1 Introduction

Fractional analysis is a generalization of classical integer-order differentiation and integration to arbitrary noninteger order. Sousa and Oliveira [1] have recently proposed a fractional differentiation operator, which they called the Ψ-Hilfer operator, unifying several different fractional operators. Stochastic fractional differential equations naturally arise in different fields such as biology, engineering, medicine, physics, and mathematics (for more applications and details, we refer to [2–12]).

We study the nonlinear Ψ-Hilfer stochastic fractional differential equation

$$ \left \{ \textstyle\begin{array}{l} ^{H}D_{0^{+}}^{\iota,\kappa;\varPsi}\mu(\varrho,\varsigma)\\\quad=\mathsf{A}\mu (\varrho,\varsigma)+\mathsf{B}\mu(\varrho,\varsigma-\boldsymbol {h}(\varrho,\varsigma))+\boldsymbol{f}(\varsigma,\mu(\varrho,\varsigma ),\mu(\varrho,\varsigma-\boldsymbol{h}(\varrho,\varsigma))), \\ \mu(\varrho,\varsigma)=\varTheta(\varrho,\varsigma),\quad \varsigma\in [-\mathit{h},0], \end{array}\displaystyle \right . $$
(1.1)

for \(\varsigma\in\varXi_{5}\) and \(\varrho\in\varUpsilon\), where \({}^{H}D_{0^{+}}^{\iota,\kappa;\varPsi}(\varrho,\cdot)\) is the Ψ-Hilfer stochastic fractional derivative operator of order \(\iota\in (0,1]\) for each \(\varrho\in\varUpsilon\) with respect to a random operator \(\varPsi\in \ell(\varUpsilon\times\varXi_{5},\mathbb {R})\) (see [1, 13]) and type \(0\leq\kappa\leq1\), \(\mu(\varrho,\varsigma)\in\mathbb{R}^{n}\), \(\boldsymbol{h}(\varrho,\varsigma)\) is a continuous map such that \(0\leq\boldsymbol{h}(\varrho,\varsigma)\leq\mathit{h}\), \(\varsigma\in \varXi_{5}=[0,\mathit{M}]\) with \(0<\mathit{M}<+\infty\), \(\varTheta(\varrho,\varsigma)\in\ell(\varUpsilon\times[-\mathit {h},0],\mathbb{R}^{n})\) is a given function, and \(\boldsymbol{f}:\varUpsilon\times\varXi_{5}\times\mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\), \(\mathsf{A}\in\mathbb{R}^{n\times n}\), and \(\mathsf{B}\in\mathbb {R}^{n\times n}\) are matrices. In this paper, we study the uniqueness of solutions for (1.1) and their Ulam–Hyers–Rassias stability with uncertainty.

2 Preliminaries

Let \(\varXi_{1}=[\mathbf{a},\mathbf{b}]\), \(\varXi_{2}=(\mathbf {a},\mathbf{b})\), \(\varXi_{3}=[-\mathit{h},0]\), \(\varXi_{4}=[-\mathit{h},\mathit{M}]\), \(\varXi_{5}=[0,\mathit{M}]\), \(\varXi_{6}=(0,1]\), \(\varXi_{7}=[0,\infty ]\), and \(\varXi_{8}=(0,\infty)\).

Definition 2.1

([14–18])

Suppose that S is a linear space and η is a fuzzy set from \(S\times\varXi_{8}\) to \(\varXi_{6}\). The ordered pair \((S,\eta)\) is said to be a fuzzy normed space (FN-space) whenever

  1. (FN1)

    \(\eta(\xi,\tau)=1\) for any \(\tau\in\varXi_{8}\) iff \(\xi=0\);

  2. (FN2)

    \(\eta(\mathbf{a} \xi,\tau)=\eta (\xi,\frac{\tau }{|\mathbf{a}|} )\) for all \(\xi\in S\), \(\tau\in\varXi_{8}\), and \(\mathbf{a} \in\mathbb{R}\) with \(\mathbf{a} \neq0\);

  3. (FN3)

    \(\eta(\xi+\zeta,\tau+\theta)\geq\min (\eta(\xi,\tau ),\eta(\zeta,\theta) )\) for all \(\xi,\zeta\in S\) and \(\tau,\theta\in\varXi_{8}\);

  4. (FN4)

    \(\eta(\xi,\cdot):\varXi_{8}\to\varXi_{6} \) is continuous.

