Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 41, 1-11; http://www.math.u-szeged.hu/ejqtde/

Solvability for second order nonlinear impulsive boundary value problems

Dandan Yang^{∗}, Gang Li School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

Abstract. In this paper, we are concerned with the solvability for a class of second order nonlinear impulsive boundary value problem. New criteria are established based on Schaefer’s fixed-point theorem. An example is presented to illustrate our main result. Our results essentially extend and complement some previous known results.

Key words: Impulsive boundary value problems; Solvability; Schaefer’s fixed-point theorem; Periodic and anti-periodic boundary conditions.

AMS(2000) Mathematics Subject classifications: 34B15,34B37.

1 Introduction

Impulsive differential equations play a very important role in understanding mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology, economics and so on, see [1,2,8,10,17]. About wide applications of the theory of impulsive differential equations to different areas, we refer the readers to monographs [5,7,18,19] and the references therein. Some resent works on periodic and anti-periodic nonlinear impulsive boundary value problems can be found in [6,12,20,21]. Recently, J. Chen, C. Tisdell, and R. Yuan in [4] studied the following first order impulsive nonlinear periodic boundary value problem

\{ \begin{matrix}−u′(t) = f(t, u), t ∈ [0, T], t ≠ t_{1}, \\ u(t_{1}^{+}) − u(t_{1}^{−}) = I(u(t_{1})), \\ u(0) = u(T), \end{matrix} (1.1)

where T > 0 and f : [0, T] × R^{n} → R^{n} is continuous on (t, u) ∈ [0, T] \ {t_{1}} × R^{n}. The authors studied the existence of solutions to the problem (1.1) in view of differential inequalities and Schaefer’s fixed-point theorem. Their results extend those of [9,14] in the sense that they allow superlinear growth in nonlinearity ∥f(t, p)∥ in ∥p∥.

∗ Corresponding author, E-mail address: ydd423@sohu.com

The project supported by the National Natural Science Foundation of China (10771212).

About further investigation, in 2007, Bai and Yang in [3] presented the existence results for the following second-order impulsive periodic boundary value problems

\{ \begin{matrix}u′′(t) = f(t, u(t), u′(t)), t ∈ [0, T] \ \{t_{1}\}, \\ u(t_{1}^{+}) = u(t_{1}^{−}) + I(u(t_{1}))), \\ u′(t_{1}^{+}) = u′(t_{1}^{−}) + J(u(t_{1})), \\ u(0) = u(T), u′(0) = u′(T). \end{matrix} (1.2)

Inspired by [3,4], in this paper, we investigate the following second order impulsive nonlinear boundary value problems

\{ \begin{matrix}−u′′(t) + p(t)u′(t) + q(t)u(t) = f(t, u(t), u′(t)), t ∈ [0, T] \ \{t_{1}, t_{2}, ..., t_{k}\}, \\ △u(t_{i}) = a_{i}, i = 1, 2, ..., k, \\ △u′(t_{i}) = b_{i}, i = 1, 2, ..., k, \\ u(0) = βu(T), u′(0) = γu′(T), \end{matrix} (1.3)

where f : [0, T]×R^{n}×R^{n} → R^{n} is continuous on (t, u, v) ∈ [0, T]\{t_{i}}×R^{n}×R^{n}, i = 1, 2, ..., k, p, q ∈ C([0, T]), a_{i}, b_{i} are constants for i = 1, 2, ..., k, β, γ are constants satisfying |β| ≥ 1, |γ| ≥ 1. Notice that our results not only extend some known results from the nonimpulsive case [16] to the impulsive case, or from single impulse [3] to multiple impulses, but also extend those of [11] in the sense that we allow superlinear growth of ∥f(t, u, v)∥ in ∥u∥ and ∥v∥. Furthermore, the impulsive boundary-value problem reduces to a periodic boundary value problem [15,22] for β = γ = 1, p = q ≡ 0, and anti-periodic boundary value problem [21] for β = γ = −1, p = q ≡ 0. Hence, the problem (1.3) can be considered as a generalization of periodic and anti-periodic boundary value problems.

We shall establish the existence of solutions for impulsive BVP (1.3) by means of well-known Schaefer’s fixed-point theorem. The rest of paper is organized as follows. In section 2, we present some definitions and lemmas, and the fixed point theorem which is key to our proof. In section 3, the new existence theorem of (1.3) is stated. An example is given in the last section to demonstrate the application of our main result.

