Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 11, 1-15; http://www.math.u-szeged.hu/ejqtde/

On the solvability of anti-periodic boundary value problems with impulse

Chuanzhi Bai ^{∗}

Department of Mathematics, Huaiyin Teachers College, Huaian, Jiangsu 223300, P R China

Abstract

In this paper, we are concerned with the existence of solutions for second order impulsive anti-periodic boundary value problem

\{ \begin{matrix}u′′(t) + f(t, u(t), u′(t)) = 0, t ≠ t_{k}, t ∈ [0, T], \\ △u(t_{k}) = I_{k}(u(t_{k})), k = 1, ⋯, m, \\ △u′(t_{k}) = I_{k}^{∗}(u(t_{k})), k = 1, ⋯, m, \\ u(0) + u(T) = 0, u′(0) + u′(T) = 0, \end{matrix}

new criteria are established based on Schaefer’s fixed-point theorem. Keywords: Anti-periodic boundary value problem; Impulsive; Schaefer’s fixed-point theorem

1. Introduction

In recent years, the solvability of the anti-periodic boundary value problems of first-order and second-order differential equations were studied by many authors, for example, we refer to [1-5] and the references therein. It should be noted that anti-periodic boundary value problems appear in physics in a variety of situations [6,7].

Impulsive differential equations, which arise in biology, physics, population dynamics, economics, etc., are a basic tool to study evolution processes that are subjected to abrupt in their states (see [8-12]). Recently, the existence results were extended to anti-periodic boundary value problems for first-order impulsive differential equations [13,14]. Very recently, Wang and Shen [15] investigated the anti-periodic boundary value problem for a class of second order differential equations by using Schauder’s fixed point theorem and the lower and upper solutions method.

Inspired by [13-15], in this paper, we investigate the anti-periodic boundary value problem for second order impulsive nonlinear differential equations of the form

∗E-mail address: czbai@hytc.edu.cn, czbai8@sohu.com

\{ \begin{matrix}u′′(t) + f(t, u(t), u′(t)) = 0, t ∈ J_{0} = J \ \{t_{1}, ⋯, t_{m}\}, \\ △u(t_{k}) = I_{k}(u(t_{k})), k = 1, ⋯, m, \\ △u′(t_{k}) = I_{k}^{∗}(u(t_{k})), k = 1, ⋯, m, \\ u(0) + u(T) = 0, u′(0) + u′(T) = 0, \end{matrix} (1.1)

where J = [0, T], 0 < t_{1} < t_{2} < ⋯ < t_{m} < T, f : [0, T] × R^{2} → R is continuous on (t, x, y) ∈

J_{0} × R^{2}, f(t_{k}^{+}, x, y) := lim_{t→t_{k}^{+}} f(t, x, y), f(t_{k}^{−}, x, y) := lim_{t→t_{k}^{−}} f(t, x, y) exist, f(t_{k}^{−}, x, y) = f(t_{k}, x, y);

△u(t_{k}) = u(t_{k}^{+}) − u(t_{k}^{−}), △u′(t_{k}) = u′(t_{k}^{+}) − u′(t_{k}^{−}); I_{k}, I_{k}^{∗} ∈ C(R, R).

To the best of the authors knowledge, no one has studied the existence of solutions for impulsive anti-periodic boundary value problem (1.1). The aim of this paper is to fill the gap in the relevant literatures.

The following Schaefer’s fixed-point theorem is fundamental in the proof of our main results.

Lemma 1.1.[16] (Schaefer) Let E be a normed linear space with H : E → E a compact operator. If the set

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

is bounded, then H has at least one fixed-point.

The paper is formulated as follows. In section 2, some definitions and lemmas are given. In section 3, we obtain a new existence theorem by using Schaefer’s fixed point theorem, and uniqueness result by using Banach’s fixed point theorem. In Section 4, an illustrative example is given to demonstrate the effectiveness of the obtained results.

2. Preliminaries

In order to define the concept of solution for (1.1), we introduce the following spaces of functions:

P C(J) = {u : J → R : u is continuous for any t ∈ J_{0}, u(t_{k}^{+}), u(t_{k}^{−}) exist, and u(t_{k}^{−}) = u(t_{k}), k = 1, ⋯, m},

P C^{1}(J) = {u : J → R : u is continuously differentiable for any t ∈ J_{0}, u′(t_{k}^{+}), u′(t_{k}^{−}) exist, and u′(t_{k}^{−}) = u′(t_{k}), k = 1, ⋯, m}.

