Ulam Stability for Fractional Partial Integro-Differential Equation with Uncertainty

Abstract

In this paper, the solvability of Darboux problems for nonlinear fractional partial integro-differential equations with uncertainty under Caputo gH-fractional differentiability is studied in the infinity domain J _∞ = [0,∞) × [0,∞). New concepts of Hyers-Ulam stability and Hyers-Ulam-Rassias stability for these problems are also investigated through the equivalent integral forms. A computational example is presented to demonstrate our main results.

1 Introduction

Fractional differential equations (DEs) and fractional integral equations can serve as an excellent tool for the description of mathematical modeling and hereditary properties of various materials and processes. In recent years, there has been a significant development in the theory of fractional DEs (see the monographs [1, 12, 20] and the references therein). When we describe a real-world phenomenon by a fractional DE, information about the behavior of a dynamical systems may be very complicated with errors and vagueness. Some authors provided a new approach to depict such physical models with the parameters and initial values defined under fuzzy fractional setting theory. One of the original works in fuzzy fractional DEs was conducted by Agarwal et al. [5] and Arshad et al. [6], where the Riemann-Liouville differentiability equipped with a fuzzy initial condition was used. This notion is a directed generalization of the fractional Riemann-Liouville derivative and the Hukuhara difference [11] (H-difference). However, there is a limitation in the H-difference, namely the fuzzy solutions increase the length of their support (see in [7, 16, 17]). In addition, the Riemann-Liouville derivative requires a quantity of the fractional derivative of unknown solution at the initial point, but it could not be measured and perhaps may not exist. In order to overcome these drawbacks, there have appeared some papers integrated the Caputo derivatives with generalized Hukuhara differentiability (gH-differentiability), called Caputo gH-differentiability, such as Allahviranloo et al. [3, 4], Hoa [9], Long et al. [14], and Mazandarani [18, 19].

In this paper, we give the notions of fuzzy Riemann-Liouville fractional integral and fuzzy Caputo gH-fractional derivative for fuzzy-valued multivariable functions. These notions will be used in the study of the existence and uniqueness of two types of global solutions of the following fractional partial integro-differential equation with uncertainty

$$^C_{gH}\mathcal{D}_k^{q} u(x,y)=f\left( x,y,u,{{\int}_{0}^{y}}m(y-s)g(u(x,s))ds\right),\ (x, y) \in J_{\infty} $$

(1)

with the initial conditions

$$\left\{\begin{array}{llllllll}u(x, 0) = \eta_{1}(x), \ x \in [0,\infty)\\ u(0, y) = \eta_{2}(y), \ y \in [0,\infty), \end{array}\right. $$

(2)

where ${~}^{C}_{gH}\mathcal {D}_{k}^{q}\ (k=1,2)$ are Caputo gH-fractional derivative operators, q = (q ₁,q ₂) ∈ (0,1]²; a fuzzy-valued mapping g : E → E is an integrable function; f ∈ C(J _∞ × E ²,E), $m\in L^{2}([0,\infty ),\mathbb {R})$.

In reality, the nonlinear fractional integro-differential wave equations (1) may be used to describe the dynamics of an extensible string [28] with fading memory, and the integral

$${{\int}_{0}^{y}}m(y - s) g(u(x, s)) ds $$

plays a special role in this description.

To the best of our knowledge, fuzzy fractional nonlinear integro-differential wave equations have not been investigated yet. This paper provides for the first time a dealing with such type of equations. Notice that our model are considered in the infinity domain under employing Caputo gH-fractional derivatives—an integrated of Caputo fractional derivative with gH-differentiability. A weighted metric with exponential functions will be employed to handle a technical difficulty when time variables tend to infinity. However, the calculation in fractional integral of exponential functions appears as the new difficulty; it is still traceless and unknown. This obstacle will be passed by using some auxiliary estimations given in Lemmas 2.2 and 2.3. Under some suitable assumptions ( H ₁ )– ( H ₅ ), the well-posedness of problem (1)–(2) will be proved in Section 3.

In 1940, Ulam [29] put a question regarding the stability of functional equation for homomorphism in front of a Mathematical Colloquium. The question was “when an approximate homomorphism from a group G ₁ to a metric group G ₂ can be approximated by an exact homomorphism?” Within the next 2 years, Hyers [10] gave an answer to the problem of Ulam for additive functions defined on Banach spaces G ₁ and G ₂. Furthermore, the result of Hyers has been generalized by Rassias [21].

A generalization of Ulam’s problem was recently proposed by replacing functional equations by DEs, integral equations, integro-differential equations, PDEs,.... We call a real PDE

$$ \phi \left( x,y,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial^{2} u}{\partial x^{2}},\frac{\partial^{2} u}{\partial y^{2}},\frac{\partial^{2} u}{\partial x\partial y},\dots\right)=0 $$

(3)

has Ulam stability, if for given 𝜖 > 0 and a function u such that

$$\left| \phi \left( x,y,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial^{2} u}{\partial x^{2}},\frac{\partial^{2} u}{\partial y^{2}},\frac{\partial^{2} u}{\partial x\partial y},\dots\right)\right|\leq \epsilon, $$

there exists a solution u _a (x,y) of (3) such that |u(x,y) − u _a (x,y)|≤ K(𝜖) and $\lim _{\epsilon \to 0} K(\epsilon ) =0$. If we replace 𝜖 and K(𝜖) by functions φ(x,y) and Φ(x,y), which do not depend on u(x,y) and u _a (x,y), then problem (3) is said to have the generalized Ulam stability. Unlike the general stability of DEs, the Ulam stability can guarantee the existence or even uniqueness of the exact solution, provided that an approximate solution with a determined error is given. Conversely, it is not difficult to see that the solution is stable for a differential equation with respect to Ulam stability. Therefore, the Ulam stability not only establishes an important foundation for the existence and uniqueness of the solution of DEs but also provides a reliable theoretical basis for approximately solving DEs.

As far we know, Obloza [23] seems to be the first author who investigated the Ulam stability of linear DEs. Thereafter, many scientists study Ulam stability of many types of DEs and PDEs. Researchers have presented their works with different approaches; for example, Abbas et al. [2] and Petru et al. [24] used the Picard operator technique to investigate some existence and Ulam type stability results for the Darboux problem associated to some partial fractional order differential inclusions. Huang and Li [8] discussed the stability of some classes of linear functional differential equations with multiple delays by combining direct method, iteration method, fixed-point method and open mapping theorem. Rezaei et al. [22] proved the Hyers-Ulam stability of a linear DE of the n th order by applying the Laplace transform method. Wang and Xu [30] investigated the Hyers-Ulam stability of two types of fractional linear DEs with Caputo fractional derivatives by replacing a given fractional DE by a fractional differential inequality and applying the Laplace transform method. Zada et al. generalized the concepts of Hyers-Ulam stability for non-autonomous linear differential systems. However, there are few published results regarding Ulam stability for fuzzy DEs. Up to now, all the existing results have been studied by Shen and Wang for some types of fuzzy linear DEs in [25, 26].

Based on the motivations stated above, in the present paper, we will discuss two types of the Ulam stability, which are Hyers-Ulam stability and generalized Hyers-Ulam-Rassias stability for fractional nonlinear integro-differential wave equation (1)–(2) in Theorems 4.1 and 4.2 of Section 4. At the end, as usual, a concrete example to illustrate our main results is presented in Section 5. In this example, Zadeh’s extension principle is used to estimate the distance between fuzzy-valued nonlinear functions u ²(x,y), (x,y) ∈ J _∞ .

2 Preliminaries

Denote by E the space of fuzzy numbers on $\mathbb {R}$, which are mappings $u: \mathbb {R} \to [0, 1]$ being normal, fuzzy convex, upper semi-continuous, and compactly supported. The α-level sets of fuzzy number u are defined by

$$[u]^{\alpha}=\left\{\begin{array}{llllllll}\{x \in \mathbb{R}: u(x)\geq \alpha\}&\text{ if } 0 <\alpha \leq1 \\ \operatorname{cl}(\operatorname{supp} u) &\text{ if } \alpha=0. \end{array}\right.$$

It is clear that α-level set of a fuzzy number is a closed and bounded interval $[u^{-}_{\alpha },u^{+}_{\alpha }],$ where $u^{-}_{\alpha }$ denotes the left-hand endpoint of [u]^α and $u^{+}_{\alpha }$ denotes the right-hand endpoint of [u]^α. The diameter of the α-level set of u is defined by $len[u]^{\alpha }=u^{+}_{\alpha }-u^{-}_{\alpha }$. Supremum metric is the most commonly used metric on E defined by

$$ d_{\infty} (u, v) = \sup \limits_{0 \leq\alpha \leq1 }\max\left\{|u^{-}_{\alpha}-v^{-}_{\alpha}|; |u^{+}_{\alpha}-v^{+}_{\alpha}|\right\},\ u, v \in E, $$

(4)

where $[u]^{\alpha }=[u^{-}_{\alpha },u^{+}_{\alpha }],\ [v]^{\alpha }=[v^{-}_{\alpha },v^{+}_{\alpha }]$. Then, (E,d _∞ ) is a complete metric space.

If there exists w ∈ E such that u = v + w, we call w = u ⊖ v the Hukuhara difference of u and v. The gH-difference (see [27]) of u and v, denoted by $u\circleddash _{gH}v$, is defined as the element w ∈ E such that $u\circleddash _{gH}v=w$ if (i) u = v + w or (ii) v = u + (−1)w.

The Zadeh’s extension principle allows a crisp mapping $f: \mathbb R \to \mathbb R$ extended to a fuzzy-valued mapping $\tilde {f}: E \to E$ defined by

$$\tilde{f}(u)(y)= \left\{\begin{array}{llllllll} \sup \limits_{x \in f^{-1}(y)}u(x)&\text{~if~} f^{-1}(y) \ne \emptyset\\ 0 &\text{~if~} f^{-1}(y)= \emptyset \end{array}\right. $$

for all $y \in \mathbb R$.

Definition 2.1

Let (X,d _X ), (Y,d _Y ) be metric spaces. A mapping f : X → Y is called continuous at x ₀ ∈ X if for arbitrary 𝜖 > 0, there exists δ > 0 such that for every x ∈ X with d _X (x,x ₀) < δ, we have d _Y (f(x),f(y)) < 𝜖. A mapping f is called continuous in X if f is continuous at all points x ∈ X.

Definition 2.2

[14] Given a mapping $f: J\subset \mathbb {R}^{2}\to E$, we say that f is gH differentiable with respect to x at (x ₀,y ₀) ∈ J if there exists an element $\frac {\partial f\left (x_{0}, y_{0}\right )}{\partial x}\in E$ such that for all h satisfying (x ₀ + h,y ₀) ∈ J, the gH-difference $f(x_{0}+h,y_{0})\circleddash _{g_{H}}f(x_{0},y_{0})$ exists and

$$\lim \limits_{h\to 0}\dfrac {f\left( x_{0}+h , y_{0}\right)\circleddash_{g_{H}}f\left( x_{0}, y_{0}\right)}{h}=\frac{\partial f\left( x_{0}, y_{0}\right)}{\partial x}.$$

In this case, $\frac {\partial f\left (x_{0}, y_{0}\right )}{\partial x}\in E$ is called a gH-derivative of f with respect to x at (x ₀,y ₀), provided the limit in the left hand side exists.

The gH-derivative of f with respect to y and higher order of fuzzy partial derivative of f at the point (x ₀,y ₀) ∈ I are defined similarly.

A fuzzy mapping $f:U\subset \mathbb {R}^{m}\to E$ is called integrably bounded if there exists an integrable function $h:U\to \mathbb [0,\infty )$, such that

$$d_{\infty}(f(\nu),\hat 0)\leq h(\nu) \ \forall \nu\in U.$$

Definition 2.3

A strongly measurable and integrable bounded fuzzy-valued function is called integrable. The fuzzy Aumann integral of $f:U\subset \mathbb {R}^{m}\to E$, denoted by $\int \limits _{U} f\left (\nu \right )d\nu$, is defined levelsetwise by the equation

$$\left[{\int}_{U} f(\nu)d\nu \right]^{\alpha} ={\int}_{U} [f(\nu)]^{\alpha}d\nu=\left[{\int}_{U} f_{\alpha}^{-}(\nu)d\nu,{\int}_{U} f_{\alpha}^{+}(\nu)d\nu\right], $$

where $[f(\nu )]^{\alpha }=[f_{\alpha }^{-}(\nu ),f_{\alpha }^{+}(\nu )]$ for all α ∈ [0,1].

For a subset $U\subset \mathbb {R}^{m}$, denote by

• C(X,Y ) the space of all fuzzy-valued continuous functions f : X → Y.

• $C_{gH}^{i,j}(U,E) (i,j=0,1)$ the set of all functions $f: U\subset \mathbb {R}^{2}\to E$ which have partial gH-derivatives up to order i with respect to x and up to order j with respect to y in U.

• L ^k(U,Y ) the set of all Lebesgue integrable functions $f: U\subset \mathbb {R}^{m}\to Y$, where Y = E or $Y\subset \mathbb {R}$, $k\in \mathbb {N}^{*}$.

Definition 2.4

Assume that $f \in C_{gH}^{1,0}(J,E), [f(x,y)]^{\alpha } = \left [f_{\alpha }^{-}(x,y), f_{\alpha }^{+}(x,y)\right ]$ for all α ∈ [0,1],(x,y) ∈ J. Let (x ₀,y ₀) ∈ J. We say that f is (i)-gH differentiable with respect to x at (x ₀,y ₀) ∈ J if

$$ \left[\frac{\partial f}{\partial x}\left( x_{0}, y_{0}\right)\right]^{\alpha}=\left[\frac{\partial f_{\alpha}^{-}}{\partial x}(x_{0}, y_{0}) , \frac{\partial f_{\alpha}^{+}}{\partial x}(x_{0}, y_{0}) \right] \ \text{ for all } 0 \leq \alpha \leq 1 $$

(5)

and that f is (ii)-gH differentiable with respect to x at (x ₀,y ₀) ∈ I if

$$ \left[\frac{\partial f}{\partial x}\left( x_{0}, y_{0}\right)\right]^{\alpha}=\left[ \frac{\partial f_{\alpha}^{+}}{\partial x}(x_{0}, y_{0}) , \frac{\partial f_{\alpha}^{-}}{\partial x} (x_{0}, y_{0}) \right]\ \text{ for all } \ 0 \leq \alpha \leq 1. $$

(6)

Denote by

• $D_{xy}u(x,y)=\frac {\partial ^{2}u(x,y)}{\partial x \partial y}$ the mixed second-order partial derivative of u.

• $C^{x}_{(i)-gH}(J,E)$ (or $C^{y}_{(i)-gH}(J,E)$) the set of all functions u which is (i)-gH differentiable with respect to x (or y) in J, respectively.

• $C^{x}_{(ii)-gH}(J,E)$ (or $C^{y}_{(ii)-gH}(J,E)$) the set of all functions u which are (ii)-gH differentiable with respect to x (or y) in J, respectively.

• $\mathcal {W}^{1}_{gH}(J,E)= \{u| (u,u_{x})\in C^{x}_{(k)-gH}(J,E)\times C^{y}_{(k)-gH}(J,E),\ k=1,2\}$.

• $\mathcal {W}^{2}_{gH}(J,E)= \{u| (u,u_{x})\in C^{x}_{(k)-gH}(J,E)\times C^{y}_{(l)-gH}(J,E),\ k=1,l=2 \text { or }k=2,l=1\}$.

Remark 2.1

[13,14,15] If $u\in \mathcal {W}^{1}_{gH}(J,E)$, then

$$[D_{xy}u(x, y)]^{\alpha}=\left[\frac{\partial^{2} u_{\alpha}^{-}}{\partial x\partial y} (x, y),\frac{\partial^{2} u_{\alpha}^{+}}{\partial x\partial y}(x, y)\right] \ \text{for } 0 \leq \alpha \leq 1.$$

If $u\in \mathcal {W}^{2}_{gH}(J,E)$, then

$$[D_{xy}u(x, y)]^{\alpha}=\left[\frac{\partial^{2} u_{\alpha}^{+}}{\partial x\partial y} (x, y),\frac{\partial^{2} u_{\alpha}^{-}}{\partial x\partial y}(x, y)\right] \ \text{for } 0 \leq \alpha \leq 1.$$

In the next part, by adapting the mixed Riemann-Liouville fractional integral notion of order q for real-valued functions f(x,y) in [1], we define fuzzy fractional integral for a fuzzy-valued function u : J → E. Firstly, using similar arguments in [6], we obtain the following Stacking lemma.

Lemma 2.1

Let q = (q ₁,q ₂) ∈ (0,1] × (0,1], u : J → E and $[u(x,y)]^{\alpha }=[u_{\alpha }^{-}(x,y), u_{\alpha }^{+}(x,y)]$ for all (x,y) ∈ J and α ∈ [0,1]. If $u_{\alpha }^{-}, u_{\alpha }^{+}\in L^{1}(J, \mathbb {R})$ , then for each (x,y) ∈ J , the family of closed interval

$$G_{\alpha}:= G_{\alpha}(x,y)=\left[\left( {~}^{RL} I_{0^{+}}^{q} u_{\alpha}^{-}\right)(x,y), \left( {~}^{RL} I_{0^{+}}^{q} u_{\alpha}^{+}\right)(x,y)\right] $$

defines afuzzy number v ∈ E such that [v]^α = G _α (x,y),where the mixed Riemann-Liouville fractional integral notion of order q for real-valuedfunctions $p \in L^{1}(J, \mathbb {R})$is defined as follows:

$${~}^{RL}I_{0^{+}}^{q} p(x,y)=\frac{1}{\Gamma(q_{1})\Gamma(q_{2})}{{\int}_{0}^{x}} {{\int}_{0}^{y}}(x-s)^{q_{1}-1}(y-t)^{q_{2}-1}p(s,t)dtds,$$

provided that the expression on the right hand side is defined for almost every (x,y) ∈ J.

Then, the following definition is well defined.

Definition 2.5

Let q = (q ₁,q ₂) ∈ (0,1] × (0,1] and u : J → E, $[u(x,y)]^{\alpha }=[u_{\alpha }^{-}(x,y),u_{\alpha }^{+}(x,y)]$ for all (x,y) ∈ J and α ∈ [0,1]. The left-sided mixed Riemann-Liouville fractional integral of order q for a fuzzy-valued function u is defined levelsetwise by

$$\left[{~}^{RL}_{F}\mathcal I_{0^{+}}^{q} u(x,y)\right]^{\alpha}=\left[\left( {~}^{RL} I_{0^{+}}^{q}u_{\alpha}^{-}\right)(x,y),({~}^{RL} I_{0^{+}}^{q}u_{\alpha}^{+}(x,y)\right] $$

and denoted by

$$ \left( {~}^{RL}_{F}\mathcal I_{0^{+}}^{q} u\right)(x,y)=\frac{1}{\Gamma(q_{1})\Gamma(q_{2})}{{\int}_{0}^{x}} {{\int}_{0}^{y}}(x-s)^{q_{1}-1}(y-t)^{q_{2}-1}u(s,t)dtds. $$

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In particular cases,

$$ \left( {~}^{RL}_{F}\mathcal I_{0^{+}}^{1} u\right)(x,y)= {{\int}_{0}^{x}} {{\int}_{0}^{y}} u(s,t)dtds, \ \text{ for } q=(1,1), \ (x,y )\in I. $$

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Proposition 2.1

[14] Let p = (p ₁,p ₂),q = (q ₁,q ₂) ∈ (0,1] × (0,1]. Then,

$${~}^{RL}_{F}\mathcal I_{0^{+}}^{p} {~}^{RL}_{F}\mathcal I_{0^{+}}^{q} u ={~}^{RL}_{F}\mathcal I_{0^{+}}^{p+q}u, $$

provided that the expressions on the right and the left hand sides are defined.

Definition 2.6

Let q = (q ₁,q ₂) ∈ [0,1) × [0,1), $u \in \mathcal W^{1}_{gH}(J,E) \cup \mathcal W^{2}_{gH}(J,E)$.

• A mapping $u \in \mathcal W^{1}_{gH}(J,E)$ is called (1)-Caputo gH-fractional differentiable of order q if the left-sided mixed Riemann-Liouville fractional integral of order 1 − q for D _{x y} u exists, and this derivative is defined by

  $${~}^{C}_{gH}\mathcal{D}_{1}^{q}u(x,y)={~}^{RL}_{F}\mathcal I_{0^{+}}^{1-q}(D_{xy}u(x,y)),\ (x,y)\in J.$$

• A mapping $u \in \mathcal W^{2}_{gH}(J,E)$ is called (2)-Caputo gH-fractional differentiable of order q if the left-sided mixed Riemann-Liouville fractional integral of order 1 − q for D _{x y} u exists, and this derivative is defined by

  $${~}^{C}_{gH}\mathcal {D}_2^{q}u(x,y) ={~}^{RL}_{F}\mathcal I_{0^{+}}^{1-q}(D_{xy}u(x,y)),\ (x,y)\in J,$$

where 1 − q = (1 − q ₁,1 − q ₂) ∈ (0,1] × (0,1].

The following technical lemmas will be used frequently in the rest of this paper.

Lemma 2.2

The equation

$$ t=e^{-\lambda t},\ \ \lambda>0 $$

(9)

has a unique solution t ₀that satisfies the following estimation

$$ t_{0}\le \frac{C}{\lambda^{\frac{1}{1+\varepsilon}}}, $$

(10)

where C > 0, ε > 0is arbitrary.

Proof

It is easy to see that the equation t = e ^−λt has a unique solution t ₀ ∈ (0,1) for given λ > 0. Taking logarithm on both sides of (9) we have lnt ₀ = −λ t ₀ or $\lambda =-\frac {\ln t_{0}}{t_{0}}=\frac 1{t_{0}}\ln \left (\frac 1{t_{0}}\right )=r\ln r$ with $r=\frac 1{t_{0}}>1$. Since lnr ≤ C r ^ε for arbitrary ε > 0, where C > 0 does not depend on r, it follows that r lnr ≤ C r ^{1 + ε} or $\lambda \le C\left (\frac {1}{t_{0}}\right )^{1+\varepsilon }$. Hence, $t_{0}\le \left (\frac {C}{\lambda }\right )^{\frac {1}{1+\varepsilon }},$ where C does not depend on t ₀. □

Lemma 2.3

Let λ > 0,x ∈ [0,a), q ∈ (0,1]be given. For all ε > 0, the following estimation

$$ {{\int}_{0}^{x}}(x-s)^{q-1}e^{\lambda s}ds \leq \frac {e^{\lambda x}}{q}\left[ 2\left( \frac{C}{\lambda^{\frac{1}{1+\varepsilon}}}\right)^{\frac{q}{2}}+\frac{1}{\lambda} \left( \frac{C}{\lambda^{\frac{1}{1+\varepsilon}}}\right)^{q}\right] $$

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holds, where C > 0doesnot depend on λ > 0,x ∈ [0,a), q ∈ (0,1].

Proof

By putting t = x − s, we have

$$I={{\int}_{0}^{x}}(x-s)^{q-1}e^{\lambda s}ds=-{{\int}_{x}^{0}}t^{q-1}e^{\lambda (x-t)}dt=e^{\lambda x} {{\int}_{0}^{x}} t^{q-1}e^{-\lambda t}dt.$$

Assume that t ₀ ∈ [0,x] is a solution of the equation t ^q−1 = e ^(1−q)λt. It is easy to see that $t_{0}^{1-q}=e^{(q-1)\lambda t_{0}}\le 1$ since q ≤ 1. It follows that t ₀ ≤ 1.

Case 1

For t ≥ t ₀, for each λ > 0, we will prove t ^q−1 ≤ e ^(1−q)λt. Indeed, let f(t) = t ^q−1 − e ^(1−q)λx, we have

$$f^{\prime}(x)=(q-1)x^{q-2}-\lambda (1-q) e^{(1-q)\lambda x}\le 0. $$

This proves that f is a nonincreasing function for all t ≥ t ₀ and then f(t) ≤ f(t ₀) = 0, that is t ^q−1 ≤ e ^(1−q)λt.

Case 2

For 0 ≤ t < t ₀ ≤ 1, we consider two functions g(t) = e ^−λt and $h(t)=t^{-\frac {q}{2}}$.

We can prove that g(t) ∈ [e ^−λ,1) and h(t) ∈ (1; + ∞) for all 0 ≤ t < t ₀ ≤ 1. Since $\max \limits _{[0,t_{0})}g(t)=1$ and h(t) is defined on (0,1],h(t) ∈ (1,∞) for all 0 < t < t ₀ ≤ 1, thus, g(t) ≤ h(t) ∀t ∈ (0,t ₀], or

$$e^{-\lambda t} < t^{-\frac{q}{2}} \ \ \text{for all} \ \ t \in (0, t_{0}].$$

From Case 1 and Case 2, we see that

$$\begin{array}{@{}rcl@{}} I&=&e^{\lambda x} \left[{\int}_{0}^{t_{0}}t^{q-1}e^{-\lambda t}dt+{\int}_{t_{0}}^{x} t^{q-1}e^{-\lambda t}dt\right] \\ &\leq& e^{\lambda x} \left[{\int}_{0}^{t_{0}}t^{q-1}t^{-\frac{q}{2}}dt+{\int}_{t_{0}}^{x} e^{\lambda (1-q)t}e^{-\lambda t}dt\right] \\ &=&e^{\lambda x} \left[{\int}_{0}^{t_{0}}t^{\frac{q}{2}-1}dt+{\int}_{t_{0}}^{x} e^{-\lambda qt}dt\right] \\ &=&e^{\lambda x} \left[\frac{2}{q}t_{0}^{\frac{q}{2}}-\frac{1}{\lambda q} (e^{-\lambda qx}-e^{-\lambda qt_{0}})\right]\\ &=&e^{\lambda x} \left[\frac{2}{q}t_{0}^{\frac{q}{2}}+\frac{1}{\lambda q} e^{-\lambda qt_{0}}-\frac{1}{\lambda q}e^{-\lambda qx}\right]. \end{array} $$

Since $e^{-\lambda t_{0}}=t_{0}$ and $\frac {1}{\lambda q}e^{-\lambda qx}>0$ for all x > 0, and hence,

$$I\le \frac{e^{\lambda x}}q \left[2t_{0}^{\frac{q}{2}}+\frac{1}{\lambda } {t_{0}^{q}}\right] \text{ for } x>0. $$

From inequation (10), we have $t_{0} \le \frac {C}{\lambda ^{\frac {1}{1+\varepsilon }}}$, then

$$2t_{0}^{\frac{q}{2}}+\frac{1}{\lambda}e^{-\lambda q t_{0}} \le 2\left( \frac{C}{\lambda^{\frac{1}{1+\varepsilon}}}\right)^{\frac{q}{2}}+\frac{1}{\lambda}\left( \frac{C}{\lambda^{\frac{1}{1+\varepsilon}}}\right)^{q}.$$

Therefore,

$$I={{\int}_{0}^{x}}(x-s)^{q-1}e^{\lambda s}ds \leq e^{\lambda x}G(\lambda q), $$

where

$$ G(\lambda q)=\dfrac1q\left[ 2\left( \frac{C}{\lambda^{\frac{1}{1+\varepsilon}}}\right)^{\frac{q}{2}}+\frac{1}{\lambda}\left( \frac{C}{\lambda^{\frac{1}{1+\varepsilon}}}\right)^{q}\right]. $$

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□

Remark 2.2

For λ > 0 large enough, G(λ q) < 1.

3 The Well-Posedness

In this section, we consider a Darboux problem (which is sometimes called a characteristic initial value problem) of nonlinear integro-differential wave equation (1)–(2) in domain J _∞ := [0,∞) × [0,∞)

$$\begin{array}{@{}rcl@{}} {~}^{C}_{gH}\mathcal{D}_k^{q} u(x,y)&=&f\left( x,y,u,{{\int}_{0}^{y}}m(y-s)g(u(x,s))ds\right),\ (x, y) \in J_{\infty}\\ u(x, 0)& =& \eta_{1}(x), \ x \in [0,\infty)\\ u(0, y) &=& \eta_{2}(y), \ y \in [0,\infty), \end{array} $$

where ${~}^{C}_{gH}\mathcal {D}_k^{q}\ (k=1,2)$ are Caputo gH-fractional derivative operators defined in Definition 2.6, q = (q ₁,q ₂) ∈ (0,1]² ; η ₁ ∈ C([0,∞),E), η ₂ ∈ C([0,∞),E) are given functions such that η ₂(y) ⊖ η ₁(0) exists for all y ∈ [0,∞) and η ₁(0) = η ₂(0) = u ₀ ∈ E; a fuzzy-valued mapping g : E → E is an integrable function; f ∈ C(J _∞ × E ²,E), $m\in L^{2}([0,\infty ),\mathbb {R})$ satisfies

$$ M:=\left( {\int}_{0}^{\infty} m^{2}(s)ds\right)^{\frac{1}{2}} <\infty. $$

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For (x,y) ∈ J _∞ , we denote

$$ \psi{(x,y)}=\eta_{1}(x)+\eta_{2}(y) \ominus \eta_{1}(0) $$

(14)

and

$$\begin{array}{@{}rcl@{}} \mathbb{G}[u](x,y)&=& {{\int}_{0}^{y}}m(y-s)g(u(x,s))ds,\ (x,y)\in J_{\infty},\\ \mathbb{F}[u](x,y)&=&f\left( x,y,u(x,y),\mathbb{G}[u](x,y)\right), \ (x,y)\in J_{\infty}. \end{array} $$

Adapting Lemma 4.4 in [15], we have the following assertions.

Lemma 3.1

Let $u\in \mathcal W^{1}_{gH}(J,E) \cup \mathcal W^{2}_{gH}(J,E)$ be a fuzzy-valued function satisfying(1)–(2)in J _∞ .

• (1) If $u\in \mathcal {W}^{1}_{gH}(J_{\infty },E)$ , then u satisfies

  $$ u(x, y)=\psi(x, y)+{~}^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[u](x,y) \text{ for all } (x, y) \in J_{\infty}, $$

  (15)

• (2) If $u\in \mathcal {W}^{2}_{gH}(J_{\infty },E)$,then u satisfies

  $$ u(x, y)=\psi(x, y) \ominus (-1){~}^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[u](x,y) \text{ for all } (x, y) \in J_{\infty}. $$

  (16)

Definition 3.1

• (1) A function u ∈ C(J _∞ ,E) satisfying integral equation (15) is called a (1)-weak solution of problem (1)–(2), with k = 1.

• (2) A function u ∈ C(J _∞ ,E) satisfying integral equation (16) is called a (2)-weak solution of problem (1)–(2), with k = 2.

Remark 3.1

Some of the following assumptions will be used throughout this paper.

• ( H ₁ ) A mapping f : J _∞ × E × E → E satisfies the Lipschitz condition with respect to two last variables, i.e., there exists a positive real number L such that

  $$d_{\infty}(f(x,y,\varphi_{1},\varphi_{2}),f(x,y,\tilde{\varphi_{1}},\tilde{\varphi_{2}})) \leq L[d_{\infty}(\varphi_{1},\tilde{\varphi_{1}})+d_{\infty}(\varphi_{2}, \tilde{\varphi_{2}})]$$

  for all $(x,y)\in J_{\infty },\ \varphi _{1}, \varphi _{2}, \tilde {\varphi _{1}}, \tilde {\varphi _{2}} \in E$ and

  $$d_{\infty} (f(x,y,\hat{0},\mathbb{G}(\hat{0})),\hat{0}) \leq \hat{M}e^{c(x+y)}$$

  for all (x,y) ∈ J _∞ , where $\hat {M},c$ are positive real numbers.

• ( H ₂ ) A mapping g : E → E is increasing and satisfies Lipschitz condition, i.e., there exists a positive real number K such that for all $\varphi , \tilde {\varphi } \in E$

  $$d_{\infty}(g(\varphi),g(\tilde{\varphi})) \leq Kd_{\infty}(\varphi,\tilde{\varphi}).$$

• ( H ₃) There exists positive real numbers M _i and c _i (i = 1,2) such that

  $$d_{\infty}(\eta_{1}(x),\hat{0}) \leq M_{1}e^{c_{1}x},\ d_{\infty}(\eta_{2}(y),\hat{0})\le M_{2}e^{c_{2}y},\ (x,y)\in J_{\infty}.$$

For λ > 0, we consider

$$C_{\lambda}(J_{\infty}, E)=\left\{u \in C(J_{\infty},E)| \sup\limits_{(x,y) \in J_{\infty}}d_{\infty}(u(x,y), \hat{0})e^{-\lambda (x+y)}<\infty\right\},$$

and the supremum weighted metric on C _λ (J _∞ ,E) is

$$ H_{\lambda}(f, g) = \sup \limits_{\left( s, t\right) \in J_{\infty}} \{d_{\infty} (f\left( s, t\right), g\left( s, t\right))e^{-\lambda \left( s + t \right)}\}. $$

(17)

Lemma 3.2

For each λ > 0, (C _λ (J _∞ ,E),H _λ )is a complete weighted metric space.

Proof

Suppose that {u _m } _m≥1 is a Cauchy sequence in C _λ (J _∞ ,E).

Step 1

The sequence {u _m } _m≥1 converges to a function u in C(J _∞ ,E).

In fact, for each 𝜖 > 0, there exists $n_{\epsilon } \in \mathbb {N}$ such that for all m,n ≥ n _𝜖 , we have H _λ (u _m ,u _n ) < 𝜖. Therefore,

$$ d_{\infty}(u_{m}(x,y),u_{n}(x,y)) <\epsilon e^{\lambda(x+y)},\ (x,y) \in J_{\infty} $$

(18)

for all m,n ≥ n _𝜖 .

It follows that for each (x,y) ∈ J _∞ , {u _m (x,y)} _m≥1 is also a Cauchy sequence in E. Since (E,d _∞ ) is a complete metric space (see in [15]), there exists u(x,y) ∈ E such that

$$ \lim \limits_{m \to \infty}d_{\infty}(u_{m}{(x,y)},u(x,y))=0. $$

(19)

For fixed (x,y) ∈ J _∞ , by taking limit on the left hand side of (18) combined with (19), we obtain

$$ d_{\infty}(u_{n}(x,y),u(x,y))< \epsilon e^{\lambda (x+y)} \text{ for } n \geq n_{\epsilon},$$

which is equivalent to

$$ d_{\infty}(u_{n}(x,y),u(x,y))e^{-\lambda (x+y)}< \epsilon $$

(20)

for n ≥ n _𝜖 ;(x,y) ∈ J _∞ . Hence,

$$\sup \limits_{(x,y) \in J_{\infty}} d_{\infty}(u_{n}(x,y),u(x,y))e^{-\lambda(x+y)}< \epsilon $$

for all n ≥ n _𝜖 . This proves that H _λ (u _n ,u) < 𝜖 hold for all n ≥ n _𝜖 . Thus, $\lim \limits _{n \to \infty } H_{\lambda }(u_{n},u) =0.$

Now, we show that u ∈ C(J _∞ ,E). In fact, since u _m is a continuous function, there exists $\delta ^{1}_{\epsilon }>0$ such that for all (x,y),(x ₀,y ₀) ∈ J _∞ satisfying the condition $|x-x_{0}|+|y-y_{0}| < \delta ^{1}_{\epsilon },$ we have

$$d_{\infty}(u_{m}(x,y),u_{m}(x_{0},y_{0}))< \frac{\epsilon}{3}.$$

Denote $\epsilon _{0}=\epsilon e^{\lambda (x_{0}+y_{0})}$. So with $\delta ^{2}_{\epsilon } = \frac {1}{\lambda }\ln \frac {\epsilon }{6\epsilon _{0}}$, if (x,y) ∈ J _∞ satisfies $|x-x_{0}|+|y-y_{0}| < \delta ^{2}_{\epsilon }$, then from (20) we have

$$d_{\infty}(u_{m}(x,y),u(x,y)) < \epsilon e^{\lambda(x+y)}=\epsilon_{0} e^{\lambda(x-x_{0}+y-y_{0})}<\epsilon_{0} e^{\delta^{2}_{\epsilon}}=\frac{\epsilon}{6} \text{ for } m \geq n_{\epsilon}.$$

Let $\delta _{\epsilon }=\min \left \{\delta ^{1}_{\epsilon },\delta ^{2}_{\epsilon }\right \}$. Whenever (x,y) ∈ J _∞ satisfies |x − x ₀| + |y − y ₀|≤ δ _𝜖 , we have

$$\begin{array}{@{}rcl@{}} d_{\infty}(u(x,y),u(x_{0},y_{0}))&\leq& d_{\infty}(u(x,y),u_{m}(x,y))+d_{\infty}(u_{m}(x,y),u_{m}(x_{0},y_{0})) \\ &&+d_{\infty}(u_{m}(x_{0},y_{0}),u(x_{0},y_{0}))\\ &\leq& \frac{\epsilon}{6} + \frac{\epsilon}{3} +\frac{\epsilon}{6} < \epsilon. \end{array} $$

That implies u is a continuous function on J _∞ .

Step 2

We now prove that u ∈ C _λ (J _∞ ,E). The remaining task is to prove that

$$\sup \limits_{(x,y) \in J_{\infty}}d_{\infty}(u(x,y),\hat{0})e^{-\lambda(x+y)}< \infty.$$

In fact, we have

$$\begin{array}{@{}rcl@{}} &&\sup \limits_{(x,y) \in J_{\infty}}d_{\infty}(u(x,y),\hat{0})e^{-\lambda(x+y)}\\ &\leq& \sup \limits_{(x,y)\in J_{\infty}} d_{\infty}(u(x,y),u_{m}(x,y))e^{-\lambda(x+y)}+\sup \limits_{(x,y)\in J_{\infty}} d_{\infty}(u_{m}(x,y),\hat{0})e^{-\lambda(x+y)}\\ &=&H_{\lambda}(u,u_{m})+\sup \limits_{(x,y)\in J_{\infty}} d_{\infty}(u_{m}(x,y),\hat{0})e^{-\lambda(x+y)}. \end{array} $$

Because u _m ∈ E _λ and $\lim _{m \to \infty } H_{\lambda }(u,u_{m}) =0$, we obtain

$$\sup \limits_{(x,y) \in J_{\infty}}d_{\infty}(u(x,y),\hat{0})e^{-\lambda(x+y)}<\infty.$$

After conducting the above three steps, we see that u _m → u ∈ C _λ (J _∞ ,E) with respect to the metric H _λ ; thus, C _λ (J _∞ ,E) is a complete metric space. □

Remark 3.2

It is easy to see that for all (x,y) ∈ J _∞ ,

$$\begin{array}{@{}rcl@{}} &&d_{\infty}(\mathbb{G}[u](x,y),\mathbb{G}[v](x,y))\\ &\leq& {{\int}_{0}^{y}}m(y-s)d_{\infty}(g(u(x,s)),g(v(x,s)))ds\\ &\le& K {{\int}_{0}^{y}}m(y-s)d_{\infty}(u(x,s),v(x,s))ds\\ &\le& K \left( {{\int}_{0}^{y}}m^{2}(y-s)ds\right)^{\frac{1}{2}}\left( {{\int}_{0}^{y}}d_{\infty}^{2}(u(x,s),v(x,s))ds\right)^{\frac{1}{2}}\\ &\le &KM\left( {{\int}_{0}^{y}}d_{\infty}^{2}(u(x,s),v(x,s))ds\right)^{\frac{1}{2}}. \end{array} $$

(21)

Furthermore,

$$\begin{array}{@{}rcl@{}} &&d_{\infty}(\mathbb{F}[u](x,y),\mathbb{F}[v](x,y)) \\ &\leq& L[ d_{\infty}(u(x,y),v(x,y))+ d_{\infty}(\mathbb{G}[u](x,y),\mathbb{G}[v](x,y))]\\ &\le& L\left[ d_{\infty}(u(x,y),v(x,y))+ KM\left( {{\int}_{0}^{y}}d_{\infty}^{2}(u(x,s),v(x,s))ds\right)^{\frac{1}{2}}\right]\\ &\le& LH_{\lambda}(u,v)\left[e^{\lambda(x+y)}+ KM\left( {{\int}_{0}^{y}}e^{2\lambda(x+s)}ds\right)^{\frac{1}{2}}\right]\\ &\le& L\left( 1+ \frac{KM}{\sqrt{2\lambda}}\right)H_{\lambda}(u,v)e^{\lambda(x+y)}. \end{array} $$

(22)

Theorem 3.1

Suppose that the assumptions ( H ₁ ) , ( H ₂ ) , ( H ₃ ) are satisfied with function f ∈ C(J _∞ × E ²,E). Then, the problem(1)–(2)has a unique (1)-weak fuzzy solution u ∈ C(J _∞ ,E). Moreover, there exist $\tilde {M}>0$ and λ ₁ > 0such that

$$ d_{\infty}(u(x,y),\hat{0}) \leq \tilde{M} e^{\lambda_{1}(x+y)} \text{ for } (x,y) \in J_{\infty}. $$

(23)

Proof

We consider the operator T ₁ : C(J _∞ ,E) → C(J _∞ ,E) by

$$T_{1}(u(x,y)):= \psi(x, y)+^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[u](x,y) \text{ for } (x,y) \in J_{\infty}.$$

Step 1

We prove that T ₁(C _λ (J _∞ ,E)) ⊂ C _λ (J _∞ ,E) for λ > 0 arbitrary.

In fact, assume that u ∈ C _λ (J _∞ ,E). Then, there exists ρ > 0 such that

$$d_{\infty}(u(s,t),\hat{0}) \leq \rho e^{\lambda (s+t)}$$

for all (s,t) ∈ J _∞ . We have for all (s,t) ∈ J _∞

$$\begin{array}{@{}rcl@{}} d_{\infty}(f(s,t,u(s,t), \mathbb{G}[u](s,t)), \hat{0})&\le& d_{\infty}(f(s,t,u(s,t), \mathbb{G}[u](s,t),f(s,t,\hat{0}, \mathbb{G}[\hat{0}]))\\ &&+~d_{\infty}(f(s,t,\hat{0}, \mathbb{G}[\hat{0}]), \hat{0}). \end{array} $$

From (22), one has

$$\begin{array}{@{}rcl@{}} &&d_{\infty}(f(s,t,u(s,t), \mathbb{G}[u](s,t)),f(s,t,\hat{0}, \mathbb{G}[\hat{0}](s,t)))\\ &=&d_{\infty}(\mathbb{F}[u](s,t),\mathbb{F}[\hat{0}](s,t))\\ &\le& L\left[ 1+ \frac{KM}{\sqrt{2\lambda}}\right]H_{\lambda}(u,\hat{0})e^{\lambda(s+t)}\\ &\le& L\left[ 1+ \frac{KM}{\sqrt{2\lambda}}\right] \rho e^{\lambda(s+t)}. \end{array} $$

(24)

From (24) and hypotheses (H ₁), one gets

$$ d_{\infty}(f(s,t,u(s,t), \mathbb{G}[u](s,t)), \hat{0}) \le L\left[ 1+ \frac{KM}{\sqrt{2\lambda}}\right]\rho e^{\lambda(s+t)}+\hat{M}e^{c(s+t)}. $$

(25)

Furthermore, from Lemma 2.3 and (25), for all λ ≥ c > 0, we have

$$\begin{array}{@{}rcl@{}} &&d_{\infty}\left( {~}^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[u](x,y), \hat{0}\right)\\ &\le& \frac{1}{\Gamma(q_{1}) \Gamma(q_{2})}{{\int}_{0}^{x}} {{\int}_{0}^{y}} (x-s)^{q_{1}-1}(y-t)^{q_{2}-1}d_{\infty}(f(s,t,u(s,t), \mathbb{G}[u](s,t)), \hat{0}) dtds\\ &\le& \frac{1}{\Gamma(q_{1}) \Gamma(q_{2})}{{\int}_{0}^{x}} {{\int}_{0}^{y}} (x-s)^{q_{1}-1}(y\,-\,t)^{q_{2}-1} \left[L\left[ 1\!+ \frac{KM}{\sqrt{2\lambda}}\right]\rho e^{\lambda(s+t)}+\hat{M}e^{c(s+t)}\right] dsdt\\ &\le& \frac{L\left[ 1+ \frac{KM}{\sqrt{2\lambda}}\right]\rho+\hat{M}}{\Gamma(q_{1}) \Gamma(q_{2})}{{\int}_{0}^{x}} {{\int}_{0}^{y}} (x-s)^{q_{1}-1}(y-t)^{q_{2}-1} e^{\lambda(s+t)} dsdt\\ &\le &\left[L\left( \rho+\frac{KM\rho}{\sqrt{2\lambda}}\right)+\hat{M}\right]G(\lambda q_{1})G(\lambda q_{2}) e^{\lambda(x+y)}\frac {1}{\Gamma(q_{1})\Gamma(q_{2})}. \end{array} $$

(26)

If λ ≥ max{c,c ₁,c ₂}, then from hypothesis ( H ₃ ) and (26), we have for all (x,y) ∈ J _∞

$$\begin{array}{@{}rcl@{}} &&d_{\infty}(T_{1}u(x,y), \hat{0})e^{-\lambda(x+y)}\\ & \leq& \left[d_{\infty}(\eta_{1}(x), \hat{0}) + d_{\infty}(\eta_{2}(y), \hat{0}) +d_{\infty}(\eta_{1}(0), \hat{0}) + d_{\infty}\left( {~}^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[u](x,y), \hat{0}\right)\right]e^{-\lambda(x+y)}\\ &\leq& 2M_{1}e^{(c_{1}-\lambda)(x+y)}+M_{2}e^{(c_{2}-\lambda)(x+y)}+\left[L\left( \rho+\frac{KM\rho}{\sqrt{2\lambda}}\right)+\hat{M}\right]\frac {G(\lambda q_{1})G(\lambda q_{2})}{\Gamma(q_{1})\Gamma(q_{2})}\\ &\leq& 2M_{1}+M_{2}+ \left[L\left( \rho+\frac{KM\rho}{\sqrt{2\lambda}}\right)+\hat{M}\right]\frac {G(\lambda q_{1})G(\lambda q_{2}) }{\Gamma(q_{1})\Gamma(q_{2})}< \infty. \end{array} $$

(27)

It shows that T ₁(u) ∈ C _λ (J _∞ ,E).

Step 2

We prove that T ₁ is a contraction mapping.

Indeed, from the property of the metric d _∞ , (22) and Lemma 2.3, we have

$$\begin{array}{@{}rcl@{}} d_{\infty}&&(T_{1}u(x,y), \ T_{1}v(x,y))\\ &&\leq \frac{1}{\Gamma(q_{1})\Gamma(q_{2})}{{\int}_{0}^{x}} {{\int}_{0}^{y}} (x-s)^{q_{1}-1}(y-t)^{q_{2}-1} d_{\infty}\left( \mathbb{F}[u](s,t),\mathbb{F}[v](s,t)\right)dtds \\ && \le \frac{L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right)}{\Gamma(q_{1}) \Gamma(q_{2})}H_{\lambda}(u,v){{\int}_{0}^{x}} {{\int}_{0}^{y}} (x-s)^{q_{1}-1}(y-t)^{q_{2}-1}e^{\lambda (s+t)}dsdt \\ && \le \frac{L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right)}{\Gamma(q_{1}) \Gamma(q_{2})}H_{\lambda}(u,v)G(\lambda q_{1})G(\lambda q_{2})e^{\lambda (x+y)}. \end{array} $$

(28)

It follows that

$$H_{\lambda}(T_{1}u,T_{1}v) \leq \frac {L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right)}{\Gamma(q_{1})\Gamma(q_{2})}G(\lambda q_{1})G(\lambda q_{2})H_{\lambda}(u,v). $$

If we choose λ ≥ max{λ ₀,c,c ₁,c ₂}, where λ ₀ is defined by (29), then

$$\frac {L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right)}{\Gamma(q_{1})\Gamma(q_{2})}G(\lambda q_{1})G(\lambda q_{2})<1.$$

Thus, T ₁ is a contraction mapping in C _λ (J _∞ ,E). Consequently, there exists a unique fuzzy-valued function u defined on J _∞ , satisfying equations (15). This is (1)-weak fuzzy solution of (1)–(2) in C(J _∞ ,E).

Finally, denote

$$\lambda_{0}:=\inf\left\{\lambda>0\text{ such that }\ \frac {\left( 1+\frac{KM}{\sqrt{2\lambda}}\right)L}{\Gamma(q_{1})\Gamma(q_{2})}G(q_{1}\lambda)G(q_{2}\lambda)<1\right\}, $$

(29)

where G(.) is defined by (12).

λ ₁ = max {λ ₀,c,c ₁,c ₂} and

$$\tilde{M}:= 2M_{1}+M_{2}+ G(\lambda_{1} q_{1})G(\lambda_{1} q_{2})\frac {\left[L\left( \rho+\frac{KM\rho}{\sqrt{2\lambda_{1}}}\right)+\hat{M}\right]}{\Gamma(q_{1})\Gamma(q_{2})}.$$

Assume that $u \in C_{\lambda _{1}}(J_{\infty }, E)$ is a fixed point of T ₁, i.e., T ₁(u) = u. Then, from (27), we have

$$d_{\infty}(u(x,y), \hat{0})e^{-\lambda_{1}(x+y)} \leq 2M_{1}+M_{2}+ \left[L\left( \rho+\frac{KM\rho}{\sqrt{2\lambda_{1}}}\right)+\hat{M}\right]\frac {G(\lambda_{1} q_{1})G(\lambda_{1}q_{2})}{\Gamma(q_{1})\Gamma(q_{2})} $$

or

$$d_{\infty}(u(x,y),\hat{0}) \leq \tilde{M} e^{\lambda_{1}(x+y)} $$

for all (x,y) ∈ J _∞ . Hence, we have inequation (23). It means that the unique (1)-weak solution u is bounded in the space (C _λ (J _∞ ,E),H _λ ), λ ≥ λ ₁. The theorem is proved completely. □

Now for each λ > 0, we consider the set

$$\hat{C}_{\lambda}(J_{\infty}, E)= \left\{u \in C_{\lambda}(J_{\infty}, E): \exists\ \psi(x, y) \ominus (-1) \,{~}^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[u](x,y) \text{ for all }(x,y)\in J_{\infty} \right\}.$$

Lemma 3.3

[14] If $m\in C([0,\infty ),\mathbb {R})$ , g ∈ C(E,E)and f ∈ C(J _∞ × E ²,E), then $(\hat {C}_{\lambda }(J_{\infty }, E), H_{\lambda })$ is a complete metric space for each positive number λ .

Theorem 3.2

Suppose that $m: [0,\infty )\to \mathbb {R}$ is continuous, g ∈ C(E,E), f ∈ C(J _∞ × E ²,E)and all assumptions ( H ₁ ) − ( H ₃ ) are fulfilled. In addition, the following assumptions are satisfied, with λ ≥ λ ₀ in(29).

( H ₄ ) :

$\hat {C}_{\lambda }(J_{\infty }, E) \ne \emptyset .$

( H ₅ ) :

If $u \in \hat {C}_{\lambda }(J_{\infty }, E)$ , then the following H-difference exists for all (x,y) ∈ J _∞

$$ \psi(x, y) \ominus (-1) \,{~}^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[\nu(u)](x,y), $$

(30)

where

$$\nu (u)(x,y) = \psi(x,y) \ominus (-1)^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[u](x,y).$$

Then, problem (1)–(2)has a unique (2)-weaksolution on J _∞ and the estimation (23)still valid.

Proof

We transform the issue into a fixed point problem. Consider an operator

$$T_{2}: \hat{C}_{\lambda}(J_{\infty}, E) \to \hat{C}_{\lambda}(J_{\infty}, E)$$

defined by

$$T_{2}(u(x,y))= \psi(x, y) \ominus (-1)^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[u](x,y).$$

By revisiting the fact that d _∞ (u ⊖ v,e ⊖ f) ≤ d _∞ (u,e) + d _∞ (v,f) and inequation (28), we have

$$\begin{array}{@{}rcl@{}} &&d_{\infty}(T_{2}u(x,y), \ T_{2}v(x,y)) \\ &\leq& \frac{1}{\Gamma(q_{1})\Gamma(q_{2})}{{\int}_{0}^{x}} {{\int}_{0}^{y}} (x-s)^{q_{1}-1}(y-t)^{q_{2}-1} d_{\infty}\left( \mathbb{F}[u](s,t),\mathbb{F}[v](s,t)\right)dtds \\ & \le& \frac{\left( 1+\frac{KM}{\sqrt{2\lambda}}\right)L}{\Gamma(q_{1}) \Gamma(q_{2})}G(\lambda q_{1})G(\lambda q_{2})e^{\lambda (x+y)}H_{\lambda}(u,v). \end{array} $$

Thus,

$$H_{\lambda}(T_{2}u,T_{2}v) \leq \frac {L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right)}{\Gamma(q_{1})\Gamma(q_{2})}G(\lambda q_{1})G(\lambda q_{2})H_{\lambda}(u,v). $$

Since λ ≥ λ ₀ then

$$\frac {\left( 1+\frac{KM}{\sqrt{2\lambda}}\right)L}{\Gamma(q_{1})\Gamma(q_{2})}G(q_{1}\lambda)G(q_{2}\lambda)<1,$$

for which T ₂ is a contraction mapping. Consequently, T ₂ has a unique fixed point $u\in \hat {C}_{\lambda }(J_{\infty }, E)$, which is a (2)-weak solution of (1)–(2) and satisfies estimation (23). □

Remark 3.3

The main difficulty in this result is whether the H-difference (30) exists and which classes of fuzzy-valued functions can satisfy this condition. We now revisit here a special class of fuzzy-valued mappings that can be voted as an example for functions satisfying Theorem 3.2.

Denote by $\mathcal {T}$ the set of all triangular fuzzy sets in E. For $u\in \mathcal {T}$, u has the parametric form $u=(u^{0}_{-},u^{1},u^{0}_{+})$. Bede and Stefanini [7] gave a sufficient condition for the existence of H-difference $u\ominus v,\ u,v\in \mathcal {T}$ as follows.

Lemma 3.4

[7] If $u,v\in \mathcal {T}$ and $len(v)\le \min \{u^{1}-u^{0}_{-},u^{0}_{+}-u^{1}\}$ , then H-difference u ⊖ v exists.

Lemma 3.5

Suppose that hypotheses ( H ₁ ) −( H ₃ ) are satisfied with $f: J_{\infty }\times \mathcal {T}^{2}\to \mathcal {T}$ . Moreover, assume that $\psi (x,y)\in \mathcal {T}$ for all (x,y) ∈ J _∞ and

$$z(x,y) \ge \frac{2}{\Gamma(q_{1})\Gamma(q_{2})}\left[L\left( \rho+KM\frac\rho{\sqrt{2\lambda}}\right)+\hat{M}\right]G(\lambda q_{1})G(\lambda q_{2}) e^{\lambda(x+y)},$$

where

$$z(x,y)=\min\{(\psi(x,y))^{1}-(\psi(x,y))^{0}_{-},(\psi(x,y))^{0}_{+}-(\psi(x,y))^{1}\}.$$

Then, theoperator T ₂is welldefined, i.e., $T_{2}(\hat {C}_{\lambda }(J_{\infty }, \mathcal T))\subset \hat {C}_{\lambda }(J_{\infty }, \mathcal T)$for all λ > 0arbitrary.

Proof

It is easy to see that for all $v\in \mathcal {T}$, one gets $len (v)=v^{0}_{+}-v^{0}_{-}\le |v^{0}_{+}|+|v^{0}_{-}|$ and $\ d_{\infty }(v,\hat {0})=\max \left \{ |v^{0}_{+}|,|v^{0}_{-}|\right \}$. Thus, $len (v)\le 2d_{\infty }(v,\hat {0})$ for all $v\in \mathcal {T}.$

For all (x,y) ∈ J _∞ , set $v(x,y)=(-1)^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb {F}[u](x,y)$. Since $f: J_{\infty }\times (\mathcal {T})^{2}\to \mathcal {T}, v(x,y)\in \mathcal {T}$, it follows from (26) that

$$\begin{array}{@{}rcl@{}} len (v(x,y))&&\le 2d_{\infty}(v(x,y),\hat{0})\le 2d_{\infty}\left( {~}^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[u](x,y),\hat{0}\right)\\ &&\le \frac{2}{\Gamma(q_{1})\Gamma(q_{2})}\left[L\left( \rho+KM\frac\rho{\sqrt{2\lambda}}\right)+\hat{M}\right]G(\lambda q_{1})G(\lambda q_{2}) e^{\lambda(x+y)}\\ && \le z(x,y). \end{array} $$

Hence, by applying Lemma 3.4, H-difference $\psi (x,y)\ominus (-1)^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb {F}[u](x,y)$ exists for every (x,y) ∈ J _∞ . □

The following result gives the continuous dependence of fuzzy solutions on initial conditions.

Theorem 3.3

Assume that $m: [0,\infty )\to \mathbb {R}$ is continuous, g ∈ C(E,E), f ∈ C(J _∞ × E ²,E)and all hypotheses ( H ₁ ) − ( H ₅ ) are satisfied. Let u and v be fuzzy solutions of the following integral equations, respectively

$$\begin{array}{@{}rcl@{}} u(x, y, \omega) & = \varphi_{1}(x, y, \omega) \ominus (-1)^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[u](x,y)\\ v(x, y, \omega) & = \varphi_{2}(x, y, \omega) \ominus (-1)^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[v](x,y), \end{array} $$

where φ ₁(x,y),φ ₂(x,y)are defined on J _∞ . Then, we have

$$H_{\lambda}(u,v) \leq \tau_{o}H_{\lambda}(\varphi_{1}, \varphi_{2}),$$

where λ ≥ max{λ ₀,c,c ₁,c ₂},

$$\tau_{o}:=\frac{\Gamma(q_{1}) \Gamma(q_{2})}{\Gamma(q_{1}) \Gamma(q_{2}) - L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right) G(\lambda q_{1}) G(\lambda q_{2})}.$$

Proof

For all (x,y) ∈ J _∞ , we have

$$\begin{array}{@{}rcl@{}} &&d_{\infty}(u(x,y), v(x,y)) \\ &\le& d_{\infty}(\varphi_{1}(x, y), \varphi_{2}(x, y))+ d_{\infty}\left( {~}^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[u](x,y),{~}^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)\right)\\ &\leq& d_{\infty}(\varphi_{1}(x, y), \varphi_{2}(x, y))\\ && +~ \frac{1}{\Gamma(q_{1})\Gamma(q_{2})}{{\int}_{0}^{x}} {{\int}_{0}^{y}} (x-s)^{q_{1}-1}(y-t)^{q_{2}-1} d_{\infty}\left( \mathbb{F}[u](s,t),\mathbb{F}[v](s,t)\right)dtds. \end{array} $$

By the assertion (22), we get

$$\begin{array}{@{}rcl@{}} &&d_{\infty}(u(x,y), v(x,y))\\ &\leq& d_{\infty}(\varphi_{1}(x, y), \varphi_{2}(x, y))+\frac{L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right) H_{\lambda}(u,v)}{\Gamma(q_{1}) \Gamma(q_{2})} {{\int}_{0}^{x}} {{\int}_{0}^{y}} (x\!-s)^{q_{1}-1}(y-t)^{q_{2}-1} e^{\lambda(s+t)}dtds\\ &\leq& d_{\infty}(\varphi_{1}(x, y), \varphi_{2}(x, y)) +\frac{L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right) H_{\lambda}(u,v)}{\Gamma(q_{1}) \Gamma(q_{2})} e^{\lambda(x+y)}G(\lambda q_{1}) G(\lambda q_{2}). \end{array} $$

Multiplying by e ^{−λ(x + y)} and taking supremum both sides of this inequation, we obtain

$$H_{\lambda}(u,v) \leq H_{\lambda}(\varphi_{1}, \varphi_{2})+\frac{L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right) H_{\lambda}(u,v)}{\Gamma(q_{1}) \Gamma(q_{2})} G(\lambda q_{1}) G(\lambda q_{2}). $$

Set $\tau _{o}:=\frac {\Gamma (q_{1}) \Gamma (q_{2})}{\Gamma (q_{1}) \Gamma (q_{2}) - L\left (1+\frac {KM}{\sqrt {2\lambda }}\right ) G(\lambda q_{1}) G(\lambda q_{2})}.$ Since λ ≥ max{λ ₀,c,c ₁,c ₂}, then τ _o > 0. We have H _λ (u,v) ≤ τ _o H _λ (φ ₁,φ ₂). It completes the proof. □

4 The Ulam Stability

Firstly, we consider the following inequation

$$ d_{\infty}\left( \!{~}^{C}_{gH}\mathcal{D}_{k}^{q} v(x,y), \mathbb{F}[v](x,y)\right)\leq \epsilon \, \, \text{for all} \, \, (x,y) \in J_{\infty}, \ k\in \{1,2\}. $$

(31)

Definition 4.1

• 1. A mapping $v \in \mathcal W_{gH}^{1}(J_{\infty }, E)$ is called a solution of in (31) with k = 1 if there exists a function h ₁ ∈ C _λ (J _∞ ,E) such that

  • (i) $d_{\infty }(h_{1}(x,y), \hat {0}) \leq \epsilon \, \text { for all} \, (x,y) \in J_{\infty },$

  • (ii) $^{C}_{gH}\mathcal {D}_{1}^{q} v(x,y)=\mathbb {F}[v](x,y) + h_{1}(x,y)\, \, \text {for all} \, (x, y) \in J_{\infty }.$

• 2. A mapping $v \in \mathcal W_{gH}^{2}(J_{\infty }, E)$ is called a solution of in (31) with k = 2 if there exists a function h ₂ ∈ C _λ (J _∞ ,E) such that

  • (i) $d_{\infty }(h_{2}(x,y), \hat {0}) \leq \epsilon \, \text { for all} \,\in J_{\infty },$

  • (ii) $^{C}_{gH}\mathcal {D}_{2}^{q} v(x,y)=\mathbb {F}[v](x,y) + h_{2}(x,y) \, \, \text {for all} \, (x, y) \in J_{\infty }.$

Lemma 4.1

1. If v is a solution of in(31)with k = 1, then it satisfies the following integral inequation

$$d_{\infty}\left( v(x,y), v(x,0) + v(0,y) \ominus v(0,0) +^{RL}_{F}\mathcal I^{q}_{0^{+}} \mathbb{F}[v](x,y)\right) \leq \frac{\epsilon x^{q_{1}}y^{q_{2}}}{\Gamma (q_{1}+1) \Gamma(q_{2}+1)} $$

(32)

for all (x,y) ∈ J _∞ .

2. If v is a solution of in (31)with k = 2,then it satisfies the following integral inequation

$$d_{\infty}\left( v(x,y), v(0,y) \!+ v(x,0)\ominus v(0,0) \!\ominus\! (-1)^{RL}_{F}\mathcal I^{q}_{0^{+}} \mathbb{F}[v](x,y)\right) \leq\! \frac{\epsilon x^{q_{1}}y^{q_{2}}}{\Gamma (q_{1}+1) \Gamma(q_{2}+1)} $$

(33)

for all (x,y) ∈ J _∞ .

Proof

• (1) Let v(x,y) be a solution of in (31) with k = 1. Then, there exists a function h ₁ ∈ C _λ (J _∞ ,E) such that

  $${~}^{C}_{gH}\mathcal{D}_{1}^{q} v(x,y)=\mathbb{F}[v](x,y) + h_{1}(x,y) \, \, \text{for all} \, (x, y) \in J_{\infty}.$$

By the definition of the fractional Caputo derivative $^{C}_{gH}\mathcal {D}_{1}^{q}v(x, y)$, we have

$${~}^{RL}_{F}\mathcal I^{1-q}_{0^{+}}D_{xy} v(x,y)=\mathbb{F}[v](x,y) + h_{1}(x,y).$$

It follows that

$${~}^{RL}_{F}\mathcal I_{0^{+}}^{q}{~}^{RL}_{F}\mathcal I^{1-q}_{0^{+}}D_{xy} v(x,y)={~}^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y) +^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y).$$

From Proposition 2.1, we have

$${{\int}_{0}^{x}} {{\int}_{0}^{y}} D_{st} v(s,t)dtds{=}^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y). $$

(34)

Because $v \in \mathcal W_{gH}^{1}(J_{\infty }, E)$, we consider two cases

Case 1

If v is (i)-gH differentiable with respect to x and $\frac {\partial v}{\partial y}$ is (i)-gH differentiable with respect to y, then from (34) one gets

$${{\int}_{0}^{x}}\left[\frac{\partial v}{\partial s}(s,y) \ominus \frac{\partial v}{\partial s}(s,0)\right] ds {=}^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y). $$

It follows that

$$ {{\int}_{0}^{x}}\frac{\partial v}{\partial s}(s,y)ds = {{\int}_{0}^{x}} \frac{\partial v}{\partial s}(s,0)ds +^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y). $$

Since v is (i)-gH differentiable w.r.t. x, one gets

$$v(x,y)\ominus v(0,y)= v(x,0)\ominus v(0,0)+^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y) $$

or

$$ v(x,y)=v(0,y)+ [v(x,0)\ominus v(0,0)]+^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y) . $$

(35)

Case 2

If v is (ii)-gH differentiable with respect to x and $\frac {\partial v}{\partial x}$ is (ii)-gH differentiable with respect to y, then

$${{\int}_{0}^{x}} \left[(-1)\frac{\partial v}{\partial s}(s,0) \ominus (-1)\frac{\partial v}{\partial s}(s,y)\right] ds =^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y)$$

or

$${{\int}_{0}^{x}} (-1)\frac{\partial v}{\partial s}(s,0) ds = {{\int}_{0}^{x}} (-1)\frac{\partial v}{\partial s}(s,y)ds +^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y). $$

Since u is (ii)-gH differentiable with respect to x, we have

$$\begin{array}{@{}rcl@{}} &&v(0,0)\ominus v(x,0)=v(0,y) \ominus v(x,y)+{~}^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+{~}^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y) \\ \Longleftrightarrow\ &&v(0,0)= v(x,0)+[v(0,y) \ominus v(x,y)]+^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y)\\ \Longleftrightarrow\ &&v(0,y) \ominus v(x,y)=v(0,0)\ominus \left[v(x,0)+^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y)\right] \\ \Longleftrightarrow\ &&v(0,y)= v(x,y)+v(0,0)\ominus \left[v(x,0)+^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y)\right]. \end{array} $$

It follows that

$$\begin{array}{@{}rcl@{}} v(x,y)&=&v(0,y)\ominus\left[v(0,0)\ominus \left( v(x,0)+^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y)\right)\right] \\ & = & v(0,x) + v(0,y) \ominus v(0,0) +^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y) +^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y)) \end{array} $$

(36)

for all (x,y) ∈ J _∞ .

Thus, from (35) and (36), we obtain

$$\begin{array}{@{}rcl@{}} &&d_{\infty}\left( v(x,y), v(x,0) + v(0,y) \ominus v(0,0) +^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)\right) \\ &\leq& d_{\infty}\left( {~}^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{1}(x,y), \hat{0}\right)\\ & \leq& \frac{\epsilon}{\Gamma(q_{1}) \Gamma(q_{2})} {{\int}_{0}^{x}} {{\int}_{0}^{y}} (x-s)^{q_{1}-1}(y-t)^{q_{2}-1} dtds \\ &\leq& \frac{\epsilon x^{q_{1}}y^{q_{2}}}{\Gamma(q_{1}+1) \Gamma(q_{2}+1)}. \end{array} $$

• (2) Let $v \in \mathcal W_{gH}^{2}(J_{\infty }, E)$ be a solution of in (31) with k = 2. There exists a function h ₂ ∈ C _λ (J _∞ ,E) such that

  $${~}^{C}_{gH}\mathcal {D}_{2}^{q} v(x,y)=\mathbb{F}[v](x,y) + h_{2}(x,y) \, \, \text{for all} \, (x, y) \in J_{\infty}.$$

Case 3

If v is (ii)-gH differentiable with respect to x and $\frac {\partial v}{\partial x}$ is (i)-gH differentiable with respect to y. Similarly, as in Case 1, we have

$$\begin{array}{@{}rcl@{}} &&(-1)v(0,y)\!\ominus\! (-1)v(x,y)\,=\, (-1)v(0,0) \ominus (\!-1)v(x,0)\!+^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)\!+^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{2}(x,y)) \\ &&\Longleftrightarrow v(0,y) \ominus v(x,y) = v(0,0) \ominus u(x,0)+ (-1)^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+(-1)^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{2}(x,y)) \\ &&\Longleftrightarrow v(0,y)= v(x,y)+ u(0,0)\ominus v(x,0)+ (-1)^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)+(-1)^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{2}(x,y))\\ &&\Longleftrightarrow v(x,y)= v(0,x) + v(0,y) \ominus v(0,0) \ominus (-1)^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y) \ominus (-1)^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{2}(x,y). \end{array} $$

Case 4

If v is (i)-gH differentiable with respect to x and $\frac {\partial v}{\partial x}$ is (ii)-gH differentiable with respect to y, we also get similar results and then we have the following estimation:

$$\begin{array}{@{}rcl@{}} &&d_{\infty}\left( v(x,y), v(x,0) + v(0,y) \ominus v(0,0) \ominus (-1)^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y)\right) \\ &\leq& d_{\infty}\left( {~}^{RL}_{F}\mathcal I_{0^{+}}^{q}h_{2}(x,y), \hat{0}\right) \\ & \leq& \frac{\epsilon}{\Gamma(q_{1}) \Gamma(q_{2})} {{\int}_{0}^{x}} {{\int}_{0}^{y}} (x-s)^{q_{1}-1}(y-t)^{q_{2}-1} dtds\\ & \leq& \frac{\epsilon x^{q_{1}}y^{q_{2}}}{\Gamma(q_{1}+1) \Gamma(q_{2}+1)}. \end{array} $$

The proof is complete. □

Definition 4.2

(1) A mapping $v \in \mathcal W_{gH}^{1}(J_{\infty }, E)$ is called a type 1 weak solution of in (31) with k = 1 if it satisfies (32).

(2) A mapping $v \in \mathcal W_{gH}^{2}(J_{\infty }, E)$ is called a type-2 weak solution of in (31) with k = 2 if it satisfies (33).

Definition 4.3

Problem (1)–(2) is called to be generalized type k Hyers-Ulam stable (k = 1,2) if there exists an increasing, continuous function $\mu :\mathbb {R}_{+}\to \mathbb {R}_{+}$, μ(0) = 0, such that for arbitrary 𝜖 > 0 and each type k weak solution v of in equation (31), there exists a (k)-weak solution u of problem (1)–(2) such that

$$H_{\lambda}(u,v) \leq \mu(\epsilon),\ \text{ for all }\lambda\ge \lambda_{0},$$

where λ ₀ is defined by (29).

Moreover, if there exists $\hat {c}>0$ such that $\mu (t)=\hat {c}t$, $t \in \mathbb {R}_{+}$, then problem (1)–(2) is said to be Hyers-Ulam stable.

Denote

$$T_{v}[u](x,y)= v(x,0)+v(0,y)\ominus v(0,0) \ominus (-1) \,^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[u](x,y),\ (x,y) \in J_{\infty}$$

and

$$\hat{C}_{v\lambda}(J_{\infty}, E):=\{u \in C_{\lambda}(J_{\infty}, E): \ T_{v}[u](x,y) \text{ exists for all }(x,y)\in J_{\infty} \}.$$

Theorem 4.1

Under the assumptions ( H ₁ ) − ( H ₃ ) , problem(1)–(2)is type 1 Hyers-Ulam stable. In addition, assume that for each type-2 weak solution v of in(31), we have

• ( H ₆ ) $\hat {C}_{v\lambda }(J_{\infty }, E)\ne \emptyset .$

• (H ₇) If $u \in \hat {C}_{v\lambda }(J_{\infty }, E)$ , then the following H-difference exists for all (x,y) ∈ J _∞

  $$ v(x,0)+v(0,y)\ominus v(0,0) \ominus (-1) \,^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[\nu(u)](x,y), $$

  (37)

where ν(u)(x,y) = T _v [u](x,y).

Then, problem (1)–(2) is type 2 Hyers-Ulam stable.

Proof

We need only prove the second part of this theorem. For arbitrary 𝜖 > 0, suppose that v ∈ C _λ (J _∞ ,E) is a type-2 weak solution of the in (31). By Theorem 3.2, we see that the problem (1)–(2), with initial conditions u(x,0) = v(x,0);u(0,y) = v(0,y), has a unique (2)-weak solution u(x,y) satisfying

$$u(x, y)=v(x,0) +v(0,y) \ominus v(0,0) \ominus (-1)^{RL}_{F}\mathcal I^{q}_{0^{+}} \mathbb{F}[u](x,y) \, \, \text{ for all } \, \, (x, y) \in J_{\infty}.$$

Then, we have

$$\begin{array}{@{}rcl@{}} &&d_{\infty}(u(x,y), v(x,y)) \\ &\leq& d_{\infty}\left( v(x,y),v(x,0) + v(0,y) \ominus v(0,0) \ominus (-1)^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[v](x,y)\right)\\ && +~ d_{\infty}((-1)^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[u](x,y),(-1)^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y))) \\ &\leq& \frac{\epsilon x^{q_{1}}y^{q_{2}}}{\Gamma(q_{1}+1) \Gamma(q_{2}+1)}\\ &&+~ \frac{1}{\Gamma(q_{1}) \Gamma(q_{2})} {{\int}_{0}^{x}} {{\int}_{0}^{y}} (x-s)^{q_{1}-1}(y-t)^{q_{2}-1}d_{\infty}(\mathbb{F}[u](s,t), \mathbb{F}[v](s,t))dtds. \end{array} $$

Using similar assertions as (22) and (28), we get

$$\begin{array}{@{}rcl@{}} &&d_{\infty}(u(x,y), v(x,y))\\ &\leq& \frac{\epsilon x^{q_{1}}y^{q_{2}}}{\Gamma(q_{1}+1) \Gamma(q_{2}+1)} +\frac{L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right) H_{\lambda}(u,v)}{\Gamma(q_{1}) \Gamma(q_{2})} e^{\lambda(x+y)}G(\lambda q_{1}) G(\lambda q_{2}). \end{array} $$

(38)

It follows that

$$\begin{array}{@{}rcl@{}} && e^{\lambda(x+y)}d_{\infty}(u(x,y), v(x,y)) \\ &\le& \frac{\epsilon x^{q_{1}}y^{q_{2}}}{e^{\lambda (x+y)}\Gamma(q_{1}+1) \Gamma(q_{2}+1)}+\frac{L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right) H_{\lambda}(u,v)}{\Gamma(q_{1}) \Gamma(q_{2})} G(\lambda q_{1}) G(\lambda q_{2}). \end{array} $$

By revisiting the fact that for λ ≥ q ₁, $\max _{x\in [0,\infty )}\frac {x^{q_{1}}}{e^{\lambda x}}=\frac {q^{q_{1}}_{1}}{\lambda e^{q_{1}}}$, taking supremum both sides of the inequality (38), we obtain

$$H_{\lambda}(u,v) \leq \frac{\epsilon q_{1}^{q_{1}}q_{2}^{q_{2}}}{(\lambda e)^{q_{1}+q_{2}}\Gamma(q_{1}+1) \Gamma(q_{2}+1)}+\frac{L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right) H_{\lambda}(u,v)}{\Gamma(q_{1}) \Gamma(q_{2})} G(\lambda q_{1}) G(\lambda q_{2}). $$

For λ ≥ λ ₀, one has

$$\hat{c}:=\frac{q_{1}^{q_{1}}q_{2}^{q_{2}}\Gamma(q_{1}) \Gamma(q_{2})}{(\lambda e)^{q_{1}+q_{2}} \Gamma(q_{1}+1) \Gamma(q_{2}+1) \left[\Gamma(q_{1}) \Gamma(q_{2}) - L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right) G(\lambda q_{1}) G(\lambda q_{2})\right]}>0.$$

This implies that

$$H_{\lambda}(u,v) \leq \hat{c}\epsilon . $$

Therefore, problem (1)–(2) is type-2 Hyers-Ulam stable. □

In the next part of this section, we will prove the Hyers-Ulam-Rassias stability property of problem (1)–(2). For Φ ∈ L ¹(J _∞ ,[0, + ∞)), consider the following inequation:

$$ d_{\infty}\left( \!{~}^{C}_{gH}\mathcal {D}_{k}^{q} v(x,y), \mathbb{F}[v](x,y)\right)\leq \Phi(x) \, \text{ for all} \, \, (x,y) \in J_{\infty}, \ k\in\left\{1,2\right\}. $$

(39)

Definition 4.4

• 1. A mapping $v \in \mathcal W_{gH}^{1}(J_{\infty }, E)$ is a solution of in (39) with k = 1 if there exists a function h ₃ ∈ C _λ (J _∞ ,E) such that

  • (i) $d_{\infty }(h_{3}(x,y), \hat {0}) \leq \Phi (x,y) \, \text {for all} \, (x,y) \in J_{\infty },$

  • (ii) ${~}^{C}_{gH}\mathcal {D}_{1}^{q} v(x,y)=\mathbb {F}[v](x,y) + h_{3}(x,y), \, \, (x, y) \in J_{\infty }.$

• 2. A mapping $v \in \mathcal W_{gH}^{2}(J_{\infty }, E)$ is a solution of in (39) with k = 2 if there exists a function h ₄ ∈ C _λ (J _∞ ,E) such that

  • (i) $d_{\infty }(h_{4}(x,y), \hat {0}) \leq \Phi (x,y)\, \text {for all} \, (x,y) \in J_{\infty },$

  • (ii) $^{C}_{gH}\mathcal {D}_{2}^{q} v(x,y)=\mathbb {F}[v](x,y) + h_{4}(x,y), \, \, (x, y) \in J_{\infty }.$

By using analogous arguments as in Lemma 4.1, we receive the following results.

Lemma 4.2

1. If v is a solution of in(39)with k = 1, then it satisfies the following integral inequation:

$$d_{\infty}\left( v(x,y), v(x,0) + v(0,y) \ominus v(0,0) +^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[v](x,y)\right) \leq^{RL}_{F}\mathcal I_{0^{+}}^{q} \Phi(x,y) $$

(40)

for all (x,y) ∈ J _∞ .

2.If v is a solutionof in (39)with k = 2,then it satisfies the following integral inequation:

$$d_{\infty}\left( v(x,y),v(x,0) + v(0,y) \ominus v(0,0) \ominus (-1)^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[v](x,y)\right) \leq^{RL}_{F}\mathcal I_{0^{+}}^{q} \Phi(x,y) $$

(41)

for all (x,y) ∈ J _∞ .

Definition 4.5

Problem (1)–(2) is said to be generalized type k Hyers-Ulam-Rassias stable (k = 1,2) with respect to Φ if there exists a real number c _f,Φ > 0 such that for each v ∈ C _λ (J _∞ ,E) satisfying integral in equation (40), there exists a (k) −weak solution u ∈ C _λ (J _∞ ,E) of problem (1)–(2) such that

$$H_{\lambda}(u,v)\leq c_{f,\Phi}\sup\limits_{(x,y)\in J_{\infty}}\Phi(x,y).$$

Theorem 4.2

1. Assume that all assumptions ( H ₁ )-( H ₃ ) are fulfilled. Moreover for each Φ ∈ L ¹(J _∞ ,[0, + ∞)), there exists m _Φ > 0, ν ≥ 0such that

$$ \left( {~}^{RL}_{F}\mathcal I_{0^{+}}^{q}\Phi\right)(x,y)\leq m_{\Phi} e^{\nu (x+y)} \Phi(x,y) \text{ for all } (x,y) \in J_{\infty}. $$

(42)

Then, problem (1)–(2)is generalized type 1 Hyers-Ulam-Rassias stable.

2.Assume that all assumptions(H ₁)-(H ₃),(H ₆)-(H ₇)and (42)hold. Then, (1)–(2)is generalized type-2 Hyers-Ulam-Rassias stable.

Proof

Let v ∈ C _λ (J _∞ ,E) be a solution of the in (39)

$$d_{\infty}\left( \!{~}^{C}_{gH}\mathcal {D}_{k}^{q} v(x,y), \mathbb{F}[v](x,y)\right)\leq \Phi(x,y),\ k\in\left\{1,2\right\} \, \text{ for all} \, \, (x,y) \in J_{\infty} $$

and by Theorem 3.2, there exists a fuzzy-valued function u ∈ C _λ (J _∞ ,E) defined by

$$u(x, y)=v(x,0) +v(0,y) \ominus v(0,0) \ominus (-1)^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[u](x,y) \ \text{for all} \ (x, y) \in J_{\infty}. $$

From Lemma 4.2, we have

$$\begin{array}{@{}rcl@{}} &&d_{\infty}(u(x,y), v(x,y))\\ & \leq& d_{\infty}\left( v(x,y),v(x,0) + v(0,y) \ominus v(0,0) \ominus (-1)\,^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[v](x,y)\right)\\ &&+~ d_{\infty}({~}^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[u](x,y),^{RL}_{F}\mathcal I_{0^{+}}^{q}\mathbb{F}[v](x,y))) \\ &&\leq^{RL}_{F}\mathcal I_{0^{+}}^{q} \Phi(x,y) \\ &&+~ \frac{1}{\Gamma(q_{1}) \Gamma(q_{2})} {{\int}_{0}^{x}} {{\int}_{0}^{y}} (x-s)^{q_{1}-1}(y-t)^{q_{2}-1}d_{\infty}(\mathbb{F}[u](s,t), \mathbb{F}[v](s,t))dtds. \end{array} $$

It follows readily from (22) that

$$\begin{array}{@{}rcl@{}} &&d_{\infty}(u(x,y), v(x,y))\\ &&\leq^{RL}_{F}\mathcal I_{0^{+}}^{q} \Phi(x,y) +\frac{L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right) H_{\lambda}(u,v)}{\Gamma(q_{1}) \Gamma(q_{2})} e^{\lambda(x+y)}G(\lambda q_{1}) G(\lambda q_{2}). \end{array} $$

(43)

From (42), we have for all λ > ν

$$e^{-\lambda (x+y)}d_{\infty}(u(x,y), v(x,y)) \leq m_{\Phi} \Phi(x,y) +\frac{L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right) H_{\lambda}(u,v)}{\Gamma(q_{1}) \Gamma(q_{2})} G(\lambda q_{1}) G(\lambda q_{2}). $$

Since λ ≥ λ ₀ then

$$c_{f,\Phi}=\frac{m_{\Phi} \Gamma(q_{1}) \Gamma(q_{2})}{\Gamma(q_{1}) \Gamma(q_{2}) - L\left( 1+\frac{KM}{\sqrt{2\lambda}}\right) G(\lambda q_{1}) G(\lambda q_{2})}>0.$$

This implies that

$$H_{\lambda}(u,v) \leq c_{f,\Phi}\sup\limits_{(x,y)\in J_{\infty}}\Phi(x,y). $$

Therefore, problem (1)–(2) is generalized type-2 Hyers-Ulam-Rassias stable. □

5 Application Example

Example 5.1

We consider the following fuzzy fractional PDE

$$ {~}^{C}_{gH}\mathcal{D}_{k}^{q} u(x,y)=\frac{1}{e^{x+y+5}+x+2}u^{2}(x,y)+{{\int}_{0}^{y}} \frac {1}{y-s+1}u(x,s)ds, $$

(44)

(x,y) ∈ [0,∞) × [0,∞), with the initial conditions

$$ \left\{\begin{array}{llllllll} u(x, 0) = C(x+1), & x \in [0,\infty)\\ u(0, y) = Ce^{y},& y \in [0,\infty), \end{array}\right. $$

(45)

where q = (q ₁,q ₂) ∈ [0,1) × [0,1) and C = (1,2,3) is a triangle fuzzy number and

$$\mathbb{F}[u](x,y)=\frac{1}{e^{x+y+5}+x+2}u^{2}(x,y)+{{\int}_{0}^{y}}\frac {1}{y-s+1}u(x,s)ds \, \, $$

for all (x,y) ∈ [0,∞) × [0,∞).

We will use Zadeh’s extension principle to prove the Lipschitz property of f.

Lemma 5.1

There exists a positive number M _o such that

$$d_{\infty}(\phi^{2}, \chi^{2}) \leq M_{o}d_{\infty}(\phi,\chi)\ \forall \phi,\chi\in E.$$

Proof

Consider a crisp mapping

$$\begin{array}{@{}rcl@{}} {f}:{\mathbb{R}}&\longrightarrow&{[0,+\infty)}\\{t}&\longmapsto&{t^{2}} \end{array} $$

By Zadeh extension principle, the following extended fuzzy mapping

$$\begin{array}{@{}rcl@{}} &&{\mathcal{F}}:{E}\longrightarrow{E}\\ &&\qquad\!{\phi}\longmapsto{\mathcal{F}(\phi)=\phi^{2}} \end{array} $$

is defined by

$$\mathcal{F}(\phi)(t)=\phi^{2}(t)= \left\{\begin{array}{llllllll} \sup \limits_{x \in f^{-1}(t)}\phi(x) &\text{if~} f^{-1}(t) \ne \emptyset\\ 0&\text{if~} f^{-1}(t)= \emptyset, \end{array}\right. $$

where $f^{-1}(t) = \{x \in \mathbb {R}: f(x) = t\}$. Therefore, we have

$$\mathcal{F}(\phi)(t)=\phi^{2}(t)= \left\{\begin{array}{llllllll} \sup \{\phi(\sqrt{t}), \phi(-\sqrt{t}): t \geq 0\} \\ \, \, \, \, \, \, \, \, \, \, 0, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, t<0. \end{array}\right. $$

The α-level set of the fuzzy number ϕ ² is defined by

$$\begin{array}{@{}rcl@{}} [\phi^{2}]^{\alpha} &=& \{t: \phi^{2}(t) \geq \alpha\}\\ &=&\{t\geq 0: \phi(\sqrt{t}) \geq \alpha\} \cup \{t\geq 0: \phi(-\sqrt{t}) \geq \alpha \}\\ &=&\{t^{2}: \phi(t) \geq \alpha\}. \end{array} $$

One has

$$\begin{array}{@{}rcl@{}} d_{H}([\phi^{2}]^{\alpha}, [\chi^{2}]^{\alpha}) &=&\max\left\{\sup \limits_{x \in [\phi^{2}]^{\alpha}} \sup \limits_{y \in [\chi^{2}]^{\alpha}} |x-y|, \sup \limits_{y \in [\chi^{2}]^{\alpha}} \sup \limits_{x \in [\phi^{2}]^{\alpha}} |y-x|\right\}\\ &=&\max\left\{\sup \limits_{t_{1} \in [\phi]^{\alpha}} \sup \limits_{t_{2} \in [\chi]^{\alpha}} |{t_{1}^{2}}-{t_{2}^{2}}|, \sup \limits_{t_{2} \in [\chi]^{\alpha}} \sup \limits_{t_{1} \in [\phi]^{\alpha}} |{t_{2}^{2}}-{t_{1}^{2}}|\right\}. \end{array} $$

On the other hand, since ϕ,χ ∈ E, without loss of generality we assume that [ϕ]^α = [a,b],[χ]^α = [c,d]. Moreover, for each $t_{1}, t_{2} \in \mathbb {R},$ we have the inequality

$$\begin{array}{@{}rcl@{}} |t_{1}+t_{2}| & \leq& |t_{1}|+|t_{2}|\\ & \leq& \max\{|a|, |b|\}+\max\{|c|, |d|\}. \end{array} $$

This implies

$$\begin{array}{@{}rcl@{}} \sup \limits_{t_{1} \in [\phi]^{\alpha}} \sup \limits_{t_{2} \in [\chi]^{\alpha}} |t_{1}+t_{2}| &\leq& \sup \limits_{t_{1} \in [\phi]^{\alpha}}|t_{1}| +\max\{|c|, |d|\}\\ &\leq& \max\{|a|, |b|\}+\max\{|c|, |d|\}:=M_{o}, \end{array} $$

and thus,

$$\begin{array}{@{}rcl@{}} d_{H}([\phi^{2}]^{\alpha}, [\chi^{2}]^{\alpha})& \leq& M_{o}\max\left\{\sup \limits_{t_{1} \in [\phi]^{\alpha}} \sup \limits_{t_{2} \in [\chi]^{\alpha}} |t_{1}-t_{2}|, \sup \limits_{t_{2} \in [\chi]^{\alpha}} \sup \limits_{t_{1} \in [\phi]^{\alpha}} |t_{2}-t_{1}|\right\}\\ &\leq& M_{o}d_{H}([\phi]^{\alpha}, [\chi]^{\alpha}). \end{array} $$

□

We define an operator

$$T: C([0,\infty)^{2}, E) \to C([0,\infty)^{2}, E)$$

by

$$Tu(x,y)=C(e^{x}+y)+^{RL}_{F}\mathcal I_{0^{+}}^{q} \mathbb{F}[u](x,y).$$

From Lemma 5.1, we have the following estimate:

$$\begin{array}{@{}rcl@{}} d_{\infty}(\mathbb{F}[u](x,y), \mathbb{F}[v](x,y)) &\leq& \frac{1}{e^{5}+2}d_{\infty}(u^{2}(x,y), v^{2}(x,y))\\ && +~d_{\infty} \left( {{\int}_{0}^{y}}\frac {1}{y-s+1}u(x,s)ds,{{\int}_{0}^{y}} \frac {1}{y-s+1}v(x,s)ds\right)\\ &\leq& \frac{M_{o}}{e^{5}+2} d_{\infty}(u(x,y), v(x,y))\\ &&+~d_{\infty} \left( {{\int}_{0}^{y}}\frac {1}{y-s+1}u(x,s)ds,{{\int}_{0}^{y}}\frac {1}{y-s+1}v(x,s)ds\right). \end{array} $$

Put $L_{o}:=\max \{\frac {M_{o}}{e^{5}+2},1\}$, we obtain

$$\begin{array}{@{}rcl@{}} &&d_{\infty}(\mathbb{F}[u](x,y), \mathbb{F}[v](x,y))\\ &\leq& L_{o} \left[d_{\infty}(u(x,y), v(x,y))+d_{\infty} \left( {{\int}_{0}^{y}} \frac {1}{y-s+1}u(x,s)ds,{{\int}_{0}^{y}} \frac {1}{y-s+1}v(x,s)ds\right)\right]. \end{array} $$

Thus, f satisfies hypothesis ( H ₁ ).

Clearly, the condition ( H ₂ ) is satisfied with g(u) = u. Moreover, we have

$$d_{\infty}(u(x,0), \hat{0}) \le e^{2x+2}$$

and

$$d_{\infty}(u(0,y), \hat{0}) \le 2 e^{y}.$$

Hence, Theorem 3.1 implies that the problem (44)–(45) has a (1)-weak unique bounded fuzzy solution on [0,∞) × [0,∞). Moreover, by Theorem 4.1, the (44)–(45) is type-1 Hyers-Ulam stable.

Furthermore, if we choose Φ(x,y) = e ^{x + y} for all (x,y) ∈ [0,∞) × [0,∞), then we can see that

$$\begin{array}{@{}rcl@{}} {~}^{RL} I^{q} \Phi(x,y)&=& \frac{1}{\Gamma(q_{1})\Gamma(q_{2})}{{\int}_{0}^{x}}(x-s)^{q_{1}-1}e^{s}ds{{\int}_{0}^{y}} (y-t)^{q_{2}-1}e^{t}dt\\ & \leq& \frac{1}{\Gamma(q_{1}+1) \Gamma(q_{2}+1)}e^{x+y}G(q_{1}) G(q_{2}). \end{array} $$

If ν = 0 and $m_{\Phi }:=\frac {G(q_{1}) G(q_{2})}{\Gamma (q_{1}+1) \Gamma (q_{2}+1)}>0,$ then

$${~}^{RL}I^{q}\Phi(x,y) \leq m_{\Phi} \Phi(x,y).$$

Therefore, the hypothesis (42) is satisfied and problem (44)-(45) is generalized type-1 Hyers-Ulam-Rassias stable with respect to Φ.

Remark 5.1

From Lemma 3.4, Lemma 3.5, by using a typical classes of fuzzy-valued functions (triangular fuzzy sets $\mathcal {T}$), we can show the existence of (2)-weak unique bounded fuzzy solution on [0,∞) × [0,∞) of problem (44)–(45), and the generalized type-2 Hyers-Ulam stability of this equation provides a reliable theoretical basis for approximately solving DEs.

6 Conclusions

In this paper, we study the global existence of weak solutions of Darboux problems for nonlinear fractional partial integro-differential equations with uncertainty under Caputo gH-fractional derivatives. Especially, we develop the Hyers-Ulam stable concepts for this problem. When we study the Hyers-Ulam-Rassias stable or Hyers-Ulam stable properties of a equation, we need not obtain the exact solutions. All the requirements are to find a function which satisfies an approximate inequation with control functions. That says, there exists a close exact solution when the problem is Hyers-Ulam-Rassias stable or Hyers-Ulam stable. So this study provides a reliable theoretical basis for approximately solving DEs and PDEs.