Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump

: No previous study has involved uncertain fractional differential equation (FDE, for short) with jump. In this paper, we propose the uncertain FDEs with jump, which is driven by both an uncertain V -jump process and an uncertain canonical process. First of all, for the one-dimensional case, we give two types of uncertain FDEs with jump that are symmetric in terms of form. The next, for the multidimensional case, when the coefﬁcients of the equations satisfy Lipschitz condition and linear growth condition, we establish an existence and uniqueness theorems of uncertain FDEs with jump of Riemann-Liouville type by Banach ﬁxed point theorem. A symmetric proof in terms of form is suitable to the Caputo type. When the coefﬁcients do not satisfy the Lipschitz condition and linear growth condition, we just prove an existence theorem of the Caputo type equation by Schauder ﬁxed point theorem. In the end, we present an application about uncertain interest rate model.


Introduction
Wiener process is a type of stationary-independent increment stochastic process with normal random increments designed by Wiener in 1923 [1]. Then stochastic differential equation (SDE, for short) was proposed by Itô in 1951 as a vital tool to model stochastic dynamic systems [2]. Following that, many areas such as noted European option pricing model [3] by SDEs and famous stochastic epidemic dynamic model hidden in the observed data [4] were developed. As all we know, The SDEs based on probability theory need a large of available sample data. However, when we lack of data or the size of sample data applied in practice are less in many situations, we need to invite some domain experts to evaluate the belief degree that each event happens.
Human uncertainty with respect to belief degrees [5] can play an important role in addressing the issue of indeterminate phenomenon. For describing the evolution of uncertain phenomenon, the uncertain differential equation (UDE, for short) was first proposed by Liu [6]. Following that, Liu [7] also proposed the concept of stability of UDEs. Later, Chen and Liu [8] proved an existence and uniqueness theorem for an UDE and Yao et al. [9] proved some stability theorems. Besides, a large and growing body of literature [10][11][12][13][14] about stability theorems for UDEs have been investigated. Further, Yao and Chen [15] first proposed Euler's method combined with 99-method to obtain the numerical solution of the UDEs. With the perfect of theory and maturity of numerical method of the UDEs, The UDEs have been successfully applied to many area such as optimal control theory [16,17], differential game theory [18,19], wave equation [20][21][22] and finance theory [23]. To understand developing process of the UDEs comprehensively, the readers can refer the book [24].
V-jumps uncertain processes proposed by Deng et al. [25] are often used to describe the evolution of uncertain phenomenon with jumps, in which the uncertain process may be caused a sudden change by emergency, such as economics crisis, outbreaks of infectious diseases, earthquake, war, etc. Here, It is needed to see that the cadlag functions [26] (right-continuous with left limits) are vital to deal with point process and related applications. The definition of V-jump uncertain process is as follows Definition 1. An uncertain process V k with respect to time k is said to be a V-jump process with parameters θ 1 and θ 2 (0 < θ 1 < θ 2 < 1) for k ≥ 0 if (i) V 0 = 0, (ii) V k has stationary and independent increments, (iii) For any given k > 0, every increment V r+k − V r is a Z jump uncertain variable ξ ∼ Z (θ 1 , θ 2 , k) for ∀r > 0, whose uncertainty distribution is Deng et al. [27] proved an existence and uniqueness of solution to UDE with V-jump under Lipschitz condition and linear growth condition on the coefficients. The uncertain differential equation with V-jump is expressed as follows where C k is an uncertain canonical process with respect to time k, V k is an uncertain V-jump process with respect to time k, and p 1 , p 2 and p 3 are some given functions.
Uncertain differential equations with V-jumps are widely applied to uncertain optimal control with V-jumps. Some related references can be seen in [28][29][30][31][32][33]. For the phenomena of complex systems, fractional differential equations (FDEs) [34] are very suitable for characterizing materials and processes with memory and genetic properties. When considering the research of uncertain complex systems, we are eagerly looking forward to having a usable mathematical tool and basic principles to model these complex systems. To better describe the uncertain complex phenomena, Zhu [35] proposed two types of uncertain fractional differential equations in the one-dimensional case, which is the Riemann-Liouville type and Caputo type, respectively. In the same year, Zhu [36] proved the existence and uniqueness of two types of uncertain fractional differential equations in the multidimensional case. The expressions of these two types of equations are as follows where D p Z k and c D p Z k denote the Riemann-Liouville type and Caputo type fractional derivative of the function Z k , respectively. C k is an uncertain canonical process with respect to time k, f , g are given functions. Based on the above uncertain FDEs, Lu et al. [37] further analyzed the solution of the uncertain linear FDE. Lu et al. [38] proposed the numerical methods for uncertain FDEs and compared some principles [39] for FDEs with the Caputo derivatives. Jin et al. [40] simulated the extreme values for solution to uncertain FDE and applied it to American stock model. To model discrete fractional calculus, Lu et al. [41] proposed uncertain fractional forward difference equations for Riemann-Liouville type. Furthermore, Lu et al. [42] investigated finite-time stability of uncertain FDEs. However, the uncertain FDEs with jump has not been studied so far. Inspired by Zhu [35,36] and Deng et al. [25,27], for describing the state of the uncertain fractional differential system with jumps more accurately, we propose uncertain FDEs with jump, which is very significant for the characterization of uncertain complex systems when meeting a sudden change by emergency.
The remainder of the paper is organized as follows. In Section 2, we recall some concepts of fractional order derivatives. Section 3 first gives two types of uncertain FDEs with jump in the one-dimensional case, then analyzes the multidimensional case, gives existence and uniqueness theorem of uncertain FDEs with jump by fixed point theorem, finally discuss an application about uncertain interest rate model. In Section 4, we give a brief conclusion.

Fractional Order Derivatives
We first recall two classes of fractional order derivatives in the one-dimensional case. Definition 2. [43] The fractional primitive of order p > 0 of a function φ : [u, v] → R is defined by where Γ is the gamma function satisfying Remark 1. [43] The properties of the gamma function are as follows: Besides, the beta function satisfying and B(p, q) = B(q, p). The relation between them is For a power function (k − u) η , it holds that Definition 4. [43] Let φ : [u, v] → R at least be a m order differentiable function. The pth Caputo fractional derivative of φ is defined by

Remark 2.
[43] For m − 1 < p ≤ m and k > 0, it holds that The pth Caputo fractional order derivative of the function φ : [0, T] → R n is defined by Meanwhile, they have the following relationship

Two Types of Uncertain FDEs with Jump in the One-Dimensional Case
Definition 5. Let C k be a canonical process and V k be a V-jump process. Suppose that f , g, h : [0, +∞) × R → R are three functions. Then is called an uncertain FDE with jump of the Riemann-Liouville type. A solution of (1) with the initial condition is an uncertain process Z k such that holds almost surely. Definition 6. Let C k be a canonical process and V k be a V-jump process. Suppose that f , g, h is called an uncertain FDE of the Caputo type. A solution of (3) is an uncertain process Z k such that We will use the following classical Mittag-Leffler function [43] , p > 0, q > 0 Theorem 1. Let C k and V k be two integrable uncertain processes.
(i) The uncertain FDE with jump The proof is similar to that of (i).
Theorem 3. Let a be a real number and µ k , ν k , σ k two functions on [0, T]. Then with the initial condition Proof. It is obvious that For Z k provided by (6), we have It follows from Theorem 3 in Ref [35] and the Mittag-Leffler function that In addition, we have We let r = s + τ(k − s), 0 ≤ τ ≤ 1 in (10), then Substituting (9), (11)-(13) into (8) yields Thus, (6) is a solution of (5) by Definition 5.
has a solution Proof. The proof of Theorem 4 is similar to that of Theorem 3, we omit here.

Remark 4.
In this part, we introduce the Riemann-Liouville type and the Caputo type of uncertain FDE with jump in the one-dimensional case. Now we state those concepts in a multidimensional case. In the next part, we will always assume p ∈ (0, 1]. Let C k = (C 1k , C 2k , · · · , C lk ) T be an l-dimensional canonical process and V k = (V 1k , V 2k , · · · , V lk ) T be an l-dimensional V-jump process.

Existence and Uniqueness of Uncertain FDEs with Jump in the Multidimensional Case
Definition 7. Let C k be a canonical process and V k be a V-jump process. Suppose that f : [0, +∞) × R n → R n , and g, h : [0, +∞) × R n → R n×l are three functions. Then, is called an uncertain FDE with jump of the Riemann-Liouville type. A solution of (16) with the initial condition is an uncertain process Z k such that holds almost surely.

Definition 8.
Let C k be a canonical process and V k be a V-jump process. Suppose that f : [0, +∞) × R n → R n , and g, h : [0, +∞) × R n → R n×l are three functions. Then, is called an uncertain FDE of the Caputo type. A solution of (18) is an uncertain process Z k such that holds almost surely.
For simplicity, we use | · | to denote a norm in R n or R n×l . Let C [u,v] denote the space of continuous R n -valued functions on [u, v], which is a Banach space with the norm Give three functions f (k, z) : [0, T] × R n → R n , g(k, z) : [0, T] × R n → R n×l and h(k, z) : [0, T] × R n → R n×l . Now we introduce the following mapping Φ on C [0,T] : for Z k ∈ C [0,T] , where z 0 is a given initial state.

Lemma 1.
For uncertain process Z k ∈ C [0,T] , the mapping Ψ defined by is sample-continuous, whereũ = u p−1 if u > 0, orũ = 1 if u = 0, and f , g and h satisfy the linear growth condition where L is a positive constant.
If the coefficients f , g and h do not satisfy the Lipschitz condition and linear growth condition, we will give the existence theorem just for continuous f , g and h as follows.

Theorem 6.
(Existence) Let f(k,z),g(k,z) and h(k,z) be continuous in Then, uncertain FDE of the Caputo type (18) has a solution Z k in k ∈ [0, T] with the crisp initial condition Z 0 = z 0 ∈ R n .
Proof. For any γ ∈ Γ, let d > 0 be a positive number such that where K γ is the Lipschitz constant of the canonical process C k , and where t = min{T, d}. It holds that Q is a closed convex set. Define a mapping Φ on Q by This means Φ(Z k (γ)) ∈ Q, and the mapping Φ is bounded uniformly in Z k (γ) ∈ Q. Besides, for 0 ≤ k 1 < k 2 ≤ t, it holds that We know that Φ is a compact mapping on Q by the Ascoli-Arzela theorem.
(k − r) p−1 g(r, Z r,i (γ))dC r (γ) It is easy to see that that Φ is continuous on Q.

Application
In this subsection, we will discuss an application of the present study. We give an uncertain interest rate model that the short interest rate Z k satisfies the following uncertain FDE with jump where m, µ, ν, and σ are positive numbers. The above model is the FDE with jump form of the model in Ref [43]. Then, the price of a zero-coupon bond is where E is the uncertain expected value [5], d is a maturity date.

Conclusions
The main goal of the current study is to prove existence and uniqueness for uncertain FDEs with jump. In addition, we give an application about uncertain interest rate model. Of course, the readers can further study more complex models by uncertain FDEs with jump, such as uncertain stock model and uncertain optimal control model. One source of weakness in our study is the lack of numerical methods, these will be the focus of our future research.
Author Contributions: All authors contributed equally and significantly in writing this article: writing-original draft preparation, Z.J., review, X.L. and C.L., investigation, Z.J., funding acquisition, X.L. and C.L. All authors read and approved the final manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.