2 Preliminaries

First, we introduce and denote the Banach space P C([0, T], R^{n}) by P C([0, T], R^{n}) = {u : [0, T] → R^{n}|u ∈ C([0, T] \ {t_{i}}, R^{n}),

u is left continuous at t = t_{i}, the right − hand limit u(t_{i}^{+}) exists}

with the norm

∥u∥_{P C} = sup_{t∈[0,T]} ∥u(t)∥,

where ∥ · ∥ is the usual Euclidean norm.

We denote the Banach space P C^{1}([0, T]; R^{n}) by

P C^{1}([0, T], R^{n}) = {u ∈ C^{1}([0, T] \ {t_{i}}, R^{n}),

u is left continuous at t ≠ t_{i}, u′(t_{i}^{+}), u′(t_{i}^{−}) exist}

with the norm

∥u∥_{P C^{1}} = max{∥u∥_{P C}, ∥u′∥_{P C}}.

The following fixed-point theorem due to Schaefer, is essential in the proof of our main result. Lemma 2.1. Let E be a normed linear space and Φ : E → E be a compact operator. Suppose that the set

S = {x ∈ E|x = λΦ(x), for some λ ∈ (0, 1)}

is bounded. Then Φ has a fixed point in E. Lemma 2.2. Assume p ∈ C[0, T], q(t) ∈ C([0, T], (−∞, 0]). Let ϕ_{1}, ϕ_{2} be the solutions of

\{ \begin{matrix}ϕ′′_{1}(t) + p(t)ϕ′_{1}(t) + q(t)ϕ_{1}(t) = 0, \\ ϕ_{1}(0) = 0, ϕ_{1}(T) = T, \end{matrix} (2.1)

and

\{ \begin{matrix}ϕ′′_{2}(t) + p(t)ϕ′_{2}(t) + q(t)ϕ_{2}(t) = 0, \\ ϕ_{2}(0) = T, ϕ_{2}(T) = 0. \end{matrix} (2.2)

Then (i) ϕ_{1} is strictly increasing on [0,T]; (ii) ϕ_{2} is strictly decreasing on [0,T].

Proof. The proof is similar to that of Lemma 2.1 in [13], so we omit it here. Remark 2.3. It follows from Lemma 2.2 that

ϕ_{1}(t)ϕ_{2}(s) ≤ ϕ_{1}(s)ϕ_{2}(s) ≤ ϕ_{1}(s)ϕ_{2}(t), 0 ≤ t ≤ s ≤ T. (2.3)

In order to prove our main results, we present a useful lemma in this section. Consider the following impulsive boundary value problem

\{ \begin{matrix}−u′′(t) + p(t)u′(t) + q(t)u(t) = h(t), t ≠ \{t_{1}, t_{2}, ..., t_{k}\}, \\ △u(t_{i}) = a_{i}, i = 1, 2, ..., k, \\ △u′(t_{i}) = b_{i}, i = 1, 2, ..., k, \\ u(0) = βu(T), u′(0) = γu′(T), \end{matrix} (2.4)

where a_{i}, b_{i} are constants for i = 1, 2, ..., k, h ∈ P C[0, T]. Lemma 2.4. For h(t) ∈ P C[0, T], the problem (2.4) has the unique solution

u(t) = Mϕ_{1}(t) + Nϕ_{2}(t) + ϕ_{1}(t) ∫_{t}^{T} \frac{1}{ρ}h(s)ϕ_{2}(s)l(s)ds

+ϕ_{2}(t) ∫_{0}^{t} \frac{1}{ρ}h(s)ϕ_{1}(s)l(s)ds + ∑_{t_{i}<t} b_{i}(t − t_{i}) + ∑_{t_{i}<t} a_{i}, (2.5)

where ϕ_{1}, ϕ_{2} satisfies (2.1), (2.2) respectively, and

l(t) = exp(∫_{0}^{t} p(s)ds), ρ := ϕ1^{′}(0), (2.6)

M = \frac{γ ∫_{0}^{T} \frac{1}{ρ}h(s)ϕ_{1}(s)l(s)dsϕ2^{′}(T) − ∫_{0}^{T} \frac{1}{ρ}h(s)ϕ_{2}(s)l(s)dsϕ1^{′}(0) + γ ∑_{t_{i}<T} b_{i}}{ϕ1^{′}(0) − γϕ1^{′}(T) + β(ϕ2^{′}(0) − γϕ2^{′}(T))}

−\frac{β ( ∑_{t_{i}<T} b_{i}(T − t_{i}) + ∑_{t_{i}<T} a_{i}) (ϕ2^{′}(0) − γϕ2^{′}(T))}{T [ϕ1^{′}(0) − γϕ1^{′}(T) + β(ϕ2^{′}(0) − γϕ2^{′}(T))]}. (2.7)

N = \frac{βγ ∫_{0}^{T} \frac{1}{ρ}h(s)ϕ_{1}(s)l(s)dsϕ2^{′}(T) − β ∫_{0}^{T} \frac{1}{ρ}h(s)ϕ_{2}(s)l(s)dsϕ1^{′}(0) + βγ ∑_{t_{i}<T} b_{i}}{ϕ1^{′}(0) − γϕ1^{′}(T) + β(ϕ2^{′}(0) − γϕ2^{′}(T))}

+\frac{β ( ∑_{t_{i}<T} b_{i}(T − t_{i}) + ∑_{t_{i}<T} a_{i}) (ϕ1^{′}(0) − γϕ1^{′}(T))}{T [ϕ1^{′}(0) − γϕ1^{′}(T) + β(ϕ2^{′}(0) − γϕ2^{′}(T))]}. (2.8)

Proof. Since ϕ_{1}, ϕ_{2} are two linearly independent solutions of the equation

−u′′(t) + p(t)u′(t) + q(t)u(t) = 0, t ∈ [0, T], (2.9)

we know the solutions of (2.9) can be presented as

u(t) = c_{1}ϕ_{1}(t) + c_{2}ϕ_{2}(t),

where c_{1}, c_{2} are any constants.

Let u^{∗} = c_{1}(t)ϕ_{1}(t) + c_{2}(t)ϕ_{2}(t) be a special solution of

−u′′(t) + p(t)u′(t) + q(t)u(t) = h(t), t ∈ [0, T]. (2.10)

Employing the method of variation of parameter, by some calculation, we get

c_{1}(t) = ∫_{t}^{T} \frac{1}{ρ}h(s)ϕ_{2}(s)l(s)ds, c_{2}(t) = ∫_{0}^{t} \frac{1}{ρ}h(s)ϕ_{1}(s)l(s)ds.

So the solution of (2.10) can be given as

u(t) = c_{1}ϕ_{1}(t) + c_{2}ϕ_{2}(t) + u^{∗}.

Next, we consider

\{ \begin{matrix}−u′′(t) + p(t)u′(t) + q(t)u(t) = h(t), t ≠ \{t_{1}, t_{2}, ..., t_{k}\}, \\ △u(t_{i}) = a_{i}, i = 1, 2, ..., k, \\ △u′(t_{i}) = b_{i}, i = 1, 2, ..., k. \end{matrix} (2.11)

It is easy to know the solution of (2.11) is as the following form

u(t) = c_{1}ϕ_{1}(t) + c_{2}ϕ_{2}(t) + u^{∗} + ∑_{t_{i}<t} b_{i}(t − t_{i}) + ∑_{t_{i}<t} a_{i}. (2.12)

Finally, we consider the solution of (2.4). Substituting (2.12) into u(0) = βu(T), u′(0) = γu′(T), we have

\{ \begin{matrix}c_{2}T − c_{1}βT − β ∑_{t_{i}<T} b_{i}(T − t_{i}) − β ∑_{t_{i}<T} a_{i} = 0, \\ γ ∫_{0}^{T} \frac{1}{ρ}h(s)l(s)ϕ_{1}(s)dsϕ2^{′}(T) + γ ∑_{t_{i}<T} b_{i} \\ − ∫_{0}^{T} \frac{1}{ρ}h(s)l(s)ϕ_{2}(s)dsϕ1^{′}(0) = c_{1}(ϕ1^{′}(0) − γϕ1^{′}(T)) + c_{2}(ϕ2^{′}(0) − γϕ2^{′}(T)). \end{matrix} (2.13)

By some calculations, we get

c_{1} = \frac{γ ∫_{0}^{T} \frac{1}{ρ}h(s)ϕ_{1}(s)l(s)dsϕ2^{′}(T) − ∫_{0}^{T} \frac{1}{ρ}h(s)ϕ_{2}(s)l(s)dsϕ1^{′}(0) + γ ∑_{t_{i}<T} b_{i}}{ϕ1^{′}(0) − γϕ1^{′}(T) + β(ϕ2^{′}(0) − γϕ2^{′}(T))}

−\frac{β ( ∑_{t_{i}<T} b_{i}(T − t_{i}) + ∑_{t_{i}<T} a_{i}) (ϕ2^{′}(0) − γϕ2^{′}(T))}{T [ϕ1^{′}(0) − γϕ1^{′}(T) + β(ϕ2^{′}(0) − γϕ2^{′}(T))]} =: M,

c_{2} = \frac{βγ ∫_{0}^{T} \frac{1}{ρ}h(s)ϕ_{1}(s)l(s)dsϕ2^{′}(T) − β ∫_{0}^{T} \frac{1}{ρ}h(s)ϕ_{2}(s)l(s)dsϕ1^{′}(0) + βγ ∑_{t_{i}<T} b_{i}}{ϕ1^{′}(0) − γϕ1^{′}(T) + β(ϕ2^{′}(0) − γϕ2^{′}(T))}

+\frac{β ( ∑_{t_{i}<T} b_{i}(T − t_{i}) + ∑_{t_{i}<T} a_{i}) (ϕ1^{′}(0) − γϕ1^{′}(T))}{T [ϕ1^{′}(0) − γϕ1^{′}(T) + β(ϕ2^{′}(0) − γϕ2^{′}(T))]} := N.

Hence, the problem (2.4) has the unique solution

u(t) = Mϕ_{1}(t) + Nϕ_{2}(t) + ϕ_{1}(t) ∫_{t}^{T} \frac{1}{ρ}h(s)ϕ_{2}(s)l(s)ds

+ϕ_{2}(t) ∫_{0}^{t} \frac{1}{ρ}h(s)ϕ_{1}(s)l(s)ds + ∑_{t_{i}<t} b_{i}(t − t_{i}) + ∑_{t_{i}<t} a_{i}.

Let f : [0, T] × R^{n} × R^{n} → R^{n} be continuous. We now introduce a mapping A : P C^{1}([0, T], R^{n}) → P C([0, T], R^{n}) defined by

Au(t) = Mϕ_{1}(t) + Nϕ_{2}(t) + ϕ_{1}(t) ∫_{t}^{T} \frac{1}{ρ}f(s, u(s), u′(s))ϕ_{2}(s)l(s)ds

+ϕ_{2}(t) ∫_{0}^{t} \frac{1}{ρ}f(s, u(s), u′(s))ϕ_{1}(s)l(s)ds + ∑_{t_{i}<t} b_{i}(t − t_{i}) + ∑_{t_{i}<t} a_{i}. (2.14)

In view of Lemma 2.4, we easily know that u is a fixed point of operator A iff u is a solution to the impulsive periodic boundary problem (1.3). Lemma 2.5. Let f : [0, T] × R^{n} × R^{n} → R^{n} be continuous. Then A : P C^{1}([0, T], R^{n}) → P C([0, T], R^{n}) is a compact map. Proof. This is similar to that of Lemma 3.2 in [4].

For convenience, let

∥ϕ_{1}∥_{0} = max_{0≤t≤T} |ϕ_{1}(t)|, ∥ϕ_{2}∥_{0} = max_{0≤t≤T} |ϕ_{2}(t)|, G_{1} = max_{0≤t≤T} |ϕ_{1}(t)ϕ_{2}(t)|,

L = max_{0≤t≤T} |l(t)|, ∥ϕ1^{′}∥_{0} = max_{0≤t≤T} |ϕ1^{′}(t)|, ∥ϕ2^{′}∥_{0} = max_{0≤t≤T} |ϕ2^{′}(t)|. (2.15)

Now we are in the position to present our main results.

3 Main results

Theorem 3.1. Suppose that f : [0, T] × R^{n} × R^{n} → R^{n} is continuous and p ∈ C([0, T], [0, +∞)), q(t) ≡ q ≤ 0, |β| ≥ 1, |γ| ≥ 1, a_{i}, b_{i} are constants for i = 1, 2, ..., k. If there exist nonnegative constants α, Q such that

∥f(t, u, v)∥ ≤ 2α⟨v, pv + qu − f(t, u, v)⟩ + Q, (t, u, v) ∈ ([0, T] \ {t_{1}, t_{2}, ..., t_{k}}) × R^{n} × R^{n}. (3.1)

Then BVP (1.3) has at least one solution.

Proof. Let u ∈ P C([0, T], R^{n}) be such that u = λAu for some λ ∈ (0, 1). That is,

\{ \begin{matrix}−u′′(t) + p(t)u′(t) + qu(t) = λf(t, u(t), u′(t)), t ∈ [0, T] \ \{t_{1}, t_{2}, ..., t_{k}\}, \\ △u(t_{i}) = λa_{i}, i = 1, 2, ..., k, \\ △u′(t_{i}) = λb_{i}, i = 1, 2, ..., k, \\ u(0) = βu(T), u′(0) = γu′(T). \end{matrix} (3.2)

By Lemma 2.5, A is a compact map. In order to utilize Lemma 2.1, next, we will show S = {u ∈ P C^{1}|u = λAu, λ ∈ (0, 1)} is bounded. By(2.3), (2.14)-(2.15) together with (3.1)-(3.2), we obtain

∥u(t)∥ = λ∥Au(t)∥

= λ∥Mϕ_{1}(t) + Nϕ_{2}(t) + \frac{1}{ρ} ∫_{t}^{T} ϕ_{1}(t)ϕ_{2}(s)l(s)f(s, u(s), u′(s))ds

+\frac{1}{ρ} ∫_{0}^{t} ϕ_{1}(s)l(s)ϕ_{2}(t)f(s, u(s), u′(s))ds + ∑_{t_{i}<t} b_{i}(t − t_{i}) + ∑_{t_{i}<t} a_{i}∥

≤ |M|∥ϕ_{1}(t)∥ + |N|∥ϕ_{2}(t)∥ + |\frac{1}{ρ}ϕ_{1}(t)ϕ_{2}(t)L| ∫_{t}^{T} λ∥f(s, u(s), u′(s))∥ds

+|\frac{1}{ρ}ϕ_{1}(t)ϕ_{2}(t)L| ∫_{0}^{t} λ∥f(s, u(s), u′(s))∥ds + ∑_{t_{i}<T} |b_{i}|(T − t_{i}) + ∑_{t_{i}<T} |a_{i}|

≤ |M|∥ϕ_{1}∥_{0} + |N|∥ϕ_{2}∥_{0} + \frac{2}{|ρ|}G_{1}L ∫_{0}^{T} (2α⟨u′(s), λp(s)u′(s) + λqu(s) − λf(s, u(s), u′(s)⟩ + Q)ds

+ ∑_{t_{i}<T} |b_{i}|(T − t_{i}) + ∑_{t_{i}<T} |a_{i}|

= \frac{2}{|ρ|}G_{1}L[∫_{0}^{T} 2α⟨u′(s), p(s)u′(s) + qu(s) − λf(t, u, u′) − (1 − λ)p(s)u′(s) − (1 − λ)qu(s)⟩ds + QT]

+|M|∥ϕ_{1}∥_{0} + |N|∥ϕ_{2}∥_{0} + ∑_{t_{i}<T} |b_{i}|(T − t_{i}) + ∑_{t_{i}<T} |a_{i}|

= \frac{2}{|ρ|}G_{1}L[∫_{0}^{T} 2α⟨u′(s), u′′(s)⟩ds−2α(1−λ) ∫_{0}^{T} ⟨u′(s), p(s)u′(s)⟩ds−2α(1−λ)q ∫_{0}^{T} ⟨u′(s), u(s)⟩ds

+QT] + |M|∥ϕ_{1}∥_{0} + |N|∥ϕ_{2}∥_{0} + ∑_{t_{i}<T} |b_{i}|(T − t_{i}) + ∑_{t_{i}<T} |a_{i}|

= \frac{2}{|ρ|}G_{1}L[α ∫_{0}^{T} \frac{d}{ds}∥u′(s)∥^{2}ds−2α(1−λ) ∫_{0}^{T} ⟨\sqrt{p(s)}u′(s), \sqrt{p(s)}u′(s)⟩ds−α(1−λ)q ∫_{0}^{T} \frac{d}{ds}∥u(s)∥^{2}ds

+QT] + |M|∥ϕ_{1}∥_{0} + |N|∥ϕ_{2}∥_{0} + ∑_{t_{i}<T} |b_{i}|(T − t_{i}) + ∑_{t_{i}<T} |a_{i}|

= \frac{2}{|ρ|}G_{1}L[α(∥u′(T)∥^{2} − ∥u′(0)∥^{2}) − 2α(1 − λ) ∫_{0}^{T} ∥\sqrt{p(s)}u′(s)∥^{2}ds − α(1 − λ)q(∥u(T)∥^{2} − ∥u(0)∥^{2})

+QT] + |M|∥ϕ_{1}∥_{0} + |N|∥ϕ_{2}∥_{0} + ∑_{t_{i}<T} |b_{i}|(T − t_{i}) + ∑_{t_{i}<T} |a_{i}|

≤ \frac{2}{|ρ|}G_{1}L[α(1 − γ^{2})∥u′(T)∥^{2} − α(1 − λ)q(1 − β^{2})∥u(T)∥^{2} + QT]

+|M|∥ϕ_{1}∥_{0} + |N|∥ϕ_{2}∥_{0} + ∑_{t_{i}<T} |b_{i}|(T − t_{i}) + ∑_{t_{i}<T} |a_{i}|

≤ \frac{2}{|ρ|}G_{1}LQT + |M|∥ϕ_{1}∥_{0} + |N|∥ϕ_{2}∥_{0} + ∑_{t_{i}<T} |b_{i}|(T − t_{i}) + ∑_{t_{i}<T} |a_{i}|.

A similar calculation yields an estimate on u′: differentiating both sides of the integration and taking norms yields, for each t ∈ [0, T], we have

∥u′(t)∥ = λ∥(Au)′(t)∥

= λ∥Mϕ1^{′}(t) + Nϕ2^{′}(t) + ∫_{t}^{T} \frac{1}{ρ}f(s, u(s), u′(s))ϕ_{2}(s)l(s)dsϕ1^{′}(t)

+ ∫_{0}^{t} \frac{1}{ρ}f(s, u(s), u′(s))ϕ_{1}(s)l(s)dsϕ2^{′}(t) + ∑_{t_{i}<t} b_{i}∥

≤ |M|∥ϕ1^{′}∥_{0} + |N|∥ϕ2^{′}∥_{0} + \frac{1}{|ρ|}(∥ϕ1^{′}∥_{0}∥ϕ_{2}∥_{0} + ∥ϕ_{1}∥_{0}∥ϕ2^{′}∥_{0})L ∫_{0}^{T} λ∥f(s, u(s), u′(s))∥ds + ∑_{t_{i}<T} |b_{i}|

≤ \frac{1}{|ρ|}(∥ϕ1^{′}∥_{0}∥ϕ_{2}∥_{0} + ∥ϕ_{1}∥_{0}∥ϕ2^{′}∥_{0})L ∫_{0}^{T} [2α⟨u′(s), λp(s)u′(s) + λqu(s)

−λf(s, u(s), u′(s)⟩ + Q]ds + |M|∥ϕ1^{′}∥_{0} + |N|∥ϕ2^{′}∥_{0} + ∑_{t_{i}<T} |b_{i}|

= \frac{1}{|ρ|}(∥ϕ1^{′}∥_{0}∥ϕ_{2}∥_{0} + ∥ϕ_{1}∥_{0}∥ϕ2^{′}∥_{0})L[∫_{0}^{T} 2α⟨u′(s), p(s)u′(s) + qu(s)

−λf(t, u, u′) − (1 − λ)p(s)u′(s) − (1 − λ)qu(s)⟩ds + QT] + |M|∥ϕ1^{′}∥_{0} + |N|∥ϕ2^{′}∥_{0} + ∑_{t_{i}<T} |b_{i}|

= \frac{1}{|ρ|}(∥ϕ1^{′}∥_{0}∥ϕ_{2}∥_{0} + ∥ϕ_{1}∥_{0}∥ϕ2^{′}∥_{0})L[∫_{0}^{T} 2α⟨u′(s), u′′(s)⟩ds − 2α(1 − λ) ∫_{0}^{T} ⟨u′(s), p(s)u′(s)⟩ds

−2α(1 − λ)q ∫_{0}^{T} ⟨u′(s), u(s)⟩ds + QT] + |M|∥ϕ1^{′}∥_{0} + |N|∥ϕ2^{′}∥_{0} + ∑_{t_{i}<T} |b_{i}|

= \frac{1}{|ρ|}(∥ϕ1^{′}∥_{0}∥ϕ_{2}∥_{0} + ∥ϕ_{1}∥_{0}∥ϕ2^{′}∥_{0})L[α ∫_{0}^{T} \frac{d}{ds}∥u′(s)∥^{2}ds − 2α(1 − λ) ∫_{0}^{T} ⟨\sqrt{p(s)}u′(s), \sqrt{p(s)}u′(s)⟩ds

−α(1 − λ)q ∫_{0}^{T} \frac{d}{ds}∥u(s)∥^{2}ds + QT] + |M|∥ϕ1^{′}∥_{0} + |N|∥ϕ2^{′}∥_{0} + ∑_{t_{i}<T} |b_{i}|

= \frac{1}{|ρ|}(∥ϕ1^{′}∥_{0}∥ϕ_{2}∥_{0} + ∥ϕ_{1}∥_{0}∥ϕ2^{′}∥_{0})L[α(∥u′(T)∥^{2} − ∥u′(0)∥^{2}) − 2α(1 − λ) ∫_{0}^{T} ∥\sqrt{p(s)}u′(s)∥^{2}ds

−α(1 − λ)q(∥u(T)∥^{2} − ∥u(0)∥^{2}) + QT] + |M|∥ϕ1^{′}∥_{0} + |N|∥ϕ2^{′}∥_{0} + ∑_{t_{i}<T} |b_{i}|

≤ \frac{1}{|ρ|}(∥ϕ1^{′}∥_{0}∥ϕ_{2}∥_{0} + ∥ϕ_{1}∥_{0}∥ϕ2^{′}∥_{0})L[α(1 − γ^{2})∥u′(T)∥^{2}

−α(1 − λ)q(1 − β^{2})∥u(T)∥^{2} + QT] + |M|∥ϕ1^{′}∥_{0} + |N|∥ϕ2^{′}∥_{0} + ∑_{t_{i}<T} |b_{i}|.

≤ \frac{1}{|ρ|}(∥ϕ1^{′}∥_{0}∥ϕ_{2}∥_{0} + ∥ϕ_{1}∥_{0}∥ϕ2^{′}∥_{0})LQT + |M|∥ϕ1^{′}∥_{0} + |N|∥ϕ2^{′}∥_{0} + ∑_{t_{i}<T} |b_{i}|.

Thus, we conclude that

∥u∥_{P C^{1}} ≤ max\{\frac{2}{|ρ|}G_{1}LQT + |M|∥ϕ_{1}∥_{0} + |N|∥ϕ_{2}∥_{0} + ∑_{t_{i}<T} |b_{i}|(T − t_{i}) + ∑_{t_{i}<T} |a_{i}|,

\frac{1}{|ρ|}(∥ϕ1^{′}∥_{0}∥ϕ_{2}∥_{0} + ∥ϕ_{1}∥_{0}∥ϕ2^{′}∥_{0})LQT + |M|∥ϕ1^{′}∥_{0} + |N|∥ϕ2^{′}∥_{0} + ∑_{t_{i}<T} |b_{i}|\}.

As a result, set S is bounded. Applying Scheafer’s fixed-point theorem, the problem (3.2) has at least one fixed point, which means that (1.3) has at least one solution. We complete the proof.

A similar discuss as Theorem 3.1 leads to the following result.

Remark 3.2. If the condition (3.1) is replaced by

∥f(t, u, v)∥ ≤ 2α⟨v, pv − f(t, u, v)⟩ + Q,  (t, u, v) ∈ ([0, T] \ {t_{1}, t_{2}, ..., t_{k}}) × R^{n} × R^{n},  (3.3)

and all the other assumptions are satisfied in Theorem 3.1, then the problem (1.3) has at least one solution.

4 An example

In this section, an example is given to highlight our main result. Consider the scalar impulsive periodic BVP given by

\{ \begin{matrix}−u′′(t) + t^{2}u′(t) − 9u(t) = f(t, u(t), u′(t)), t ∈ [0, 1] \ \{t_{1}, t_{2}, ..., t_{k}\}, \\ △u(t_{i}) = a_{i}, i = 1, 2, ..., k, \\ △u′(t_{i}) = b_{i}, i = 1, 2, ..., k, \\ u(0) = u(1), u′(0) = u′(1), \end{matrix} (4.1)

where 0 < t_{1} < ⋯ < t_{k} < 1, f(t, u, v) = \frac{8}{π} arctan u + (1 − t^{2})v^{2} − v^{3}, p(t) = t^{2}, and q = −9. We claim that (4.1) has at least one solution.

Proof. Let T = 1, β = γ = 1, and f(t, u, u′) = (1 − t^{2})u^{′2} − u^{′3} + \frac{8}{π} arctan u. It is easy to check that

x^{4} − 2x^{3} − x^{2} − 4x + 12 ≥ 0,  x ≥ 0.  (4.2)

And we see that

|f(t, u, v)| = |\frac{8}{π} arctan u + (1 − t^{2})v^{2} − v^{3}|

≤ \frac{8}{π} × \frac{π}{2} + |v|^{2} + |v|^{3}

≤ 4 + |v|^{2} + |v|^{3},  (t, u, v) ∈ [0, 1] × R^{2}.  (4.3)

On the other hand, for α = \frac{1}{2}, Q = 16, we have

2α⟨v, pv − f(t, u, v)⟩ + Q

= v(t^{2}v − (1 − t^{2})v^{2} + v^{3} − \frac{8}{π} arctan u) + 16

= v^{4} − (1 − t^{2})v^{3} + t^{2}v^{2} − \frac{8}{π} arctan uv + 16

≥ |v|^{4} − |v|^{3} − 4|v| + 16, (t, u, v) ∈ [0, 1] × R^{2}. (4.4)

In view of (4.2), we have

|v|^{4} − |v|^{3} − 4|v| + 16 ≥ |v|^{3} + |v|^{2} + 4. (4.5)

By (4.3)-(4.5), we obtain that

∥f(t, u, v)∥ ≤ 2α⟨v, pv − f(t, u, v)⟩ + Q.

Thus, condition (3.3) holds. By Remark 3.2, we conclude that the solvability of (4.1) follows.

Acknowledgement

The authors would like to thank the anonymous referee for careful reading of the manuscript and providing valuable suggestions to improve the quality of this paper.

[1] R. P. Agarwal and D. O’Regan, Multiple nonnegative solutions for second order impulsive differential equations, Appl. Math. Comput., 114(2000), 51-59.

[2] C. Bai, Existence of solutions for second order nonlinear functional differential equations with periodic boundary value conditions, Int. J. Pure and Appl. Math., 16(2004), 451-462.

[3] C. Bai and D. Yang, Existence of solutions for second-order nonlinear impulsive differential equations with periodic boundary value conditions, Boundary Value Problems, (2007), Article ID41589, 13pages.

[4] J. Chen, C. C. Tisdell and R. Yuan, On the solvability of periodic boundary value problems with impulse, J. Math. Anal. Appl., 331(2007), 902-912.

[5] M. Choisy, J. F. Guegan and P. Rohani, Dynamics of infectious diseases and pulse vaccination: Teasing apart the embedded resonance effects, Physica D: Nonlinear Phenomena, 22( 2006), 26-35.

[6] W. Ding, M. Han and J. Mi, Periodic boundary value problem for the second-order impulsive functional differential equations, Comput. Math. Appl., 50(2005), 491-507.

[7] A. d’Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. Math. Lett., 18(2005), 729-732.

[8] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.

[9] J. Li, J. J. Nieto and J. Shen, Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl., 325(2007), 226-236.

[10] X. Liu (Ed.), Advances in impulsive differential equations, Dynamics Continuous, Discrete & Impulsive Systems, Series A, Math. Anal., 9(2002), 313-462.

[11] Y. Liu, Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations, J. Math. Anal. Appl., 327(2007), 435-452.

[12] Z. Luo and J. J. Nieto, New results of periodic boundary value problem for impulsive integro-differential equations, Nonlinear Anal., 70(2009), 2248-2260.

[13] R. Ma and H.Wang, Positive solutions of nonlinear three-point boundary value problems, J. Math. Anal. Appl., 279(2003), 216-227.

[14] J. J. Nieto, Periodic boundary value problems for first-order impulsive ordinary differential equations, Nonlinear Anal., 51(2002), 1223-1232.

[15] I. Rachunkvá and M.Tvrdý, Existence results for impulsive second-order periodic problems, Nonlinear Anal., 59(2004), 133-146.

[16] M. Rudd and C. C. Tisdell, On the solvability of two-point, second-order boundary value problems, Appl. Math. Lett., 20(2007), 824-828.

[17] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.

[18] S. Tang, L. Chen, Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biolo., 44(2002), 185-199.

[19] W. Wang, H. Wang and Z. Li, The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy, Chaos, Solitons & Fractals, 32(2007), 1772-1785.

[20] K. Wang, A new existence result for nonlinear first-order anti-periodic boundary-value problems, Appl. Math. Letters, 21(2008), 1149-1154.

[21] Y. Xing and V. Romanvski, On the solvability of second-order impulsive differential equations with antiperiodic boundary value conditions, Boundary Value Problems, (2008), Article ID 864297, 18pages.

[22] M. Yao, A. Zhao and J. Yan, Periodic boundary value problems of second-order impulsive differential equations, Nonlinear Anal., 70(2009), 262-237.

(Received February 17, 2009)