On Goursat problem for fuzzy delay fractional hyperbolic partial differential equations

Abstract

In this paper, we consider Goursat boundary value problems for fuzzy delay fractional partial differential equations under Caputo gH-derivatives. Firstly, the unique solvability of the problems in finite domain is considered. Secondly, by restricting the force function and boundary conditions by exponential growth and using fixed point approach, the existence and the Ulam-Hyers stability of fuzzy mild solutions of the problem in infinite domain are investigated in two types with different geometric behaviors. It is necessary to implement that in many engineering applications, e.g. numerical solutions of a fuzzy dynamical system, the object of optimization problems or the trajectory of a economical dynamics etc., where finding the exact solution is more difficult than solving its approximate solutions, Ulam-Hyers stability is quite useful. Finally, as usual some examples are given to illustrate the obtained theoretical results.

Keywords

Goursat problems,fuzzy delay partial differential equations,Ulam stability,Caputo gH-differentiability

1. Introduction

Because of the fact that multi-variable functions are often faced simultaneously when observing phenomena, partial differential equations (PDEs) are ideal in dealing with real-life problems. One of the most well-known classes of PDEs are the Goursat problems. Goursat problems are used to describe physical situations involving exchange of heat or mass between a stationary medium and moving medium, which occurs in various fields of study such as in engineering, physics, and applied mathematics, see [12, 30].

It is well-known that, deterministic PDEs are not always the best option when dealing with real-life phenomena containing uncertainty. For example, when modelling certain dynamic phenomenon using PDEs, the model is not always precise. This is due to the incomplete knowledge or vague information on the dynamic system. For instance, the initial value may contain fuzziness or the parameters of equations are measured by devices inherited error. Moreover, in some control problems the value of parameters of systems are described in the set of linguistics terms. Hence, fuzzy differential equations and fuzzy partial differential equations appeared as the new and efficient tools to model many real world phenomena. Buckley and Feuring [10] first introduced fuzzy PDEs by incorporating PDEs and fuzzy set theory in one setting. This subject has soon become an active research directions promoted by Allahviranloo et. al. [8], Bertone et al [9], Gouyandeh et al. [11] and Long et al. [17 –19], and Prakash et al [20].

Recently, PDEs of fractional order with delay appear frequently in applications such as mathematical models, control theory, climate models, etc. This subject has been studied extensively in the last decade, see [1 –3]. Some scientists studied fuzzy framework corresponding to deterministic delay equations. However, most of these works are considered for the fuzzy delay ordinary differential equations, see for example [13–16 , 32]. Some authors have been studied numerical methods and applied fuzzy fractional differential equations in real world problems, see [4–7 , 21–24]. In addition, some researches on the Ulam stability for fuzzy differential equations are presented in [25 –29]. When we fuzzify the models containing uncertain phenomena, which have lag time in the independent variables, it turns out that the research on fuzzy fractional PDEs with delay has not been investigated.

Motivated by this fact, in this paper, we consider for the first time Goursat boundary value problems for a fuzzy fractional hyperbolic equation under Caputo generalized Hukuhara derivatives reduced to the canonical forms. Two kinds of mild solutions of these problems are defined in sense of Caputo-gH differentiability. For each type of derivatives, we establish and investigate the existence and uniqueness of fuzzy mild solutions. The special of the problem is in the technique of calculating fractional integral in fuzzy sense. We have mutated fractional hyberbolic equation into systems of integral equations corresponding to two general types of gH-derivatives. In addition, finite delays occurring in the problem are also processed through the construction of appropriate metrics (section 3). Finally, based on idea of fixed point theorem, we will discuss the Ulam-Hyers stability in Theorems 4.1 of Section 4. Some application examples are given in Section 5.

2. Preliminaries

Denote $R_{F}$ by the set of fuzzy numbers $v : ℝ \to [0, 1]$ . The κ-level sets, denoted by $[v]^{κ} = [v_{κ}^{-}, v_{κ}^{+}]$ , of a fuzzy number v is defined by ${x \in ℝ : v (x) \geq κ}$ if 0 < κ ≤ 1 and $\bar{{x : v (x) > 0}}$ if κ = 0 .

Supremum metric on $R_{F}$ is defined by $\begin{matrix} d_{\infty} (u, v) = sup_{0 \leq κ \leq 1} max {| u_{κ}^{-} - v_{κ}^{-} |; | u_{κ}^{+} - v_{κ}^{+} |} . \end{matrix}$

Then, $(R_{F}, d_{\infty})$ is a complete metric space.

For $f, g \in R_{F}$ , we denote h = f ⊖ g, the Hukuhara difference of f and g, if $h \in R_{F}$ such that f = g + h. And we denote by f_{circleddashgH}g, the gH-difference of f and g, if there exists $h \in R_{F}$ such that f = g + h or g = f + (-1) ⊙ h.

Definition 2.1. Suppose that $(T, d_{T})$ and $(Y, d_{Y})$ are metric spaces of fuzzy numbers or fuzzy-valued functions. A function $h : T \to Y$ is continuous at $t_{0} \in T$ if for every ε > 0 there exists an δ > 0 such that for all $t \in T : d_{T} (t, t_{0}) < δ,$ we have $d_{Y} (h (t), h (t_{0})) < ε .$ The set of all continuous functions $f : T \to Y$ is denoted by $C (T, Y)$ .

Definition 2.2. [17] A mapping $f : J \subset ℝ^{2} \to R_{F}$ is called gH-differentiable with respect to (w.r.t.) x at (x₀, y₀) ∈ J if for all h be such that (x₀ + h, y₀) ∈ J, the gH-difference f (x₀ + h, y₀) _{circleddash_gH}f (x₀, y₀) exists and $\begin{matrix} lim_{h \to 0} \frac{1}{h} (f (x_{0} + h, y_{0}) ⊝_{g H} f (x_{0}, y_{0})) \end{matrix}$ exists and equal to $\frac{\partial f}{\partial x} (x_{0}, y_{0}) \in R_{F} .$

Similar to the gH-derivative of f w.r.t. y and higher order derivatives. The set of all gH-differentiable functions up to order i w.r.t. x and up to order j w.r.t. y in J is denoted by $C_{gH}^{i, j} (J, R_{F})$ (i, j = 0, 1).

Definition 2.3. [18] Assume that $f \in C_{gH}^{1, 0} (J, R_{F}) .$ f is called (i)-gH differentiable w.r.t. x at (x₀, y₀) ∈ J if $\begin{matrix} {[\frac{\partial f}{\partial x} (x_{0}, y_{0})]}^{κ} & = & [\frac{\partial f_{κ}^{-}}{\partial x} (x_{0}, y_{0}), \frac{\partial f_{κ}^{+}}{\partial x} (x_{0}, y_{0})], \\ 0 \leq κ \leq 1, \end{matrix}$ and f is called (ii)-gH differentiable w.r.t. x at (x₀, y₀) ∈ I if $\begin{matrix} [\frac{\partial f}{\partial x} (x_{0}, y_{0})]^{κ} & = & [\frac{\partial f_{κ}^{+}}{\partial x} (x_{0}, y_{0}), \frac{\partial f_{κ}^{-}}{\partial x} (x_{0}, y_{0})], \\ 0 \leq κ \leq 1, \end{matrix}$ where $[f (x, y)]^{κ} = [f_{κ}^{-} (x, y), f_{κ}^{+} (x, y)]$ , κ ∈ [0, 1], (x, y) ∈ J.

Denote

- $W_{gH}^{1} (J, R_{F})$ by the set containing all functions $w : J \to R_{F}$ which satisfy

1) w is (i) gH-differentiable w.r.t. x and w_x is (i) gH-differentiable w.r.t. y, or

2) w is (ii) gH-differentiable w.r.t. x and w_x is (ii) gH-differentiable w.r.t. y,

- $W_{gH}^{2} (J, R_{F})$ by the set of all functions $w : J \to R_{F}$ which satisfy

1) w is (i) gH-differentiable w.r.t. x and w_x is (ii) gH-differentiable w.r.t. y, or

2) w is (ii) gH-differentiable w.r.t. x and w_x is (i) gH-differentiable w.r.t. y.

Remark 2.1. [18]

If $w \in W_{gH}^{1} (J, R_{F})$ then the mixed second order gH-derivative of w w.r.t. x and y, $D_{xy}^{1} w (x, y)$ , is defined by its levelsetwise $\begin{matrix} [D_{xy}^{1} w (x, y)]^{κ} & = & [\frac{\partial^{2} w_{κ}^{-}}{\partial x \partial y} (x, y), \frac{\partial^{2} w_{κ}^{+}}{\partial x \partial y} (x, y)], \\ 0 \leq κ \leq 1 . \end{matrix}$

If $u \in W_{gH}^{2} (J, R_{F})$ then the mixed second order gH-derivative of w w.r.t. x and y, $D_{xy}^{2} w (x, y)$ , is defined by its levelsetwise $\begin{matrix} [D_{xy}^{2} w (x, y)]^{κ} & = & [\frac{\partial^{2} w_{κ}^{+}}{\partial x \partial y} (x, y), \frac{\partial^{2} w_{κ}^{-}}{\partial x \partial y} (x, y)], \\ 0 \leq κ \leq 1 . \end{matrix}$

Definition 2.4. [18] Let p = (p₁, p₂) ∈ (0, 1] × (0, 1] and $w : J \to R_{F}$ . The mixed Riemann - Liouville fractional integral of order p for w is defined by its levelsetwise $\begin{matrix} [_{F}^{RL} I_{0^{+}}^{p} w (x, y)]^{κ} \\ = [^{RL} I_{0^{+}}^{p} w_{κ}^{-} (x, y),^{RL} I_{0^{+}}^{p} w_{κ}^{+} (x, y)], \end{matrix}$ where $\begin{matrix} I_{0^{+}}^{p} h (x, y) \\ = \frac{1}{Γ (p_{1}) Γ (p_{2})} \int_{0}^{x} \int_{0}^{y} (x - s)^{p_{1} - 1} (y - t)^{p_{2} - 1} h (s, t) dtds \end{matrix}$ is the mixed Riemann - Liouville fractional integral notion of order p for a real-valued function h (x, y) provided that the right hand side does exist. Then we denote $\begin{matrix} _{F}^{RL} I_{0^{+}}^{p} w (x, y) \\ = \frac{1}{Γ (p_{1}) Γ (p_{2})} \int_{0}^{x} \int_{0}^{y} (x - s)^{p_{1} - 1} (y - t)^{p_{2} - 1} w (s, t) dtds \end{matrix}$ for (x, y) ∈ J.

Definition 2.5. [18] Let q = (q₁, q₂) ∈ [0, 1) × [0, 1), $u \in W_{gH}^{k} (J, R_{F})$ , k ∈ {1, 2}. The Caputo gH-derivatives of order q of w are defined by $C_{g H} D_{k}^{q} w (x, y) = F_{R L} ℐ_{0^{+}}^{1 - q} (D_{x y}^{k} w (x, y)), (x, y) \in J,$ where 1 - q = (1 - q₁, 1 - q₂) ∈ (0, 1] × (0, 1], provided that the fractional integrals in the right hand side exist.

3. The solvability of Goursat problems

3.1. Fuzzy fractional functional hyperbolic equations in finite domain

3.1.1 Statement of the problem

Denote by J_r = [- r, a] × [- r, b], P = [- r, 0] × [- r, 0] and ${\hat{J}}_{r} = J_{r} ∖ (0, a] \times (0, b]$ . In this section we investigate the existence and uniqueness for following Goursat problem $\begin{array}{l} \underset{g H}{C} D_{k}^{q} w (x, y) \\ = F (x, y, w_{(x, y)}), k = 1, 2, (x, y) \in J_{r} \end{array}$ (3.1) with initial conditions ${\begin{matrix} w (x, y) = φ (x, y), (x, y) \in {\hat{J}}_{r} \\ w (x, 0) = η_{1} (x), x \in [0, a] \\ w (0, y) = η_{2} (y), y \in [0, b] \end{matrix}$ (3.2) where $F : J_{0} \times C (P, R_{F}) \to R_{F}$ is given function with no switching point; $η_{1} (.) \in C ([0, a], R_{F})$ , $η_{2} (.) \in C ([0, b], R_{F})$ and $φ (., .) \in C ({\hat{J}}_{r}, R_{F})$ are given functions satisfing $φ (0, 0) = η_{1} (0) = η_{2} (0) = w_{0}, η_{2} (y) ⊖ w_{0}, η_{1} (x) ⊖ w_{0} \in R_{F} for all x \in [0, a], y \in [0, b]$ and w_(x,y) (s, t) = w (x + s, y + t) , (s, t) ∈ P is the time lag represents the history of the state from time (x - r, y - r) up to the present time (x, y).

For convenience in presentation, we denote $\begin{matrix} γ (x, y) & = & η_{2} (y) + [η_{1} (x) ⊖ w_{0}] \\ = & η_{1} (x) + [η_{2} (y) ⊖ w_{0}] . \end{matrix}$

3.1.2 The solvability of the problem

For σ > 0 we consider $\begin{matrix} C_{σ} (J_{r}, R_{F}) = \\ {w \in C (J_{r}, R_{F}) | sup_{(s, t) \in J_{r}} d_{\infty} (w (s, t), \hat{0}) e^{- σ (s + t)} < + \infty} \end{matrix}$ and the supremum weighted metric on $C_{σ} (J_{r}, R_{F})$ is $\begin{matrix} D_{σ} (w, v) = sup_{(s, t) \in J_{r}} {d_{\infty} (w (s, t), v (s, t)) e^{- σ (s + t)}} . \end{matrix}$ Then $(C_{σ} (J_{r}, R_{F}), D_{σ})$ is a complete metric space.

Lemma 3.1. Suppose that $F \in C (J_{0} \times C (P, R_{F}),$ $R_{F})$ and $u \in W_{gH}^{1} (J_{0}, R_{F}) \cup W_{gH}^{2} (J_{0}, R_{F})$ satisfisfying Problem (3.1)-(3.2). Then, w satisfies either $w (x, y) = γ (x, y) + \underset{F}{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)})$ (3.3) or $w (x, y) = γ (x, y) ⊖ {(- 1)}_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) .$ (3.4)

Proof. With $φ, φ \in C (P, R_{F})$ , we denote $\begin{matrix} d_{C}^{0} (φ, φ) : = sup_{(ϖ, θ) \in P} d_{\infty} (φ (ϖ, θ), φ (ϖ, θ), \end{matrix}$ then $d_{c}^{0}$ is a metric on $C (P, R_{F}) .$ From the definition of Caputo gH-derivatives of w, by taking integral order q two sides of (3.1) and using property $_{F}^{R L} ℐ_{0 +}^{p}_{F}^{R L} ℐ_{0 +}^{q} w (x, y) =_{F}^{R L} ℐ_{0 +}^{p + q} w (x, y)$ (see in Proposition 3.1 [18]), we have $\begin{matrix} \int_{0}^{x} \int_{0}^{y} D_{st}^{k} w (s, t) dtds \\ = I_{0 +}^{q} F (x, y, w_{(x, y)}), k \in {1, 2} . \end{matrix}$ If $u \in W_{gH}^{1} (J_{0}, R_{F})$ then $\begin{matrix} _{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) \\ = \int_{0}^{x} \frac{\partial}{\partial s} (w (s, y) ⊖ w (s, 0)) d s \end{matrix}$ (3.5) for the case w and $\frac{\partial w}{\partial x}$ are both (i)- gH differentiable and $\int_{0}^{x} \frac{\partial}{\partial s} w (s, y) = (- 1) w (x, 0) ⊖ (- 1) w (x, y) .$ (3.6) for the case w and $\frac{\partial w}{\partial x}$ are both (ii)- gH differentiable.