Let \((S,\eta)\) be an FN-space. A sequence \(\lbrace\xi_{n} \rbrace\subset S\) is fuzzy convergent to \(\xi\in S\) in \((S,\eta)\) if for any \(\tau>0\) and \(0<\epsilon<1\), there exists a positive integer \(N_{\epsilon}\) such that \(\eta(\xi_{n} - \xi, \tau)>1-\epsilon\) for \(n \geq N_{\epsilon}\). A sequence \(\lbrace\xi_{n} \rbrace\subset S\) is fuzzy Cauchy in \((S,\eta)\) if for any \(\tau>0\) and \(0<\epsilon<1\), there exists a positive integer \(N_{\epsilon}\) such that \(\eta(\xi_{n} - \xi_{m} , \tau )>1-\epsilon\) for \(n,m \geq N_{\epsilon}\). An FN-space is Banach if every Cauchy sequence in it is convergent. A Banach FN-space is shortly called an FB-space. Consider the normed space \((S,\|\cdot\|)\). Then

$$\begin{aligned} \eta(\xi,\tau)=\exp \biggl(-\frac{ \Vert \xi \Vert }{\tau} \biggr) \end{aligned}$$

for \(\tau\in\varXi_{8}\) defines a fuzzy norm, and the ordered pair \((S,\eta)\) is an FN-space.

Consider the probability measure space \((\varUpsilon, \varXi_{8}, \xi)\) and let \((T,{\boldsymbol {B}}_{T} )\) and \((S,{\boldsymbol {B}}_{S} )\) be Borel measureable spaces, where T and S are FB-spaces. If \(\{\varrho: \mathcal{F}(\varrho,\xi)\in B\}\in\varXi _{8}\) for all \(\xi\in T\) and \(B\in {\boldsymbol {B}}_{S}\), we say that \(\mathcal{F}:\varUpsilon\times T\to S\) is a random operator. A random operator \(\mathcal{F}:\varUpsilon \times T\to S\) is said to be linear if \(\mathcal{F}(\varrho, \mathbf{a} \xi_{1}+\mathbf{b} \xi_{2})=\mathbf{a} \mathcal{F}(\varrho,\xi _{1})+ \mathbf{b} \mathcal{F}(\varrho, \xi_{2})\) almost everywhere for all \(\xi_{1},\upsilon_{2}\in T\) and scalars \(\mathbf{a},\mathbf{b}\), and bounded if there exists a nonnegative real-valued random variable \(M(\varrho)\) such that

$$\eta \bigl(\mathcal{F}(\varrho,\xi_{1})-\mathcal{F}(\varrho,\xi _{2}),M(\varrho)\tau \bigr)\ge\eta (\xi_{1}- \upsilon_{2},\tau ) $$

almost everywhere for all \(\xi_{1},\xi_{2}\in T\), \(\tau\in\varXi_{8}\), and \(\varrho\in\varUpsilon\).

The subject of approximation of functional equations in several spaces by direct techniques and fixed point techniques have been studied by some researchers, for instance, fuzzy Menger normed algebras [19], fuzzy metric spaces [20, 21], FN spaces [22], non-Archimedean random Lie \(C^{*}\)-algebras [23], and random multinormed space [24–29]. Some stability results for fractional differential and integral equations have been discussed in [26, 30–38].

Theorem 2.2

(The alternative of fixed point)

Let\((T, \rho)\)be a complete\(\varXi_{7}\)-valued metric space, and let\(\varLambda: T \rightarrow T\)be a strictly contractive function with Lipschitz constant\(\iota<1\). Then for every given element\(\xi \in T\), either

$$\begin{aligned} \rho \bigl(\varLambda^{n}\xi,\varLambda^{n+1}\xi \bigr) = \infty \end{aligned}$$

for each\(n\in\mathbb{N}\), or there is\(n_{0}\in\mathbb{N}\)such that

  1. (i)

    \(\rho (\varLambda^{n}\xi,\varLambda^{n+1}\xi )<\infty\)for all\(n\ge n_{0}\);

  2. (ii)

    the fixed point\(\zeta^{*}\)ofΛis the limit point of the sequence\(\{\varLambda^{n} \xi \}\);

  3. (iii)

    in the set\(V= \{\zeta\in T \mid\rho(\varLambda ^{n_{0}}\xi,\zeta)<\infty \}\), \(\zeta^{*}\)is the unique fixed point ofΛ;

  4. (iv)

    \((1-\iota)\rho (\zeta,\zeta^{\ast} ) \le\rho (\zeta,\varLambda\zeta)\)for every\(\zeta\in V\).

