Existence of local solutions for differential equations with arbitrary fractional order

In this paper, we establish sufﬁcient conditions for the existence of local solutions for a class of Cauchy type problems with arbitrary fractional order. The results are established by the application of the contraction mapping principle and Schaefer’s ﬁxed point theorem. An example is provided to illustrate the applicability of the results.

fractional orders are investigated by many authors (see [2,22]). The general theory of Cauchy fractional differential equations is deeply introduced in the monograph [1] and in the survey [28]. In fact, the equivalent Volterra integral equation to Cauchy problem for nonlinear fractional differential equations introduced in the cited articles is essential to prove the existence of such systems. However, the generalization idea of existence problems to arbitrary fractional order with arbitrary inner point as initial condition has not been investigated by the researchers. Motivated by these ideas, we study in this paper the existence of a solution to the Cauchy problem C D α t 0 x(t) = f (t, x(t)), t ∈ [t 0 , θ) ∪ (θ, T ] x (k) (θ ) = x k , k = 0, 1, 2, . . . , n − 1, θ ∈ J, (1.1) at any inner point θ of a finite interval J = [t 0 , T ] involving the Caputo fractional derivative C D α t 0 , where α ∈ (n − 1, n], n ∈ N, and f is a given continuous function. The inner points of the interval involved in the problem can be used as impulses in a physical approach or sometimes nonlocal boundary condition, hence the problem may be considered as a case of nonlocal fractional differential model. However, the used technique of obtaining the solution of the problem is new compared with any previous works ([1]: Section 3.4.2) and the results on arbitrary fractional ordered differential equations generalize the existing problems.

Preliminaries
We introduce in this section some basic definitions and properties of fractional calculus (see [1]) which will be used in this paper.
Next, we introduce the Caputo fractional derivative.

Definition 2.2
The Caputo fractional derivative of x is defined as: In what follows, we assume that f and x are continuous functions, such that f and C D α t 0 x are fractional integrable of any order less than or equal α.
The compositions between the Caputo fractional derivative and fractional integrals are given by the following lemma.
The following result will be used in the proof of the main theorem in the next section.

Lemma 2.4
Let (u n ) be a sequence of real numbers and n, k ∈ N, such that 0 ≤ k ≤ n − 1. If v is a positive real number, then The left-hand side of Eq. (2.1) can be rearranged as: Hence, by Binomial expansion, the inner sum can be reduced to for all m ≥ 1, by which the result is obtained.

Existence problems for linear case
Consider the linear fractional differential equation (3.1) We introduce next the basic idea in this article, namely, the solution of ( 3.1) as an integral form.
which can be integrated n times to have that can be reduced to Using binomial expansion, we have In accordance with (3.3), Eq. (3.2) follows. Now, let n − 1 < α < n, Lemma 2.3, implies that In accordance with the given conditions in (3.1), Eq. (3.5) can be rewritten in the following array form (assuming 0 which can be algebraically solved to obtain, for r = 1, 2, . . . , n, Alternatively, for m = 0, 1, 2 · · · , n − 1, it can be rewritten as: In accordance with (3.6) and (3.4), we deduce that The terms of this double summation can be rearranged to have by which the solution of (3.1) is Next, let t 0 < θ ≤ T . Then binomial expansion can be applied to obtain On the other hand, applying the operator C D α t 0 , n − 1 < α ≤ n to Eq. (3.2), and using Lemma 2.3, we get Eq. (3.1) which completes the proof.
The following is a direct result of Theorem 3.1.

Corollary 3.2 Let c be any real number. Then the fractional differential system
is equivalent to In particular, if c = 0, then (3.9) is equivalent to

Existence problems for nonlinear cases
We investigate in this section the existence of a local solution for the fractional systems (1.1) by applying Banach's and Schaefer's fixed point theorems.
be a fractional integrable function of order α > 0 that satisfies the following hypothesis: (H1) There exists a positive constant A such that for any t ∈ J h and x, y ∈ Y h . Moreover, let B = sup t∈J f (t, 0) and C = max{A, B}.
In accordance with Theorem 3.1, the fractional nonlinear system is equivalent to the integral equation Accordingly, we define the operator on Y h as follows: The next hypothesis is essential to state and prove the first main result in this section.
(H2) Let θ and r be positive real numbers such that It is obviously by (H2) that maps into itself. Next, let x, y ∈ . Then since γ < 1, then is a contraction mapping on . Hence, has a fixed point which is the unique solution to (4.1).
Next, we show the existence of a local solution for the Cauchy problem be a fractional integrable function of order α > 0. Hence, by Theorem 3.1, the system (??) is equivalent to the Fredholm-Volterra integral equation We need to modify the hypothesis (H2) as the following: (H3) Let β and r be positive real numbers such that The proof of the next result is similar to that one of Theorem 4.1, hence it is omitted.

Corollary 4.2
Assume that (H1) and (H3) are satisfied. Then, there exists a unique solution for the fractional system (4.4) in Y h 1 .
be the Banach space of all continuous real valued functions J h 2 , such that C D α t 0 y exists. Next result concerns with the existence of a local solution for the Cauchy problem which is equivalent to Volterra integral equation (see Eq. (3.8)) The hypothesis (H2) will be replaced by the following: (H4) Let η and r be positive real numbers such that The last result is devoted to solve the existence problem of the fractional system (4.1) which has equivalent integral Eq. (4.2). We define the operator Theorem 4.4 [3] If is a closed bounded convex subset of a Banach space X and : → is completely continuous, then has a fixed point in .
Theorem 4.5 [3] Let X be a Banach space. Assume that : X → X is completely continuous operator and the set V = {x ∈ X : x = μ x, 0 < μ < 1} is bounded. Then, has a fixed point in X.
The last result can be introduced now. We give an example to explain the applicability of the above results.
Example 4.7 Consider the following nonlinear fractional differential equation 10 0