Green’s function for Sturm-Liouville-type boundary value problems of fractional order impulsive differential equations and its application

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth, and involves derivatives of fractional order. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional differential equations in comparison with classical integer-order models. An excellent account in the study of fractional differential equations can be found in [1–5]. For the basic theory and recent development of the subject, we refer to a text by Lakshmikantham et al. [6]. For more details and examples, see [7–23] and the references therein.

Integer-order impulsive differential equations have become important in recent years as mathematical models of phenomena in both physical and social sciences. There has been a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments; see, for instance, [24–26]. Recently, the boundary value problems of impulsive differential equations of integer order have been studied extensively in the literature (see [27–36]).

On the other hand, impulsive differential equations of fractional order play an important role in theory and applications, see [37–46] and the references therein. However, as pointed out in [38, 39], the fractional impulsive differential equations have not been addressed so extensively and many aspects of these problems are yet to be explored. For example, the theory using Green’s function to express the solution of fractional impulsive differential equations has not been investigated till now. Now, in this paper, we shall study the expression of the solution of fractional impulsive differential equations by using Green’s function.

where ${}^c\mathbf{D}_{0+}^q$ is the Caputo fractional derivative, $J=[0,1]$, $\omega (t):J\to R+$ is a continuous function, $f:J\times R\times R\to R$ is a continuous function, ${\mathcal{I}}_k,{\mathcal{J}}_k:R\to R$ are continuous functions, ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2\in R$ and ${\alpha }_1{\alpha }_2+{\alpha }_1{\beta }_2+{\alpha }_2{\beta }_1\ne 0$. $\mathrm{\Delta }x{|}_{t=t_k}=x(t_k^{+})-x(t_k^{-})$ with $x(t_k^{+})={lim}_{h\to 0^{+}}x(t_k+h)$, $x(t_k^{-})={lim}_{h\to 0^{-}}x(t_k+h)$, $k=1,2,\dots ,n$, $0=t_0<t_1<\cdots <t_n<t_{n+1}=1$. $\mathrm{\Delta }{x}^{\prime }{|}_{t=t_k}$ has a similar meaning for $x^{\prime }(t)$.

Some special cases of (1.1) have been investigated. For example, Bai and Lü [13] considered problem (1.1) with ${\mathcal{I}}_k\equiv 0$ and ${\mathcal{J}}_k\equiv 0$. By using the fixed point theorem in cones, they proved some existence and multiplicity results of positive solutions of problem (1.1).

where ${\mathcal{I}}_k,{\mathcal{L}}_k\in C[0,1]$, $k=1,2,\dots ,n$.

Thus, it is an interesting problem, and so it is worthwhile to study. We will give the answers in the following sections.

The organization of this paper is as follows. In Section 2, we present the expression and properties of Green’s function associated with problem (1.1). In Section 3, we give some preliminaries about the operator and the fixed point theorem. In Section 4, we get some existence results for problem (1.1) by means of some standard fixed point theorems. The final section of the paper contains two examples to illustrate our main results.

where ${}^c\mathbf{D}_{0+}^q$ is the Caputo fractional derivative, $\sigma (t)\in C[0,1]$, ${\mathcal{I}}_k,{\mathcal{J}}_k:R\to R$ are continuous functions, ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2\in R$ and ${\alpha }_1{\alpha }_2+{\alpha }_1{\beta }_2+{\alpha }_2{\beta }_1\ne 0$. $\mathrm{\Delta }x{|}_{t=t_k}=x(t_k^{+})-x(t_k^{-})$ with $x(t_k^{+})={lim}_{h\to 0^{+}}x(t_k+h)$, $x(t_k^{-})={lim}_{h\to 0^{-}}x(t_k+h)$, $k=1,2,\dots ,n$, $0=t_0<t_1<\cdots <t_n<t_{n+1}=1$. $\mathrm{\Delta }{x}^{\prime }{|}_{t=t_k}$ has a similar meaning for $x^{\prime }(t)$.

where η is defined in (2.4).

Substituting (2.7), (2.8) into (2.6), we obtain (2.2). This completes the proof. □

Remark 2.2 The expression of the solution of problem (2.1) is simpler than that of [37–46].

From (2.3), we can prove the following results.

From the proof of Theorem 2.1 we have the following results.

Proposition 2.3 The solution of fractional impulsive differential equations can be expressed by Green’s function, and it is not Green’s function of the corresponding fractional differential equations, but Green’s function of the corresponding integer order differential equations.

In this section, we give some preliminaries for discussing the solvability of problem (1.1) as follows.

Definition 3.1 A function $x\in P{C}^1[J,R]\cap C^2[J^{\prime },R]$ with its Caputo derivative of order q existing on J is a solution of problem (1.1) if it satisfies (1.1).

We give the following hypotheses:

(H₁) $\omega :J\to [0,+\mathrm{\infty })$ is a continuous function, and there exists $t_0\in J$ such that $\omega (t_0)>0$;

(H₂) $f:J\times R\times R\to R$ is a continuous function;

(H₃) ${\mathcal{I}}_k,{\mathcal{J}}_k:R\to R$ are continuous functions.

It follows from Theorem 2.1 that:

Using Lemma 3.1, problem (1.1) reduces to a fixed point problem $x=Tx$, where T is given by (3.2). Thus problem (1.1) has a solution if and only if the operator T has a fixed point.

From (3.2) and Lemma 3.1, it is easy to obtain the following result.

Lemma 3.2 Assume that (H₁)-(H₃) hold. Then $T:P{C}^1[J,R]\to P{C}^1[J,R]$ is completely continuous.

Proof Note that the continuity of f, ω, ${\mathcal{I}}_k$ and ${\mathcal{J}}_k$ together with $G_1(t,s)$ and $G_{1s}^{\prime }(t,s)$ ensures the continuity of T.

It follows from (3.3), (3.5) and (3.6) that T is equicontinuous on all subintervals $(t_k,t_{k+1}]$, $k=1,2,\dots ,n$. Thus, by the Arzela-Ascoli theorem, the operator $T:P{C}^1[J,R]\to P{C}^1[J,R]$ is completely continuous. □

To prove our main results, we also need the following two lemmas.

Lemma 3.3 (See [47, 48]) (Schauder fixed point theorem)

Let D be a nonempty, closed, bounded, convex subset of a B-space X, and suppose that $T:D\to D$ is a completely continuous operator. Then T has a fixed point $x\in D$.

Lemma 3.4 (See [47, 48]) (Leray-Schauder fixed point theorem)