Definition 2.3

(One-parameter Mittag-Leffler function)

The Mittag-Leffler function is given by the series

$$\begin{aligned} \mathcal{E}_{\vartheta}(\varpi)=\sum_{i=0}^{\infty} \frac{\varpi ^{i}}{\varGamma(\vartheta i+1)}, \end{aligned}$$

where \(\vartheta\in\mathbb{C}\), \(\operatorname{Re}(\vartheta)>0\), and Γ is the gamma function given by

$$\begin{aligned} \varGamma(\varpi)= \int_{0}^{\infty}e^{-\varsigma} \varsigma^{\varpi -1}\,d\varsigma \end{aligned}$$

for \(\operatorname{Re}(\varpi)>0\). In particular, if \(\vartheta=1\), then we have

$$\begin{aligned} \mathcal{E}_{1}(\varpi)=\sum_{k=0}^{\infty} \frac{\varpi^{k}}{\varGamma (k+1)}=\sum_{k=0}^{\infty} \frac{\varpi^{k}}{ k!}=e^{\varpi}. \end{aligned}$$

Definition 2.4

([1, 39])

Consider \(\iota>0\) and the increasing and positive monotone random operator \(\varPsi(\varrho,\varsigma)\) on \(\varUpsilon\times(\mathbf{a},\mathbf{b}]\) with continuous derivative random operator \(\varPsi'(\varrho,\varsigma)\) on \(\varUpsilon \times\varXi_{2}\). Define the LR (left-right) stochastic fractional integrals of a random operator f and random operator Ψ on \(\varUpsilon \times\varXi_{1}\) by

$$\begin{aligned} \mathcal{I}_{\mathbf{a}^{+}}^{\iota;\varPsi}\boldsymbol{f}(\varrho , \varsigma):=\frac{1}{ \varGamma(\iota)} \int_{\mathbf{a}}^{\varsigma}\varPsi'(\varrho, \upsilon ) \bigl(\varPsi(\varrho,\varsigma)-\varPsi(\varrho,\upsilon) \bigr)^{\iota -1}\boldsymbol{f}(\varrho,\upsilon)\,d\upsilon \end{aligned}$$

for all \(\varsigma\in\varXi_{2}\) and \(\varrho\in\varUpsilon\).

Definition 2.5

([1, 40])

Consider \(n\in\mathbb {N}^{+}\) and let \(n-1<\iota<n\). Let \(\varXi_{1}\) be the interval such that \(-\infty\leq\mathbf{a}<\mathbf{b}\leq+\infty\), and let \(\boldsymbol{f},\varPsi\in\ell^{n}(\varUpsilon\times\varXi_{1},\mathbb {R})\) be two random operators, where Ψ is increasing, and \(\varPsi'(\varrho,\varsigma)\neq0\) for all \(\varsigma\in\varXi_{1}\) and \(\varrho\in\varUpsilon\). Define the L-Ψ-Hilfer stochastic fractional derivative operator \({}^{H}D_{\mathbf{a}^{+}}^{\iota,\kappa;\varPsi}(\varrho,\cdot)\) of order ι and type \(0\leq\kappa\leq1\) by

$$\begin{aligned} ^{H}D_{\mathbf{a}^{+}}^{\iota,\kappa;\varPsi}\boldsymbol{f}(\varrho , \varsigma):=\mathcal{I}_{\mathbf{a}^{+}}^{\kappa(n-\iota); \varPsi} \biggl( \frac{1}{\varPsi'(\varrho,\varsigma)}\frac{d}{d\varsigma } \biggr)^{n} \mathcal{I}_{\mathbf{a}^{+}}^{(1-\kappa)(n-\iota); \varPsi}\boldsymbol{f}(\varrho,\varsigma). \end{aligned}$$

We define the R-Ψ-Hilfer stochastic fractional derivative operator as in [1].

Lemma 2.6

([1])

If\(\boldsymbol{f}\in\ell_{\delta;\varPsi}^{1}(\varUpsilon\times\varXi _{1})\), \(0<\iota<1\), \(0\leq\kappa\leq 1\), and\(\delta=\iota+\kappa(1-\iota)\), then

$$\begin{aligned} \mathcal{I}_{\mathbf{a}^{+}}^{\iota;\varPsi} {{}^{H}D_{\mathbf{a}^{+}}^{\iota,\kappa;\varPsi}} \boldsymbol{f}(\varrho ,\varsigma)=\boldsymbol{f}(\varrho,\varsigma)- \frac{(\varPsi(\varrho ,\varsigma)-\varPsi(\varrho,\mathbf{a}))^{\delta-1}}{\varGamma(\delta )}\mathcal{I}_{\mathbf{a}^{+}}^{(1-\kappa)(1-\iota);\varPsi}\boldsymbol {f}( \varrho,\mathbf{a}). \end{aligned}$$

Let \(\varXi_{1}\) (\(-\infty<\mathbf{a}<\mathbf{b}<+\infty\)), and let \(\eta(\cdot,\tau)_{\ell (\varUpsilon\times\varXi_{1} )}\) denote the fuzzy norm of \(\mu=(\mu_{1}(\varrho,\varsigma),\mu_{2}(\varrho,\varsigma),\ldots,\mu _{n}(\varrho,\varsigma))^{T}\in\mathbb{R}^{n}\) on \(\varUpsilon\times\varXi_{1}\) defined by

$$\begin{aligned} \eta \bigl(\mu(\varrho,\varsigma),\tau \bigr)=\min_{1\leq i\leq n}\eta _{E} \bigl(\mu_{i}(\varrho,\varsigma),\tau \bigr), \end{aligned}$$

where \(\eta(\cdot,\tau)_{E}\) denotes the Euclidean fuzzy norm of \(\mu_{i}(\varrho,\varsigma)\in\mathbb{R}\) on \(\varUpsilon\times\varXi_{1}\). Denote the space of continuous random operators by \(\ell(\varUpsilon \times\varXi_{1})\) and define \(\mu\in\mathbb{R}^{n}\) on \(\varUpsilon\times\varXi_{1}\) by

$$\begin{aligned} \eta \bigl(\mu(\varrho,\varsigma),\tau \bigr)_{\ell(\varUpsilon\times \varXi_{1})} =\min _{\varsigma\in\varXi_{1}}\eta \bigl(\mu(\varrho ,\varsigma),\tau \bigr). \end{aligned}$$

The weighted space \(\ell_{1-\delta;\varPsi}(\varUpsilon\times\varXi _{1},\mathbb{R}^{n})\) of random operators μ on \(\varUpsilon\times\varXi_{1}\) is defined by

$$\begin{aligned} \ell_{1-\delta;\varPsi}(\varUpsilon\times\varXi_{1})=\bigl\{ \mu: \varUpsilon \times\varXi_{1}\rightarrow\mathbb{R}^{n}: \bigl(\varPsi(\varrho,\varsigma )-\varPsi(\varrho,0)\bigr)^{1-\delta}\mu( \varrho,\varsigma)\in\ell (\varUpsilon\times\varXi_{1})\bigr\} \end{aligned}$$

for \(\delta=\iota+\kappa(1-\iota)\), with the norm

$$\begin{aligned} \eta \bigl(\mu(\varrho,\varsigma),\tau \bigr)_{\ell_{1-\delta;\varPsi }(\varUpsilon\times\varXi_{1})} =\eta \bigl( \bigl(\varPsi(\varrho,\varsigma )-\varPsi(\varrho,0)\bigr)^{1-\delta}\mu( \varrho,\varsigma),\tau \bigr)_{\ell (\varUpsilon\times\varXi_{1})}. \end{aligned}$$

Definition 2.7

([39])

We say that system (1.1) has the Ulam–Hyers–Rassias stability if for each continuously differentiable random operator \(\nu (\varrho,\varsigma)\in \mathbb{R}^{n}\) satisfying

$$\begin{aligned} & \eta \bigl({}^{H}D_{0^{+}}^{\iota,\kappa;\varPsi} \nu(\varrho,\varsigma )-\mathsf{A}\nu(\varrho,\varsigma)-\mathsf{B}\nu\bigl( \varrho,\varsigma -\boldsymbol{h}(\varrho,\varsigma)\bigr)-\boldsymbol{f}\bigl( \varrho,\varsigma,\nu (\varrho,\varsigma),\nu\bigl(\varrho,\varsigma- \boldsymbol{h}(\varrho ,\varsigma)\bigr)\bigr),\tau \bigr) \\ &\quad\geq\varphi(\varsigma,\tau), \end{aligned}$$
(2.1)

where \(\varphi(\varsigma,\tau)\in\varXi_{6}\) is a continuous fuzzy set for all \(\varsigma\in\varXi_{1}\), \(\tau\in\varXi_{8}\), and \(\varrho \in\varUpsilon\), there exist a solution \(\mu(\varrho,\varsigma)\in \mathbb{R}^{n}\) of system (1.1) and a constant \(\mathbf{C}>0\) such that

$$\begin{aligned} \eta \bigl(\mu(\varrho,\varsigma)-\nu(\varrho,\varsigma),\tau \bigr)\geq \varphi \biggl(\varsigma,\frac{\tau}{\mathbf{C}} \biggr), \end{aligned}$$

where C is independent of \(\mu(\varrho,\varsigma)\) and \(\nu (\varrho,\varsigma)\). If \(\varphi(\varrho,\varsigma)\) is fixed in the above inequalities, then we get the Ulam–Hyers stability with uncertainty of system (1.1).

