Solvability of impulsive (n,n-p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n,n-p)$$\end{document} boundary value problems for higher order fractional differential equations

We present a new general method for converting an impulsive fractional differential equation to an equivalent integral equation. Using this method and employing a fixed point theorem in Banach space, we establish existence results of solutions for a boundary value problem of impulsive singular higher order fractional differential equation. An example is presented to illustrate the efficiency of the results obtained. A conclusion section is given at the end of the paper.


Introduction
Fractional differential equation is a generalization of ordinary differential equation to arbitrary non-integer orders. Fractional differential equations, therefore, find numerous applications in different branches of physics, chemistry and biological sciences such as visco-elasticity, feed back amplifiers, electrical circuits, electro-analytical chemistry, fractional multipoles and neuron modelling. The reader may refer to the books and monographs [1][2][3] for fractional calculus and developments on fractional differential and fractional integro-differential equations with applications.
On the other hand, the theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such characteristics arise naturally and often; for example, phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics. For an introduction of the basic theory of impulsive differential equation, we refer the reader to [4].
Solvability of boundary value problems for higher order ordinary differential equations were investigated by many authors. For example, in [5][6][7][8][9][10][11][12][13][14][15][16], the following ðn; n À kÞ type problems were studied: In [17,18], the following more general boundary value problems were studied: where k 2 IN nÀ1 1 ; q 2 IN k 0 . In [6,19,20], authors studied existence of solutions of the following problems: On the one hand, it is interesting to generalize results on boundary value problems for higher order ordinary differential equations; in mentioned papers, in [21], authors studied existence of solutions of the following boundary value problem for higher order fractional differential equation In [22], solutions of the following problem were presented: On the other hand, higher order fractional differential equations have applications such as the fractional order elastic beam equations see [23], the fractional order viscoelastic material model see [24], the fractional viscoelastic model see [25][26][27] and so on. There has been no papers concerned with the solvability of boundary value problems for higher order impulsive fractional differential equations since it is difficult to convert an impulsive fractional differential equation to an equivalent integral equation.
To fill this gap, in this paper, we discuss the following two boundary value problems for nonlinear impulsive singular fractional differential equation where (a) n À 1\a\n, n is a positive integer, c D Ã 0 þ is the Caputo fractional derivative of orders Ã with starting point 0, A function x : ð0; 1 ! IR is said to be a solution of (1.6) or and x satisfies all equations in (1.6) or (1.7), respectively. In [28], a general method for converting an impulsive fractional differential equation to an equivalent integral equation was presented. We present a new method (Lemma 2.2) for converting BVP (1.6) to an equivalent integral equation in this paper. We shall construct a weighted Banach space and apply the Leray-Schauder nonlinear alternative to obtain the existence of at least one solution of (1.6) and (1.7), respectively. Our results are new and naturally complement the literature on fractional differential equations.
The paper is outlined as follows. ''Preliminaries'' contains some preliminary results. Main results are presented in ''Main results''. In ''Examples'', we give an example to illustrate the efficiency of the results obtained. A conclusion section is given at the end of the paper.

Preliminaries
For the convenience of the readers, we shall state the necessary definitions from fractional calculus theory.
Step 2. We prove that x satisfies ( Then for t 2 ðt iþ1 ; t iþ2 we have CðaÀ1Þ hðvÞdv ðnÞ ds Cðn À aÞ CðaÀ1Þ hðvÞdv ðnÞ ds Cðn À aÞ UðtÞ þ Similarly to Step 1 we can get that UðtÞ: Then there exist constants jxðtÞj: Proof The proof is standard and omitted. h Proof The proof is standard and omitted.
For x 2 X, denote f x ðtÞ ¼ f ðt; xðtÞÞ and I jx ðt s Þ ¼ I j ðt s ; xðt s ÞÞ. Denote Then jMj 6 ¼ 0 and jNj ¼ 1. One has for a determinant ja i;j j ðnÀkÞÂðnÀkÞ that where A i;nÀj is the algebraic cofactor of a i;nÀj : ð2:5Þ Suppose that ja i;j j 1. It is easy to show that where M ij and N ij are the algebraic cofactors of m ij and n ij , respectively. M Ã and N Ã are the adjoint matrix of M and N, respectively. From (2.5) and (2.6), we know that jM ij j Cðn À kÞ and jN ij j CðnÞ. h if and only if Suppose that x 2 X and x is a solution of (2.7). By Lemma 2.2, we know that there exist constants c vj 2 IR such that ðt À sÞ aÀ1 CðaÞ f u ðsÞds; t 2 ðt s ; t sþ1 ; s 2 IN 0 : ð2:12Þ By Definition 2, we have Hence,  From above discussion, we know that x 2 X and x satisfies (2.7) if (2.8) holds. The proof is completed. h Remark 2.1 It is easy to see from Lemma 2.6 that x 2 X is a solution of (2.10) if and only if x satisfies that there exists constants d vs 2 IR such that Math Sci (2016) 10:71-81 77 In [28], authors have proved this result but our proof of Lemma 2.6 is different from that in [28]. Now, we define the operator T 1 on X by ð2:16Þ Remark 2.2 By Lemma 2.5, we know that T 1 : X ! X is well defined and x 2 X is a solution of system (1.6) if and only if x 2 X is a fixed point of the operator T 1 .

Lemma 2.6
The operator T 1 : X ! X is completely continuous.
Proof The proof is standard and is omitted, one may see [21]. Proof Similarly to the proof of Lemma 2.5, we get Lemma 2.7. Now, we define the operator T 2 on X by Lemma 2.8 The operator T 2 : X ! X is completely continuous.
Proof The proof is standard and is omitted, one may see [21]. h

Main results
In this section, we are ready to present the main theorems. We need the following assumptions: a i jxj r i t p ð1 À tÞ q ; t 2 ð0; 1Þ; x 2 IR; Cðn À kÞ Cði þ 1ÞjjMjj Bða À ðn þ l À k À jÞ þ q; p þ 1Þ Cða À ðn þ l À k À jÞÞ þ a 0 Bða þ q; p þ 1Þ CðaÞ ; Cðn À kÞ Cði þ 1ÞjjMjj Bða À ðn þ l À k À jÞ þ q; p þ 1Þ Cða À ðn þ l À k À jÞÞ a u M u r r u r: ð3:4Þ Then T 1 X r X r . So T 1 has a fixed point in X r . Then BVP (1.6) has a solution. We consider the following three cases: M u r ru r ¼ 0, we can choose r [ 0 sufficiently small such that (3.4) holds. Then T 1 X r X r . So T 1 has a fixed point in X r . Then BVP(1.6) has a solution.
M u \1, we can choose r [ 0 sufficiently small such that (3.4) holds. Then T 1 X r X r . So T 1 has a fixed point in X r . Then BVP (1.6) has a solution.
1=r u 0 . Then we have by the inequality in (iii) that M u r r u r: Then T 1 X r X r . So T 1 has a fixed point in X r .

Examples
To illustrate the usefulness of our main result, we present an example that Theorem 3.1 can readily apply. where a i ; A i ði ¼ 0; 1Þ are nonnegative constants.

Conclusion
In this paper, we discuss the solvability of two classes of boundary value problems or higher order fractional differential equations involving the Caputo fractional derivatives. Using some fixed point theorems in Banach spaces, we establish sufficient conditions for the existence of solutions of these kinds of problems.
In recent years, there have been several kinds of fractional derivatives proposed such as the Riemann-Liouville fractional derivative, the Hadamard fractional derivative, etc., see [29,30]. Hence, it is interesting to study the existence and uniqueness of solutions of boundary value problems for other kinds of fractional differential equations. It is also interesting to find the similar properties and the difference properties between these different kinds of fractional differential equations.
The fixed point theorems in Banach spaces [31] are main tools for investigating the solvability of boundary value problems for fractional differential equations. It needs to find other methods for finding solutions for these kinds of problems.
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