Continuous dependence of uncertain fractional differential equations with Caputo’s derivative

Abstract

Uncertain fractional differential equation driven by Liu process plays an important role in describing uncertain dynamic systems. This paper investigates the continuous dependence of solution on the parameters and initial values, respectively, for uncertain fractional differential equations involving the Caputo fractional derivative in measure sense. Several continuous dependence theorems are obtained based on uncertainty theory by employing the generalized Gronwall inequality, in which the coefficients of uncertain fractional differential equation are required to satisfy the Lipschitz conditions. Several illustrative examples are provided to verify the validity of the obtained results.

Keywords

Uncertainty theory fractional differential equation Caputo derivative continuous dependence

1 Introduction

The description of indeterministic phenomena was mainly taken by probability and fuzzy set theory for a long period. However, some domain experts have to be invited to provide their belief degrees for an indeterminate event when lacking adequate sample data for decision-making. A lot of surveys imply that the belief degree from domain experts depends heavily on human personal knowledge and preference while concerning the indeterminate factors. Human uncertainty always leads to inherent ambiguity in researches if these factors were described as random variables or fuzzy variables (see Liu [1]). Thereby it may have many negative effects on optimal research results. In order to describe this indeterminacy mathematically, uncertainty theory was proposed by Liu [2] based on uncertain measure which may be interpreted as a measure of the belief degree for an indeterminate event. For the sake of depicting uncertain dynamic systems, uncertain differential equation (UDE) was introduced by Liu [3]. Then the existence and uniqueness theorems of solution to UDE were studied by Chen and Liu [5] when the coefficients satisfy the linear growth and Lipschitz conditions. In recent years, the topics related to the numerical methods and the properties of solutions for UDEs have been widely discussed, respectively, more details may refer to [6 –11] and the references therein. Based on uncertain differential equation, many researches have been done to deal with uncertain systems, such as uncertain finance (Chen et al. [12]), multi-period portfolio selection problem (Li et al. [13]) and uncertain optimal control (Zhu [14]), etc. More theoretical knowledge and applications related to uncertain differential equations may be found in monographs [1, 15].

However, the complex nonlinear equations and artificial empirical parameters are needed to provide when applying ordinary differential equations to model complex dynamic systems, which may be incompatible with the actual work. Fractional calculus provides an excellent tool for describing the memory and historical characteristics of systems and has been widely used in system modelling in recent years, such as viscoelasticity, dynamics, economics and control. More basic theory and applications about fractional calculus can be seen in [16 –19]. Ye et al. [20] presented a generalized Gronwall inequality and applied it to study the dependent problems of solution to fractional differential equation. For the sake of depicting the complex systems with memory effects and disturbed by uncertainty noise, Agarwal et al. [21] first introduced the concept of fuzzy fractional differential equation with the Riemann-Liouville derivative. Allahviranloo et al. [22] proposed the concept of the Caputo type of fuzzy fractional differential equation under the Generalized Hukuhara differentiability. In recent years, the study on the fuzzy fractional calculus and fuzzy fractional differential equations based on fuzzy set theory has made great progress, such as the explicit solutions [23, 24], numerical methods [25] and their applications [26 –28] etc. Meanwhile, the investigations on the stochastic fractional differential equations driven by Brownian motion have also been developed based on probability theory, see [29 –31] and the references therein.

The concepts of uncertain fractional differential equations driven by Liu process were proposed by Zhu [32] based on uncertainty theory and a new interest rate model was put forward as an application. The existence and uniqueness theorems of solutions to uncertain fractional differential equations were derived in Zhu [33], in which the coefficients of uncertain fractional system are required to satisfy the linear growth condition and Lipschitz condition. Lu et al. [34] extended Zhu’s concepts to order n - 1 < p ≤ n, where n is a positive integer. After that, Lu and Zhu [35] derived the inverse uncertainty distribution of solution to an uncertain fractional differential equation and a numerical method was provided for solving these equations. After that, the finite stability almost surely of solution to uncertain difference equation was investigated by Lu and Zhu [36] and a numerical approach was provided in Lu and Zhu [37] with the help of the proposed comparison theorems. Moreover, Wang and Zhu [38] studied the analytic solutions for uncertain fractional differential equations with delays. The uncertain fractional differential equations have also been applied to study the evolution laws of uncertain financial markets, see [39, 40], and the references therein.

The study of the continuous dependent properties of solution on initial values and parameters is one of the most important topics in the study of differential equations, which mainly investigates the effects on the continuity of the solution with the small changes of the initial values and the parameters of coefficients, respectively. Due to the properties of fractional-order operator, it brings great difficulties in investigating the properties of UFDEs, while its memory and hereditary (nonlocality) properties depict the real problems perfectly. To our best knowledge, the issue on the continuous dependence of solution on initial values and parameters for uncertain fractional differential equations has been not yet presented.

Since the integer-order derivatives of the state at the initial point need to be specified when employing the Caputo type of FDEs to describe the systems, while the fractional derivatives (or integrals) of the state at the initial point have to be specified which have unclear physical meaning when using the Riemann-Liouville cases, this paper mainly investigates the continuous dependent properties of solution on initial values and parameters for the Caputo type of uncertain fractional differential equations. The rest of this paper is organized as follows. Some basic concepts and resultant lemmas in uncertainty theory and fractional differential equation are reviewed in Section 2. The continuous dependence theorems are derived in Section 3 based on uncertainty theory with respect to the parameters and initial conditions, respectively. Several illustrative examples are provided in Section 4 to explain the continuous dependent results. A brief conclusion is made in Section 5.

2 Preliminaries

Let Γ be a nonempty set, and $L$ be a σ-algebra over Γ. Each element $Λ \in L$ is called an event. An uncertain measure is a set function $M : L \to [0, 1]$ if it satisfies the following axioms: (i) (normality axiom) $M {Γ} = 1$ for the universal set Γ; (ii) (duality axiom) $M {Λ} + M {Λ^{c}} = 1$ for any event Λ; (iii) (subadditivity axiom) $M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}}$ for every countable sequence of events Λ₁, Λ₂, ⋯. The triplet $(Γ, L, M)$ is called an uncertainty space. Besides, a product uncertain measure $M$ is defined by Liu [4] for providing the operational law: (product axiom) Let $(Γ_{k}, L_{k}, M_{k})$ be uncertainty spaces for k = 1, 2, ⋯. Then the product uncertain measure $M$ is an uncertain measure satisfying $M {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} M_{k} {Λ_{k}},$ (1) where Λ_k are arbitrarily chosen events from $L_{k}$ for k = 1, 2, ⋯, respectively.

An uncertain variable is a measurable function ξ from an uncertainty space $(Γ, L, M)$ to the set of real numbers. The uncertainty distribution $Φ : ℝ \to [0, 1]$ of an uncertain variable ξ is defined by $Φ (x) = M {ξ \leq x}$ for $x \in ℝ$ .

Let T be an index set and let $(Γ, L, M)$ be an uncertainty space. An uncertain process is a measurable function from $T \times (Γ, L, M)$ to the set of real numbers, ie., for each t ∈ T and any Borel set B of real numbers, the set {X_t ∈ B} = {γ ∣ X_t (γ) ∈ B} is an event. An uncertain process X_t is said to have independent increments if X_{t
₀}, X_{t
₁} - X_{t
₀}, X_{t
₂} - X_{t
₁}, ⋯ , X_{t
_k} - X_{t
_k-1} are independent uncertain variables for any times t₀ < t₁ < ⋯ < t_k. An uncertain process X_t is said to have stationary increments if for any given t > 0, the increments X_s+t - X_s are identically distributed uncertain variables for all s > 0.

A Liu process was defined by Liu [4], which satisfies: (i) C₀ = 0 and almost all sample paths are Lipschitz continuous; (ii) C_t has stationary and independent increments; (iii) every increment C_s+t - C_s is a normal uncertain variable with expected value 0 and variance t². Based on Liu process, uncertain calculus was defined by Liu [4]. For any partition of closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is written as $Δ = max_{1 \leq i \leq k} | t_{i + 1} - t_{i} |$ . Then the uncertain integral of X_t with respect to C_t is defined as $\begin{matrix} \int_{a}^{b} X_{t} d C_{t} = lim_{Δ \to 0} \sum_{i = 1}^{k} X_{t_{i}} (C_{t_{i + 1}} - C_{t_{i}}) \end{matrix}$ provided that the limit exists almost surely and is finite. In this case, the uncertain process X_t is said to be integrable.

Lemma 1. (Yao et al. [9]) Let C_t be a Liu process on an uncertainty space ( $Γ, L, M$ ). Then there exists an uncertain variable K such that K_γ is a Lipschitz constant of the sample path C_t (γ) for each γ ∈ Γ, and $lim_{x \to + \infty} M {γ \in Γ ∣ K_{γ} \leq x} = 1 .$ (2)

It is clear that for any ɛ > 0, there exists a real number H_ɛ > 0 such that $M {γ \in Γ ∣ K_{γ} \leq H_{ɛ}} \geq 1 - ɛ$ by Lemma 1.

Lemma 2. (Liu [2]) Let ξ be an uncertain variable and f be a negative function. If h is decreasing on (- ∞ , 0] and increasing on [0, + ∞), then for any given t > 0, we have

$M {| ξ | \geq t} \leq \frac{E [h (ξ)]}{h (t)} .$ (3)

For continuous function f (t) with $f^{'} (t) \in L^{1} (ℝ^{+})$ . The Caputo fractional derivative with order 0 < α < 1 of f (t) is given by $_{a}^{c} D_{t}^{α} f (t) = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{- α} f' (s) d s,$ where Γ (.) is the Gamma function defined by $Γ (z) = \int_{0}^{\infty} s^{z - 1} e^{- s} ds$ . Denote $_{0}^{c} D_{t}^{α}$ by the abbreviation ^cD^α.

Lemma 3. (Generalized Gronwall inequality [20]) Suppose that α > 0, a (t) is nonnegative and local integrable on [0, T], T< + ∞. g (t) is a nonnegative, nondecreasing continuous function defined on [0, T), $g (t) \leq \hat{M}$ (constant). u (t) is nonnegative and local integrable function on [0, T] which satisfies the inequality $\begin{matrix} u (t) \leq a (t) + g (t) \int_{0}^{t} (t - s)^{α - 1} u (s) d s \end{matrix}$ for 0 ≤ t < T. Then $u (t) \leq a (t) + \int_{0}^{t} [\sum_{n = 1}^{\infty} \frac{(g (t) Γ (α))^{n}}{Γ (n α)} (t - s)^{n α - 1} a (s)] d s .$ (4) If a (t) is a nondecreasing function on [0, T], then the inequality $u (t) \leq a (t) E_{α} (g (t) Γ (α) t^{α})$ (5) holds, where E_α,β (z) is the Mittag-Leffler function with the form $E_{α, β} (z) = \sum_{k = 0}^{+ \infty} \frac{z^{k}}{Γ (k α + β)}$ . Especially when β = 1, E_α (z) = E_α,1 (z).

3 Continuous dependence theorems

The Caputo type of uncertain fractional differential equation with the form

$_{t_{0}}^{c} D_{t}^{α} X_{t} = f (t, X_{t}) + g (t, X_{t}) \frac{d C_{t}}{d t}$ (6) with initial value X_t|_t=t₀ = x₀, has an equivalent integral equation

$\begin{matrix} X_{t} = & x_{0} + \frac{1}{Γ (α)} \int_{t_{0}}^{t} (t - s)^{α - 1} f (s, X_{s}) d s \\ + \frac{1}{Γ (α)} \int_{t_{0}}^{t} (t - s)^{α - 1} g (s, X_{s}) d C_{s} \end{matrix}$ (7) The existence and uniqueness theorems of solutions for the system (6) were derived by Zhu [33], in which the coefficients of uncertain fractional system (6) satisfy the global Lipschitz condition and linear growth condition with the Lipschitz constant L.

We first focus on the continuous dependence on initial value (t₀, x₀) of solution to uncertain fractional system (6). To simplify matters, we assume that the coefficients f and g satisfy the Assumption 1.

Assumption 1. For the uncertain fractional system (6), the coefficients f (t, x) and g (t, x) are global Lipschitz continuous and satisfy the linear growth condition with respect to x on $[0, T] \times ℝ$ with constant L, i.e. $\begin{matrix} | f (t, x) - f (t, y) | \lor | g (t, x) - g (t, y) | \leq L | x - y |, \\ | f (t, x) | \lor | g (t, x) | \leq L (1 + | x |) \end{matrix}$ for any $x, y \in ℝ$ and t ∈ [0, T], where the operator ∨ means the maximum operator.

Theorem 1. Assume Y_t is a solution of the uncertain fractional system $_{τ}^{c} D_{t}^{α} Y_{t} = f (t, Y_{t}) + g (t, Y_{t}) \frac{d C_{t}}{d t}$ (8) with initial value Y_τ = y₀. Then under the Assumption 1, $lim_{(τ, y_{0}) \to (t_{0}, x_{0})} M {sup_{t \in [t_{0} \lor τ, T]} | Y_{t} - X_{t} | > ε} = 0$ (9) holds for any given number ε > 0.

Proof. It follows from (7) that the uncertain fractional system (61) is equivalent with the integral equation

$\begin{matrix} Y_{t} = & y_{0} + \frac{1}{Γ (α)} \int_{τ}^{t} (t - s)^{α - 1} f (s, X_{s}) d s \\ + \frac{1}{Γ (α)} \int_{τ}^{t} (t - s)^{α - 1} g (s, X_{s}) d C_{s} . \end{matrix}$ (10)

Without loss of generality, suppose that 0 ≤ t₀ < τ < t₀ + δ (δ is a small constant). For each γ ∈ {γ ∣ K_γ ≤ H_ɛ}, from the Assumption 1, we have

$\begin{matrix} | X_{t} & (γ) | \leq | x_{0} | + \frac{1}{Γ (α)} \int_{t_{0}}^{t} (t - s)^{α - 1} L (1 + | X_{s} (γ) |) d s \\ + \frac{1}{Γ (α)} \int_{t_{0}}^{t} (t - s)^{α - 1} L (1 + | X_{s} (γ) |) | d C_{s} (γ) | \\ \leq & | x_{0} | + \frac{1}{Γ (α)} \int_{t_{0}}^{t} (t - s)^{α - 1} L (1 + | X_{s} (γ) |) d s \\ + \frac{K_{γ}}{Γ (α)} \int_{t_{0}}^{t} (t - s)^{α - 1} L (1 + | X_{s} (γ) |) d s \\ \leq & | x_{0} | + \frac{(1 + K_{γ}) L}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} d s \\ + \frac{(1 + K_{γ}) L}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} | X_{s} (γ) | d s \\ \leq & | x_{0} | + \frac{(1 + K_{γ}) {LT}^{α}}{Γ (α + 1)} \\ + \frac{(1 + K_{γ}) L}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} | X_{s} (γ) | d s \end{matrix}$ By Lemma 3, it can be derived that

$\begin{matrix} | X_{t} (γ) | \leq & [| x_{0} | + \frac{(1 + K_{γ}) {Lt}^{α}}{Γ (α + 1)}] E_{α} (L (1 + K_{γ}) t^{α}) \\ \leq & [| x_{0} | + \frac{(1 + K_{γ}) {LT}^{α}}{Γ (α + 1)}] E_{α} (L (1 + K_{γ}) T^{α}) \\ \leq & [| x_{0} | + \frac{(1 + H_{ɛ}) {LT}^{α}}{Γ (α + 1)}] E_{α} (L (1 + H_{ɛ} T^{α})) \end{matrix}$ for t ∈ [t₀, T] and γ ∈ {γ ∣ K_γ ≤ H_ɛ}, which implies that there exists a constant M_ɛ > 0 such that |f (t, X_t (γ)) | ≤ M_ɛ and |g (t, X_t (γ)) | ≤ M_ɛ hold for t ∈ [0, T] and γ ∈ {γ ∣ K_γ ≤ H_ɛ}, due to the continuity of f and g on $[0, T] \times ℝ$ . Denote V_t (γ) = |Y_t (γ) - X_t (γ) |, we calculate

$\begin{matrix} V_{t} & (γ) = | y_{0} - x_{0} | + | \frac{1}{Γ (α)} \int_{τ}^{t} (t - s)^{α - 1} f (s, Y_{s} (γ)) d s \\ + \frac{1}{Γ (α)} \int_{τ}^{t} (t - s)^{α - 1} g (s, Y_{s} (γ)) d C_{s} (γ) \\ - \frac{1}{Γ (α)} \int_{t_{0}}^{t} (t - s)^{α - 1} f (s, X_{s} (γ)) d s \\ - \frac{1}{Γ (α)} \int_{t_{0}}^{t} (t - s)^{α - 1} g (s, X_{s} (γ)) d C_{s} (γ) | \\ \leq & | y_{0} - x_{0} | + \frac{1}{Γ (α)} \int_{t_{0}}^{τ} (t - s)^{α - 1} | f (s, X_{s} (γ)) | d s \\ + \frac{1}{Γ (α)} \int_{t_{0}}^{τ} (t - s)^{α - 1} | g (s, X_{s} (γ)) | | d C_{s} (γ) | \\ + \frac{1}{Γ (α)} \int_{τ}^{t} (t - s)^{α - 1} | f (s, Y_{s} (γ)) - f (s, X_{s} (γ)) | d s \\ + \frac{1}{Γ (α)} \int_{τ}^{t} (t - s)^{α - 1} | g (s, Y_{s} (γ)) - g (s, X_{s} (γ)) | | d C_{s} (γ) | . \end{matrix}$ Since f (t, x) and g (t, x) satisfy the Assumption 1 and |f (t, X_t (γ)) | ≤ M_ɛ and |g (t, X_t (γ)) | ≤ M_ɛ hold for t ∈ [0, T] and γ ∈ {γ ∣ K_γ ≤ H_ɛ}, by employing the Lemma 1, we obtain

$\begin{matrix} V_{t} ( & γ) \leq | y_{0} - x_{0} | + \frac{(1 + K_{γ}) M_{ɛ}}{Γ (α)} \int_{t_{0}}^{τ} (t - s)^{α - 1} d s \\ + \frac{(1 + K_{γ}) L}{Γ (α)} \int_{τ}^{t} (t - s)^{α - 1} V_{s} (γ) d s \\ = & a (t) + \frac{(1 + K_{γ}) L}{Γ (α)} \int_{τ}^{t} (t - s)^{α - 1} V_{s} (γ) d s \end{matrix}$ where $a (t) = | y_{0} - x_{0} | + \frac{(1 + K_{γ}) M_{ɛ}}{Γ (α + 1)} [(t - t_{0})^{α} - (t - τ)^{α}]$ . By Lemma 3, we have

$\begin{matrix} V_{t} & (γ) = | Y_{t} (γ) - X_{t} (γ) | \\ \leq & a (t) + \int_{τ}^{t} [\sum_{n = 1}^{\infty} \frac{((1 + K_{γ}) L)^{n}}{Γ (n α)} (t - s)^{n α - 1} a (s)] d s \\ = & a (t) + \sum_{n = 1}^{\infty} \frac{((1 + K_{γ}) L)^{n}}{Γ (n α)} [\int_{τ}^{t} (t - s)^{n α - 1} | y_{0} - x_{0} | d s \\ + \frac{(1 + K_{γ}) M_{ɛ}}{Γ (α + 1)} \int_{τ}^{t} (t - s)^{n α - 1} (s - t_{0})^{α} d s \\ - \frac{(1 + K_{γ}) M_{ɛ}}{Γ (α + 1)} \int_{τ}^{t} (t - s)^{n α - 1} (s - τ)^{α} d s] \\ \leq & a (t) + \sum_{n = 1}^{\infty} \frac{((1 + K_{γ}) L)^{n}}{Γ (n α)} [\int_{0}^{t} (t - s)^{n α - 1} | y_{0} - x_{0} | d s \\ + \frac{(1 + K_{γ}) M_{ɛ}}{Γ (α + 1)} \int_{t_{0}}^{t} (t - s)^{n α - 1} (s - t_{0})^{α} d s \\ - \frac{(1 + K_{γ}) M_{ɛ}}{Γ (α + 1)} \int_{τ}^{t} (t - s)^{n α - 1} (s - τ)^{α} d s] \\ = & a (t) + | y_{0} - x_{0} | \sum_{n = 1}^{\infty} \frac{((1 + K_{γ}) {Lt}^{α})^{n}}{Γ (n α + 1)} + (1 + K_{γ}) M_{ɛ} \\ \times [\sum_{n = 1}^{\infty} \frac{((1 + K_{γ}) L)^{n} ((t - t_{0})^{n α + α} - (t - τ)^{n α + α})}{Γ (n α + α + 1)}] \\ = & | y_{0} - x_{0} | E_{α} ((1 + K_{γ}) {Lt}^{α}) \\ + (1 + K_{γ}) M_{ɛ} [(t - t_{0})^{α} E_{α, α + 1} ((1 + K_{γ}) L (t - t_{0})^{α}) \\ - (t - τ)^{α} E_{α, α + 1} ((1 + K_{γ}) L (t - τ)^{α})] \\ \leq & | y_{0} - x_{0} | E_{α} ((1 + H_{ɛ}) {Lt}^{α}) \\ + (1 + H_{ɛ}) M_{ɛ} [(t - t_{0})^{α} E_{α, α + 1} ((1 + H_{ɛ}) L (t - t_{0})^{α}) \\ - (t - τ)^{α} E_{α, α + 1} ((1 + H_{ɛ}) L (t - τ)^{α})] ≜ B_{t} (τ, y_{0}) . \end{matrix}$ Due to the continuity of B_t (τ, y₀) with respect to t in the interval [t₀, T], there exists a t^* ∈ [t₀, T] such that B_t (τ, y₀) ≤ B_{t
^*} (τ, y₀) holds for any t ∈ [t₀, T]. It is clear that $lim_{(τ, y_{0}) \to (t_{0}, x_{0})} B_{t^{*}} (τ, y_{0}) = 0$ holds. That is, for any ε > 0, there exist a δ′ (ε) >0 and a small deleted neighbourhood U^o (P₀) = {(τ, y) ∣0 < (τ - t₀) ² + (y - x₀) ² ≤ δ′ (ε)} of the point P₀ (t₀, x₀) such that V_t (γ) ≤ B_{t
^*} (τ, y₀) ≤ ε holds for any (τ, y₀) ∈ U^o (P₀) when γ ∈ {γ ∣ K_γ ≤ H_ɛ}, which implies that

$\begin{matrix} {γ ∣ K_{γ} \leq H_{ɛ}} \subset {γ ∣ sup_{t \in [t_{0} \lor τ, T]} | Y_{t} (γ) - X_{t} (γ) | \leq ε} . \end{matrix}$ Thus we obtain $\begin{matrix} M & {sup_{t \in [t_{0} \lor τ, T]} | Y_{t} - X_{t} | \leq ε} \\ \geq M {γ \in Γ ∣ K_{γ} \leq H_{ɛ}} > 1 - ɛ \end{matrix}$ holds for any ɛ > 0, provided that (t, y₀) ∈ U^o (P₀). Then we have $\begin{matrix} lim_{(τ, y_{0}) \to (t_{0}, x_{0})} M & {sup_{t \in [t_{0} \lor τ, T]} | Y_{t} - X_{t} | \leq ε} = 1 \end{matrix}$ which completes the proof of Theorem 1.□

In system engineering, when the small change of parameters of the coefficients in UFDEs makes the system state change greatly, it means that the system is more sensitive on these parameters. Then the effects of the parameters on the system control results should be fully taken into account in the subsequent system modeling. To study the continuity of the solution on parameters for UFDEs, we consider the uncertain integral equation

$\begin{matrix} X_{t, λ} = & x_{0} + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} f (s, X_{s, λ}, λ) d s \\ + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} g (s, X_{s, λ}, λ) d C_{s}, \end{matrix}$ (11) where the coefficients f (t, x, λ), g (t, x, λ) all depend on the parameter λ.

For the convenience of discussion, some essential assumptions of the coefficients f and g should be proposed.

Assumption 2. Let [a, b] be arbitrary real interval and a ≤ b. The coefficients $f (t, x, λ) : [0, T] \times ℝ \times [a, b]$ and $g (t, x, λ) : [0, T] \times ℝ \times [a, b]$ are continuous and satisfy the linear growth condition and global Lipschitz condition in x, i.e. there exists a positive constant L such that

$\begin{matrix} | f (t, x, λ) | \lor | g (t, x, λ) | \leq L (1 + | x |) \\ | f (t, x, λ) - f (t, y, λ) | \lor | g (t, x, λ) - g (t, y, λ) | \leq L | x - y |, \end{matrix}$ for any $x, y \in ℝ$ , t ∈ [0, T] and λ ∈ [a, b].

It is clear that the Eq.(11) has a unique solution X_t,λ on interval [0, T] under Assumption 2. Now we study the continuity of X_t,λ with respect to λ.

Theorem 2. Under Assumption 2, $lim_{λ \to λ_{0}} M {sup_{t \in [0, T]} | X_{t, λ} - X_{t, λ_{0}} | > ε} = 0$ (12) holds for any given number ε > 0, where λ₀ ∈ [a, b].

Proof. For γ ∈ {γ ∣ K_γ ≤ H_ɛ}, we prove the solution X_λ (γ) of the uncertain fractional integral equation

$\begin{matrix} X_{t, λ} (γ) \\ = x_{0} + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} f (s, X_{s, λ} (γ), λ) d s \\ + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} g (s, X_{s, λ} (γ), λ) d C_{s} (γ) \end{matrix}$ (13) is bonded on [0, T]. Indeed, it follows from the Lemma 1 that for any t ∈ [0, T] $\begin{matrix} | X_{t, λ} (γ) | \\ \leq & | x_{0} | + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} | f (s, X_{s, λ} (γ), λ) | d s \\ + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} | g (s, X_{s, λ} (γ), λ) | | d C_{s} (γ) | \\ \leq & | x_{0} | + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} | f (s, X_{s, λ} (γ), λ) | d s \\ + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} | g (s, X_{s, λ} (γ), λ) | K_{γ} d s \\ \leq & | x_{0} | + \frac{L}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} (1 + | X_{s, λ} (γ) |) d s \\ + \frac{K_{γ} L}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} (1 + | X_{s, λ} (γ) |) d s \\ = & | x_{0} | + \frac{L (1 + K_{γ})}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} d s \\ + \frac{L (1 + K_{γ})}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} | X_{s, λ} (γ) | d s \\ \leq & | x_{0} | + \frac{{LT}^{α} (1 + K_{γ})}{Γ (α + 1)} \\ + \frac{L (1 + K_{γ})}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} | X_{s, λ} (γ) | d s . \end{matrix}$

Denoting $A_{λ} = | x_{0} | + \frac{{LT}^{α} (1 + K_{γ})}{Γ (α + 1)}$ , by the generalized Gronwall inequality in Lemma 3, $\begin{matrix} | X_{t, λ} (γ) | \leq & A_{λ} E_{α} ((1 + K_{γ}) {Lt}^{α}) \\ \leq & A_{λ} E_{α} ((1 + K_{γ}) {LT}^{α}) < \infty \end{matrix}$ holds for any t ∈ [0, T]. It means that X_t,λ (γ) is bounded on [0, T]. We calculate

$\begin{matrix} X_{t, λ} - X_{t, λ_{0}} \\ = \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} [f (s, X_{s, λ}, λ) - f (s, X_{s, λ_{0}}, λ_{0})] d s \\ + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} [g (s, X_{s, λ}, λ) - g (s, X_{s, λ_{0}}, λ_{0})] d C_{s} \\ = Y_{t, λ} + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} [f (s, X_{s, λ}, λ) - f (s, X_{s, λ_{0}}, λ)] d s \\ + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} [g (s, X_{s, λ}, λ) - g (s, X_{s, λ_{0}}, λ)] d C_{s} \end{matrix}$ (14) where

$\begin{matrix} Y_{t, λ} \\ = \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} [f (s, X_{s, λ_{0}}, λ) - f (s, X_{s, λ_{0}}, λ_{0})] d s \\ + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} [g (s, X_{s, λ_{0}}, λ) - g (s, X_{s, λ_{0}}, λ_{0})] d C_{s} . \end{matrix}$ Denote

$\begin{matrix} F (s, λ, γ) & = f (s, X_{s, λ_{0}} (γ), λ) - f (s, X_{s, λ_{0}} (γ), λ_{0}), \\ G (s, λ, γ) & = g (s, X_{s, λ_{0}} (γ), λ) - g (s, X_{s, λ_{0}} (γ), λ_{0}) . \end{matrix}$ Then we have $\begin{matrix} | Y_{t, λ} & (γ) | \leq \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} | F (s, λ, γ) | d s \\ + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} | G (s, λ, γ) | | d C_{s} (γ) | \\ \leq & \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} | F (s, λ, γ) | d s \\ + \frac{K_{γ}}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} | G (s, λ, γ) | d s . \end{matrix}$ Since the coefficients f (t, x, λ) and g (t, x, λ) are continuous, there exists a t^* ∈ [0, T] such that

$\begin{matrix} | Y_{t, λ} & (γ) | \leq \frac{1}{Γ (α)} \int_{0}^{t^{*}} (t^{*} - s)^{α - 1} | F (s, λ, γ) | d s \\ + \frac{K_{γ}}{Γ (α)} \int_{0}^{t^{*}} (t^{*} - s)^{α - 1} | G (s, λ, γ) | d s \end{matrix}$ (15) holds for any t ∈ [0, T]. Setting the right side of the inequality (15) as H (λ, γ), it follows from the Assumption 2 that

$\begin{matrix} | & X_{t, λ} (γ) - X_{t, λ_{0}} (γ) | \leq H (λ, γ) + \frac{1}{Γ (α)} \\ \times & [\int_{0}^{t} (t - s)^{α - 1} | f (s, X_{s, λ} (γ), λ) - f (s, X_{s, λ_{0}} (γ), λ) | d s \\ + \int_{0}^{t} (t - s)^{α - 1} | g (s, X_{s, λ} (γ), λ) - g (s, X_{s, λ_{0}} (γ), λ) | | d C_{s} (γ) |] \\ \leq & H (λ, γ) + \frac{L (1 + K_{γ})}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} | X_{s, λ} (γ) - X_{s, λ_{0}} (γ) | d s . \end{matrix}$ By Lemma 3, we obtain

$\begin{matrix} | X_{t, λ} (γ) - X_{t, λ_{0}} (γ) | \leq H (λ, γ) E_{α} ((1 + K_{γ}) {LT}^{α}) . \end{matrix}$ holds for t ∈ [0, T] and γ ∈ Γ. Since X_t,λ (γ) is bounded on [0, T], by Lebesgue dominated convergence theorem, we obtain

$\begin{matrix} lim_{λ \to λ_{0}} H (λ, γ) \\ = lim_{λ \to λ_{0}} \frac{1}{Γ (α)} \int_{0}^{t^{*}} (t^{*} - s)^{α - 1} | F (s, λ, γ) | d s \\ + lim_{λ \to λ_{0}} \frac{K_{γ}}{Γ (α)} \int_{0}^{t^{*}} (t^{*} - s)^{α - 1} | G (s, λ, γ) | d s \\ = \frac{1}{Γ (α)} \int_{0}^{t^{*}} (t^{*} - s)^{α - 1} [lim_{λ \to λ_{0}} | F (s, λ, γ) |] d s \\ + \frac{K_{γ}}{Γ (α)} \int_{0}^{t^{*}} (t^{*} - s)^{α - 1} [lim_{λ \to λ_{0}} | G (s, λ, γ) |] d s \\ = 0 . \end{matrix}$ (16) Thus for any ε > 0 and γ ∈ {γ ∣ K_γ ≤ H_ɛ}, there exists a small δ > 0 such that $G (λ, γ) \leq \frac{ε}{E_{α} ((1 + H_{ɛ}) {LT}^{α})}$ holds for λ satisfying |λ - λ₀| ≤ δ. Then by employing the Lemma 1, we have

$| X_{t, λ} (γ) - X_{t, λ_{0}} (γ) | \leq ε$ (17) holds for any t ∈ [0, T], under the conditions K_γ ≤ H_ɛ and |λ - λ₀| ≤ δ. Then we obtain

$\begin{matrix} {γ ∣ K_{γ} \leq H_{ɛ}} \subset {sup_{t \in [0, T]} | X_{t, λ} (γ) - X_{t, λ_{0}} (γ) | \leq ε} \end{matrix}$ provided that |λ - λ₀| ≤ δ. Moreover, $\begin{matrix} M & {sup_{t \in [0, T]} | X_{t, λ} (γ) - X_{t, λ_{0}} (γ) | \leq ε} \\ \geq M {γ ∣ K_{γ} \leq H_{ɛ}} > 1 - ɛ \end{matrix}$ provided that |λ - λ₀| ≤ δ. It implies $\begin{matrix} lim_{λ \to λ_{0}} M & {sup_{t \in [0, T]} | X_{t, λ} (γ) - X_{t, λ_{0}} (γ) | \leq ε} = 1, \end{matrix}$ which completes the proof of Theorem 1.□

4 Illustrative examples

In this section, we give two examples to show the continuous dependent results of solution to uncertain fractional differential equation on initial values and parameters, respectively.

Example 1. To show the continuity of solution to uncertain fractional differential equation on initial values, we consider the following initial value problems

$_{t_{0}}^{c} D_{t}^{α} X_{t} = {aX}_{t} + σ_{t} \frac{d C_{t}}{d t}$ (18) and

$X_{t} |_{t = t_{0}} = x_{0},$ (19)

It is easy to find that the coefficients f (t, x) = ax and g (t, x) = σ_t satisfy the Lipschitz condition. Thus the solution of the initial value problem (18) is continuity in measure with respect to initial value (t₀, x₀) by Theorem 1. Now, we verify this result.

It follows from Theorem 2 in Lu et al. [34] that the analytic solutions of these two initial value problems can be written as

$\begin{matrix} X_{t} & = x_{0} E_{α, 1} ({at}^{α}) + \int_{t_{0}}^{t} (t - s)^{α - 1} E_{α, α} (a (t - s)^{α}) σ_{s} d C_{s} \\ Y_{t} & = y_{0} E_{α, 1} ({at}^{α}) + \int_{τ}^{t} (t - s)^{α - 1} E_{α, α} (a (t - s)^{α}) σ_{s} d C_{s}, \end{matrix}$ respectively. Without loss of generality, assume τ > t₀, then we have

$\begin{matrix} | X_{t} - Y_{t} | \leq & | x_{0} - y_{0} | E_{α, 1} ({at}^{α}) \\ + | \int_{t_{0}}^{τ} (t - s)^{α - 1} E_{α, α} (a (t - s)^{α}) σ_{s} d C_{s} | \end{matrix}$ Taking $ξ_{t} = \int_{t_{0}}^{τ} (t - s)^{α - 1} E_{α, α} (a (t - s)^{α}) σ_{s} d C_{s}$ , then ξ_t is an normal uncertain process with the expected value 0 and variance $\int_{t_{0}}^{τ} (t - s)^{α - 1} E_{α, α} (a (t - s)^{α}) | σ_{s} | d s$ , and

$\begin{matrix} {| X_{t} - Y_{t} | \geq ɛ} \subset {| x_{0} - y_{0} | E_{α, 1} ({at}^{α}) + | ξ_{t} | \geq ɛ} . \end{matrix}$ By Lemma 2 with h (x) = x², we have

$\begin{matrix} M & {| X_{t} - Y_{t} | \geq ɛ} \\ \leq & M {| x_{0} - y_{0} | E_{α, 1} ({at}^{α}) + | ξ_{t} | \geq ɛ} \\ = & M {| ξ_{t} | \geq ɛ - | x_{0} - y_{0} | E_{α, 1} ({at}^{α})} \\ \leq & \frac{E [{| \int_{t_{0}}^{τ} (t - s)^{α - 1} E_{α, α} (a (t - s)^{α}) σ_{s} d C_{s} |}^{2}]}{{(ɛ - | x_{0} - y_{0} | E_{α, 1} ({at}^{α}))}^{2}} \\ = & \frac{{(\int_{t_{0}}^{τ} (t - s)^{α - 1} E_{α, α} (a (t - s)^{α}) | σ_{s} | d s)}^{2}}{{(ɛ - | x_{0} - y_{0} | E_{α, 1} ({at}^{α}))}^{2}} \to 0 \end{matrix}$ as (τ, y₀) → (t₀, x₀), which implies the continuity of solution on initial values.

Similarly, the solution of initial value problem (18) with a parameter a can also be proved to be continuous with respect to a in measure, if the coefficients are regarded as f (t, x, a) = ax and g (t, x, a) = σ_t.

Example 2. Consider the nonlinear uncertain fractional differential equation

$_{t_{0}}^{c} D_{t}^{α} X_{t} = exp (- t) X_{t} + \frac{1}{1 + t^{2}} sin X_{t} \frac{d C_{t}}{d t}, t \geq t_{0}$ (20) with initial value X_{t
₀} = x₀ and t₀ ≥ 0. Since the coefficients f (t, x) = exp(- t) x and $g (t, x) = \frac{1}{1 + t^{2}} sin x$ satisfy the linear growth condition

$\begin{matrix} | f (t, x) | \lor | g (t, x) | = | e^{- t} x | \lor | \frac{1}{1 + t^{2}} sin x | < 1 + | x | \end{matrix}$ and the Lipschitz condition

$\begin{matrix} | f (t, x) - f (t, y) | & = e^{- t} | x - y | \leq | x - y |, \\ | g (t, x) - g (t, y) | & = \frac{1}{1 + t^{2}} | sin x - sin y | \leq | x - y | \end{matrix}$ for $x, y \in ℝ$ and t ∈ [0, T], it follows from Theorem 1 that the solution of the initial value problem (43) is continuous in measure sense with respect to the initial point (t₀, x₀).

Example 3. Consider the nonlinear uncertain fractional differential equation

$^{c} D^{α} X_{t} = \sqrt{1 + λ^{2} X_{t}^{2}} + exp (- λ t^{2} - X_{t}^{2}) \frac{d C_{t}}{d t}$ (21) with initial value X_t|_t=0 = x₀, where λ ∈ [0, 1]. Since the coefficients $f (t, x, λ) = \sqrt{1 + λ^{2} x^{2}}$ and g (t, x, λ) = exp(- λt² - x²) satisfy $\begin{matrix} | f ( & t, x, λ) - f (t, y, λ) | \\ = | \sqrt{1 + λ^{2} x^{2}} - \sqrt{1 + λ^{2} y^{2}} | \leq | λ | | x - y | \\ \leq | x - y |, \end{matrix}$ and $\begin{matrix} | g (t & , x, λ) - g (t, y, λ) | \\ = | exp (- λ t^{2} - x^{2}) - exp (- λ t^{2} - y^{2}) | \\ \leq exp (- λ t^{2}) | x - y | \leq | x - y | \end{matrix}$ for $x, y \in ℝ$ and t ∈ [0, T], it follows from Theorem 1 that the solution of the initial value problem (42) is continuous in measure sense with respect to λ ∈ [0, 1].

Example 4. Consider the nonlinear uncertain fractional differential equation $_{t_{0}}^{c} D_{t}^{α} X_{t} = λ arctan X_{t} + ln (1 + λ X_{t}^{2}) \frac{d C_{t}}{d t}$ (22) with initial value X_t|_t=t₀ = x₀, where 1 ≤ λ ≤ λ₀ (a constant). Since the coefficients f (t, x, λ) = λ arctan x and g (t, x, λ) = ln(1 + λx²) satisfy the Lipschitz condition $\begin{matrix} | f (t, x, λ) & - f (t, y, λ) | = λ | arctan x - arctan y | \\ \leq & λ | x - y | \leq λ_{0} | x - y |, \\ | g (t, x, λ) & - g (t, y, λ) | = | ln (1 + λ x^{2}) - ln (1 + λ y^{2}) | \\ = & | \frac{2 λ ξ}{1 + λ ξ^{2}} | | x - y | \leq \sqrt{λ} | x - y | \\ \leq & \sqrt{λ_{0}} | x - y | \end{matrix}$ hold for $x, y \in ℝ$ , t ∈ [0, T] and 1 ≤ λ ≤ λ₀, based on Lagrange mean value theorem, where ξ ∈ (x, y). Meanwhile $\begin{matrix} | f (t, x, λ) | \leq & λ_{0} (1 + | x |) \\ | g (t, x, λ) | \leq & λ (1 + | x |) \leq λ_{0} (1 + | x |) \end{matrix}$ hold for $x, y \in ℝ$ and t ∈ [0, T], due to the fact 1 ≤ λ ≤ λ₀. Then the solution to UFDE (44) is continuous in measure sense with respect to the parameter λ ∈ [1, λ₀] by employing Theorem 1. Moreover, the solution to UFDE (44) is continuous in measure with respect to the initial point (t₀, x₀) based on Theorem 1 while setting the λ as a constant satisfying 1 ≤ λ ≤ λ₀.

5 Conclusion

This paper presented the continuous dependence theorems of solution with respect to parameters and initial points for uncertain fractional differential equation based on uncertainty theory. The results show that the solutions of these equations are continuous dependent with respect to the parameters and initial points in measure sense when the coefficients of these equations satisfy the Lipschitz condition. Future work will focus on the stability analysis for uncertain fractional differential equations.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 61673011) and Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. KYCX18_0371).

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