From (3.5) and $\frac{\partial w}{\partial x}$ is (i)-gH differentiable w.r.t. y, one gets $\begin{array}{l} _{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) \\ = \int_{0}^{x} \frac{\partial}{\partial s} w (s, y) d s ⊖ \int_{0}^{x} \frac{\partial}{\partial s} w (s, 0) d s \\ = w (x, y) ⊖ w (0, y) ⊖ [w (x, 0) ⊖ w (0, 0)] \\ = w (x, y) ⊖ γ (x, y) . \end{array}$ Therefore $w (x, y) = γ (x, y) +_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)})$ (3.7) for all (x, y) ∈ J₀.

From (3.6) and $\frac{\partial w}{\partial x}$ is (ii)-gH differentiable w.r.t. y, we have $\begin{array}{l} _{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) \\ = [w (0, 0) ⊖ w (x, 0)] ⊖ [w (0, y) ⊖ w (x, y)] . \end{array}$ This implies $w (0, 0) ⊖ w (x, 0) =_{F}^{R L} ℐ_{0 +}^{q} F (x, y,$ w_(x,y)) + w (0, y) ⊖ w (x, y) . Thus we obtain (3.3) again.

If $w \in W_{gH}^{2} (J_{0}, R_{F})$ then by using similar arguments, we have $\begin{array}{l} _{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) \\ = \int_{0}^{x} \frac{\partial}{\partial s} w (s, y) d s ⊖ \int_{0}^{x} \frac{\partial}{\partial s} w (s, 0) d s . \end{array}$

Suppose that $\frac{\partial w}{\partial x}$ is (ii)-gH differentiable w.r.t. y, we obtain $[(- 1) w (0, y) ⊖ (- 1) w (x, y)] ⊖ [- 1 w (0, 0) ⊖ (- 1) w (x, 0)] =_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) .$ This follows $w (0, y) = w (x, y) + [(- 1) I_{0^{+}}^{q} F (x, y, w_{(x, y)}) + w (0, 0) ⊖ w (x, 0)] .$

Therefore

$w (x, y) = w (0, y) ⊖ [{(- 1)}_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) +$ $w (0, 0) ⊖ w (x, 0)] = w (0, y) ⊖ [{(- 1)}_{F}^{R L} ℐ_{0 +}^{q} F (x, y,$ $w_{(x, y)}) + w (0, 0) ⊖ w (x, 0)] = w (0, y) ⊖ {(- 1)}_{F}^{R L}$ $I_{0 +}^{q} F (x, y, w_{(x, y)}) ⊖ w (0, 0) + w (x, 0) = γ (x, y) ⊖$ ${(- 1)}_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) .$

The case when w is (ii)-gH differentiable w.r.t. x and $\frac{\partial w}{\partial x}$ is (i)-gH differentiable w.r.t. y, by doing similarly obtain the desired results. □

From Lemma 3.1.2, mild solutions of the Problem (3.1)-(3.2) are defined as follows.

Definition 3.1. A function $u \in C (J_{r}, R_{F})$ is called 1) a mild solution in type 1 of the Problem (3.1)-(3.2) if $w (x, y) = φ (x, y), (x, y) \in {\hat{J}}_{r}$ and it satisfies integral equation (3.4) for all (x, y) ∈ J₀,

2) a mild solution in type 2 of the Problem (3.1)-(3.2) if $w (x, y) = φ (x, y), (x, y) \in {\hat{J}}_{r}$ and it satisfies integral equation (3.4) for all (x, y) ∈ J₀.

Theorem 3.1. Suppose that $F \in C (J_{0} \times C (P, R_{F}), R_{F})$ satifies Lipschitz condition w.r.t. third variable and Lipschitz coefficient L ∈ (0, Γ (q₁ + 1) Γ (q₂ + 1)), i.e. $d_{\infty} (F (x, y, φ_{1}), F (x, y, φ_{2})) \leq L d_{C}^{0} (φ_{1}, φ_{2})$ (3.8)for all $φ_{1}, φ_{2} \in C (P, R_{F})$ . Then problem (3.1)-(3.2) has only one solution in type 1.

Proof. The mild solution in type 1 of the Problem (3.1)-(3.2) (if it exists) is the fixed point of the operation $N_{1} : C (J_{r}, R_{F}) \to C (J_{r}, R_{F})$ , which is definied by $\begin{array}{l} N_{1} (w (x, y)) \\ = { \begin{matrix} γ (x, y) +_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) if (x, y) \in J_{0} \\ φ (x, y) if (x, y) \in {\hat{J}}_{r} \end{matrix} \end{array}$

To prove the existence of the solution of the problem, we will prove that N₁ (x, y) is a contraction mapping.

For $w, v \in C (J_{r}, R_{F})$ , from Lipschitz condition we have $\begin{array}{l} d_{\infty} (N_{1} (w (x, y)), N_{1} (v (x, y))) \\ \leq d_{\infty} (_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}),_{F}^{R L} ℐ_{0 +}^{q} F (x, y, v_{(x, y)})) \\ \leq L_{F}^{R L} ℐ_{0 +}^{q} d_{C}^{0} (w_{(x, y)}, v_{(x, y)}) \\ \leq \frac{L}{Γ (q_{1}) Γ (q_{2})} \int_{0}^{x} \int_{0}^{y} {(x - s)}^{q_{1} - 1} {(y - t)}^{q_{2} - 1} \times \\ \times \sup_{(θ, θ^{'}) \in P} d_{\infty} (w (s + θ, t + θ^{'}), v (s + θ, t + θ^{'})) d t d s \\ \leq \frac{L}{Γ (q_{1}) Γ (q_{2})} \int_{0}^{x} \int_{0}^{y} {(x - s)}^{q_{1} - 1} {(y - t)}^{q_{2} - 1} \times \\ \times D_{σ} (w, v) e^{σ (s + t)} d t d s . \end{array}$ From Remark 2.2 and Lemma 2.3 in [17], we can see $\int_{0}^{x} (x - s)^{q_{1} - 1} e^{σ s} ds \leq \frac{e^{σ x}}{q_{1}} G (σ, q_{1}), \int_{0}^{y} (y - t)^{q_{1} - 1} e^{σ t} dt \leq \frac{e^{σ y}}{q_{2}} G (σ, q_{2}),$ where G (σ, q) is defined small when σ is big enough. Then for σ big enough, we receive $\begin{matrix} d_{\infty} (N_{1} (w (x, y)), N_{1} (v (x, y))) \\ \leq \frac{L D_{σ} (w, v) e^{σ (x + y)}}{Γ (q_{1} + 1) Γ (q_{2} + 1)} . \end{matrix}$ Multiplying both with e^-σ(x+y) then taking supremum, we have $\begin{matrix} D_{σ} (N_{1} (w), N_{1} (v)) \\ \leq \frac{L D_{σ} (w, v)}{Γ (q_{1} + 1) Γ (q_{2} + 1)} . \end{matrix}$ (3.9) When $(x, y) \in {\hat{J}}_{r},$ d_∞ (N₁ (w (x, y)) , N₁ (v (x, y))) =0. Therefore $\begin{matrix} D_{σ} (N_{1} (w), N_{1} (v)) \\ \leq \frac{L D_{σ} (w, v)}{Γ (q_{1} + 1) Γ (q_{2} + 1)} . \end{matrix}$ (3.10) for every (x, y) ∈ J_r. Since $\frac{L}{Γ (q_{1} + 1) Γ (q_{2} + 1)} < 1$ , N₁ is a contraction mapping. According to the Banach fixed point theorem, N₁ has only one fixed point. And this is a mild solution in type 1 of Problem (3.1) - (3.2). □

To study the existence of solution in type 2 of this problem, we set

${\tilde{C}}_{σ} (J_{r}, ℛ_{ℱ}) = {w \in C (J_{r}, ℛ_{ℱ}) w (x, y) = γ (x, y) ⊖ {(- 1)}_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) exist o n J_{0}} .$

Theorem 3.2. Suppose that $F \in C (J_{0} \times C (P, R_{F}), R_{F})$ satifies Lipschitz condition with Lipschitz coefficient L ∈ (0, Γ (q₁ + 1) Γ (q₂ + 1)) and ${\tilde{C}}_{σ} (J_{r}, R_{F}) \neq \emptyset$ then Problem (3.1)-(3.2) has a unique mild solution in type 2.

Proof. We consider the operator N₂ denified by $\begin{array}{l} N_{2} (w (x, y)) \\ = { \begin{matrix} γ (x, y) ⊖ {(- 1)}_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) (x, y) \in J_{0} \\ φ (x, y) with (x, y) \in {\hat{J}}_{r} \end{matrix} \end{array}$

Apply inequality d_∞ (w ⊖ v, w ⊖ e) = d_∞ (w, w) +d_∞ (v, e) with u, v, w, e are fuzzy numbers, we have $\begin{array}{l} d_{\infty} (N_{2} (w (x, y)), N_{2} (v (x, y))) \\ \leq d_{\infty} (_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}),_{F}^{R L} ℐ_{0 +}^{q} F (x, y, v_{(x, y)})) . \end{array}$ Similar to the proof of Theorem 3.1.2, we will also point out that $\begin{matrix} D_{σ} (N_{2} (w), N_{2} (v)) \leq \frac{{LD}_{σ} (w, v)}{Γ (q_{1} + 1) Γ (q_{2} + 1)}, \end{matrix}$ with each (x, y) ∈ J_r. Since $\frac{L}{Γ (q_{1} + 1) Γ (q_{2} + 1)} < 1$ , N₂ is a contraction mapping. According to the Banach fixed point theorem, N₂ has only one fixed point. And this is a mild solution type in type 2 of Problem (3.1)-(3.2). □

3.2. Fuzzy fractional functional hyperbolic equations in infinite domain

Denote by $J_{r}^{\infty} = [- r, \infty) \times [- r, \infty)$ and ${\hat{J}}_{r}^{\infty} = J_{r}^{\infty} \ [0, \infty) \times [0, \infty)$ . In this part, we will study the existence of the solution of the Problem (3.1)-(3.2) in infinite domain $J_{r}^{\infty}$ with the boundary conditions ${\begin{matrix} w (x, y) = φ (x, y), (x, y) \in {\hat{J}}_{r}^{\infty} \\ w (x, 0) = η_{1} (x), x \in [0, \infty) \\ w (0, y) = η_{2} (y), y \in [0, \infty) . \end{matrix}$ (3.11)

For σ > 0,we denote $C_{σ} (J_{r}^{\infty}, R_{F}) = {w \in C$ $(J_{r}^{\infty}, R_{F}) | sup_{(s, t) \in J_{r}^{\infty}} d_{\infty} (w (s, t), \hat{0}) e^{- σ (s + t)} < + \infty}$ and $\begin{matrix} H_{σ} (w, v) \\ = sup_{(s, t) \in J_{r}^{\infty}} {d_{\infty} (w (s, t), v (s, t)) e^{- σ (s + t)}} . \end{matrix}$ Then $(C_{σ} (J_{r}^{\infty}, R_{F}), H_{σ})$ is also a complete metric space.

We consider Problem (3.1)-(3.11) under the following assumptions

( H ₁) $F : J_{0}^{\infty} \times C (P, R_{F}) \to R_{F}$ satisfies the Lipschitz condition with L ∈ (0, Γ (q₁ + 1) Γ (q₂ + 1)) $d_{\infty} (F (x, y, ν), F (x, y, ν^{'})) \leq {Ld}_{C}^{0} (ν, ν^{'})$ (3.12) for all $ν, ν^{'} \in C (P, R_{F}), (x, y) \in J_{0}^{\infty}$ . ( H ₂) $d_{\infty} (F (s, t, \hat{0}), \hat{0}) \leq M_{1} e^{c_{1} (s + t)}$ for all $(s, t) \in J_{0}^{\infty}$ , where M₁, c₁ are positive real numbers. ( H ₃) $d_{\infty} (η_{1} (x), \hat{0}) \leq M_{2} e^{c_{2} x},$ $d_{\infty} (η_{2} (y), \hat{0}) \leq M_{3} e^{c_{3} y}$ , where M_i and c_i (i = 2, 3) are positive real numbers, for all x ∈ [0, ∞), y ∈ [0, ∞).

Lemma 3.2. Assume that (H₁) and (H₂) hold for all $(x, y) \in J_{0}^{\infty}$ . Then for all $(x, y) \in J_{0}^{\infty}$ , we have $\begin{matrix} Γ (q_{1} + 1) & Γ {(q_{2} + 1)}_{F}^{R L} ℐ_{0 +}^{q} d_{\infty} (F (x, y, w_{(x, y)}), \hat{0}) \\ \leq M_{1} e^{c_{1} (x + y)} + ρ L e^{σ (x + y)} . \end{matrix}$ (3.13)

Proof. For all $(x, y) \in J_{0}^{\infty}$ , we have $\begin{array}{l} Γ (q_{1}) Γ {(q_{2})}_{F}^{R L} ℐ_{0 +}^{q} d_{\infty} (F (x, y, w_{(x, y)}), \hat{0}) \\ \leq \int_{0}^{x} \int_{0}^{y} {(x - s)}^{q_{1} - 1} {(y - t)}^{q_{2} - 1} \times \\ \times d_{\infty} (F (s, t, w_{(s, t)}), F (s, t, \hat{0})) d t d s \\ + \int_{0}^{x} \int_{0}^{y} {(x - s)}^{q_{1} - 1} {(y - t)}^{q_{2} - 1} \\ \times d_{\infty} (F (s, t, \hat{0}), \hat{0}) d t d s . \end{array}$ Applying hypothesis (H2) and Lemma 2.3 ([17]), one has $\begin{matrix} \int_{0}^{x} \int_{0}^{y} (x - s)^{q_{1} - 1} (y - t)^{q_{2} - 1} d_{\infty} (F (s, t, \hat{0}), \hat{0}) dtds \\ \leq \int_{0}^{x} \int_{0}^{y} (x - s)^{q_{1} - 1} (y - t)^{q_{2} - 1} M_{1} e^{c_{1} (x + y)} dxdy \\ \leq \frac{M_{1}}{q_{1} q_{2}} e^{c_{1} (x + y)} . \end{matrix}$ It implies from hypothesis (H1) that $\begin{matrix} _{F}^{R L} ℐ^{q} & d_{\infty} (F (x, y, w_{(x, y)}), \hat{0}) \leq_{L F}^{R L} ℐ^{q} d_{C}^{0} (w_{(x, y)}, \hat{0}) \\ + \frac{M_{1}}{Γ (q_{1} + 1) (Γ (q_{1} + 1)} e^{c_{1} (x + y)} . \end{matrix}$ (3.14)

With $(θ, θ^{'}) \in P, (s, t) \in J_{0}^{\infty},$ there exists a positive number ρ satisfying $\begin{matrix} d_{\infty} (w (s + θ, y + θ^{'}), \hat{0}) & \leq & ρ e^{σ (s + t + θ + θ^{'})} \\ \leq & ρ e^{σ (s + t)} . \end{matrix}$ Therefore, $\begin{matrix} d_{C}^{0} (w_{(s, t)}, \hat{0}) & = & \sup_{(θ, θ^{'}) \in P} d_{\infty} (w (s + θ, t + θ^{'}), \hat{0}) \\ \leq & ρ e^{σ (s + t)} . \end{matrix}$ This implies $\begin{array}{l} _{F}^{R L} ℐ^{q} d_{C}^{0} (w_{(x, y)}, \hat{0}) \\ \leq \frac{1}{Γ (q_{1}) Γ (q_{2}))} \int_{0}^{x} \int_{0}^{y} {(x - s)}^{q_{1} - 1} {(y - t)}^{q_{2} - 1} \\ \times ρ e^{σ (s + t)} d t d s \\ \leq \frac{ρ L}{Γ (q_{1}) Γ (q_{2})} \int_{0}^{x} {(x - s)}^{q_{1} - 1} e^{σ s} d s \\ \times \int_{0}^{y} {(y - t)}^{q_{1} - 1} e^{σ t} d t \\ \leq \frac{ρ L}{Γ (q_{1} + 1) Γ (q_{2} + 1)} e^{σ (x + y)} . \end{array}$ (3.15)

From (3.15)-(3.16), we recieve (3.13). It ends the proof. □

Lemma 3.3.Assume that (H₁) , (H₂) and (H₃) hold for all $(x, y) \in J_{r}^{\infty}$ . Then $N_{3} (w (x, y)) \in C_{σ} (J_{r}^{\infty}, R_{F})$ for all $w (x, y)) \in C_{σ} (J_{r}^{\infty}, R_{F})$ where the operator N₃ (w (x, y)) is denified by $\begin{array}{l} N_{3} (w (x, y)) \\ = {\begin{array}{l} γ (x, y) +_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) if (x, y) \in J_{0}^{\infty} \\ φ (x, y) if (x, y) \in {\hat{J}}_{r}^{\infty} . \end{array} \end{array}$

Proof. $\begin{array}{l} d_{\infty} (N_{3} (w (x, y), \hat{0})) \\ \leq d_{\infty} (γ (x, y), \hat{0}) + d_{\infty} (_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}), \hat{0}) \\ \leq d_{\infty} (γ (x, y), \hat{0}) +_{F}^{R L} ℐ_{0 +}^{q} d_{\infty} (F (x, y, w_{(x, y)}), \hat{0}) . \end{array}$ From Lemma 3.2, we have $d_{\infty} (N_{3} (w (x, y), \hat{0})) \leq d_{\infty} (γ (x, y), \hat{0}) + \frac{ρ {Le}^{σ (x + y)} + M_{1} e^{c_{1} (x + y)}}{Γ (q_{1} + 1) Γ (q_{2} + 1)} .$

This implies $\begin{matrix} d_{\infty} (N_{3} (w (x, y), \hat{0})) e^{- σ (x + y)} \\ \leq d_{\infty} (γ (x, y), \hat{0}) e^{- σ (x + y)} \\ + \frac{L ρ}{Γ (q_{1} + 1) Γ (q_{2} + 1)} + \frac{M_{1} e^{(c_{1} - σ) (x + y)}}{Γ (q_{1} + 1) Γ (q_{2} + 1)} . \end{matrix}$ We have $\begin{matrix} d_{\infty} (γ (x, y), \hat{0}) e^{- σ (x + y)} \\ \leq [d_{\infty} (w (x, 0), \hat{0}) + d_{\infty} (w (0, y), \hat{0}) \\ + d_{\infty} (w_{0}, \hat{0})] e^{- σ (x + y)} \\ \leq M_{2} e^{(c_{2} - σ) (x + y)} + M_{3} e^{(c_{3} - σ) (x + y)} \\ + d_{\infty} (w_{0}, \hat{0}) e^{- σ (x + y)} . \end{matrix}$ (3.16) Choosing σ > max {c₁, c₂, c₃}, it leads to $\begin{matrix} d_{\infty} (γ (x, y), \hat{0}) e^{- σ (x + y)} \leq M_{2} + M_{3} + ρ . \end{matrix}$ (3.17) From (3.16)-(3.17), we can see that $\begin{matrix} d_{\infty} & (N_{3} (w (x, y), \hat{0})) e^{- σ (x + y)} \leq M_{2} + M_{3} \\ + ρ + \frac{L ρ}{Γ (q_{1} + 1) Γ (q_{2} + 1)} + \frac{M_{1} e^{(c_{1} - σ) (x + y)}}{Γ (q_{1} + 1) Γ (q_{2} + 1)} \\ < + \infty . \end{matrix}$ It is obvious that if N₃ (w (x, y)) = φ (x, y) then $N_{3} (w (x, y)) \in C_{σ} (J_{r}^{\infty}, R_{F}) .$ Therefore N₃ (w (x, y)) $\in C_{σ} (J_{r}^{\infty}, R_{F})$ for every $(x, y) \in J_{r}^{\infty}$ . □

Lemma 3.4.If F satisfies the conditions (H₁) - (H₂) - (H₃) then we have $\begin{matrix} d_{\infty} & (_{F}^{R L} ℐ^{q} F (x, y, w_{(x, y)}),_{F}^{R L} ℐ^{q} F (x, y, v_{(x, y)})) \\ \leq \frac{L H_{σ} (w, v)}{Γ (q_{1} + 1) Γ (q_{2} + 1)} e^{σ (x + y)} \end{matrix}$ for all $w, v \in C_{σ} (J_{r}^{\infty}, R_{F}), (x, y) \in J_{0}^{\infty}$ .

Proof. It follows from (H₁) that $\begin{array}{l} d_{\infty} (_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}),_{F}^{R L} ℐ_{0 +}^{q} F (x, y, v_{(x, y)})) \\ \leq_{F}^{R L} ℐ_{0 +}^{q} d_{\infty} (F (x, y, w_{(x, y)}), F (x, y, v_{(x, y)})) \\ \leq L_{F}^{R L} ℐ_{0 +}^{q} d_{C}^{0} (w_{(x, y)}, v_{(x, y)}) \\ \leq L_{F}^{R L} ℐ_{0 +}^{q} \sup_{(θ, θ^{'}) \in P} d_{\infty} (w (x + θ, y + θ^{'}), \\ v (x + θ, y + θ^{'})) \\ \leq L_{F}^{R L} ℐ_{0 +}^{q} \sup_{(ω, ω^{'}) \in [x - r, x] \times [y - r, y]} d_{\infty} (w (ω, ω^{'}), \\ v (ω, ω^{'}) \\ \leq L_{F}^{R L} ℐ_{0 +}^{q} (\sup_{(ω, ω^{'}) \in J_{r}^{\infty}} d_{\infty} (w (ω, ω^{'}), v (ω, ω^{'}))) . \end{array}$ In another way $\begin{matrix} sup_{(ω, ω^{'}) \in J_{r}^{\infty}} d_{\infty} (w (ω, ω^{'}), v (ω, ω^{'}) \\ \leq H_{σ} (w, v) e^{σ (ω + ω^{'})} . \end{matrix}$ This implies $\begin{array}{l} d_{\infty} (_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}),_{F}^{R L} ℐ_{0 +}^{q} F (x, y, v_{(x, y)})) \\ \leq L_{F}^{R L} ℐ_{0 +}^{q} (H_{σ} (w, v) e^{σ (x + y)}) . \end{array}$ (3.18) On the other hand, $\begin{array}{l} L_{F}^{R L} ℐ_{0 +}^{q} (H_{σ} (w, v) e^{σ (x + y)}) \\ = \frac{L H_{σ} (w, v)}{Γ (q_{1}) Γ (q_{1})} \int_{0}^{x} {(x - s)}^{q_{1} - 1} e^{σ s} d s \times \\ \times \int_{0}^{y} {(y - t)}^{q_{2} - 1} e^{σ t} d t \\ \leq \frac{L H_{σ (w, v)}}{Γ (q_{1} + 1) Γ (q_{2} + 1)} e^{σ (x + y)} . \end{array}$ (3.19) From (3.18)-(3.19), we have $\begin{array}{l} d_{\infty} (_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}),_{F}^{R L} ℐ_{0 +}^{q} F (x, y, v_{(x, y)})) \\ \leq \frac{L H_{σ} (w, v)}{Γ (q_{1} + 1) Γ (q_{2} + 2)} e^{σ (x + y)} \end{array}$ for all $(x, y) \in J_{0}^{\infty} .$ □

Theorem 3.3. Suppose that the assumptions (H₁) - (H₃) are satisfied. Then Problem (3.1)-(3.11) has a unique mild solution in type 1.

Proof. From Lemma 3.2, N₃ is well-defined on $C_{σ} (J_{r}^{\infty}, R_{F})$ . Moreover, with $u, v \in C_{σ} (J_{r}^{\infty}, R_{F})$ , we have $\begin{array}{l} d_{\infty} (N_{3} (w_{(x, y)}), N_{3} (v_{(x, y)})) \\ \leq d_{\infty} (_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}),_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)})) . \end{array}$ Applying Lemma 3.4, we have $\begin{matrix} d_{\infty} (N_{3} (w_{(x, y)}), N_{3} (v_{(x, y)})) \\ \leq \frac{{LH}_{σ} (w, v)}{Γ (q_{1} + 1) Γ (q_{2} + 1)} e^{σ (x + y)} . \end{matrix}$ Therefore, $\begin{matrix} sup_{(x, y) \in J_{\infty}} d_{\infty} (N_{3} (w_{(x, y)}), N_{3} (v_{(x, y)})) e^{- σ (x + y)} \\ \leq \frac{L H_{σ} (w, v)}{Γ (q_{1} + 1) Γ (q_{2} + 1)} . \end{matrix}$ It leads to $H_{σ} (N_{3} (w), N_{3} (v)) \leq \frac{L H_{σ} (w, v)}{Γ (q_{1} + 1) Γ (q_{2} + 1)} .$ With $(x, y) \in {\hat{J}}_{r}^{\infty}$ , d_∞ (N₃ (w_(x,y)) , N₃ (v_(x,y))) =0. Thus, for each $(x, y) \in J_{r}^{\infty}$ , we get $\begin{matrix} H_{σ} (N_{3} (w), N_{3} (v)) \leq \frac{L H_{σ} (w, v)}{Γ (q_{1} + 1) Γ (q_{2} + 1)} . \end{matrix}$ Because $\frac{L}{Γ (q_{1} + 1) Γ (q_{2} + 1)} < 1$ , N₃ (w (x, y)) is a contraction mapping. Consequently, N₃ (w (x, y)) has a unique fixed point $w \in C_{σ} (J_{r}^{\infty}, R_{F}),$ which is the mild solution in type 1 of Problem (3.1)-(3.11). □

Theorem 3.4. Suppose that all asummptions (H₁) - (H₂) - (H₃) are fulfilled. Futhermore, ${\hat{C}}_{σ} (J_{r}^{\infty}, R_{F}) = {w (x, y) \in C_{σ} (J_{r}^{\infty}, R_{F})$ $| w (x, y) = γ (x, y) ⊖ {(- 1)}_{F}^{R L} F (x, y, w_{(x, y)})} \neq \emptyset .$

Then Problem (3.1)-(3.11) has a unique mild solution in type 2.

Proof. We consider the operator $\begin{array}{l} N_{4} (w (x, y)) \\ = {\begin{array}{l} γ (x, y) ⊖ {(- 1)}_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) if (x, y) \in {\hat{J}}^{\infty} \\ φ (x, y) if (x, y) \in {\hat{J}}_{r}^{\infty} . \end{array} \end{array}$ Using the similar method in the proof of Theorem 3.2, we prove that N₄ is a contraction mapping from ${\hat{C}}_{σ} (J_{r}^{\infty}, R_{F})$ to ${\hat{C}}_{σ} (J_{r}^{\infty}, R_{F})$ , which has an unique fixed point. That is the mild solution in type 2 of the problem. □

4. Ulam-Hyers stability

Definition 4.1.A solution of following in equation $d_{\infty} (_{g H}^{C} D_{k}^{q} v (x, y), F (x, y, v_{(x, y)})) \leq ϵ$ (4.20) is a function $v \in W_{gH}^{k} (J_{\infty}, R_{F})$ , which satisfies $\begin{array}{l} _{g H}^{C} D_{k}^{q} v (x, y) \\ = F (x, y, v_{(x, y)}) + h_{k} (x, y), (x, y) \in J_{0}^{\infty}, \end{array}$ where $h_{k} \in C_{σ} (J_{0}^{\infty}, R_{F})$ and $d_{\infty} (h_{k} (x, y), \hat{0}) \leq ε$ for all $(x, y) \in J_{0}^{\infty} .$

Lemma 4.1.

For k = 1, if g is a solution of (4.20), then it satisfies $\begin{matrix} d_{\infty} (g (x, y), g (x, 0) + g (0, y) ⊖ g (0, 0) + \\ +_{F}^{RL} I_{0^{+}}^{q} F (x, y, g_{(x, y)})) \\ \leq \frac{ε x^{q_{1}} y^{q_{2}}}{Γ (q_{1} + 1) Γ (q_{2} + 1)} \end{matrix}$ (4.21)for all $(x, y) \in J_{0}^{\infty}$ .

For k = 2, if g is a solution of (4.20), then it satisfies $\begin{matrix} d_{\infty} (g (x, y), g (0, y) + g (x, 0) ⊖ g (0, 0) ⊖ \\ ⊖ (- 1)_{F}^{RL} I_{0^{+}}^{q} F [v] (x, y)) \\ \leq \frac{ε x^{q_{1}} y^{q_{2}}}{Γ (q_{1} + 1) Γ (q_{2} + 1)} \end{matrix}$ (4.22)for all $(x, y) \in J_{0}^{\infty}$ .

Proof. Suppose that g is a solution of (4.20), then there exists a function $h_{k} \in C_{σ} (J_{0}^{\infty}, E)$ such that $\begin{array}{l} _{g H}^{C} D_{k}^{q} g (x, y) \\ = F (x, y, g_{(x, y)}) + h_{k} (x, y) forall (x, y) \in J_{0}^{\infty} . \end{array}$ By doing the same arguments in the Proof of Lemma 3.1, we have $\begin{matrix} g (x, y) & = & g (0, y) + [g (x, 0) ⊖ g (0, 0)] \\ +_{F}^{RL} I_{0^{+}}^{q} F (x, y, g_{(x, y)}) \\ +_{F}^{RL} I_{0^{+}}^{q} h_{1} (x, y), (k = 1), \end{matrix}$ (4.23) or with (k = 2) $\begin{matrix} g (x, y) & = & g (0, x) + g (0, y) ⊖ g (0, 0) ⊖ \\ (- 1)_{F}^{RL} I_{0^{+}}^{q} F [g] (x, y) ⊖ \\ (- 1)_{F}^{RL} I_{0^{+}}^{q} h_{2} (x, y) . \end{matrix}$ (4.24) From (4.23), we have $\begin{matrix} d_{\infty} (g (x, y), g (x, 0) + g (0, y) ⊖ g (0, 0) \\ +_{F}^{RL} I_{0^{+}}^{q} F (x, y, g_{(x, y)})) \\ \leq d_{\infty} (_{F}^{RL} I_{0^{+}}^{q} h_{1} (x, y), \hat{0}) \\ \leq \frac{ε}{Γ (q_{1}) Γ (q_{2})} \int_{0}^{x} \int_{0}^{y} (x - s)^{q_{1} - 1} (y - t)^{q_{2} - 1} dtds \\ \leq \frac{ε x^{q_{1}} y^{q_{2}}}{Γ (q_{1} + 1) Γ (q_{2} + 1)} . \end{matrix}$ From (4.24), one gets $\begin{matrix} d_{\infty} (g (x, y), & g (x, 0) + g (0, y) ⊖ g (0, 0) \\ +_{F}^{RL} I_{0^{+}}^{q} F (x, y, g_{(x, y)})) \\ \leq d_{\infty} (_{F}^{RL} I_{0^{+}}^{q} h_{2} (x, y), \hat{0}) \\ \leq \frac{ε x^{q_{1}} y^{q_{2}}}{Γ (q_{1} + 1) Γ (q_{2} + 1)} . \end{matrix}$ It completes the proof. □

Definition 4.2.

1) Mapping $v \in W_{gH}^{1} (J_{0}^{\infty}, R_{F})$ satisfying (4.21) is called a mild solution in type 1 of in equation (4.20) with k = 1.

2) Mapping $v \in W_{gH}^{2} (J_{0}^{\infty}, R_{F})$ satisfying (4.22) is called a mild solution in type 2 of in equation (4.20) with k = 2.

Definition 4.3. If there exists $\hat{c} > 0$ such that for arbitrary ε > 0, for each mild solution g in type k of in equation (4.20), there exists a mild solution w in type k of equation (3.1) with initial conditions ${\begin{matrix} w (x, 0) = g (x, 0), x \in [0, \infty) \\ w (0, y) = g (0, y), y \in [0, \infty) . \end{matrix}$ (4.25) such that $H_{σ} (w, g) \leq \hat{c} ε$ then problem (3.1)-(3.11) is called Ulam-Hyers stable in type k (k = 1, 2).

Denote T [g] (x, y) = g (x, 0) + g (0, y) ⊖ g (0, 0) $⊖ {(- 1)}_{F}^{R L} ℐ_{0^{+}}^{q} F (x, y, g_{(x, y)}), (x, y) \in J_{0}^{\infty}$ and $\begin{matrix} {\hat{C}}_{g σ} (J_{0}^{\infty}, W_{gH}^{1} (J_{0}^{\infty}, R_{F})) \\ : = {v \in C_{σ} (J_{0}^{\infty}, W_{gH}^{1} (J_{0}^{\infty}, R_{F})) : \exists T [g] (x, y)} . \end{matrix}$

Theorem 4.1. Assume that hypotheses (H₁) - (H₃) are satisfied. Then problem (3.1)-(3.11) is Ulam-Hyers stable in type 1. In addition, assume that for each mild solution g in type 2 of inequation (4.20) $\begin{matrix} {\hat{C}}_{g σ} & (J_{0}^{\infty}, W_{gH}^{1} (J_{0}^{\infty}, R_{F})) \neq \emptyset, \end{matrix}$ then problem (3.1)-(3.11) is Ulam-Hyers stable in type 2.

Proof. For arbitrary ε > 0, suppose that $g \in C_{σ} (J_{0}^{\infty}, W_{gH}^{1} (J_{0}^{\infty}, R_{F}))$ is a mild solution in type 1 of the inequation (4.20). By Theorem 3.1, we can see that problem (3.1)-(4.25) has a unique mild solution in type 1 satisfying $w (x, y) = N_{5} (w (x, y)),$ where $\begin{array}{l} N_{5} (w (x, y)) & = & g (x, 0) + g (0, y) ⊖ g (0, 0) \\ +_{F}^{R L} ℐ_{0 +}^{q} F (x, y, w_{(x, y)}) \\ if (x, y) \in J_{0}^{\infty} . \end{array}$

Then we have $\begin{matrix} d_{\infty} (w (x, y), g (x, y)) \\ \leq d_{\infty} (N_{5} (w (x, y)), N_{5} (g (x, y))) \\ + d_{\infty} (N_{5} (g (x, y)), g (x, y)) . \end{matrix}$ We have $\begin{array}{l} d_{\infty} (N_{5} (w (x, y)), N_{5} (g (x, y))) \\ \leq d_{\infty} (_{F}^{R L} ℐ^{q} F (x, y, w_{(x, y)}), \\ \underset{F}{R L} ℐ^{q} F (x, y, g_{(x, y)})) . \end{array}$ From Lemma 3.4, we receive $\begin{matrix} d_{\infty} & (_{F}^{R L} ℐ^{q} F (x, y, w_{(x, y)}),_{F}^{R L} ℐ^{q} F (x, y, g_{(x, y)})) \\ \leq \frac{L H_{σ} (w, g)}{Γ (q_{1} + 1) Γ (q_{2} + 2)} e^{σ (x + y)} . \end{matrix}$ Multiplying both sides with e^-σ(x+y) then taking supremum we obtain $\begin{matrix} H_{σ} (w, g) \leq \frac{{LH}_{σ} (w, v)}{Γ (q_{1} + 1) Γ (q_{2} + 1)} + H_{σ} (g, N_{3} g) . \end{matrix}$ It leads to $\begin{matrix} H_{σ} (w, g) \leq ε \frac{Γ (q_{1} + 1) Γ (q_{2} + 1)}{Γ (q_{1} + 1) Γ (q_{2} + 1) - L} . \end{matrix}$ Because $\frac{L}{Γ (q_{1} + 1) Γ (q_{2} + 1)} < 1$ , we can see that $\hat{c} : = \frac{Γ (q_{1} + 1) Γ (q_{2} + 1)}{Γ (q_{1} + 1) Γ (q_{2} + 1) - L} > 0 .$ Then $H_{σ} (w, v) \leq ε \hat{c} .$ Therefore, equation (3.1) is Ulam-Hyers stable in type 1. □

5. Application examples

Example 5.1. We consider the the following model $\begin{array}{l} _{g H}^{C} D_{1}^{q} w (x, y) & = & \frac{1}{4} w (x - 1, y - 1) \\ + \frac{K}{16} (3 x y + \frac{11}{2} x + \frac{11}{2} y + \frac{35}{4}) \end{array}$ (5.26) in [0, 2] × [0, 2] with the conditions ${\begin{matrix} w (x, 0) = \frac{K}{4} (x + \frac{3}{2}), x \in [0, 2] \\ w (0, y) = \frac{K}{4} (y + \frac{3}{2}), y \in [0, 2] \end{matrix}$ (27)

Suppose that K is a triangular fuzzy number with [K] ^κ = [κ, 1 - κ] , κ ∈ [0, 1] . For $c_{1} = 2, M_{1} = \frac{1}{16},$ $c_{2} = c_{3} = \frac{1}{4}, M_{2} = M_{3} = 1$ , if we choose σ > max {c₁, c₂, c₃} =1 then all the conditions of Theorem 3.1 are satisfied. By applying the Theorem 3.1, the problem (5.26)-(5.27) has a mild solution in type 1 and it is generalized Ulam-Hyer stable. Indeed, we can check that $w (x, y) = {\begin{matrix} \frac{K}{4} (x + \frac{3}{2}) (y + \frac{3}{2}) & for (x, y) \in [0, 2]^{2} \\ Kxy & for (x, y) \in [- 1, 2]^{2} ∖ [0, 2]^{2} \end{matrix}$ is a mild solution in type 1 of the problem in [-1, 2] × [-1, 2] .

Example 5.2. We consider the the following model $\begin{matrix} _{g H}^{C} D_{1}^{q} & w (x, y) = \frac{π}{8} w (x - 1, y - 1) \\ + \frac{K π}{16} (x + y + (\sqrt{x y} - 1)^{2}) \end{matrix}$ (5.28) on the domain (x, y) ∈ [0, ∞) × [0, ∞) with the conditions $\begin{matrix} {\begin{matrix} w (x, 0) = K \sqrt{x + 1}, x \in [0, \infty) \\ w (0, y) = K \sqrt{y + 1}, x \in [0, \infty) \\ w (x, y) = K (sin x + sin y), (x, y) \in [- 1, \infty)^{2} ∖ [0, \infty)^{2} \end{matrix} \end{matrix}$ (5.29) [K] ^κ = [κ, 1 - κ] , κ ∈ [0, 1] is a triangular fuzzy number.

It is easy to see that $F (x, y, w_{(x, y)}) = \frac{π}{8} w (x - 1, y - 1) + \frac{K π}{16} (x + y + (\sqrt{xy} - 1)^{2})$ satisfies Lipschitz condition with Lipschitz constant is $L = \frac{π}{8}$ , since $\begin{matrix} d_{\infty} (F (x, y, w_{(x, y)}), F (x, y, g_{(x, y)})) \\ \leq \frac{π}{8} d_{\infty} (w_{(x, y)}, g_{(x, y)}) . \end{matrix}$ From the following estimation $\begin{matrix} d_{\infty} (F (x, y, \hat{0}), \hat{0}) \\ = d_{\infty} (\frac{K π}{16} (x + y + (\sqrt{xy} - 1)^{2}, \hat{0}) \\ = \frac{π}{16} (x + y + (\sqrt{xy} - 1)^{2}) d_{H} ([K]^{κ}, [0]^{κ}) \\ \leq \frac{π}{16} e^{x + y}, \end{matrix}$ we implies that the condition (H₃) is satisfied with $c_{1} = 1, M_{1} = \frac{π}{16} .$ By doing the same arguments we have $\begin{matrix} d_{\infty} (η_{1} (x), \hat{0}) \\ \leq e^{2} e^{\frac{x}{3}} and d_{\infty} (η_{2} (y), \hat{0}) \leq e^{2} e^{\frac{y}{3}} . \end{matrix}$ Thus (H₂) - (H₃) are satisfied with $c_{2} = c_{3} = \frac{1}{3}, M_{2} = M_{3} = e^{2} .$ If we choose σ > max {c₁, c₂, c₃} =1 then all the conditions of Theorem 3.3 are satisfied. We can see that $\frac{L}{Γ (\frac{3}{2}) Γ (\frac{3}{2})} < 1$ , then by applying the Theorem 3.1 and Theorem 4.1, the problem (5.28)-(5.29) has a mild solution in type 1. It is easy to see that $w (x, y) = K \sqrt{(x + 1) (y + 1)}$ for (x, y) ∈ [0, ∞) × [0, ∞) and w (x, y) = K (sin x + sin y) for (x, y) ∈ [-1, ∞) × [-1, ∞) ∖ [0, ∞) × [0, ∞) is a mild solution in type 1 of the problem and the problem is Ulam-Hyer stable.

Example 5.3. Consider following model with Caputo-gH derivatives in type 2 $\begin{matrix} _{g H}^{C} D_{2}^{q} w (x, y) = \frac{1}{6} w (x - 1, y - 1) \\ + K (\frac{64}{9 π} x^{\frac{3}{2}} x^{\frac{3}{2}} + x y - x - y + \frac{1}{2}) \end{matrix}$ (5.30) on the domain (x, y) ∈ [0, ∞) × [0, ∞) with the conditions ${\begin{matrix} w (x, 0) = \frac{1}{2} K, x \in [0, \infty) \\ w (0, y) = \frac{1}{2} K, y \in [0, \infty) \\ w (x, y) = K (1 + xy), (x, y) \in [- 1, \infty)^{2} ∖ [0, \infty)^{2} \end{matrix}$ (31) where [K] ^κ = [κ, 1 - κ] , κ ∈ [0, 1] .

We can see that $F (x, y, w_{(x, y)}) = \frac{1}{6} w (x - 1, y - 1) + K (\frac{64}{9 π} x^{\frac{3}{2}} x^{\frac{3}{2}} + xy - x - y + \frac{1}{2})$ satisfies Lipschitz condition with Lipschitz constant is $L = \frac{1}{6}$ and $\begin{matrix} d_{\infty} (F (x, y, \hat{0}), \hat{0}) \leq 3 e^{\frac{3}{2} (x + y)} . \end{matrix}$ Therefore (H₁) - (H₃) are satisfied with $a_{1} = 3, M - 1 = \frac{32}{,} c_{2} = c_{3} = 1, M_{2} = M_{3} = 1 .$ Applying the Theorem 3.2 and Theorem 4.1, the problem (5.30)-(5.31) has a mild solution in type 2 and it is generalized Ulam-Hyer stable. We can see that $\begin{matrix} w (x, y) = {\begin{matrix} \frac{1}{2} K ⊖ Kxy & for (x, y) \in [0, \infty)^{2} \\ K (1 + xy) & for (x, y) \in [- 1, \infty)^{2} ∖ [0, \infty)^{2} . \end{matrix} \end{matrix}$ is a mild solution in type 2 of the problem.

6. Conclusions

The existence and uniqueness of mild solutions for a class of fuzzy partial differential equations with delays in finite and infinite domains are considered. The obtained results are gained by fixed point approach in the weighted metric space. Moreover, we have also investigated the Ulam-Hyers stability of problems. It is worth to mention that the Ulam stability can be used instead of Lyapunov stability in some control problems and it is quite useful in many applied problems where finding the exact solution is more difficult than solving its approximate solutions.

Footnotes

Acknowledgments

The author would like to thank Editor-in-chiefs Prof. Langari; Associate editor Prof. Allahviranloo; the anonymous referees for their helpful comments and valuable suggestions, which have greatly improved the paper.

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