Skip to main content

Ulam–Hyers–Rassias stability for nonlinear Ψ-Hilfer stochastic fractional differential equation with uncertainty


We consider a nonlinear Cauchy problem involving the Ψ-Hilfer stochastic fractional derivative with uncertainty, and we give a stability result. Using fixed point theory, we are able to provide a fuzzy Ulam–Hyers–Rassias stability for the considered nonlinear stochastic fractional differential equations.


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 [212]).

We study the nonlinear Ψ-Hilfer stochastic fractional differential equation

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

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.


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


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 [2429]. Some stability results for fractional differential and integral equations have been discussed in [26, 3038].

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


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


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

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

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

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



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

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

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

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

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


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


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

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

Ulam–Hyers–Rassias stability with uncertainty

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

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

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 .


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

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

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\(\ell(\varUpsilon\times\varXi_{4})\).


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

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

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


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

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

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

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

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 \(\varUpsilon\times[0,20]\).


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.


  1. Sousa, J., Oliveira, E.: On the Ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)

    MathSciNet  Article  Google Scholar 

  2. Doungmo Goufo, E.F., Kumar, S., Mugisha, S.B.: Similarities in a fifth-order evolution equation with and with no singular kernel. Chaos Solitons Fractals 130, Article ID 109467 (2020)

    MathSciNet  Article  Google Scholar 

  3. Ghanbari, B., Kumar, S., Kumar, R.: A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative. Chaos Solitons Fractals 133, Article ID 109619 (2020)

    MathSciNet  Article  Google Scholar 

  4. Kumar, S., Kumar, R., Cattani, C., Samet, B.: Chaotic behaviour of fractional predator–prey dynamical system. Chaos Solitons Fractals 135, Article ID 109811 (2020)

    MathSciNet  Article  Google Scholar 

  5. Kumar, S., Kumar, R., Agarwal, R.P., Samet, B.: A study of fractional Lotka–Volterra population model using Haar wavelet and Adams–Bashforth–Moulton methods. Math. Methods Appl. Sci. 43, 5564–5578 (2020)

    Article  Google Scholar 

  6. Kumar, S., Ghosh, S., Samet, B., Doungmo Goufo, E.F.: An analysis for heat equations arises in diffusion process using new Yang–Abdel–Aty–Cattani fractional operator. Math. Methods Appl. Sci. 43, 6062–6080 (2020)

    Article  Google Scholar 

  7. Kumar, S., Ahmadian, A., Kumar, R., Kumar, D., Singh, J., Baleanu, D., Salimi, M.: An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets. Mathematics 8(4), Article ID 558 (2020)

    Article  Google Scholar 

  8. Alshabanat, A., Jleli, M., Kumar, S., Samet, B.: Generalization of Caputo–Fabrizio fractional derivative and applications to electrical circuits. Front. Phys. 8, Article ID 64 (2020)

    Article  Google Scholar 

  9. Kumar, S., Ghosh, S., Lotayif, M.S., Samet, B.: A model for describing the velocity of a particle in Brownian motion by Robotnov function based fractional operator. Alex. Eng. J. 59, 1435–1449 (2020)

    Article  Google Scholar 

  10. Malkowsky, E., Rakočević, V.: Advanced Functional Analysis. CRC Press, Boca Raton (2019)

    MATH  Book  Google Scholar 

  11. Ćirić, L.B.: Some recent results in metrical fixed point theory (2003)

  12. Todorčević, V.: Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics. Springer, Cham (2019)

    MATH  Book  Google Scholar 

  13. Luo, D., Shah, K., Luo, Z.: On the novel Ulam–Hyers stability for a class of nonlinear ψ-Hilfer fractional differential equation with time-varying delays. Mediterr. J. Math. 16(5), Article ID 112 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  14. Bag, T., Samanta, S.K.: Finite dimensional intuitionistic fuzzy normed linear spaces. Ann. Fuzzy Math. Inform. 8(2), 245–257 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Hadžić, O., Pap, E.: Fixed Point Theory in Probabilistic Metric Spaces. Mathematics and Its Applications, vol. 536. Kluwer Academic, Dordrecht (2001)

    MATH  Book  Google Scholar 

  16. Font, J.J., Galindo, J., Macario, S., Sanchis, M.: Mazur–Ulam type theorems for fuzzy normed spaces. J. Nonlinear Sci. Appl. 10(8), 4499–4506 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  17. Bînzar, T., Pater, F., Nădăban, S.: On fuzzy normed algebras. J. Nonlinear Sci. Appl. 9(9), 5488–5496 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  18. Saadati, R., Vaezpour, S.M.: Some results on fuzzy Banach spaces. J. Appl. Math. Comput. 17(1–2), 475–484 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  19. Mirmostafaee, A.K.: Perturbation of generalized derivations in fuzzy Menger normed algebras. Fuzzy Sets Syst. 195, 109–117 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  20. Naeem, R., Anwar, M.: Jessen type functionals and exponential convexity. J. Math. Comput. Sci. 17(3), 429–436 (2017)

    MATH  Article  Google Scholar 

  21. Park, C., Shin, D.Y., Saadati, R., Lee, J.R.: A fixed point approach to the fuzzy stability of an AQCQ-functional equation. Filomat 30(7), 1833–1851 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  22. Park, C., Anastassiou, G.A., Saadati, R., Yun, S.: Functional inequalities in fuzzy normed spaces. J. Comput. Anal. Appl. 22(4), 601–612 (2017)

    MathSciNet  Google Scholar 

  23. Kang, J.I., Saadati, R.: Approximation of homomorphisms and derivations on non-Archimedean random Lie \(C^{\ast}\)-algebras via fixed point method. J. Inequal. Appl. 2012, Article ID 251 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Agarwal, R.P., Saadati, R., Salamati, A.: Approximation of the multiplicatives on random multi-normed space. J. Inequal. Appl. 2017, Article ID 204 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  25. Cho, Y.J., Rassias, T.M., Saadati, R.: Stability of Functional Equations in Random Normed Spaces. Springer Optimization and Its Applications, vol. 86. Springer, Berlin (2013)

    MATH  Book  Google Scholar 

  26. El-Moneam, M.A., Ibrahim, T.F., Elamody, S.: Stability of a fractional difference equation of high order. J. Nonlinear Sci. Appl. 12(2), 65–74 (2019)

    MathSciNet  Article  Google Scholar 

  27. Park, C., Kim, S.O., Alaca, C.: Stability of additive-quadratic ρ-functional equations in Banach spaces: a fixed point approach. J. Nonlinear Sci. Appl. 10(3), 1252–1262 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  28. Saadati, R., Park, C.: Approximation of derivations and the superstability in random Banach -algebras. Adv. Differ. Equ. 2018, Article ID 418 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  29. Madadi, M., Saadati, R., De la Sen, M.: Stability of unbounded differential equations in Menger k-normed spaces: a fixed point technique. Mathematics 8(3), Article ID 400 (2020)

    Article  Google Scholar 

  30. Constantinescu, C.D., Ramirez, J.M., Zhu, W.R.: An application of fractional differential equations to risk theory. Finance Stoch. 23(4), 1001–1024 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  31. El-Sayed, A.M.A., Gaafar, F.M.: Existence and uniqueness of solution for Sturm–Liouville fractional differential equation with multi-point boundary condition via Caputo derivative. Adv. Differ. Equ. 2019, Article ID 46 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  32. Jiang, J., O’Regan, D., Xu, J., Fu, Z.: Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions. J. Inequal. Appl. 2019, Article ID 204 (2019)

    MathSciNet  Article  Google Scholar 

  33. Sene, N.: Stability analysis of the generalized fractional differential equations with and without exogenous inputs. J. Nonlinear Sci. Appl. 12(9), 562–572 (2019)

    MathSciNet  Article  Google Scholar 

  34. Sene, N.: Global asymptotic stability of the fractional differential equations. J. Nonlinear Sci. Appl. 13(3), 171–175 (2020)

    MathSciNet  Google Scholar 

  35. Agarwal, R.P., Hristova, S., O’Regan, D.: Lyapunov functions to Caputo reaction–diffusion fractional neural networks with time-varying delays. J. Math. Comput. Sci. 18(3), 328–345 (2018)

    MATH  Article  Google Scholar 

  36. Ahmed Hamdy, M., El-Borai Mahmoud, M., El-Owaidy, H.M., Ghanem, A.S.: Null controllability of fractional stochastic delay integro-differential equations. J. Math. Comput. Sci. 19(3), 143–150 (2019)

    Article  Google Scholar 

  37. Ali, A., Shah, K., Li, Y. Khan, R.A.: Numerical treatment of coupled system of fractional order partial differential equations. J. Math. Comput. Sci. 19(2), 74–85 (2019)

    Article  Google Scholar 

  38. Ndolane, S.: Exponential form for Lyapunov function and stability analysis of the fractional differential equations. J. Math. Comput. Sci. 18(4), 388–397 (2018)

    MATH  Article  Google Scholar 

  39. Vanterler da C. Sousa, J., Capelas de Oliveira, E.: Ulam–Hyers stability of a nonlinear fractional Volterra integro-differential equation. Appl. Math. Lett. 81, 50–56 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  40. Vanterler da C. Sousa, J., Capelas de Oliveira, E.: On a new operator in fractional calculus and applications. J. Fixed Point Theory Appl. 20(3), Article ID 96 (2018)

    MATH  Article  Google Scholar 

Download references


The authors are thankful to the area editor and three anonymous referees for giving valuable comments and suggestions, which helped to improve the final version of this paper.

Availability of data and materials

Not applicable.


Not applicable.

Author information

Authors and Affiliations



All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Corresponding author

Correspondence to Reza Saadati.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chaharpashlou, R., Saadati, R. & Atangana, A. Ulam–Hyers–Rassias stability for nonlinear Ψ-Hilfer stochastic fractional differential equation with uncertainty. Adv Differ Equ 2020, 339 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


  • 54C40
  • 14E20
  • 46E25
  • 47H10
  • 20C20


  • Fuzzy controller function
  • Random operator
  • Ulam–Hyers–Rassias stability
  • Ψ-Hilfer stochastic fractional derivative random operator
  • Time-varying delays