Existence, Uniqueness, and E_q–Ulam-Type Stability of Fuzzy Fractional Differential Equation

In real-life phenomena, numerous physical processes are used to present fractional-order sets that may change with space and time. The operations of differentiation and integration of fractional order are authorized by fractional calculus. The fractional order may be taken on imaginary and real values [1,2,3]. The theory of fuzzy sets is continuously drawing the attention of researchers. This is mainly due to its extended adaptability in various fields including mechanics, engineering, electrical, processing signals, thermal system, robotics, control, signal processing, and in several other areas [4,5,6,7,8,9,10]. Therefore, it has been a topic of increasing concern for researchers during the past few years.

Fuzzy fractional differential equations appeared for the first time in 2010 when an idea of the solution was initially proposed by Agarwal et al. [11]. However, the Riemann–Liouville H derivative based on the strongly generalizing Hukuhara differentiability [12,13] was defined by Allahviranloo and Salahshour [14,15]. They worked on solutions to Cauchy problems under this kind of derivative.

In the above, $q\in (0,1]$, through using Laplace transforms [13] and Mittag–Leffler functions [12]. By using fractional hyperbolic functions and the properties of these functions, Chehlabi et al. obtain some new results [16]. More latest studies on fuzzy fractional differential equations can be found through references [17,18,19,20,21,22].

In 1940, Ulam promoted the Ulam stability. Lately, Hyers and Rassias used this concept of stability. Since then, in mathematical analysis and differential equations, the Ulam-type stability has had great significance. In fractional differential equations, $E_{\alpha }$–Ulam-type stabilities were promoted by Wang in 2014 [23].

In the above equation, $\alpha \in (0,1]$ and $\lambda >0$. Shen studied the Ulam stability under the generalization of Hukuhara differentiability of a first-order linear fuzzy differential equation in 2015 [24]. Later, Shen et al. investigated the Ulam stability of a nonlinear fuzzy fractional equation with the help of fixed-point techniques in 2016 [25], by focusing on the initial condition where ${}_0^{RL}{D}_{0^{+}}^q$ denoted Riemann–Liouville H derivative with respect to order $q\in (0,1],f:(0,T]\times {\mathbb{E}}^1\to {\mathbb{E}}^1,T\in {\mathbb{R}}_{+}$, and $\lambda \in \mathbb{R}$.

More results can be observed that are related to Ulam-type stability in [26,27,28]. Motivated by the above-cited papers, we aim to deal with fuzzy fractional differential equations of the form, with initial conditions

Here, ${}_0^c{D}_{0^{+}}^q$ denotes the Caputo derivative of order $q\in (1,2],f:(0,T]\times {\mathbb{E}}^1\to {\mathbb{E}}^1,T\in {\mathbb{R}}_{+}$ and $\lambda \in \mathbb{R}$.

This paper focuses on facilitating, with as few conditions as possible, to assure the uniqueness and existence of a solution to Cauchy problems (1) and (2). It establishes a link between fuzzy fractional differential equations and the Ulam-type stability, which enhances and generalizes some familiar outputs in the existing literature.

Assume that $P_k(\mathbb{R})$ denotes the collection of all nonempty convex and compact subsets of $\mathbb{R}$ and define sums and scalar products in $P_k(\mathbb{R})$ in the usual manner. Let A and B be two nonempty bounded subsets in $\mathbb{R}$. The distance between A and B is defined through the Hausdorff metric,

In the above equality, $\Vert x\Vert$ stands for the usual Euclidean norm in $\mathbb{R}$. Now it is well known that the metric D turns the space $(P_k(\mathbb{R}),D)$ into a complete and separable metric space [26].

Denote where (1)–(4) stands for the following properties of the function u:

For $1<q⩽2$, denote ${\left[u\right]}^q=\overline{\{t\in \mathbb{R}|u(t)⩾q\}}$. Now, from (1)–(4), it follows that the q-level set ${\left[u\right]}^q\in P_k(\mathbb{R})\forall 1⩽q⩽2$.

Define $\underline{u}$ as the lower branch and $\overline{u}$ as the upper branch of the fuzzy number $u\in {\mathbb{E}}^1$. The set ${\left[u\right]}^q=\{t\in \mathbb{R}|u(t)\ge q\}:=[{\underline{u}}^q,{\overline{u}}^q]$ is known as the q-level set of fuzzy number u, where $q\in (1,2]$. The length of q-level set is calculated as diam ${\left[u\right]}^q={\overline{u}}^q-{\underline{u}}^q$.

Let $C^{\mathbb{E}}[a,b]$ and $L^{\mathbb{E}}[a,b]$ be spaces for all continuous and Lebesgue integrable fuzzy-valued functions on $[a,b]$, respectively. Moreover, $(C^{\mathbb{E}}[a,b],D)$ stands for the complete metric space, where

In this part, existence and uniqueness of solutions to the Cauchy problem in (1) and (2) are discussed. We can start with the lemma given below.

By applying Lemma 4 and Remark 4 with $f(t,u(t))\oplus B(t)C(t)$ instead of $f(t)$, it follows that the Cauchy problem in (1) and (2) possesses an integral version. In case $\lambda ⩾1$ and the function $t\mapsto u(t),t\in [0,T]$ is assumed to be ${}^c$[(i)-GH]-differentiable, then the function u satisfies

In case $\lambda ⩽1$ and the function $t\mapsto u(t)$ is supposed to be ${}^c$[(ii)-GH]-differentiable, then the function u satisfies

In case $\lambda <1$ and the function $t\mapsto u(t)$ is ${}^c$[(i)-GH]-differentiable, then the function u satisfies

In case $\lambda >1$ and the function $t\mapsto u(t)$ is ${}^c$[(ii)-GH]-differentiable, then the function u satisfies

We should formulate the basic assumptions before initiating our main work: