Hyers-Ulam stability for nonlocal fractional partial integro-differential equation with uncertainty

Abstract

In this paper, we study nonlocal problems for fractional partial intergro-differential equations with uncertainty in the framework of partially ordered generalized metric spaces of fuzzy valued functions. Based on generalized contractive-like property over comparable items, which is weaker than the Lipschitz condition, we prove the global existence of mild solutions on the infinite domain J_∞ = [0, ∞) × [0, ∞). Moreover, Hyers-Ulam stability of this problem is given with the help of Perov-like fixed point theorem.

Keywords

Fuzzy fractional partial integro-differential equation convergent matrix vector-valued metric partially ordered generalized metric spaces gH-differentiability

1 Introduction

In [5], Agarwal et al. combined differential equations of fractional order with uncertainty to introduce a new type of dynamical system $\begin{matrix} D^{q} x (t) = f (t, x (t)), t \in (0, T], \\ lim_{t \to 0^{+}} t^{1 - q} x (t) = x_{0} \in E, \end{matrix}$ where f : (0, T] × E → E is fuzzy-valued continuous functions, q ∈ (0, 1] is the fractional order, and E be the set of fuzzy numbers. They formulated the Riemann-Liouville differentiability notion as the base to define the concept of fuzzy fractional differential equations. We notice that the differentiability of the fuzzy-valued functions f in the fractional case was not covered in this work. To extend this, Allahviranloo et al. [2] introduced the Riemann-Liouville H-differentiability of fuzzy-valued functions, Mazandarani and Kamyad [23] introduced the Caputo type fuzzy fractional derivative for the solution of fuzzy fractional differential equation. Since then, the fuzzy fractional differential equation has been attracting more and more attentions due to its capable of modeling many processes in physics, chemistry and engineering operating under uncertain environment, see for instance [1 , 31]. Furthermore, it appeared in the theory of the control of dynamical systems, when the controlled system or the controller is described by an fractional differential equation [8]. For a more recent significant contributions in calculation in fuzzy-valued functions, we can see [6 , 32].

In [24], Nieto and Rodríguez-López developed some extensions of the classical Banach fixed point theorem in partially ordered sets to study the existence and uniqueness of solution for fuzzy differential equations. Follow up this work, Villamizar-Roa et al. [33] study the existence and uniqueness of solution for fuzzy initial value problems in the setting of a generalized Hukuhara derivative and by using some recent results of fixed point of weakly contractive mappings on partially ordered sets. In [4, 17] the authors applied the contractive-like mapping principle to study fuzzy fractional partial differential equations and interval-valued fractional integro-differential equations in partially ordered metric spaces. More generally, in [21] we generalized the results [3, 14] on the existence of coincidence points for a pair of mappings. Then we investigated the existence and uniqueness of fuzzy solutions to a boundary value problem for a class of fuzzy partial hyperbolic equation under generalized Hukuhara derivatives.

In this paper, we consider two associated types of fuzzy fractional partial integro-differential equations $\begin{matrix} _{gH}^{C} D_{k}^{q} u (x, y) =_{F}^{RL} I_{0^{+}}^{1 - q} f (x, y, u, u_{x}, u_{y}), k = 1, 2 \end{matrix}$ (1.1) $(x, y) \in J_{\infty} = [0, \infty) \times [0, \infty)$ with the nonlocal condition ${\begin{matrix} u (0, y) \oplus \sum_{j \in K_{1}} a_{j} ⊙ u (x_{j}, y) \\ = φ_{1} (y) \oplus \sum_{j \in K_{2}} a_{j} ⊙ u (x_{j}, y) \\ u (x, 0) \oplus \sum_{j \in Q_{1}} b_{j} ⊙ u (x, y_{j}) \\ = φ_{2} (x) \oplus \sum_{j \in Q_{2}} b_{j} ⊙ u (x, y_{j}) \\ K_{1} \cup K_{2} = Q_{1} \cup Q_{2} = {1, 2, . . ., n} \\ K_{1} \cap K_{2} = Q_{1} \cap Q_{2} = \emptyset \end{matrix}$ (1.2)

Here we recall the notion Caputo gH-derivatives $_{gH}^{C} D_{k}^{q} u (x, y)$ of order q = (q₁, q₂) ∈ (0, 1] ² with respect to x, y of function u (see [18], Definition 3.2) and $_{F}^{RL} I_{0^{+}}^{1 - q} f (x, y, u, u_{x}, u_{y})$ is the mixed Riemann-Liouville fractional integral of order 1 - q (see [18], Definition 3.1); 0 < x₁ ≤ x₂ ≤ … ≤ x_n < 1 and 0 < y₁ ≤ y₂ ≤ … ≤ y_n < 1 are given points; a_j, b_j are real positive numbers that $\tilde{a} : = \sum_{j \in K_{2}} a_{j} - \sum_{j \in K_{1}} a_{j} \geq 0$ and $\tilde{b} : = \sum_{j \in Q_{2}} a_{j} - \sum_{j \in Q_{1}} a_{j} \geq 0$ ; f : J_∞ × E³ → E is given function and $φ_{1}, φ_{2} \in C_{gH}^{1} ((0, \infty), E)$ -the space of all continuous gH-differentiable functions.

Let us remind that, the nonlocal conditions was first established for abstract Cauchy problem in Banach space by Byszewski and Lakshmikantham [11, 12]. These nonlocal conditions can be used to describe of motion phenomena with better effect than the classical conditions. After that, there are many significant researches on various classes of nonlocal differential equations, see for instance [7 , 34].

Base on the classification of gH-differentiability, we define four types of (k)-mild (k = 1, ... , 4) solutions (Definition 3.1). Each one can be considered as the fixed point of corresponding integral equations defined in Lemma 3.1. Then we use Perov-like fixed point theorem [9] in the framework of partially order completely generalized metric spaces of fuzzy-valued functions to study the existence of mild solutions of Problem (1.1)–(1.2). To this end, we first recall the framework $C = (C (J_{\infty}, E))^{3}$ from Lemma 2.3 in [20], which are partially order completely generalized metric spaces. Under Hypothesis (G1) on the weakly contractive of f combining with matrix convergent to zero technique, we prove the existence of (1)-mild solutions in Theorem 3.1. The other cases will be gained similarly in Theorems 3.2, 3.3 and 3.4. The existence of an upper or a lower solution is the sufficient condition to ensure the solvability of the problem. Moreover, the Hyers-Ulam stability is considered in meaning of partial ordered metric spaces $C = (C (J_{\infty}, E))^{3}$ . Finally, some examples are given to demonstrate our theoretical results.

2 Preliminaries

2.1 Partially ordered metric space

Let (X, ≲ _X) be an order set and Xⁿ be the n-times Cartesian product of X. Consider x = (x₁, x₂, …, x_n) , y = (y₁, y₂, …, y_n) ∈ Xⁿ, we define an order relation ≲_{X
ⁿ} in Xⁿ by x ≲ _{X
ⁿ}y if and only if x_i ≲ _Xy_i, for all i = 1, 2, …, n. The inverse order relation ≳_{X
ⁿ} is defined similarly.