P C(J) and P C^{1}(J) are Banach space with the norms :

∥u∥_{P C} = sup_{t∈J} |u(t)|, and

∥u∥_{P C^{1}} = max{∥u∥_{P C}, ∥u′∥_{P C}}. A solution to the impulsive BVP (1.1) is a function u ∈ P C^{1}(J) ∩ C^{2}(J_{0}) that satisfies (1.1)

for each t ∈ J.

Consider the following impulsive BVP with p ≥ 0, q > 0

\{ \begin{matrix}−u′′(t) + pu′(t) + qu(t) = σ(t), t ∈ J_{0}, \\ △u(t_{k}) = I_{k}(u(t_{k})), k = 1, ⋯, m, \\ △u′(t_{k}) = I_{k}^{∗}(u(t_{k})), k = 1, ⋯, m, \\ u(0) + u(T) = 0, u′(0) + u′(T) = 0, \end{matrix} (2.1)

where σ ∈ P C(J).

For convenience, we set I_{k} = I_{k}(u(t_{k})), I_{k}^{∗} = I_{k}^{∗}(u(t_{k})),

r_{1} := \frac{p + \sqrt{p^{2} + 4q}}{2} > 0, r_{2} := \frac{p − \sqrt{p^{2} + 4q}}{2} < 0. (2.2)

Lemma 2.1. u ∈ P C^{1}(J) ∩ C^{2}(J_{0}) is a solution of (2.1) if and only if u ∈ P C^{1}(J) is a solution of the impulsive integral equation

u(t) = ∫_{0}^{T} G(t, s)σ(s)ds + ∑_{k=1}^{m}[G(t, t_{k})(−I_{k}^{∗}) + W (t, t_{k})I_{k}], (2.3)

where

G(t, s) = \frac{1}{r_{1} − r_{2}} \{ \begin{matrix}\frac{e^{r_{2}(t−s)}}{1+e^{r_{2}T}} − \frac{e^{r_{1}(t−s)}}{1+e^{r_{1}T}} , & 0 ≤ s < t ≤ T, \\ \frac{e^{r_{1}(T +t−s)}}{1+e^{r_{1}T}} − \frac{e^{r_{2}(T +t−s)}}{1+e^{r_{2}T}} , & 0 ≤ t ≤ s ≤ T, \end{matrix} (2.4)

and

W (t, s) = \frac{1}{r_{1} − r_{2}} \{ \begin{matrix}\frac{r_{1}e^{r_{2}(t−s)}}{1+e^{r_{2}T}} − \frac{r_{2}e^{r_{1}(t−s)}}{1+e^{r_{1}T}} , & 0 ≤ s < t ≤ T, \\ \frac{r_{2}e^{r_{1}(T +t−s)}}{1+e^{r_{1}T}} − \frac{r_{1}e^{r_{2}(T +t−s)}}{1+e^{r_{2}T}} , & 0 ≤ t ≤ s ≤ T. \end{matrix} (2.5)

Proof. If u ∈ P C^{1}(J) ⋂ C^{2}(J_{0}) is a solution of (2.1), setting

v(t) = u′(t) − r_{2}u(t), (2.6) then by the first equation of (2.1) we have

v′(t) − r_{1}v(t) = −σ(t), t ≠ t_{k}. (2.7) Multiplying (2.7) by e^{−r_{1}t} and integrating on [0, t] and (t_{1}, t], respectively, we get

e^{−r_{1}t}v(t) − v(0) = − ∫_{0}^{t} σ(s)e^{−r_{1}s}ds, 0 ≤ t ≤ t_{1},

e^{−r_{1}t}v(t) − e^{−r_{1}t_{1}}v(t_{1}^{+}) = − ∫_{t_{1}}^{t} σ(s)e^{−r_{1}s}ds, t_{1} < t ≤ t_{2}.

So

v(t) = e^{r_{1}t}[v(0) − ∫_{0}^{t} e^{−r_{1}s}σ(s)ds + e^{−r_{1}t_{1}}(I_{1}^{∗} − r_{2}I_{1}), t_{1} < t ≤ t_{2}.

