$, which is unique in the set

$$V=\{\alpha \in Y:\rho (\alpha ,\nu )<\infty \}.$$

(ii) ${\Lambda }^n\nu \to {\nu }_0$ in $(Y,\rho )$ as $n\to \infty$.

(iii) using (17) we obtain:

$$\begin{array}{c}\hfill \rho (\nu ,{\nu }_0)\le \frac{1}{1-rL}\rho (\Lambda \nu ,\nu )\le \frac{1}{1-rL},\end{array}$$

which implies the validity of (5) for each $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$. □

3. Application

Example 1.

Consider positive real numbers K and L such that $LK\in \Pi$, $K<\frac{b(b-a)}{a}$. Assume that $\Xi =[a,b]$. For an arbitrary polynomial $p(\gamma ,\mu )$, we let a continuously differentiable $RO$ $\nu :ϝ\times \Xi \to \mathbb{R}$ to satisfy:

$$\begin{array}{c}\hfill \eta \left({\int }_0^{\mu }[F(\gamma ,\xi ,\nu (\gamma ,\xi ))-{\nu }^{{}^{\prime }}(\gamma ,\xi )]d\xi ,\tau \right)\ge E_q(\mu ,\tau ),\end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$. If we set $F(\gamma ,\mu ,\nu )=L\nu (\gamma ,\mu )+p(\gamma ,\mu )$, where $E_q$ is a $FML$ function,

$$\begin{array}{c}\hfill \underset{\xi \in \Xi }{inf}{E}_q(\xi ,\frac{\tau }{(b-a)})\ge E_q\left(\mu ,\frac{\tau }{K}\right),\end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$, then, by Theorem 4, there is a unique continuous $RO$ ${\nu }_0:ϝ\times \Xi \to \mathbb{R}$ such that:

$$\begin{array}{c}\hfill {\nu }_0(\gamma ,\mu )=\nu (\gamma ,0)+{\int }_0^{\mu }\left(L\nu (\gamma ,\xi )+p(\gamma ,\xi )\right)d\xi ,\end{array}$$

and

$$\begin{array}{c}\hfill \eta \left(\nu (\gamma ,\mu )-{\nu }_0(\gamma ,\mu ),\tau \right)\ge E_q\left(\mu ,(1-rL)\tau \right),\end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$.

Example 2.

Consider that r and L are positive constants with $rL\in \Pi$ and

$$\Xi =\left\{\mu \in \mathbb{R}\mid |\mu -c|\le r,\mathit{for}\mathit{some}c\in \mathbb{R}\right\}.$$

Let a continuous discrete random function $\nu :ϝ\times \Xi \to \mathbb{R}$ satisfy the following inequality:

$$\begin{array}{c}\hfill \eta \left({\int }_c^{\mu }[F(\gamma ,\xi ,\nu (\gamma ,\xi ))-{\nu }^{{}^{\prime }}(\gamma ,\xi )]d\xi ,\tau \right)\ge E_q(\mu ,\tau ),\end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$, where $p(\gamma ,\mu )$ is a polynomial. If we set $F(\gamma ,\mu ,\nu )=L\nu (\gamma ,\mu )+p(\gamma ,\mu )$, where $E_q$ is a $FML$ function,

$$\begin{array}{c}\hfill \underset{\xi \in \Xi }{inf}{E}_q(\xi ,\frac{\tau }{2r})\ge E_q\left(\mu ,\frac{\tau }{r}\right),\end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$, $\gamma \in ϝ$, then, by Theorem 2 there exists a unique random operator ${\nu }_0:ϝ\times \Xi \to \mathbb{R}$ such that:

$$\begin{array}{c}\hfill {\nu }_0(\gamma ,\mu )=\nu (\gamma ,0)+{\int }_0^{\mu }\left(L\nu (\gamma ,\xi )+p(\gamma ,\xi )\right)d\xi ,\end{array}$$

and

$$\begin{array}{c}\hfill \eta \left(\nu (\gamma ,\mu )-{\nu }_0(\gamma ,\mu ),\tau \right)\ge E_q\left(\mu ,(1-rL)\tau \right),\end{array}$$

for any $\mu \in \Xi$, $\tau \in \Delta$ and $\gamma \in ϝ$.

4. Conclusions

In this paper we introduced a new fuzzy controller function to stabilize the $SDE$ of the form ${\nu }^{{}^{\prime }}(\gamma ,\mu )=F\left(\gamma ,\mu ,\nu (\gamma ,\mu )\right)$. By adopting the $FP$ technique, we proved the fuzzy $MLHUS$ stability of the $SDE$. Some examples were given to illustrate the theoretical findings and to show the effectiveness of the method. Extension of the method to SDEs of different types will be further investigated.