Definition 2.1. [28] Suppose that X is a nonempty set. Mapping $d : X \times X \to ℝ_{+}^{n}$ is called a vector-valued metric on X if it satisfies

$d (u, v) ≳_{ℝ^{n}} \hat{0}$ for all u, v ∈ X and $d (u, v) = \hat{0} \Rightarrow u = v,$ where $\hat{0} = (0, 0, . . ., 0) \in ℝ^{n}$ ;

d (u, v) = d (v, u), for all u, v∈ X ;

$d (u, w) ≲_{ℝ^{n}} d (u, v) + d (v, w)$ , for all u, v, w ∈ X.

We call the pair (X, d) is a generalized metric space in the sense of Perov.

Definition 2.2. [29] (X, d, ≲ _X) is called a partially order generalized metric space if

(X, d) is a generalized metric space in the sense of Perov;

(X, ≲ _X) is a partially ordered set.

On the partially ordered set (X, ≲ _X), we say f : X → X is monotone nondecreasing mapping if x, y ∈ X, x ≲ _Xy implies f (x) ≲ _Xf (y).

A square matrix M with nonnegative elements is said to be convergent to zero (convergent matrix) if M^k → 0 as k → ∞.

Proposition 2.1. [27, 29] The property of being convergent to zero of matrix M is equivalent to each of the following conditions

the eigenvalues of M are located inside the unit disc of the complex plane;

I - M is nonsingular and (I - M) ^-1 has nonnegative elements.

Lemma 2.1. [20] Suppose that A and B are square matrices of the same order and A converges to zero matrix. Then A + B also converges to zero matrix if all elements of matrix B = (b_ij (λ)) depend on λ > 0 and $\lim_{λ \to \infty} b_{i j} (λ) = 0, \forall i, j .$

We denote X_{≲_X} by the set of all comparable pairs in (X, ≲ _X), i.e., $X_{≲_{X}} = {(x, y) \in X \times X : x ≲_{X} y or y ≲_{X} x} .$

Perov-like fixed point theorem in partially ordered complete generalized metric spaces was stated as follows.

Theorem 2.1. [[9], Theorem 3.2] Suppose that (X, d, ≲ _X) is a complete generalized metric space and operator T : X → X is continuous and satisfies

for each (x, y) ∉ X_{≲_X} there exists z ∈ X such that both (x, z) , (y, z)∈ X_{≲_X} ;

if (x, y) ∈ X_{≲_X} then (T (x) , T (y)) ∈ X_{≲_X};

there exists x₀ ∈ X such that (x₀, T (x₀)) ∈ X_{≲_X};

there exits a square matric A which converges to zero such that $d (T (x), T (y)) \leq Ad (x, y), \forall (x, y) \in X_{≲_{X}} .$

Then T has a fixed point in X, i.e., there exists a point u ∈ X such that T (u) = u.

2.2 Partially ordered metric space of fuzzy-valued functions

Denote E by the space of all fuzzy sets on $ℝ$ that are nonempty subsets ${(x, u (x)) : x \in ℝ}$ in $ℝ \times [0, 1]$ of certain functions u: $ℝ \to [0, 1]$ being normal, fuzzy-convex, upper semi-continuous and compact-supported. The α-level sets of fuzzy set u are defined by $[u]^{α} = {\begin{matrix} {x \in ℝ : u (x) \geq α}, & if 0 < α \leq 1, \\ cl (supp u) & if α = 0 \end{matrix}$

The space E has a nice property [10] that every u ∈ E have the parametric form denoted by $[u]^{α} = [u_{α}^{-}, u_{α}^{+}]$ , α ∈ [0, 1], with the diameter of its $len ([u])^{α} = u_{α}^{+} - u_{α}^{-} .$ It is well-known that (E, d_∞) is a complete metric space with $d_{\infty} (u, v) = sup_{0 \leq α \leq 1} d_{H} ({[u]}^{α}, {[v]}^{α}), for all u, v \in E .$

The addition ⊕ and the multiplication by a scalar ⊙ can be defined levelsetwise. For any u, v ∈ E and real number λ ${[u \oplus v]}^{α} = {[u]}^{α} + {[v]}^{α} and {[λ ⊙ u]}^{α} = λ {[u]}^{α} .$

If there exists w ∈ E such that u = v ⊕ w, we call w = u ⊖ v the H-difference of u and v. If u ⊖ v exists, then $[u ⊖ v]^{α} = [u_{α}^{-} - v_{α}^{-}, u_{α}^{+} - v_{α}^{+}]$ for all 0 ≤ α ≤ 1.

Let $x, y \in E, [x]^{α} = [x_{α}^{-}, x_{α}^{+}]$ , $[y]^{α} = [y_{α}^{-}, y_{α}^{+}], α \in [0, 1] .$ An order relation in E is defined as follows [24] for all α ∈ [0, 1] $x ≲_{E} y if and only if x_{α}^{-} \leq y_{α}^{-} and x_{α}^{+} \leq y_{α}^{+} .$

Lemma 2.2. [21, 24] Some properties of partial order ≲_E are:

If x ≲ _Ey, then x + z ≲ _Ey + z, forall z ∈ E.

If nondecreasing sequence {x_n} ⊂ E converges to x in E, then x_n ≲ _Ex, for all $n \in ℕ .$

Every pair of elements of E has an upper bound and a lower bound.

Denote C (J_∞, E) by the space of all fuzzy-valued continuous functions defined on J_∞ and $C = C (J_{\infty}, E)^{3}$ is the 3 times Cartesian product of C (J_∞, E). We define the partial order induced by $\begin{matrix} f, g \in C (J_{\infty}, E), f ≲_{C (J_{\infty}, E)} g \\ \Leftrightarrow f (x, y) ≲_{E} g (x, y), \end{matrix}$ for all (x, y) ∈ J_∞.

For u, v ∈ E, the gH-difference of u and v denoted by u_⊝gHv, is defined as the element w ∈ E such that $\begin{matrix} u ⊝_{gH} v = w \Leftrightarrow (i) u = v + w or \\ (ii) v = u + (- 1) ⊙ w . \end{matrix}$

Notice that if u_⊝gHv and u ⊖ v exist, then u_⊝gHv = u ⊖ v.

Base on gH-difference notion, Bede and Stefanini [10] defined the notion gH-differentiability. In this paper, we denote $C_{gH}^{i, j} (J_{\infty}, E)$ by the space of all functions f : J_∞ → E which have partial gH-derivatives up to order i with respect to x and up to order j with respect to y in J_∞, see [18, 19] for more details.

3 The existence of mild solutions

For convenience in representation, with $z = (u, v, w) \in C$ we denote $\begin{matrix} 𝕃 (z) (x, y) \\ = (\tilde{b} ⊙ u (x, y_{j}), \tilde{b} ⊙ v (x, y_{i}), \tilde{a} ⊙ w (x_{j}, y)), \\ 𝕀 (f) (z) (x, y) \\ = (\int_{0}^{y} w (x, t) dt, \int_{0}^{y} f (x, t, z) dt, \int_{0}^{x} f (s, y, z) ds), \\ 𝔹 (f) (z) (x, y) \\ = (⊖ (- 1) ⊙ \int_{0}^{y} w (x, t) dt, \int_{0}^{y} f (x, t, z) dt, \\ \int_{0}^{x} f (s, y, z) ds), \end{matrix}$ and $Φ (x, y) = (φ_{2} (x), φ_{2}^{'} (x), φ_{1}^{'} (y)) .$

Here, we suppose that φ₁, φ₂ ∈ C¹ ((0, ∞) , E) are gH-differentiable continuous satisfied suitable condition $φ_{1} (0) \oplus \tilde{a} ⊙ u (x_{i}, 0) = φ_{2} (0) \oplus \tilde{b} ⊙ u (0, y_{j}) .$

Lemma 3.1. [Sketch to Lemma 4.4 in [16]] Assume that $u \in C_{gH}^{1, 1} (J_{\infty}, E)$ satisfies problem (1.1)-(1.2).

If u is (i)-gH differentiable w.r.t. x and y, u_x is (i)-gH differentiable w.r.t. y and u_y is (i)-gH differentiable w.r.t. x with no switching point, then z = (u, u_x, u_y) is a solution of integral equation $z = Φ \oplus 𝕃 (z) \oplus 𝕀 (f) (z) .$ (3.3)

If u is (ii)-gH differentiable w.r.t. x and y, u_x is (ii)-gH differentiable w.r.t. y and u_y is (ii)-gH differentiable w.r.t. x with no switching point, then z = (u, u_x, u_y) is a solution of integral equation $z = Φ \oplus 𝕃 (z) ⊖ (- 1) ⊙ 𝕀 (f) (z) .$ (3.4)

If u is (i)-gH differentiable w.r.t. x and y, u_x is (ii)-gH differentiable w.r.t. y and u_y is (ii)-gH differentiable w.r.t. x with no switching point, then z = (u, u_x, u_y) is a solution of integral equation $z = Φ \oplus 𝕃 (z) ⊖ (- 1) ⊙ 𝔹 (f) (z) .$ (3.5)

If u is (ii)-gH differentiable w.r.t. x and y, u_x is (i)-gH differentiable w.r.t. y and u_y is (i)-gH differentiable w.r.t. x with no switching point, then z = (u, u_x, u_y) is a solution of integral equation $z = Φ \oplus 𝕃 (z) \oplus 𝔹 (f) (z) .$ (3.6)

Definition 3.1.

A solution $z = (u, u_{x}, u_{y}) \in C$ of (3.3) is called a type (1)-mild solution of problem (1.1)-(1.2).

A solution $z = (u, u_{x}, u_{y}) \in C$ of (3.4) is called a type (2)-mild solution of problem (1.1)-(1.2).

A solution $z = (u, u_{x}, u_{y}) \in C$ of (3.5) is called a type (3)-mild solution of problem (1.1)-(1.2).

A solution $z = (u, u_{x}, u_{y}) \in C$ of (3.6) is called a type (4)-mild solution of problem (1.1)-(1.2).

Remark 3.1. We split the domain J_∞ into four subdomains: J_∞ = J₁ ∪ J₂ ∪ J₃ ∪ J₄, where J₁ = [0, x_n]× [0, y_n] ; J₂ = [x_n, + ∞) × [0, y_n] ; J₃ = [0, x_n]× [y_n, + ∞) ; J₄ = [x_n, + ∞) × [y_n, + ∞) , x_n ≤ 1, y_n ≤ 1. For fixed λ > 1, we construct suitable metric for each subdomain as follows $\begin{matrix} H_{1} (u, v) \\ = sup_{(x, y) \in J_{1}} {d_{\infty} (u (x, y), v (x, y))}, \\ H_{2} (u, v) \\ = sup_{(x, y) \in J_{2}} {e^{- λ (x - x_{n})} d_{\infty} (u (x, y), v (x, y))}, \\ H_{3} (u, v) \\ = sup_{(x, y) \in J_{3}} {e^{- λ (y - y_{n})} d_{\infty} (u (x, y), v (x, y))}, \\ H_{4} (u, v) \\ = sup_{(x, y) \in J_{4}} {e^{- λ (x + y - x_{n} - y_{n})} d_{\infty} (u (x, y), v (x, y))} . \end{matrix}$

Set $\begin{matrix} H (u, v) \\ = max {H_{1} (u, v); H_{2} (u, v); H_{3} (u, v); H_{4} (u, v)} . \end{matrix}$

Now for $z = (u, v, w), \bar{z} = (\bar{u}, \bar{v}, \bar{w}) \in C,$ we denote $\hat{H} (z, \bar{z}) = (H (u, \bar{u}), H (v, \bar{v}), H (w, \bar{w})) .$

We can see that $(C, \hat{H}, ≲_{C})$ is a partially order completely generalized metric space.

Denote $\begin{matrix} \hat{C} ([0, \infty)) \\ = {φ : [0, \infty) \to [0, \infty) : φ (t) = 0 \Leftrightarrow t = 0} . \end{matrix}$

A nondecreasing continuous function ψ in $\hat{C} ([0, \infty))$ is called an altering distance function on [0, ∞). We need following hypothesis.

Hypothesis (G1). Assume that f is weakly contractive over comparable elements, i.e., for some altering distance function ψ and $φ \in \hat{C} ([0, \infty))$ , the following estimation holds

$\begin{matrix} ψ (H (f (x, y, z), f (x, y, \bar{z}))) \\ \leq ψ (D (z, \bar{z})) - φ (D (z, \bar{z})), \end{matrix}$ (3.7) for all $z = (u, v, w) ≲_{C} \bar{z} = (\bar{u}, \bar{v}, \bar{w}),$ where $D (z, \bar{z}) = aH (u, \bar{u}) + bH (v, \bar{v}) + cH (w, \bar{w})),$ and a, b, c are positive numbers such that $M : = (\begin{matrix} \tilde{b} & 0 & 1 \\ a & \tilde{a} + b & c \\ a & b & \tilde{a} + c \end{matrix})$ converges to zero.

Lemma 3.2. Under hypothesis (G1) the following inequality

$\begin{matrix} H (f (x, y, z), f (x, y, \bar{z})) \\ \leq aH (u, \bar{u}) + bH (v, \bar{v}) + cH (w, \bar{w}) \end{matrix}$ (3.8) holds for all $z = (u, v, w) ≲_{C} \bar{z} = (\bar{u}, \bar{v}, \bar{w})$ in $C .$

Proof. In a proof by contradiction, we assume that there exists $z ≲_{C} \bar{z}$ that $D (z, \bar{z}) < H (f (x, y, z), f (x, y, \bar{z})),$ for some (x, y) ∈ J_∞. Due to the nondecreasing property of ψ, we imply $ψ (D (z, \bar{z})) \leq ψ (H (f (x, y, z), f (x, y, \bar{z}))) .$

On the other hand, we have $\begin{matrix} ψ (H (f (x, y, z), f (x, y, \bar{z}))) \\ \leq ψ (D (z, \bar{z})) - φ (D (z, \bar{z})) \\ \leq ψ (D (z, \bar{z})), for all z ≲_{C} \bar{z} . \end{matrix}$

Thus, we have $ψ (D (z, \bar{z})) = ψ (H (f (x, y, z), f (x, y, \bar{z}))) .$ Then, we imply that $0 \leq - φ (D (z, \bar{z}))$ or $φ (D (z, \bar{z})) = 0$ .

Because $φ \in \hat{C} ([0, \infty))$ , we imply that $D (z, \bar{z}) = 0 .$ Hence $ψ (H (f (x, y, z), f (x, y, \bar{z}))) \leq ψ (D (z, \bar{z})) = 0 .$

This implies $H (f (x, y, z), f (x, y, \bar{z})) = 0$ , that leads to contradiction. □

Definition 3.2.

A fuzzy-valued function $μ \in C_{gH}^{1, 1} (J_{\infty}, E)$ is called an (k)-lower (k = 1, 2) solution of problem (1.1)-(1.2) if $\begin{matrix} _{gH}^{C} D_{k}^{q} μ (x, y) \\ ≲_{C (J_{\infty}, E)}_{F}^{RL} I_{0^{+}}^{1 - q} f (x, y, μ, μ_{x}, μ_{y}) \end{matrix}$ and ${\begin{matrix} μ (0, y) \oplus \tilde{a} ⊙ μ (x_{i}, y) ≲_{C (J_{\infty}, E)} φ_{1} (y) \\ μ (x, 0) \oplus \tilde{b} ⊙ μ (x, y_{j}) ≲_{C (J_{\infty}, E)} φ_{2} (x) \end{matrix}$

A fuzzy-valued function $μ \in C_{gH}^{1, 1} (J_{\infty}, E)$ is called an (k)-upper (k = 1, 2) solution of problem (1.1)-(1.2) if $\begin{matrix} _{F}^{RL} I_{0^{+}}^{1 - q} f (x, y, μ, μ_{x}, μ_{y}) \\ ≲_{C (J_{\infty}, E)}_{gH}^{C} D_{k}^{q} μ (x, y) \end{matrix}$ and ${\begin{matrix} φ_{1} (y) ≲_{C (J_{\infty}, E)} μ (0, y) \oplus \tilde{a} ⊙ μ (x_{i}, y) \\ φ_{2} (x) ≲_{C (J_{\infty}, E)} μ (x, 0) \oplus \tilde{b} ⊙ μ (x, y_{j}) \end{matrix}$

Remark 3.2. To prove the existence of mild solutions of problem (1.1)-(1.2), we transfer them into Perov-like fixed points problem by considering the vector-valued operator $𝕋 = (T_{1}, T_{2}, T_{3})$ defined by $\begin{matrix} T_{1} (z) (x, y) & = & φ_{2} (x) \oplus \tilde{b} ⊙ u (x, y_{j}) \\ \oplus \int_{0}^{y} w (x, t) dt, \\ T_{2} (z) (x, y) & = & φ_{2}^{'} (x) \oplus \tilde{b} ⊙ v (x, y_{j}) \\ \oplus \int_{0}^{y} f (x, t, z) dt, \\ T_{3} (z) (x, y) & = & φ_{1}^{'} (y) \oplus \tilde{a} ⊙ w (x_{j}, y) \\ \oplus \int_{0}^{x} f (s, y, z) ds . \end{matrix}$

We can decomposite $𝕋$ into three terms $𝕋 (z) = Φ \oplus 𝕃 (z) \oplus 𝕀 (f) (z), z \in C .$

Lemma 3.3. Under hypothesis (G1), the inequality $\hat{H} (𝕋 (z), 𝕋 (\bar{z})) \leq M \hat{H} (z, \bar{z})$ (3.9) holds for all $z ≲_{C} \bar{z}$ in $C$ .

Proof. We will prove inequality (3.9) in each subdomain J_i, i = 1, ... , 4. We need to prove the case when (x, y) ∈ J₁ and (x, y) ∈ J₂, the orther cases are proven similarly. Indeed, if (x, y) ∈ J₁ we have $\begin{matrix} d_{\infty} (T_{2} (z), T_{2} (\bar{z})) \\ = d_{\infty} (φ_{2}^{'} (x) \oplus \tilde{b} ⊙ v (x, y_{j}) \oplus \int_{0}^{y} f (x, t, z) dt, \\ φ_{2}^{'} (x) \oplus \tilde{b} ⊙ \bar{v} (x, y_{j}) \oplus \int_{0}^{y} f (x, t, \bar{z}) dt) \\ \leq \tilde{b} d_{\infty} (v, \bar{v}) + \int_{0}^{y} d_{\infty} (f (x, t, z), f (x, t, \bar{z})) dt . \end{matrix}$

By taking supremum both sides of this inequality on J₁ and applying Lemma 3.2 we have $\begin{matrix} H_{1} (T_{2} (z), T_{2} (\bar{z})) \\ \leq \tilde{b} H_{1} (v, \bar{v}) + H_{1} (f (x, t, z), f (x, t, \bar{z})) y \\ \leq {ay}_{n} H_{1} (u, \bar{u}) + (\tilde{b} + {by}_{n}) H_{1} (v, \bar{v}) \\ + {cy}_{n} H_{1} (w, \bar{w}) . \end{matrix}$

Therefore

$\begin{matrix} H_{1} (T_{2} (z), T_{2} (\bar{z})) \leq a H_{1} (u, \bar{u}) \\ + (\tilde{b} + b) H_{1} (v, \bar{v}) + {cH}_{1} (w, \bar{w}) . \end{matrix}$ (3.10)

Similarly, we have $H_{1} (T_{1} (z), T_{1} (\bar{z})) \leq \tilde{b} H_{1} (u, \bar{u}) + H_{1} (w, \bar{w}),$ (3.11) and

$\begin{matrix} H_{1} (T_{3} (z), T_{3} (\bar{z})) \leq a H_{1} (u, \bar{u}) \\ + {bH}_{1} (v, \bar{v}) + (\tilde{a} + c) H_{1} (w, \bar{w}) . \end{matrix}$ (3.12)

By combination (3.10), (3.11) and (3.12), we imply

$\begin{matrix} (\begin{matrix} H_{1} (T_{1} (z), T_{1} (\bar{z})) \\ H_{1} (T_{2} (z), T_{2} (\bar{z})) \\ H_{1} (T_{3} (z), T_{3} (\bar{z}) \end{matrix}) \\ ≲_{ℝ^{3}} (\begin{matrix} \tilde{b} & 0 & 1 \\ a & \tilde{b} + b & c \\ a & b & \tilde{a} + c \end{matrix}) (\begin{matrix} H_{1} (u, \bar{u}) \\ H_{1} (v, \bar{v}) \\ H_{1} (w, \bar{w}) \end{matrix}) . \end{matrix}$ (3.13)

When (x, y) ∈ J₂, we have

$\begin{matrix} \int_{0}^{y} d_{\infty} (f (x, t, z), f (x, t, \bar{z})) dt \\ = \int_{0}^{y} d_{\infty} (f (x, t, z), f (x, t, \bar{z})) e^{- λ (x - x_{n})} e^{λ (x - x_{n})} dt \\ \leq sup (d_{\infty} (f (x, t, z), f (x, t, \bar{z})) e^{- λ (x - x_{n})}) \\ \int_{0}^{y} e^{λ (x - x_{n})} dt \\ \leq H_{2} (f (x, t, z), f (x, t, \bar{z}) e^{λ (x - x_{n})} y_{n} . \end{matrix}$ (3.14)

Doing in similar way, we obtain

$\begin{matrix} \int_{0}^{x} d_{\infty} (f (x, t, z), f (x, t, \bar{z})) dt \\ \leq H_{2} (f (x, t, z), f (x, t, \bar{z}) \int_{0}^{x} e^{λ (s - x_{n})} ds \\ \leq H_{2} (f (x, t, z), f (x, t, \bar{z}) \frac{1}{λ} (e^{λ (x - x_{n})} - e^{- λ x_{n}}) \\ \leq H_{2} (f (x, t, z), f (x, t, \bar{z}) \frac{1}{λ} e^{λ (x - x_{n})} \\ \leq H_{2} (f (x, t, z), f (x, t, \bar{z}) e^{λ (x - x_{n})} \end{matrix}$ (3.15)

Therefore $\begin{matrix} d_{\infty} (T_{2} (z), T_{2} (\bar{z})) \\ \leq \tilde{b} d_{\infty} (v, \bar{v}) + H_{2} (f (x, t, z), f (x, t, \bar{z}) e^{λ (x - x_{n})} y_{n} . \end{matrix}$

Divide both sides by e^λ(x-x_n), then taking supremum on J₂, we obtain $\begin{matrix} H_{2} (T_{2} (z), T_{2} (\bar{z})) \\ \leq \tilde{b} H_{2} (v, \bar{v}) + H_{2} (f (x, t, z), f (x, t, \bar{z}) y_{n} \end{matrix}$ or

$\begin{matrix} H_{2} (T_{2} (z), T_{2} (\bar{z})) \leq a H_{2} (u, \bar{u}) \\ + (\tilde{b} + b) H_{2} (v, \bar{v}) + {cH}_{2} (w, \bar{w}) . \end{matrix}$ (3.16)

Similarly, we have

$\begin{matrix} (\begin{matrix} H_{2} (T_{1} (z), T_{1} (\bar{z})) \\ H_{2} (T_{2} (z), T_{2} (\bar{z})) \\ H_{2} (T_{3} (z), T_{3} (\bar{z}) \end{matrix}) \\ ≲_{ℝ^{3}} (\begin{matrix} \tilde{b} & 0 & 1 \\ a & \tilde{b} + b & c \\ a & b & \tilde{a} + c \end{matrix}) (\begin{matrix} H_{2} (u, \bar{u}) \\ H_{2} (v, \bar{v}) \\ H_{2} (w, \bar{w}) \end{matrix}) . \end{matrix}$ (3.17)

Thus, we imply that $\hat{H} (𝕋 (z), 𝕋 (\bar{z})) \leq M \hat{H} (z, \bar{z}), \forall z ≲_{C} \bar{z} in C .$ (3.18)

□

Theorem 3.1. Assume that hypothesis (G1) is fulfilled. Then the existence of (1)-lower (or (1)-upper) solution $u \in C_{gH}^{1, 1} (J_{\infty}, E)$ implies the existence of at least one type (1)-mild solution of problem (1.1)-(1.2).

Proof. We recall that $(C, H, ≲_{C})$ is a partially ordered completely generalized metric space and operator $𝕋$ is continuous. Thus from Lemma 3.3 we imply that operator $𝕋$ satisfies conditions (1), (2) and (4) of Theorem 2.1. Now we will validate condition (3).

Without lost of generality, we assume that $u \in C_{gH}^{1, 1} (J_{\infty}, E)$ is a (1)-lower solution of the problem (1.1)-(1.2) then we have $_{gH}^{C} D_{1}^{q} u (x, y) ≲_{C (J_{\infty}, E)}_{F}^{RL} I_{0^{+}}^{1 - q} f (s, t, u, u_{x}, u_{y}),$ (3.19) with nonlocal condition ${\begin{matrix} φ_{1} (y) ≲_{C (J_{\infty}, E)} u (0, y) \oplus \tilde{a} ⊙ u (x_{i}, y) \\ φ_{2} (x) ≲_{C (J_{\infty}, E)} u (x, 0) \oplus \tilde{b} ⊙ u (x, y_{i}) \end{matrix}$ (3.20)

From (3.19), we can see that _kD_xyu (x, y) ≲ _{C(J_∞,E)}f (x, y, u, u_x, u_y).

We recall the notions $v (x, y) = \frac{\partial u}{\partial x} (x, y), w (x, y) = \frac{\partial u}{\partial y} (x, y)$ and z = (u, v, w) we have $\frac{\partial v}{\partial y} = \frac{\partial w}{\partial x} ≲_{C (J_{\infty}, E)} f (x, y, z), (x, y) \in J_{\infty} .$ (3.21)

Firstly, we consider the case when u is (i)-gH differentiable w.r.t. x and y, u_x is (i)-gH differentiable w.r.t. y and u_y is (i)-gH differentiable w.r.t. x. From (3.21) we have $[\frac{\partial v}{\partial y}]_{α}^{-} (x, y) \leq f_{α}^{-} (x, y, z), (x, y) \in J_{\infty} .$ (3.22)

Then by taking integral both side of (3.22) we have $v_{α}^{-} (x, y) = v_{α}^{-} (x, 0) + \int_{0}^{y} f_{α}^{-} (x, y, z) dt .$

On the other hand $\begin{matrix} v_{α}^{-} (x, 0) & = & [\frac{\partial u}{\partial x}]_{α}^{-} (x, 0) \\ \leq & φ_{2}^{'} (x) \oplus \tilde{b} ⊙ [\frac{\partial u}{\partial x}]_{α}^{-} (x, y_{i}) \\ \leq & φ_{2}^{'} \oplus \tilde{b} ⊙ v_{α}^{-} (x, y_{j}) . \end{matrix}$

It follows $\begin{matrix} v_{α}^{-} (x, y) & \leq & φ_{2}^{'} (x) \oplus \tilde{b} ⊙ v_{α}^{-} (x, y_{j}) \\ + \int_{0}^{y} f_{α}^{-} (x, t, z) dt . \end{matrix}$

Similarly, we have $\begin{matrix} v_{α}^{+} (x, y) & \leq & φ_{2}^{'} (x) \oplus \tilde{b} ⊙ v_{α}^{+} (x, y_{j}) \\ + \int_{0}^{y} f_{α}^{+} (x, t, z) dt . \end{matrix}$

Thus

$\begin{matrix} v (x, y) ≲_{C (J_{\infty}, E)} φ_{2}^{'} (x) \oplus \tilde{b} ⊙ v (x, y_{j}) \\ \oplus \int_{0}^{y} f (x, t, z) dt . \end{matrix}$ (3.23)

Doing similarly argument, we receive $u (x, y) ≲_{C (J_{\infty}, E)} φ_{2} \oplus \tilde{b} ⊙ u (x, y_{j}) \oplus \int_{0}^{y} w (x, t) dt$ (3.24) and

$\begin{matrix} w (x, y) ≲_{C (J_{\infty}, E)} φ_{1}^{'} (y) \oplus \tilde{a} ⊙ w (x_{J}, y) \\ \oplus \int_{0}^{x} f (s, y, z) ds \end{matrix}$ (3.25)

From inequalities (3.23), (3.24) and (3.25), we can see that the existence of (1)-lower solution implies the relation $z (x, y) ≲_{C} 𝕋 (z) (x, y) .$ (3.26)

The case u is (ii)-gH differentiable w.r.t. x and y, u_x is (ii)-gH differentiable w.r.t. y, u_y is (ii)-gH differentiable w.r.t. x we can prove relation (3.26) again.

Hence, all the conditions of Theorem 2.1 are satisfied. Thus the operator $𝕋$ has a fixed point $z \in C$ , that is (1)-mild solution of problem (1.1)-(1.2). □

Remark 3.3. To prove the existence of (2)-mild solution, we must supply more conditions. We consider the operator $ℙ = (P_{1}, P_{2}, P_{3})$ with the decomposition as follows $ℙ (z) = Φ \oplus 𝕃 (z) ⊖ (- 1) ⊙ 𝕀 (f) (z) .$ (3.27)

Define the solution space $\begin{matrix} \hat{C} & = & {z \in C : Φ (x, y) \oplus 𝕃 (z) (x, y) ⊖ (- 1) \\ ⊙ 𝕀 (f) (z) (x, y) exist for all (x, y) \in J_{\infty}} . \end{matrix}$

From Lemma 5.1 in [18], $(\hat{C}, H, ≲_{\hat{C}})$ is a completely generalized order metric space.

Theorem 3.2. Assume that hypothesis (G1) is fulfilled and

$\hat{C} \neq \emptyset,$

If $z \in C$ , then $ℙ (z) \in \hat{C} .$

Then the existence of (2)-lower solution (or (2)-upper solution) implies the existence of type (2)-mild solution of problem (1.1)-(1.2).

Proof. Doing the same arguments as in the proof of Theorem 3.1, we obtain the same results. For example, for (x, y) in J₁, apply Lemma 3.2 [16] we have $\begin{matrix} d_{\infty} (P_{2} (z), P_{2} (\bar{z})) \\ = d_{\infty} (φ_{2}^{'} (x) \oplus \tilde{b} ⊙ v (x, y_{j}) ⊖ (- 1) \\ ⊙ \int_{0}^{y} f (x, t, z) dt, φ_{2}^{'} (x) \oplus \tilde{b} \\ ⊙ v (x, y_{j}) ⊖ (- 1) ⊙ \int_{0}^{y} f (x, t, \bar{z}) dt) \\ \leq d_{\infty} (\tilde{b} ⊙ v (x, y_{j}), \tilde{b} \bar{v} (x, y_{j})) \\ + d_{\infty} (\int_{0}^{y} f (x, t, z) dt, \int_{0}^{y} f (x, t, \bar{z}) dt) \\ \leq \tilde{b} H_{1} (v, \bar{v}) + \int_{0}^{y} H_{1} (f (x, t, z), f (x, t, \bar{z})) dt . \end{matrix}$

Therefore

$\begin{matrix} H_{1} (P_{2} (z), P_{2} (\bar{z})) & \leq & a H_{1} (u, \bar{u}) + (\tilde{b} + b) H_{1} (v, \bar{v}) \\ + {cH}_{1} (w, \bar{w}) \end{matrix}$ (3.28) and similarly $\begin{matrix} (\begin{matrix} H_{1} (P_{1} (z), P_{1} (\bar{z})) \\ H_{1} (P_{2} (z), P_{2} (\bar{z})) \\ H_{1} (P_{3} (z), P_{3} (\bar{z}) \end{matrix}) \\ ≲_{ℝ^{3}} (\begin{matrix} \tilde{b} & 0 & 1 \\ a & \tilde{b} + b & c \\ a & b & \tilde{a} + c \end{matrix}) (\begin{matrix} H_{1} (u, \bar{u}) \\ H_{1} (v, \bar{v}) \\ H_{1} (w, \bar{w}) \end{matrix}) . \end{matrix}$

Thus $\hat{H} (ℙ (z), ℙ (\bar{z})) ≲_{ℝ^{3}} M \hat{H} (z, \bar{z}), \forall z ≲_{C} \bar{z} .$ (3.29)

Applying analogous arguments as in the proof of Theorem 3.1 we receive a fixed point $z \in C$ of the operator $ℙ$ , that is type (2)-mild solution of problem (1.1)-1.2). □

Remark 3.4. If we consider the operator $ϒ (z) = Φ \oplus 𝕃 (z) ⊖ (- 1) ⊙ 𝔹 (f) (z)$ on the space $\begin{matrix} \tilde{C} & = & {z \in C : Φ (x, y) \oplus 𝕃 (z) (x, y) ⊖ (- 1) \\ ⊙ 𝔹 (f) (z) (x, y) exist for all (x, y) \in J_{\infty}} \end{matrix}$ and $Θ (z) = Φ \oplus 𝕃 (z) \oplus 𝔹 (f) (z)$ on $\begin{matrix} \bar{C} & = & {z \in C : Φ (x, y) \oplus 𝕃 (z) (x, y) \\ \oplus 𝔹 (f) (z) (x, y) exist for all (x, y) \in J_{\infty}} . \end{matrix}$

We can prove the existence of type (3) and type (4)-mild solution of the problem (1.1)-(1.2) in similar way.

Theorem 3.3. Assume that hypothesis (G1) is fulfilled and

$\tilde{C} \neq \emptyset .$

If $z \in C$ , then $ϒ (z) \in \tilde{C} .$

Then the existence of (2)-lower (or (2)-upper) solution of problem (1.1)-(1.2) implies the existence of (3)-mild solution of the problem.

Theorem 3.4. Assume that hypothesis (G1) is fulfilled and

$\bar{C} \neq \emptyset .$

If $z \in C$ , then $Θ (z) \in \bar{C} .$

Then the existence of (1)-lower (or (1)-upper) solution of problem (1.1)-(1.2) implies the existence of (4)-mild solution of the problem.

4 Hyers-Ulam stability

In this section, we will study the Hyers-Ulam stability of integral system $z = Φ \oplus 𝕃 (z) \oplus 𝕀 (f) (z),$ (4.30) which is one of the integral forms of nonlocal problem (1.1)-(1.2).

Set $𝔽_{1} [z] = Φ \oplus 𝕃 (z) \oplus 𝕀 (f) (z) .$

Definition 4.1. For each ɛ > 0, a mapping $z \in C$ is a solution of inequation $\hat{H} (z (x, y), 𝔽_{1} [z] (x, y)) \leq ɛ, (x, y) \in J_{\infty}$ (4.31) if there exists a function $h \in C$ such that

$\hat{H} (h (x, y), \hat{0}) \leq ɛ, for all (x, y) \in J_{\infty},$

$z (x, y) = 𝔽_{1} [z] (x, y) \oplus h (x, y), for all (x, y) \in J_{\infty} .$

Definition 4.2. Equation (4.30) is called generalized Hyers-Ulam stable if there exists an increasing, continuous function $φ : ℝ_{+} \to ℝ_{+}$ , φ (0) =0, such that for arbitrary ɛ > 0 and each solution $\hat{z}$ of the inequality (4.31) there exists a solution z of problem (4.30) satisfied $\hat{H} (z, \hat{z}) \leq φ (ɛ) .$

In special case, if there exists c > 0 such that $φ (t) = ct, t \in ℝ_{+}$ . Then, equation (4.30) is said to be Hyers-Ulam stable.

Theorem 4.1. Assume that hypothesis (G1) is fulfilled and $\tilde{a}$ , $\tilde{b} < 1$ . Then the equation (4.30) is Ulam-Hyer stable.

Proof. Assume that $\bar{z} \in C$ satisfies (4.31), it means that $\bar{z}$ is a solution of the following fuzzy integral system $\bar{z} = Φ \oplus 𝕃 (\bar{z}) \oplus 𝕀 (\bar{z}) \oplus h,$ (4.32) where h (x, y) = (h₁ (x, y) , h₂ (x, y) , h₃ (x, y)).

Because hypothesis (G1) is fulfilled, from Theorem 3.1, there exists a solution $z \in C$ satisfied equation (4.30). It means that $z = Φ (x, y) \oplus 𝕃 (z) \oplus 𝕀 (z) .$ (4.33)

If z = (u, v, w), $\bar{z} = (\bar{u}, \bar{v}, \bar{w})$ are comparable, for (x, y) ∈ J₁ we have $\begin{matrix} d_{\infty} (w (x, y), \bar{w} (x, y)) \leq \tilde{a} d_{\infty} (w (x, y), \bar{w} (x, y)) \\ + \int_{0}^{x} d_{\infty} (f (s, y, z), f (s, y, \bar{u}, \bar{v}, \bar{w})) ds \\ + d_{\infty} (h_{1} (x, y), \hat{0}) . \end{matrix}$

Hence, $\begin{matrix} H_{1} (w, \bar{w}) \\ \leq \tilde{a} H_{1} (w, \bar{w}) + H (f (s, y, z), f (s, y, \bar{u}, \bar{v}, \bar{w})) + ɛ \end{matrix}$ or $\begin{matrix} H (w, \bar{w}) \\ \leq aH (u, \bar{u}) + bH (v, \bar{v}) + (c + \tilde{a}) H (w, \bar{w}) + ɛ . \end{matrix}$

Doing the same with $d_{\infty} (v, \bar{v})$ , we have $\begin{matrix} H (v, \bar{v}) \\ \leq aH (u, \bar{u}) + (b + \tilde{b}) H (v, \bar{v}) + cH (w, \bar{w}) + ɛ \end{matrix}$ and $H (u, \bar{u}) \leq \tilde{b} H (u, \bar{u}) + H (w, \bar{w}) + ɛ .$

For (x, y) ∈ J₂, by using estimation (3.14)-(3.15) and repeating all above arguments we receive $\begin{matrix} d_{\infty} (w (x, y), \bar{w} (x, y)) \leq \tilde{a} d_{\infty} (w (x, y), \bar{w} (x, y)) \\ + H_{2} (f (x, t, z), f (x, t, \bar{z}) e^{λ (x - x_{n})} y_{n} \\ + d_{\infty} (h_{1} (x, y), \hat{0}) . \end{matrix}$

Since x_n ≤ 1, dividing two sides to e^λ(x-x_n) then taking supremum, we receive $\begin{matrix} H_{2} (w (x, y), \bar{w} (x, y)) \leq \tilde{a} H_{2} (w, \bar{w}) \\ + H_{2} (f (s, y, z), f (s, y, \bar{u}, \bar{v}, \bar{w})) ds + ɛ . \end{matrix}$

Thus $\begin{matrix} H (w, \bar{w}) \\ \leq aH (u, \bar{u}) + bH (v, \bar{v}) + (c + \tilde{a}) H (w, \bar{w}) + ɛ . \end{matrix}$

By doing in similar way, we have $\begin{matrix} H (v, \bar{v}) \\ \leq aH (u, \bar{u}) + (b + \tilde{b}) H (v, \bar{v}) + cH (w, \bar{w}) + ɛ . \end{matrix}$ and $H (u, \bar{u}) \leq \tilde{b} H (u, \bar{u}) + H (w, \bar{w}) + ɛ .$ Doing the same arguments as in J₂, finally we have $\begin{matrix} (\begin{matrix} H (u, \bar{u}) \\ H (v, \bar{v}) \\ H (w, \bar{w}) \end{matrix}) \\ ≲_{ℝ^{3}} (\begin{matrix} \tilde{b} & 0 & 1 \\ a & b + \tilde{b} & c \\ a & b & c + \tilde{a} \end{matrix}) (\begin{matrix} H (u, \bar{u}) \\ H (v, \bar{v}) \\ H (w, \bar{w}) \end{matrix}) \\ + (\begin{matrix} ɛ \\ ɛ \\ ɛ \end{matrix}) . \end{matrix}$

It implies $(I - M) \hat{H} (z, \bar{z}) ≲_{ℝ^{3}} (\begin{matrix} ɛ \\ ɛ \\ ɛ \end{matrix}) .$

Since M converges to zero, then from Proposition 2.1, I - M is invertible and (I - M) ^-1 has nonnegative elements. So $\hat{H} (z, \bar{z}) ≲_{ℝ^{3}} (I - M)^{- 1} (\begin{matrix} ɛ \\ ɛ \\ ɛ \end{matrix})$ (4.34) holds for all comparable pairs $z, \bar{z}$ .

If z and $\bar{z}$ are not comparable, there exists $\tilde{z} = (\tilde{u}, \tilde{v}, \tilde{w})$ such that $z, \tilde{z}$ and $\bar{z}, \tilde{z}$ are comparable.

Because $z, \tilde{z}$ are comparable, from (4.34) we have $\hat{H} (z, \tilde{z}) ≲_{ℝ^{3}} (I - M)^{- 1} (\begin{matrix} ɛ \\ ɛ \\ ɛ \end{matrix})$ and $\hat{H} (\tilde{z}, \bar{z}) ≲_{ℝ^{3}} (I - M)^{- 1} (\begin{matrix} ɛ \\ ɛ \\ ɛ \end{matrix}) .$

This implies $\hat{H} (z, \bar{z}) ≲_{ℝ^{3}} (I - M)^{- 1} (\begin{matrix} 2 ɛ \\ 2 ɛ \\ 2 ɛ \end{matrix}) .$ (4.35)

From (4.34) and (4.35), equation (4.30) is generalized Hyers-Ulam. It completes the proof. □

5 Examples

Example 5.1. In this example, we consider nonlocal problem $_{gH}^{C} D_{1}^{q} u (x, y) = K_{F}^{RL} I_{0^{+}}^{1 - q} (\frac{1}{2 y} u_{x} + \frac{1}{4} u)$ (5.36) with nonlocal condition ${\begin{matrix} u (0, y) \oplus \frac{1}{8} u (\frac{1}{8}, y) = 1 \oplus \frac{1}{8} e^{\frac{\sqrt{y}}{8}} \\ u (x, 0) \oplus \frac{1}{8} u (x, \frac{1}{8}) = 1 \oplus \frac{1}{8} e^{\frac{\sqrt{x}}{8}} \end{matrix}$ (5.37) where [K] ^α = [α, 2 - α] , α ∈ [0, 1] is a triangular fuzzy number.

We have $f (x, y, z) = \frac{1}{2 y} u_{x} + \frac{1}{4} u, (x, y) \in (0, + \infty) \times (0, + \infty)$ ; $φ_{1} (y) = 1 \oplus \frac{1}{8} e^{\frac{\sqrt{y}}{8}}$ and $φ_{2} (x) = 1 \oplus \frac{1}{8} e^{\frac{\sqrt{x}}{8}} .$

With $z = (u, v, w), \bar{z} = (\bar{u}, \bar{v}, \bar{w})$ , set $\begin{matrix} t & = & D (z, \bar{z}) = \frac{1}{8} d_{\infty} (u, \bar{u}) \\ + \frac{1}{8} d_{\infty} (v, \bar{v}) + \frac{1}{8} d_{\infty} (w, \bar{w}) \end{matrix}$ and ψ (t) =6t, φ (t) =2t. We can prove that $d_{\infty} (f (x, y, u, v, w), f (x, y, \bar{u}, \bar{v}, \bar{w})) \leq ψ (t) - φ (t) .$

Indeed, $d_{\infty} (f (x, y, u, v, w), f (x, y, \bar{u}, \bar{v}, \bar{w})) \leq \frac{1}{2} d (u, \bar{u}) + \frac{1}{4} d (v, \bar{v})$ and $\begin{matrix} ψ (t) - φ (t) \\ = 4 (\frac{1}{8} d_{\infty} (u, \bar{u}) + \frac{1}{8} d_{\infty} (v, \bar{v}) + \frac{1}{8} d_{\infty} (w, \bar{w})) \\ = \frac{1}{2} (d_{\infty} (u, \bar{u}) + d_{\infty} (v, \bar{v}) + d_{\infty} (w, \bar{w})) \\ \geq \frac{1}{2} d (u, \bar{u}) + \frac{1}{4} d (v, \bar{v}) . \end{matrix}$

Because $a = \frac{1}{8}, b = \frac{1}{8}, c = \frac{1}{8}, \tilde{a} = \frac{1}{8}, \tilde{b} = \frac{1}{8}$ . We consider matrix $M = (\begin{matrix} \frac{1}{8} & 0 & 1 \\ \frac{1}{8} & \frac{1}{4} & \frac{1}{8} \\ \frac{1}{8} & \frac{1}{8} & \frac{1}{4} \end{matrix}) .$

It is easy to see that matrix M has three eigenvalues λ₁ = 0, 100, λ₂ = -0, 567 and λ₃ = 0, 163 which are located inside the unit disc of the complex plane. So M converges to zero matrix. Then by applying the Theorems 3.1 and 4.1, the problem (5.36)-(5.37) has at least one (1)-mild solution and this solution is generalized Ulam-Hyer stable.

Example 5.2. In this example, we consider following partial integro-differential equation $_{gH}^{C} D_{2}^{q} u (x, y) = K_{F}^{RL} I_{0^{+}}^{1 - q} \frac{u_{x} u_{y}}{u},$ (5.38) with nonlocal initial ${\begin{matrix} u (0, y) \oplus \frac{1}{16} u (\frac{π}{6}, y) = \frac{- 1}{16} sin y \oplus \frac{1}{8} u (\frac{π}{2}, y) \\ u (x, 0) \oplus \frac{1}{16} u (x, \frac{π}{6}) = \frac{- 1}{16} sin x \oplus \frac{1}{8} u (x, \frac{π}{2}) \end{matrix}$ (5.39) where [K] ^α = [α, 2 - α] , α ∈ [0, 1] is a triangular fuzzy number.

We can see that $f (x, y, z) = \frac{u_{x} u_{y}}{u}, (x, y) \in (0, + \infty) \times (0, + \infty)$ ; $φ_{1} (y) = \frac{- 1}{16} sin y$ and $φ_{2} (x) = \frac{- 1}{16} sin x .$

With z = z (u, v, w) and $\bar{z} (\bar{u}, \bar{v}, \bar{w})$ , we set $\begin{matrix} t & = & D (z, \bar{z}) = \frac{1}{16} d_{\infty} (u, \bar{u}) \\ + \frac{1}{16} d_{\infty} (v, \bar{v}) + \frac{1}{16} d_{\infty} (w, \bar{w}) \end{matrix}$ and ψ (t) =2 + | sin t|, φ (t) =1 + | cos t|. We can see that $d_{\infty} (f (x, y, z), f (x, y, \bar{u}, \bar{v}, \bar{w})) \leq ψ (t) - φ (t) .$

Indeed, $d_{\infty} (f (x, y, z), f (x, y, \bar{u}, \bar{v}, \bar{w})) \leq 1$ and $ψ (t) - φ (t) = 1 + | sin t | - | cos t | \geq 1$

With $a = \frac{1}{16}, b = \frac{1}{16}, c = \frac{1}{16}, \tilde{a} = \frac{1}{16}, \tilde{b} = \frac{1}{16}$ , we consider the matric $M = (\begin{matrix} \frac{1}{16} & 0 & 1 \\ \frac{1}{16} & \frac{1}{8} & \frac{1}{16} \\ \frac{1}{16} & \frac{1}{16} & \frac{1}{8} \end{matrix})$

Matric M has 3 eigenvalues: λ₁ = 0, 3800, λ₂ = 0, 0625, λ₃ = -0, 1326, which are located inside the unit disc of the complex plane. So M convergent to zero matrices. Then by applying the Theorem 3.2, the problem (5.38)-(5.39) has at least one (2)-mild solution and it is generalized Ulam-Hyer stable.

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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