In the same way, we can obtain that

v(t) = e^{r_{1}t} [v(0) − ∫_{0}^{t} e^{−r_{1}s}σ(s)ds + ∑_{0<t_{k}<t} e^{−r_{1}t_{k}}(I_{k}^{∗} − r_{2}I_{k})] , t ∈ J, (2.8)

where v(0) = u′(0) − r_{2}u(0). Multiplying (2.6) by e^{−r_{2}t} and integrating on [0, t_{k}] and (t_{k}, t] (t_{k} < t ≤ t_{k+1}), respectively, similar to the proof of (2.8), we have

u(t) = e^{r_{2}t} [u(0) + ∫_{0}^{t} v(s)e^{−r_{2}s}ds + ∑_{0<t_{k}<t} e^{−r_{2}t_{k}}I_{k}] , t ∈ J. (2.9)

By some calculation, we get

∫_{0}^{t} v(s)e^{−r_{2}s}ds = \frac{1}{r_{1} − r_{2}} [v(0)(e^{(r_{1}−r_{2})t} − 1) − ∫_{0}^{t}(e^{(r_{1}−r_{2})t} − e^{(r_{1}−r_{2})s})σ(s)e^{−r_{1}s}ds

+ ∑_{0<t_{k}<t} (e^{(r_{1}−r_{2})t} − e^{(r_{1}−r_{2})t_{k}}) e^{−r_{1}t_{k}}(I_{k}^{∗} − r_{2}I_{k})] .

(2.10)

Substituting (2.10) into (2.9), we obtain

u(t) = \frac{1}{r_{1} − r_{2}} [(r_{1}u(0) − u′(0))e^{r_{2}t} + (u′(0) − r_{2}u(0))e^{r_{1}t}

+ ∫_{0}^{t}(e^{r_{2}(t−s)} − e^{r_{1}(t−s)})σ(s)ds + ∑_{0<t_{k}<t} e^{r_{1}(t−t_{k})}(I_{k}^{∗} − r_{2}I_{k})

− ∑_{0<t_{k}<t} e^{r_{2}(t−t_{k})}(I_{k}^{∗} − r_{1}I_{k})] , t ∈ [0, T],

(2.11)

u′(t) = \frac{1}{r_{1} − r_{2}} [r_{2}(r_{1}u(0) − u′(0))e^{r_{2}t} + r_{1}(u′(0) − r_{2}u(0))e^{r_{1}t}

+ ∫_{0}^{t}(r_{2}e^{r_{2}(t−s)} − r_{1}e^{r_{1}(t−s)})σ(s)ds + ∑_{0<t_{k}<t} r_{1}e^{r_{1}(t−t_{k})}(I_{k}^{∗} − r_{2}I_{k})

− ∑_{0<t_{k}<t} r_{2}e^{r_{2}(t−t_{k})}(I_{k}^{∗} − r_{1}I_{k})] , t ∈ [0, T].

(2.12)

In view of u(0) + u(T) = 0, u′(0) + u′(T) = 0, we have

u′(0) − r_{2}u(0) = \frac{1}{1 + e^{r_{1}T}} [∫_{0}^{T} e^{r_{1}(T −s)}σ(s)ds − ∑_{0<t_{k}<T} e^{r_{1}(T −t_{k})}(I_{k}^{∗} − r_{2}I_{k})] ,  (2.13)

r_{1}u(0) − u′(0) = \frac{1}{1 + e^{r_{2}T}} [− ∫_{0}^{T} e^{r_{2}(T −s)}σ(s)ds + ∑_{0<t_{k}<T} e^{r_{2}(T −t_{k})}(I_{k}^{∗} − r_{1}I_{k})] .  (2.14)

Substituting (2.13) and (2.14) into (2.11), by routine calculation, we can get (2.3).

Conversely, if u is a solution of (2.3), then direct differentiation of (2.3) gives −u′′(t) = σ(t) −pu′(t) − qu(t), t ≠ t_{k}. Moreover, we obtain △u|_{t=t_{k}} = I_{k}(u(t_{k})), △u′|_{t=t_{k}} = I_{k}^{∗}(u(t_{k})), u(0) + u(T) = 0 and u′(0) + u′(T) = 0. Hence, u ∈ P C^{1}(J) ∩ C^{2}(J_{0}) is a solution of (2.1) □

Define a mapping A : P C^{1}(J) → P C^{1}(J) by

Au(t) = ∫_{0}^{T} G(t, s)[f(s, u(s), u′(s)) + pu′(s) + qu(s)]ds

+ ∑_{k=1}^{m}[G(t, t_{k})(−I_{k}^{∗}) + W (t, t_{k})I_{k}], t ∈ [0, T]. (2.15)

In view of Lemma 2.1, we easily know that u is a fixed point of operator A if and only if u is a solution to the impulsive boundary value problem (1.1).

Lemma 2.2 If u ∈ P C^{1}(J) and u(0) + u(T) = 0, then

∥u∥_{P C} ≤ \frac{1}{2} (∫_{0}^{T} |u′(s)|ds + ∑_{k=1}^{m} |△u(t_{k})|) .

Proof. Since u ∈ P C^{1}(J), we have

u(t) = u(0) + ∑_{0<t_{k}<t} △u(t_{k}) + ∫_{0}^{t} u′(s)ds. (2.16)

Set t = T, we obtain from u(0) + u(T) = 0 that

u(0) = −\frac{1}{2} (∑_{k=1}^{m} △u(t_{k}) + ∫_{0}^{T} u′(s)ds) . (2.17)

Substituting (2.17) into (2.16), we get

|u(t)| = |\frac{1}{2} (∫_{0}^{t} u′(s)ds − ∫_{t}^{T} u′(s)ds) + \frac{1}{2} ( ∑_{0<t_{k}<t} △u(t_{k}) − ∑_{t≤t_{k}} △u(t_{k}))|

≤ \frac{1}{2} (∫_{0}^{t} |u′(s)|ds + ∫_{t}^{T} |u′(s)|ds) + \frac{1}{2} ( ∑_{0<t_{k}<t} |△u(t_{k})| + ∑_{t≤t_{k}} |△u(t_{k})|)

= \frac{1}{2} (∫_{0}^{T} |u′(s)|ds + ∑_{k=1}^{m} |△u(t_{k})|) .

The proof is complete. □

It is easy to check that

|G(t, s)| ≤ \frac{1}{r_{1} − r_{2}} (\frac{1}{1 + e^{r_{2}T}} − \frac{1}{1 + e^{r_{1}T}} ) := G_{1}. (2.18)

Since

\frac{∂}{∂t}G(t, s) = \frac{1}{r_{1} − r_{2}} \{ \begin{matrix}\frac{−r_{1}e^{r_{1}(t−s)}}{1+e^{r_{1}T}} + \frac{r_{2}e^{r_{2}(t−s)}}{1+e^{r_{2}T}} , & 0 ≤ s < t ≤ T \\ \frac{r_{1}e^{r_{1}(T +t−s)}}{1+e^{r_{1}T}} + \frac{−r_{2}e^{r_{2}(T +t−s)}}{1+e^{r_{2}T}} , & 0 ≤ t ≤ s ≤ T, \end{matrix}

and

\frac{∂}{∂t}W (t, s) = \frac{1}{r_{1} − r_{2}} \{ \begin{matrix}\frac{r_{1}r_{2}e^{r_{2}(t−s)}}{1+e^{r_{2}T}} − \frac{r_{2}r_{1}e^{r_{1}(t−s)}}{1+e^{r_{1}T}} , & 0 ≤ s < t ≤ T, \\ \frac{r_{1}r_{2}e^{r_{1}(T +t−s)}}{1+e^{r_{1}T}} − \frac{r_{1}r_{2}e^{r_{2}(T +t−s)}}{1+e^{r_{2}T}} , & 0 ≤ t ≤ s ≤ T, \end{matrix}

we obtain by r_{1} ≥ −r_{2} > 0 that

|\frac{∂}{∂t}G(t, s)| ≤ \frac{r_{1}}{r_{1} − r_{2}} \{ \begin{matrix}\frac{e^{r_{1}(t−s)}}{1+e^{r_{1}T}} + \frac{e^{r_{2}(t−s)}}{1+e^{r_{2}T}} , & 0 ≤ s < t ≤ T \\ \frac{e^{r_{1}(T +t−s)}}{1+e^{r_{1}T}} + \frac{e^{r_{2}(T +t−s)}}{1+e^{r_{2}T}} , & 0 ≤ t ≤ s ≤ T, \end{matrix}

|W (t, s)| ≤ \frac{r_{1}}{r_{1} − r_{2}} \{ \begin{matrix}\frac{e^{r_{1}(t−s)}}{1+e^{r_{1}T}} + \frac{e^{r_{2}(t−s)}}{1+e^{r_{2}T}} , & 0 ≤ s < t ≤ T \\ \frac{e^{r_{1}(T +t−s)}}{1+e^{r_{1}T}} + \frac{e^{r_{2}(T +t−s)}}{1+e^{r_{2}T}} , & 0 ≤ t ≤ s ≤ T, \end{matrix}

and

|\frac{∂}{∂t}W (t, s)| ≤ \frac{−r_{1}r_{2}}{r_{1} − r_{2}} \{ \begin{matrix}\frac{e^{r_{2}(t−s)}}{1+e^{r_{2}T}} + \frac{e^{r_{1}(t−s)}}{1+e^{r_{1}T}} , & 0 ≤ s < t ≤ T, \\ \frac{e^{r_{1}(T +t−s)}}{1+e^{r_{1}T}} + \frac{e^{r_{2}(T +t−s)}}{1+e^{r_{2}T}} , & 0 ≤ t ≤ s ≤ T. \end{matrix}

Let h(x) = Be^{r_{1}x} + Ce^{r_{2}x}, for x ∈ [0, T], where B, C are two nonnegative constants. Obviously, h′′(x) ≥ 0, that is, h is a convex function on [0, T]. Thus,

h(x) ≤ max{h(0), h(T)} = max{B + C, Be^{r_{1}T} + Ce^{r_{2}T} }. (2.19) By (2.19), we easily obtain that

|\frac{∂}{∂t}G(t, s)| ≤ \frac{r_{1}}{r_{1} − r_{2}} (\frac{e^{r_{1}T}}{1 + e^{r_{1}T}} + \frac{e^{r_{2}T}}{1 + e^{r_{2}T}} ) := G_{2}, |W (t, s)| ≤ G_{2}, (2.20)

and

|\frac{∂}{∂t}W (t, s)| ≤ \frac{−r_{1}r_{2}}{r_{1} − r_{2}} (\frac{e^{r_{1}T}}{1 + e^{r_{1}T}} + \frac{e^{r_{2}T}}{1 + e^{r_{2}T}} ) := G_{3}. (2.21)

3.Main results

Throughout this section, we assume that EJQTDE, 2009 No. 11, p. 6

(H_{1}) 3 ∫_{0}^{T} c(t)dt + 6pT < 2;

(H_{2}) There exist constant 0 < η < 1 and functions a, b, c, h ∈ C(J, [0, +∞)) such that

|f(t, u, v)| ≤ a(t)|u| + b(t)|u|^{η} + c(t)|v| + h(t); (H_{3}) There exist nonnegative constants α_{k}, β_{k}, γ_{k}, δ_{k} (k = 1, 2, ⋯, m) such that

|I_{k}(u)| ≤ α_{k}|u| + β_{k}, |I_{k}^{∗}(u)| ≤ γ_{k}|u| + δ_{k}, k = 1, ⋯ m. For convenience, let

a_{1} = \frac{3 (∫_{0}^{T} a(t)dt + 2qT + ∑_{i=1}^{m} γ_{i})}{2 − 3 ∫_{0}^{T} c(t)dt − 6pT}, a_{2} = \frac{3 ∫_{0}^{T} b(t)dt}{2 − 3 ∫_{0}^{T} c(t)dt − 6pT} ,

(3.1)

a_{3} = \frac{3 (∫_{0}^{T} h(t)dt + ∑_{i=1}^{m} δ_{i})}{2 − 3 ∫_{0}^{T} c(t)dt − 6pT}.

Theorem 3.1. Suppose that (H_{1}) − (H_{3}) hold. Further assume that

\frac{b_{1}T}{2(2 − c^{∗})} + \sqrt{\frac{b_{1}T}{2(2 − c^{∗})}} ∑_{i=1}^{m} α_{i} + \frac{m}{4} ∑_{i=1}^{m} α_{i}^{2} < 1, (3.2)

where

b_{1} = ∫_{0}^{T} a(t)dt + \frac{1}{2} ∫_{0}^{T} c(t)dt + ∑_{i=1}^{m}[(p + a_{1})α_{i} + γ_{i}],

a_{1} as in (3.1) and c^{∗} = max_{t∈J} c(t) < 2. Then BVP (1.1) has at least one solution. Proof. It is easy to check by Arzela-Ascoli theorem that the operator A is completely continuous. Assume that u is a solution of the equation

u = λAu, λ ∈ (0, 1). Then,

u′′(t) = λ(Au)′′(t) = λ[−f(t, u(t), u′(t)) − pu′(t) − qu(t) + p(Au)′(t) + q(Au)(t)]

= −λf(t, u(t), u′(t)) − p(λ − 1)u′(t) − q(λ − 1)u(t), (3.3)
−u(t)u′′(t) = λu(t)f(t, u(t), u′(t)) + p(λ − 1)u(t)u′(t) + q(λ − 1)u^{2}(t)
≤ λu(t)f(t, u(t), u′(t)) + p(λ − 1)u(t)u′(t). (3.4)

Integrating (3.3) from 0 to T, we get that

u′(T) − u′(0) = ∫_{0}^{T} u′′(t)dt + ∑_{i=1}^{m} I_{i}^{∗}

= −λ ∫_{0}^{T} f(t, u(t), u′(t))dt − p(λ − 1) ∫_{0}^{T} u′(t)dt − q(λ − 1) ∫_{0}^{T} u(t)dt + ∑_{i=1}^{m} I_{i}^{∗}. (3.5)

In view of u′(0) + u′(T) = 0, we obtain by (3.5) that

|u′(0)| ≤ \frac{1}{2} ∫_{0}^{T} |f(t, u(t), u′(t))|dt + p ∫_{0}^{T} |u′(t)|dt + q ∫_{0}^{T} |u(t)|dt + \frac{1}{2} ∑_{i=1}^{m} |I_{i}^{∗}|. (3.6)

Integrating (3.3) from 0 to t, we obtain that

u′(t) − u′(0) = ∫_{0}^{t} u′′(s)ds + ∑_{0<t_{i}<t} I_{i}^{∗}

= −λ ∫_{0}^{t} f(s, u(s), u′(s))ds − p(λ − 1) ∫_{0}^{t} u′(s)ds − q(λ − 1) ∫_{0}^{t} u(s)ds + ∑_{0<t_{i}<t} I_{i}^{∗}.

(3.7)

From (3.6), (3.7) and assumptions (H_{2}), (H_{3}), we have

|u′(t)| ≤ |u′(0)| + ∫_{0}^{T} |f(s, u(s), u′(s))|ds + 2p ∫_{0}^{T} |u′(s)|ds + 2q ∫_{0}^{T} |u(s)|ds + ∑_{i=1}^{m} |I_{i}^{∗}|

≤ \frac{3}{2} ∫_{0}^{T} (a(t)|u(t)| + b(t)|u(t)|^{η} + c(t)|u′(t)| + h(t))dt + 3p ∫_{0}^{T} |u′(t)|dt

+3q ∫_{0}^{T} |u(t)|dt + \frac{3}{2} ∑_{i=1}^{m}(γ_{i}∥u∥_{P C} + δ_{i})

≤ \frac{3}{2} (∥u∥_{P C} ∫_{0}^{T} a(t)dt + ∥u∥_{P C}^{η} ∫_{0}^{T} b(t)dt + ∥u′∥_{P C} ∫_{0}^{T} c(t)dt + ∫_{0}^{T} h(t)dt)

+3pT ∥u′∥_{P C} + 3qT ∥u∥_{P C} + \frac{3}{2} ∑_{i=1}^{m}(γ_{i}∥u∥_{P C} + δ_{i}),

that is,

∥u′∥_{P C} ≤ (\frac{3}{2} ∫_{0}^{T} a(t)dt + 3qT + \frac{3}{2} ∑_{i=1}^{m} γ_{i}) ∥u∥_{P C}

+∥u∥_{P C}^{η} ∫_{0}^{T} b(t)dt + (\frac{3}{2} ∫_{0}^{T} c(t)dt + 3pT ) ∥u′∥_{P C} + \frac{3}{2} ∫_{0}^{T} h(t)dt + \frac{3}{2} ∑_{i=1}^{m} δ_{i}.

Thus, in view of assumption (H_{1}), we have ∥u′∥_{P C} ≤ a_{1}∥u∥_{P C} + a_{2}∥u∥_{P C}^{η} + a_{3}, (3.8)

EJQTDE, 2009 No. 11, p. 8 where a_{1}, a_{2}, a_{3} are as in (3.1). Integrating (3.4) from 0 to T, we get that

− ∫_{0}^{T} u(t)u′′(t)dt ≤ λ ∫_{0}^{T} u(t)f(t, u(t), u′(t))dt + p(λ − 1) ∫_{0}^{T} u(t)u′(t)dt. (3.9) In view of u(0) + u(T) = 0, u′(0) + u′(T) = 0, we have

∫_{0}^{T} u(t)u′(t)dt = \frac{1}{2} ∫_{0}^{T} d(u^{2}(t))

= \frac{1}{2} [∫_{0}^{t_{1}} d((u′(t))^{2}) + ∫_{t_{1}}^{t_{2}} d(u^{2}(t)) + ⋯ + ∫_{t_{m}}^{T} d(u^{2}(t))]

= \frac{1}{2} [(u^{2}(t_{1} − 0) − u^{2}(0)) + (u^{2}(t_{2} − 0) − u^{2}(t_{1} + 0)) + ⋯ + (u^{2}(T) − u^{2}(t_{m} + 0))]

= \frac{1}{2} [(u^{2}(t_{1} − 0) − u^{2}(t_{1} + 0)) + (u^{2}(t_{2} − 0) − u^{2}(t_{2} + 0)) + ⋯ + (u^{2}(t_{m} − 0) − u^{2}(t_{m} + 0))]

= \frac{1}{2} [−(u(t_{1} − 0) + u(t_{1} + 0))I_{1} − (u(t_{2} − 0) + u(t_{2} + 0))I_{2} − ⋯ − (u(t_{m} − 0) + u(t_{m} + 0))I_{m}] ,

(3.10)

and

∫_{0}^{T} u(t)u′′(t)dt = ∫_{0}^{T} u(t)d(u′(t)) = ∫_{0}^{t_{1}} u(t)d(u′(t)) + ∫_{t_{1}}^{t_{2}} u(t)d(u′(t)) + ⋯ + ∫_{t_{m}}^{T} u(t)d(u′(t)) = u(t)u′(t)|_{0}^{t_{1}} − ∫_{0}^{t_{1}}(u′(t))^{2}dt + u(t)u′(t)|_{t_{1}}^{t_{2}} − ∫_{t_{1}}^{t_{2}}(u′(t))^{2}dt + ⋯ + u(t)u′(t)|_{t_{m}}^{T} − ∫_{t_{m}}^{T} (u′(t))^{2}dt

= u(t_{1} − 0)u′(t_{1} − 0) − u(0)u′(0) + u(t_{2} − 0)u′(t_{2} − 0) − u(t_{1} + 0)u′(t_{1} + 0)

+ ⋯ + u(T)u′(T) − u(t_{n} + 0)u′(t_{n} + 0) − ∫_{0}^{T} (u′(t))^{2}dt = u(t_{1} − 0)u′(t_{1} − 0) − u(t_{1} + 0)u′(t_{1} + 0) + ⋯

+u(t_{n} − 0)u′(t_{m} − 0) − u(t_{m} + 0)u′(t_{m} + 0) − ∫_{0}^{T} (u′(t))^{2}dt = u(t_{1} − 0)u′(t_{1} − 0) − u(t_{1} − 0)u′(t_{1} + 0) + u(t_{1} − 0)u′(t_{1} + 0) − u(t_{1} + 0)u′(t_{1} + 0)

+ ⋯ + u(t_{m} − 0)u′(t_{m} − 0) − u(t_{m} − 0)u′(t_{m} + 0)

+u(t_{m} − 0)u′(t_{m} + 0) − u(t_{m} + 0)u′(t_{m} + 0) − ∫_{0}^{T} (u′(t))^{2}dt

= −u(t_{1} − 0)I_{1}^{∗} − u′(t_{1} + 0)I_{1} − ⋯ − u(t_{m} − 0)I_{n}^{∗} − u′(t_{m} + 0)I_{m} − ∫_{0}^{T} (u′(t))^{2}dt. (3.11) Substituting (3.10) and (3.11) into (3.9), we obtain by (H_{2}), (H_{3}) and (3.8) that

∫_{0}^{T} (u′(t))^{2}dt ≤ λ ∫_{0}^{T} u(t)f(t, u(t), u′(t))dt − u(t_{1} − 0)I_{1}^{∗} − u′(t_{1} + 0)I_{1} − ⋯ − u(t_{m} − 0)I_{m}^{∗}

−u′(t_{m} + 0)I_{m} + \frac{p(1 − λ)}{2} [(u(t_{1} − 0) + u(t_{1} + 0))I_{1} + ⋯ + (u(t_{m} − 0) + u(t_{m} + 0))I_{m}]

≤ λ ∫_{0}^{T} u(t)f(t, u(t), u′(t))dt + ∥u∥_{P C} ∑_{i=1}^{m} |I_{i}^{∗}| + ∥u′∥_{P C} ∑_{i=1} |I_{i}| + p(1 − λ)∥u∥_{P C} ∑_{i=1}^{m} |I_{i}|

≤ λ ∫_{0}^{T} u(t)f(t, u(t), u′(t))dt + ∥u∥_{P C} ∑_{i=1}^{m}(p|I_{i}| + |I_{i}^{∗}|) + ∥u′∥_{P C} ∑_{i=1}^{m} |I_{i}|

≤ ∫_{0}^{T} (a(t)u^{2}(t) + b(t)|u(t)|^{1+η} + \frac{1}{2}c(t)(u^{2}(t) + (u′(t))^{2}) + h(t))dt

+ ∑_{i=1}^{m}[p(α_{i}∥u∥_{P C} + β_{i}) + γ_{i}∥u∥_{P C} + δ_{i}]∥u∥_{P C} + ∑_{i=1}^{m}(α_{i}∥u∥_{P C} + β_{i})∥u′∥_{P C}

≤ ∥u∥_{P C}^{2} ∫_{0}^{T} a(t)dt + ∥u∥_{P C}^{1+η} ∫_{0}^{T} b(t)dt + \frac{1}{2}∥u∥_{P C}^{2} ∫_{0}^{T} c(t)dt + \frac{c^{∗}}{2} ∫_{0}^{T} (u′(t))^{2}dt

+ ∫_{0}^{T} h(t)dt + ∑_{i=1}^{m}(pα_{i} + γ_{i})∥u∥_{P C}^{2} + ∑_{i=1}^{m}((pβ_{i} + δ_{i})∥u∥_{P C}

+a_{1} ∑_{i=1}^{m} α_{i}∥u∥_{P C}^{2} + a_{2} ∑_{i=1}^{m} α_{i}∥u∥_{P C}^{1+η} + a_{2} ∑_{i=1}^{m} β_{i}∥u∥_{P C}^{η} + ∑_{i=1}^{m}(a_{1}β_{i} + a_{3}α_{i})∥u∥_{P C} + a_{3} ∑_{i=1}^{m} β_{i}.

Thus,

∫_{0}^{T} (u′(t))^{2}dt ≤ \frac{1}{1 − c^{∗}/2} [b_{1}∥u∥_{P C}^{2} + b_{2}∥u∥_{P C}^{1+η} + b_{3}∥u∥_{P C} + b_{4}∥u∥_{P C}^{η} + b_{5}], (3.12)

where

b_{1} = ∫_{0}^{T} a(t)dt + \frac{1}{2} ∫_{0}^{T} c(t)dt + ∑_{i=1}^{m}((p + a_{1})α_{i} + γ_{i}), b_{2} = ∫_{0}^{T} b(t)dt + a_{2} ∑_{i=1}^{m} α_{i},

(3.13)

b_{3} = ∑_{i=1}^{m}((p + a_{1})β_{i} + a_{3}α_{i} + δ_{i}), b_{4} = a_{2} ∑_{i=1}^{m} β_{i}, b_{5} = ∫_{0}^{T} b(t)dt + a_{3} ∑_{i=1}^{m} β_{i}.

By Lemma 2.2 and (3.12), we have

∥u∥_{P C}^{2} ≤ \frac{1}{4} (∫_{0}^{T} |u′(t)|dt)^{2} + \frac{1}{2} ∫_{0}^{T} |u′(t)|dt ∑_{i=1}^{m} |I_{i}| + \frac{1}{4} (∑_{i=1}^{m} |I_{i}|)^{2}

≤ \frac{T}{4} ∫_{0}^{T} (u′(t))^{2}dt + \frac{\sqrt{T}}{2} (∫_{0}^{T} (u′(t))^{2}dt)^{1/2} ∑_{i=1}^{m} |I_{i}| + \frac{1}{4}m ∑_{i=1}^{m} |I_{i}|^{2}

≤ \frac{T}{2(2 − c^{∗})} [b_{1}∥u∥_{P C}^{2} + b_{2}∥u∥_{P C}^{1+η} + b_{3}∥u∥_{P C} + b_{4}∥u∥_{P C}^{η} + b_{5}]

+\sqrt{\frac{T}{2(2 − c^{∗})}} [b_{1}∥u∥_{P C}^{2} + b_{2}∥u∥_{P C}^{1+η} + b_{3}∥u∥_{P C} + b_{4}∥u∥_{P C}^{η} + b_{5}]^{1/2} ∑_{i=1}^{m}(α_{i}∥u∥_{P C} + β_{i})

+\frac{m}{4} ∑_{i=1}^{m}(α_{i}^{2}∥u∥_{P C}^{2} + 2α_{i}β_{i}∥u∥_{P C} + β_{i}^{2}).

It follows from the above inequality and condition (3.2) that there exists M_{1} > 0 such that ∥u∥_{P C} ≤ M_{1}. Thus, we get by (3.8) that

∥u′∥_{P C} ≤ a_{1}M_{1} + a_{2}M_{1}^{η} + a_{3} := M_{2}. (3.14)

Thus, ∥u∥_{P C^{1}} ≤ max{M_{1}, M_{2}}. It follows from Lemma 1.1 that BVP (1.1) has at least one solution. The proof is complete. □

Corollary 3.2. Assume that (H_{1}), (H_{2}) hold. Suppose that there exist nonnegative constants β_{k}, δ_{k} (k = 1, 2, ⋯, m) such that

(H_{4}) |I_{k}(u)| ≤ β_{k}, |I_{k}^{∗}(u)| ≤ δ_{k}, k = 1, ⋯ m,

holds. