On deﬁnition of solution of initial value problem for fractional di ﬀ erential equation of variable order

: We propose a new deﬁnition of continuous approximate solution to initial value problem for di ﬀ erential equations involving variable order Caputo fractional derivative based on the classical deﬁnition of solution of integer order (or constant fractional order) di ﬀ erential equation. Some examples are presented to illustrate these theoretical results.


Introduction
In this paper, we research a continuous approximate solution of the following variable order initial value problem where 0 < p(t, x(t)) < 1, u 0 ∈ R, p(t, x(t)) and f (t, x(t)) are given real-valued functions, C D p(t,x(t)) 0+ denotes variable order Caputo fractional derivative defined by C D p(t,x(t)) 0+ x(t) = I 1−p(t,x(t)) 0+ x (t), (1.2) and I 1−p(t,x(t)) 0+ is variable order Riemann-Liouville fractional integral defined by x(t) = t 0 (t − s) −p(t,x(t)) Γ(1 − p(t, x(t)) x(s)ds, t > 0. (1.3) For details, please refer to [1,2]. Fractional calculus has been acknowledged as an extremely powerful tool in describing the natural behavior and complex phenomena of practical problems due to its applications in [3][4][5][6][7]. However, the constant fractional order calculus is not the ultimate tool to model the phenomena in nature. Therefore, variable order fractional calculus is proposed. Moreover, variable order fractional differential equations provide better descriptions for nonlocal phenomena with varying dynamics than constant order differential equations and are extensively researched in [1,2,. Among these, there are many works dealing with numerical methods for some class of variable fractional order differential equations, for instance, [1, 2, 8-10, 12-19, 21-26, 30, 31]. In particular, variable order fractional boundary value problems are considered by numerical method base on reproducing kernel theory in [30,31].
There are several definitions of variable order fractional integrals and derivatives in [1,2]. We notice that when the order p(t) is a constant function p, variable order Riemann-Liouville fractional derivative and integral are exactly constant order fractional derivative and integral. It is well known that the Riemann-Liouville fractional integral has the law of exponents, i.e. I α 0+ I β 0+ (·) = I β 0+ I α 0+ (·) = I α+β 0+ (·), α > 0, β > 0. Based on the law of exponents, we can obtain some properties which are associated with fractional derivative and integral. For this reason, fractional order differential equations are transformed into equivalent integral equations. Thus some results of nonlinear functional analysis (for instance, some fixed point theorems) have been applied to considering the existence of solution of fractional order differential equations (see, e.g. [3,20,[32][33][34] and the references therein). However, the law of exponents doesn't hold for variable order fractional integral. For example, in [21][22][23][24], I g(t) 0+ I h(t) 0+ (·) I h(t) 0+ I g(t) 0+ (·), I g(t) 0+ I h(t) 0+ (·) I h(t)+g(t) where h(t) and g(t) are both general nonnegative functions. Then we will consider the properties which are interrelated with variable order fractional integral and variable order fractional derivative by given some examples.
ds, thus, By the same way, we get ds, As a result, we deduce I p(t) . Example 1.2 shows that the law of exponents of the variable order Riemann-Liouville fractional integral doesn't hold when the order is piecewise constant function defined in the same partition. ds, is different with the result of constant order fractional derivative and integral, that is, On the other hand, we have which illustrates that C D p(t) 0+ I p(t) 0+ is different with the result of constant order fractional derivative and integral, that is, On the other hand, for 2 ≤ t ≤ 6, we get Example 1.4 verifies the properties (1.4) and (1.5) of constant order fractional calculus is impossible for I p(t) 0+ C D p(t) 0+ f (t) and C D p(t) 0+ I p(t) 0+ f (t) when the order is a piecewise function. Hence, we can claim that variable order fractional integral defined by (1.3) has no law of exponents. Moreover, for general functions p(t) and f (t), the representations of C D p(t) are not clear. These obstacles make it difficult for us to transform variable order fractional differential equation into equivalent integral equation. As a result, it is almost impossible that some nonlinear functional analysis classical methods such as fixed point theorems are applied to prove the existence of solution of the corresponding integral equation. To the best of our knowledge, there are few works ( [8,23,24,27]) to deal with the existence of solutions to variable order fractional differential equations.
In [8], authors discussed the existence of solution for a generalized fractional differential equation with non-autonomous variable order operators t is the variable order Riemann-Liouville fractional differential operator defined as follows In [8] , authors claimed that, by [35], the initial value problem (1.7) is equivalent to the integral equation which is needed in the general form given in (1.8) and the scalar a is used for characterizing the initial period such that for a < t < c, the initial information is given, and for t < a we consider f (t) = 0. Given a, one can develop a closed analytical formula for Ψ [35]. However, in our opinion, there is no theoretical basis for this assertion. Because there are not contents related to variable order fractional integral and derivative in [35].
In [27], authors considered the existence results of solution to the initial value problem (1.
By means of Arzela-Ascoli theorem, authors obtained that the following sequence existed a subsequence still denoted by the sequence {x n } which uniformly converged to a continuous function x * . Set x(t) = t 0 x * (s)ds+u 0 , then authors obtained the problem (1.1) existed one solution x(t). But, we find that it has fatal errors in these analysis procedure, that is the result of uniformly bounded of sequence {x n }. This is easy to be overlooked. In our opinion the sequence {x n } is non-uniformly bounded. As a result, the existence result of solution to the initial value problem (1.1) is not obtained by Arzela-Ascoli theorem. On the other hand, it is almost impossible to transform the initial value problem (1.1) into equivalent integral equation.
Base on these facts, how to deal with the existence of solution of variable order fractional differential equations is a principal problem to be solved. In this paper, according to the classical definition of solution of integer order(or constant fractional order) differential equation, we propose a new definition of continuous approximate solution to the problem (1.1) under two kinds of variable order p(t, x(t)) and p(t).
The paper is organized as follows. In section 2, a new definition of approximate solution to the problem (1.1) for variable order p(t, x(t)) is proposed and we provide an example to demonstrate the definition. Section 3 is devoted to introduce another definition of approximate solution for p(t), then an example is given to illustrate the theoretical result.

Definition of approximate solution for p(t, x(t))
Throughout this section, we assume that We begin with definitions and characters of solution of integer order and constant fractional order.
Remark 2.1. According to the definition of solution x(t) of differential equation, it should be defined in the interval, in which the equation is satisfied. For instance, a function x(t) is called a solution of the following initial value problem Remark 2.2. Fractional operators are typical nonlocal operators, which can well describe the memory and global correlation of physical processes. Hence, for initial value problem of fractional order differential equations, its solution in given interval is affected by the state of solution in the preceding intervals. We start off by analyzing the problem (1.1) based on the facts above. Let We consider the initial value problem defined in the interval [0, T ] as following be a solution of the initial value problem (2.2)(by standard way, we know that the initial value problem (2.2) exists continuous solution in [0, T ] under some assumptions on f ). Since x 1 (t) is right continuous at point 0, then for arbitrary small ε > 0, there exists δ 01 > 0 such that And because p(t, x 1 (t)) is right continuous at point (0, x 1 (0)) = (0, u 0 ), then together with (2.3) and (2.1), for the above ε, there exists δ 0 > 0 such that If δ 0 < T , we take δ 0 T 1 and continue next procedure. Otherwise, we take T 1 = T and end this procedure.
We assume that δ 0 < T , and then let In order to consider the existence of solution to (1.1) in the interval [T 1 , T 2 ], we let By (2.6) and integration by parts, we denote Hence, we may consider the initial value problem defined in the interval [T 1 , T ] as following where x 1 (t) is the solution of the initial value problem (2.2) and ϕ x 1 is the function defined by (2.7). Let x 2 ∈ C 1 (T 1 , T ]∩C[T 1 , T ] be a solution of the problem (2.8) (by standard way, we know that the problem (2.8) exists continuous solution in [T 1 , T ] under some assumptions on f ). Since x 2 (t) is right continuous at point T 1 , then for the above ε, there exists δ 11 > 0 such that (2.9) And because p(t, x 2 (t)) is right continuous at point (T 1 , x 2 (T 1 )) = (T 1 , x 1 (T 1 )), then together with (2.9), for the above ε, there exists δ 1 > 0 such that (2.10) If T 1 + δ 1 < T , we take T 1 + δ 1 T 2 and continue next procedure. Otherwise, we take T 2 = T and end this procedure. Obviously, according to (2.10) and (2.5), we obtain We assume that T 1 + δ 1 < T , and then let We let (2.13) Thus, by (2.13) and integration by parts, we have We may consider the initial value problem in the interval [T 2 , T ] as following where x i (t) is the solution of the initial value problems (2.2) and (2.8) respectively, and φ x i (i = 1, 2) is the function defined by (2.14). Let x 3 ∈ C 1 (T 2 , T ]∩C[T 2 , T ] be a solution of the problem (2.15)(by standard way, we know that the problem (2.15) exists continuous solution in [T 2 , T ] under some assumptions on f ). Since x 3 (t) is right continuous at point T 2 , then for the above ε, there exists δ 21 > 0 such that (2.16) And because p(t, x 3 (t)) is right continuous at point (T 2 , x 3 (T 2 )) = (T 2 , x 2 (T 2 )), then together with (2.16), for the above ε, there exists δ 2 > 0 such that If T 2 + δ 2 < T , we take T 2 + δ 2 T 3 and continue next procedure. Otherwise, we take T 3 = T and end this procedure. Obviously, according to (2.17) and (2.12), it holds We assume T 2 + δ 2 < T , and then let We let (2.20) By (2.20) and integration by parts, we get We may consider the initial value problem in the interval [T 3 , T ] as following where x i (t) are solutions of the initial value problems (2.2), (2.8) and (2.15) respectively, and ω x i (i = 1, 2, 3) is the function defined by (2.21). Since [0, T ] is a finite interval, we continue this procedure and could finish it by finite steps. That is, there exists δ n * −2 > 0, δ n * −1 > 0 (n * ∈ N) such that T n * −2 + δ n * −2 T n * −1 < T , T n * −1 + δ n * −1 ≥ T T n * . Then we have intervals [0, T 1 ], [T 1 , T 2 ], [T 2 , T 3 ], · · · , [T n * −2 , T n * −1 ], [T n * −1 , T ], and solutions and From the arguments above, we obtain a function x * ∈ C[0, T ] defined by x n * (t), T n * −1 ≤ t ≤ T.
, · · · , n * ) respectively, then the function x ∈ C[0, T ] defined by is called an approximate solution of the initial value problem (1.1). Remark 2.3. If x 1 (t), x 2 (t), · · · , x n * (t) are both unique, then we say x(t) defined by (2.27) is unique approximate solution of the initial value problem (1.1).
Remark 2.5. From Definition 2.1, we notice that approximate solution x(t) of the initial value problem (1.1) in interval [T 2 , T 3 ] is x 3 (t) which is a solution of the initial value problem (2.15). Obviously, the state of x 3 (t) is affected by the state of x 1 (t) and x 2 (t). That is, the state of x(t) in interval [T 2 , T 3 ] is affected by the state of x(t) in interval [0, T 2 ]. Hence, Definition 2.1 is suitable and reasonable according to Remark 2.2.
Remark 2.6. In our previous analysis, we chose functions (2.6), (2.13), etc, so that we obtain the initial value problems (2.8), (2.15), etc. Such a choice must meet the following three reasons at the same time. The first reason is operability, for instance, choosing function (2.13) enable us to calculate functions φ x 1 (t), φ x 2 (t), and obtain the initial value problem (2.15) defined in [T 2 , T ]. The second reason is for fitting Remark 2.1 and Remark 2.5. If we take x (t) = x 1 (T 1 ) for 0 ≤ t ≤ T 1 (here x 1 (t) is the solution of the initial value problem (2.2)), we may easily calculate function ϕ , and then have the initial value problem (2.8) with such ϕ x 1 (t). However, we see that the state of solution of problem (2.8) is only affected by x 1 (T 1 ), but not affected by the state of x 1 (t), 0 ≤ t ≤ T 1 . The third reason is the rationality of the obtained equation. For instance, according to the definition the Caputo fractional derivative C D p 2 , and then obtain initial value problem (2.8) with such ϕ x 1 (t). However, we can't obtain the existence of solution of initial value problem (2.8) with this ϕ x 1 (t).
Example 2.1. According to Definition 2.1, we consider the approximate solution of the following initial value problem By the definition of the variable order Caputo fractional derivative, there is no way to obtain explicit expression of its solution, even hardly conventional methods to study the existence of its solution. Next, according to Definition 2.1, we try to seek its approximate solution.

Definition of approximate solution for p(t)
In this section, we study the initial value problem (1.1) for p(t, x(t)) ≡ p(t). Since p(t) is a function of one variable, we may propose another definition of approximate solution of the initial value problem (1.1) such that we can simplify the analysis process in the section 2.
The following result is crucial for us to propose another definition of approximate solution of the initial value problem (1.1) for the particular order p(t).
Similar to the analysis in the section 2, we consider approximate solution of the initial value problem (1.1) in the following sense: if p(t) and α(t) satisfy (3.2), then solution x(t) of the following initial value problem is called the approximate solution of the initial value problem (1.1). We start off by analyzing the problem (3.9), and then propose a new definition of continuous approximate solutions to the initial value problem (1.1) with p(t, x(t)) = p(t).
For the initial value problem (3.9) in the interval [0, T 1 ], by (3.1), we have the initial value problem (3.10) By similar analysis in section 2, we may consider the initial value problem defined in the interval [T 1 , T 2 ] as following (3.11) where x 1 (t) is the solution of the initial value problem (3.10) and in which x (s) = s 0 x 1 (τ)dτ. Using the same method, we may consider the initial value problem defined in the interval [T 2 , T 3 ] as following (3.12) where x 1 (t) is the solution of the initial value problem (3.10), x 2 (t) is the solution of the initial value problem (3.11) and φ x i (t) is the function defined by in which x (s) = s T i−1 x i (τ)dτ, i = 1, 2, T 0 = 0. Similarly, we may consider the initial value problem defined in the interval [T i−1 , T i ] as following where x 1 (t) is the solution of the initial value problem (3.10), x 2 (t) is the solution of the initial value problem (3.11), x 3 (t) is the solution of the initial value problem (3.12), x i (t) is the solution of the initial value problem (3.13) and ψ x j is the function defined by in which x (s) = s T j−1 x j (τ)dτ, j = 1, 2, · · · , i − 1, i = 4, · · · , n * , T 0 = 0, T n * = T . Based on the arguments above, we propose the definition of solution to the initial value problem (3.9), which is crucial in our work.
The following is the definition of the approximate solution of the initial value problem (1.1) with p(t, x(t)) ≡ p(t).
Definition 3.2. We call the solution of initial value problem (3.9) defined by Definition 3.1 is an (unique) approximate solution of initial value problem (1.1) with p(t, x(t)) = p(t) if α(t) is defined by Lemma 3.1.

Remark 3.2.
For the initial value problem (1.1) with p(t, x(t)) ≡ p(t), its approximate solution defined by Definition 3.2 is consistent with by Definition 2.1.
Example 3.1. We consider the following initial value problem for linear equation By the definition of the variable order Caputo fractional derivative, there is no way to obtain explicit expression of its solution, even hardly conventional methods to study the existence of its solution. Next, we try to seek its approximate solution in the sense of Definition 3.2.