Fractional Differential Equations with Fractional Impulsive and Nonseparated Boundary Conditions

and Applied Analysis 3 if and only ifx is a solution of the impulsive fractional boundary value problem c D α x (t) = y (t) , t ∈ J 󸀠 , 1 < α < 2, Δx (t k ) = I k (x (t − k )) , Δ ( c D γ x (t k )) = I ∗ k (x (t − k )) , k = 1, 2, . . . , m, a 1 x (0) + b 1 x (T) = c 1 , a 2 x 󸀠 (0) + b 2 x 󸀠 (T) = c 2 . (12) Proof. For 1 < α < 2, by Lemma 3, we know that a general solution of the equation cDαx(t) = y(t) on each interval J k (k = 0, 1, 2, . . . , m) is given by x (t) = I α y (t) + d k + e k t


Introduction
The subject of fractional differential equations has recently evolved as an interesting and popular field of research.In fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes.More and more researchers have found that fractional differential equations play important roles in many research areas, such as physics, chemical technology, population dynamics, biotechnology, and economics (see [1][2][3][4]).Hence, there are many literatures devoted to solving fractional differential equations through theoretical analysis or numerical methods (e.g., see [5][6][7][8][9]).
The theory of impulsive differential equations of integer order has found its extensive applications in realistic mathematical modeling of a wide variety of practical situations and has emerged as an important area of investigation in recent years.For the general theory and applications of impulsive differential equations, see [10][11][12][13][14][15] and so forth.However, impulsive differential equations of fractional order have not been much studied and many aspects of these equations are yet to be explored.For some recent work on impulsive fractional differential equations, we can refer to [16][17][18][19][20][21][22][23][24] and the references therein.
We note that, as pointed out in the papers [22][23][24], the concept of piecewise continuous solutions used in some already published works to handle the impulsive fractional differential equations is not appropriate (see counterexamples given in Lemma 3.1 of [22], Section 1 in [23], and Section 3 in [24]).The papers on this topic cited above except [22] all deal with the Caputo derivative and the impulsive conditions only involve integer order derivatives.Here we study the fractional differential equations with fractional impulsive conditions and nonseparated boundary conditions [25][26][27].
The rest of the paper is organized as follows.In Section 2 we introduce some notations and definitions needed in the following sections and give the appropriate formula of solutions for our problems.In Section 3 we present the existence results for the problems (1), ( 2) and ( 1), (3).Two examples are presented in Section 4 to illustrate the results.Concluding remarks are provided in Section 5. Definition 1 (see [4]).The Riemann-Liouville fractional integral of order  for a function  is defined as

Let us set 𝐽
provided the integral exists.
Definition 2 (see [4]).For a continuous function , the Caputo derivative of order  is defined as where [] denotes the integer part of the real number .

Lemma 6. Let 𝑦 ∈ 𝑃𝐶(𝐽, R). A function 𝑥 is a solution of the fractional integral equation
where with  and  defined by (11) and if and only if  is a solution of the impulsive fractional boundary value problem Proof.The proof is similar to that of Lemma 5. Let the notations be given as in the proof of Lemma 5. Applying the boundary conditions in (35), from the relations ( 13), (15), and (17), we have After a direct computation, we obtain that with  and  defined by (11) and The remaining part of proof is the same as that of Lemma 5.
Remark 7. We note that the solution expression (32) of the problem (35) does not depend on the parameter  2 appearing in the boundary conditions.Thus, by Lemma 6, we conclude that the parameter  2 is of arbitrary nature of the problem (35).
Remark 8. Our approach for the construction of solutions for impulsive fractional differential equations is general, which provides an effective way to deal with such problems.Taking the problem (1) for example, on each interval   , we use From the counterexamples given in Lemma 3.1 of [22], Section 1 in [23], and Section 3 in [24], we know that if the case (II) was chosen to construct solutions, the solution formula obtained in the fractional integral equation form is not equivalent to the original impulsive fractional differential equation.This is the main difference between impulsive fractional differential equations and impulsive ordinary differential equations.Observe that fractional calculus has memory property.
Theorem 9 (nonlinear alternative of Leray-Schauder type [28]).Let  be a Banach space,  a nonempty convex subset of , and  a nonempty open subset of  with 0 ∈ .Suppose that  :  →  is a continuous and compact map.Then either (a)  has a fixed point in  or (b) there exist a  ∈  (the boundary of ) and  ∈ (0, 1) with  = ().
Theorem 10 (Schaefer fixed point theorem [29]).Let  be a normed space and  a continuous mapping of  into  which is compact on each bounded subset  of .Then either (I) the equation  =  has a solution for  = 1 or (II) the set of all such solutions , for 0 <  < 1, is unbounded.

Main Results
This section deals with the existence and uniqueness of solutions for the problem (1), ( 2) and the problem (1), (3). where Here   0 ,   0 ,   , and   mean that  0 ,  0 , , and  defined in Lemma 5 are related to  ∈ (, R).It is obvious that  is well defined because of the continuity of ,   , and  *  and that the problem (1), (2) has solutions if and only if the operator equation  =  has fixed points.
For the sake of convenience, we set where Then the impulsive fractional boundary value problem (1), ( 2) has at least one solution on .
Proof.We will show that the operator  defined by (39) satisfies the assumptions of the nonlinear alternative of Leray-Schauder type.From Lemma 11, the operator  : (, R) → (, R) is continuous and completely continuous.
Let  ∈ (, R) be such that () = ()() for some  ∈ (0, 1).Then using the computations in the proof that  maps bounded sets into bounded sets in Lemma 11, we have Consequently, we have Then, in view of condition (46), there exists  > 0 such that ‖‖ ̸ = .Let us set The operator  :  → (, R) is continuous and compact.
From the choice of the set , there is no  ∈  such that  =  for some  ∈ (0, 1).
Therefore, by (54), the operator  is a contraction mapping on (, R).Then it follows from Banach's fixed point theorem that the problems (1), ( 2) has a unique solution on .This completes the proof.
Next we will state some existence results for the problems (1), (3) without proofs since these are similar to the ones obtained for the problems (1), (2) above.
Theorem 15.Let (H1) hold and there exists a constant  > 0 such that where where  and ℎ : [0, ] × R → R are given continuous functions.

Examples
In this section, we give two simple examples to show the applicability of our results.(63)