Remark 2.8

A random operator \(\nu(\varrho,\varsigma)\) is a solution of (2.1) if and only if there is a random operator \(\varTheta\in\ell(\varUpsilon\times\varXi_{5},\mathbb{R}^{n})\) such that

  • \(\eta(\varTheta(\varrho,\varsigma),\tau)\geq\varphi(\varsigma ,\tau)\);

  • \({}^{H}D_{0^{+}}^{\iota,\kappa;\varPsi}\nu(\varrho,\varsigma)=\mathsf{A}\nu (\varrho,\varsigma)+\mathsf{B}\nu(\varrho,\varsigma-\boldsymbol {h}(\varrho,\varsigma))+\boldsymbol{f}(\varrho,\varsigma,\nu(\varrho ,\varsigma),\nu(\varrho,\varsigma-\boldsymbol{h}(\varrho,\varsigma )))+ \varTheta(\varrho,\varsigma)\).

Lemma 2.9

Let\(\boldsymbol{f}:\varUpsilon\times\varXi_{5}\times\mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\)be a continuous nonlinear random operator. Then the solution of system (1.1) is a continuous random operator\(\mu(\varrho,\varsigma):\varUpsilon\times\varXi_{4}\rightarrow\mathbb {R}^{n}\)satisfying

$$ \mu(\varrho,\varsigma)=\left \{ \textstyle\begin{array}{l} \frac{\varTheta(\varrho,0)}{\varGamma(\delta)\varGamma(2-\delta)}+\frac {1}{\varGamma(\iota)}\int_{0}^{\varsigma}\varPsi'(\varrho,\upsilon )(\varPsi(\varrho,\varsigma)-\varPsi(\varrho,\upsilon))^{\iota -1}[\mathsf{A}\mu(\varrho,\upsilon)\\ \quad{}+\mathsf{B}\mu(\varrho,\upsilon-\boldsymbol{h}(\varrho,\upsilon ))]+\boldsymbol{f}(\varrho,\varsigma,\mu(\varrho,\varsigma),\mu(\varrho ,\varsigma-\boldsymbol{h}(\varrho,\varsigma))), \\ \varTheta(\varrho,\varsigma),\quad \varsigma\in\varXi_{3}. \end{array}\displaystyle \right . $$
(2.2)

Proof

Let

$$\begin{aligned} \boldsymbol{g}(\varrho,\varsigma)=\mathsf{A}\mu(\varrho,\varsigma )+\mathsf{B} \mu\bigl(\varrho,\varsigma-\boldsymbol{h}(\varrho,\varsigma )\bigr)+\boldsymbol{f} \bigl(\varrho,\varsigma,\mu(\varrho,\varsigma),\mu\bigl(\varrho ,\varsigma- \boldsymbol{h}(\varrho,\varsigma)\bigr)\bigr). \end{aligned}$$

From Lemma 2.6 and (1.1), for \(\varsigma\geq0\) and \(\varrho\in\varUpsilon\), we get

$$\begin{aligned} \mu(\varrho,\varsigma) &=\frac{(\varPsi(\varrho,\varsigma)-\varPsi(\varrho,0))^{\varrho -1}}{\varGamma(\delta)} \\ &\quad\times\frac{1}{\varGamma(1-\delta)} \int_{0}^{\varsigma }\varPsi'(\varrho, \upsilon) \bigl(\varPsi(\varrho,\varsigma)-\varPsi(\varrho,\upsilon)\bigr)^{-\varrho}\mu(\varrho,0)\,d \upsilon+\mathcal {I}_{0^{+}}^{\iota;\varPsi}g(\varsigma) \\ &=\frac{\varTheta(\varrho,0)}{\varGamma(\delta)\varGamma(2-\delta )}\\ &\quad+\frac{1}{\varGamma(\iota)} \int_{0}^{\varsigma}\varPsi'(\varrho , \upsilon) \bigl(\varPsi(\varrho,\varsigma)-\varPsi(\varrho,\upsilon) \bigr)^{\iota -1}\boldsymbol{g}(\varrho,\upsilon)\,d\upsilon,\quad \varsigma\geq0 \end{aligned}$$
(2.3)

when \(\varsigma\in\varXi_{3}\) and \(\mu(\varrho,\varsigma)=\varTheta (\varrho,\varsigma)\). □

We denote the set of all eigenvalues of A defined as in system (1.1) by \(\lambda(\mathsf{A})\) and set \(\lambda_{\max}(\mathsf{A})=\max\{\operatorname{Re}(\lambda):\lambda\in\lambda(\mathsf {A})\}\) and \(\Vert \mathsf{A} \Vert =\sqrt{\lambda_{\max}(\mathsf{A}^{T}\mathsf {A})}\); also, we denote the set of all nonnegative bounded random operators on \(\varUpsilon\times\varXi_{5}\) by \(\mathcal{B}^{+}(\varXi_{5})\).

  1. (H1)

    For a nonlinear random operator \(\boldsymbol{f}: \varUpsilon\times\varXi_{5}\times\mathbb{R}^{n}\times\mathbb {R}^{n}\rightarrow\mathbb{R}^{n}\), there is a positive map \(\boldsymbol{l}(\varsigma)\in\mathcal {B}^{+}(\varXi_{5})\) such that

    $$\begin{aligned} &\eta \bigl(\boldsymbol{f}(\varrho,\varsigma,\mu_{1}, \nu_{1})-\boldsymbol {f}(\varrho,\varsigma,\mu_{2}, \nu_{2}),\tau \bigr)_{\ell(\varUpsilon \times\varXi_{5})} \\ &\quad\geq\min \biggl(\eta \biggl(\mu_{1}-\mu_{2}, \frac{\tau}{\boldsymbol {l}(\varsigma)} \biggr)_{\ell(\varUpsilon\times\varXi_{5})},\eta \biggl(\nu _{1}- \nu_{2},\frac{\tau}{\boldsymbol{l}(\varsigma)} \biggr)_{\ell (\varUpsilon\times\varXi_{5})} \biggr); \end{aligned}$$

    moreover,

    $$\begin{aligned} \eta \bigl(\mathsf{A}\mu(\varrho,\varsigma),\tau \bigr)_{\ell(\varUpsilon \times\varXi_{5})} \geq \eta \biggl(\mu(\varrho,\varsigma),\frac{\tau}{\mathbf{a}} \biggr)_{\ell (\varUpsilon\times\varXi_{5})} \end{aligned}$$

    and

    $$\begin{aligned} \eta \bigl(\mathsf{B}\mu\bigl(\varrho,\varsigma-\boldsymbol{h}(\varrho , \varsigma)\bigr),\tau \bigr)_{\ell(\varUpsilon\times\varXi_{5})} \geq \eta \biggl(\mu\bigl(\varrho, \varsigma-\boldsymbol{h}(\varrho,\varsigma)\bigr),\frac {\tau}{\mathbf{b}} \biggr)_{\ell(\varUpsilon\times\varXi_{5})} \end{aligned}$$

    for all \(\tau\in\varXi_{8}\) and \(\varrho\in\varUpsilon\), where \(\Vert \mathsf{A} \Vert =\mathbf{a}\), \(\Vert \mathsf{B} \Vert =\mathbf{b}\), and \(\sup_{\upsilon\in[0,\varsigma]}\boldsymbol{l}(\upsilon)=\overline {\boldsymbol{l}}\).

3 Ulam–Hyers–Rassias stability with uncertainty

Using Remark 2.8 and Lemma 2.9, for \(\varsigma\in\varXi_{5}\), we get

$$\begin{aligned} \nu(\varrho,\varsigma) &=\frac{\varTheta(\varrho,0)}{\varGamma(\delta)\varGamma(2-\delta )}+\frac{1}{\varGamma(\iota)} \int_{0}^{\varsigma}\varPsi'(\varrho , \upsilon) \bigl(\varPsi(\varrho,\varsigma) \\ &\quad-\varPsi(\varrho,\upsilon)\bigr)^{\iota-1} \bigl(\mathsf{A}\nu( \varrho ,\upsilon)+\mathsf{B}\nu\bigl(\varrho,\upsilon-\boldsymbol{h}(\varrho , \upsilon)\bigr) \\ &\quad+\boldsymbol{f}\bigl(\varrho,\upsilon,\nu(\varrho, \upsilon),\nu\bigl(\varrho ,\upsilon-\boldsymbol{h}(\varrho,\upsilon)\bigr)\bigr) \bigr)\,d\upsilon. \end{aligned}$$
(3.1)

Theorem 3.1

Assume that (H1) holds and

$$\eta \bigl(\bigl(\varPsi(\varrho,\mathit{\varsigma})-\varPsi(\varrho ,0) \bigr)^{\iota},\tau \bigr)\geq\eta \bigl(\varTheta(\varrho,\varsigma), \tau \bigr). $$

Then (1.1) has the Ulam–Hyers–Rassias stability with uncertainty on\(\ell(\varUpsilon\times\varXi_{4})\)when\(\eta (\varTheta(\varrho ,\varsigma),\tau )\)is increasing on\(\varUpsilon\times\varXi_{5}\)as in Remark2.8 .

Proof

For all \(\varsigma\in\varXi_{5}\), \(\tau\in\varXi_{8}\), and \(\varrho \in\varUpsilon\), using (2.2) and (3.1), we have

figure a

Hence, if \(\eta ((\varPsi(\varrho,\mathit{\varsigma})-\varPsi(\varrho ,0))^{\iota},\tau )\geq\eta (\varTheta(\varrho,\mathit{\varsigma }),\tau )\) and \(\mathbf{C}=\frac{\eta ((\nu-\mu),\frac{\tau }{\mathbf{a}+\mathbf{b}+\overline{\boldsymbol{l}}} )}{\varGamma (\iota+1)}\), then we have

$$\begin{aligned} \eta (\mu-\nu,\tau )_{\ell(\varUpsilon\times\varXi_{5})} \geq\varphi\biggl( \mathit{\varsigma},\frac{\tau}{\mathbf{C}}\biggr). \end{aligned}$$
(3.2)

Now by Definition 2.7, (1.1) has the Ulam–Hyers–Rassias stability with uncertainty on \(\varUpsilon\times \varXi_{5}\). □

Now we consider the new condition for constant \(\varphi_{0}\):

$$\begin{aligned} \text{(H2)}\quad\varphi(\varsigma,\tau)=\varphi_{0} \end{aligned}$$

for all \(\tau\in\varXi_{8}\).

Theorem 3.2

Let (H1) and (H2) hold. Suppose that

$$\eta \bigl(\bigl(\varPsi(\varrho,\mathit{\varsigma})-\varPsi(\varrho ,0) \bigr)^{\iota},\tau \bigr)\geq\eta \bigl(\varTheta(\varrho,\varsigma), \tau \bigr). $$

Then (1.1) has the Ulam–Hyers stability with uncertainty on\(\ell(\varUpsilon\times\varXi_{4})\).

Proof

From (2.2) and (3.1), for all \(\varsigma\in\varXi_{5}\) and \(\varrho\in \varUpsilon\), we have

figure b

Hence, if \(\eta ((\varPsi(\varrho,\mathit{\varsigma})-\varPsi(\varrho ,0))^{\iota},\tau )\geq\eta (\varTheta(\varrho,\mathit{\varsigma }),\tau )\) and \(\mathbf{C}=\frac{\eta ((\nu-\mu),\frac{\tau }{\mathbf{a}+\mathbf{b}+\overline{\boldsymbol{l}}} )}{\varGamma (\iota+1)}\), then

$$\begin{aligned} \eta (\mu-\nu,\tau )_{\ell(\varUpsilon\times\varXi_{5})} \geq\varphi_{0}. \end{aligned}$$
(3.3)

Thus (1.1) has the Ulam–Hyers stability with uncertainty on \(\varUpsilon\times\varXi_{5}\). □

4 Application

Now we apply our result to the following dynamic fractional-order equation systems with time-varying delay:

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} ^{H}D_{0^{+}}^{\iota,\kappa;\varPsi}\mu(\varrho,\varsigma) \\\quad=\mathsf{A}(\varsigma)\mu(\varrho,\varsigma)+\mathsf{B}(\varsigma)\mu (\varrho,\varsigma-\boldsymbol{h}(\varrho,\varsigma))+D(\varsigma )\mathcal{G}(\varrho,\varsigma)+\boldsymbol{f}(\varrho,\cdot),&\varsigma\in \varXi_{5}, \\ \mu(\varrho,\varsigma)=\varTheta(\varrho,\varsigma),& \varsigma\in\varXi _{3}, \end{array}\displaystyle \right . $$
(4.1)

where \(\mathcal{G}(\varrho,\varsigma)\in\mathbb{R}^{m}\), and \(\mathsf {A}(\varsigma)\), \(\mathsf{B}(\varsigma)\in\mathbb{R}^{n\times n}\), and \(D(\varsigma)\in \mathbb{R}^{n\times m}\) are random operator matrices such that \(\sup_{\varsigma\in \varXi_{5}}( \Vert \mathsf{A} \Vert + \Vert \mathsf{B} \Vert )<\infty\).

Corollary 4.1

Suppose that\(\Vert \mathsf{A} \Vert =\tilde{\mathbf{a}}\), \(\Vert \mathsf{B} \Vert =\tilde{\mathbf{b}}\), and there is\(\varTheta\in\ell(\varUpsilon\times\varXi_{5},\mathbb{R}^{n})\)such that

$$\begin{aligned} \int_{0}^{\varsigma}\varPsi'(\varrho, \upsilon) \bigl(\varPsi(\varrho,\varsigma )-\varPsi(\varrho,\upsilon) \bigr)^{\iota-1}\eta \bigl(\varTheta(\varrho ,\upsilon),\tau \bigr) > \varphi\biggl(\varsigma,\frac{\tau}{\mathit{M}}\biggr). \end{aligned}$$

Let (H1) hold. Then system (4.1) has the Ulam–Hyers–Rassias stability with uncertainty if

$$\eta \bigl(\bigl(\varPsi(\varrho,\mathit{\varsigma})-\varPsi(\varrho ,0) \bigr)^{\iota},\tau \bigr)\geq\eta \bigl(\varTheta(\varrho,\varsigma), \tau \bigr). $$

Also, (H2) implies that system (4.1) has the Ulam–Hyers stability with uncertainty.

Example 4.2

Suppose that for each \(\varrho\in\varUpsilon\), \(\varPsi(\varrho ,\varsigma)=\ln(\varsigma+1)\), \(\iota=0.2\), \(\kappa\rightarrow 1\), \(\mathit{M}=20\), and \(\boldsymbol{h}(\varrho,\varsigma)=2\sin \varsigma\). Let \(\varTheta(\varrho,\varsigma)=(\varsigma, \sqrt {\varsigma}+1)^{T}\), and let \(\boldsymbol{f}(\varrho,\varsigma,\mu(\varrho,\varsigma),\mu (\varrho,\varsigma-\boldsymbol{h}(\varrho,\varsigma)))=0.1\sin\mu (\varrho,\varsigma)+0.1\cos\mu(\varrho,\varsigma-2\sin\varsigma)\) with \(\mu(\varrho,\varsigma)=(\mu_{1}(\varrho,\varsigma),\mu_{2}(\varrho ,\varsigma))^{T}\). Consider system (1.1) with

$$\begin{aligned}& \mathsf{A}=\left ( \textstyle\begin{array}{c@{\quad}c} \frac{1}{5} & 0 \\ 0 & \frac{1}{6} \end{array}\displaystyle \right ), \end{aligned}$$
(4.2)
$$\begin{aligned}& \mathsf{B}=\left ( \textstyle\begin{array}{c@{\quad}c} \frac{1}{3} & 0 \\ \frac{1}{6} & 0 \end{array}\displaystyle \right ). \end{aligned}$$
(4.3)

We have \(\Vert \mathsf{A} \Vert =\mathbf{a}=\frac{1}{25}\), \(\Vert \mathsf{B} \Vert =\mathbf{b}=\frac{5}{36}\), and \(\overline {\boldsymbol{l}}=0.1\). By calculation, \((\varPsi(\varrho,\mathit{\varsigma})-\varPsi(\varrho ,0))^{\iota}\approx1.240\), so all the conditions in Theorem 3.1 are satisfied. Then (1.1) has the fuzzy Ulam–Hyers–Rassias stability with uncertainty on \(\varUpsilon\times[0,20]\). Moreover, the maximum value of \(\eta (\varTheta(\varrho,\varsigma ),\tau )\approx 0.977\) if \(\varphi(\varrho,\varsigma)\leq0.977\) is a constant continuous fuzzy set, and all the conditions in Theorem 3.2 hold, which implies that (1.1) has the fuzzy Ulam–Hyers stability with uncertainty on \(\varUpsilon\times[0,20]\).

5 Conclusion

In this paper, we considered a kind of stochastic differential equations involving the Ψ-Hilfer stochastic fractional derivative operator. A fuzzy control function helped us to make stable the stochastic differential equation (1.1). Using the fixed point method, we investigated the Ulam–Hyers–Rassias stability for the nonlinear Ψ-Hilfer stochastic fractional differential equation (1.1) with uncertainty.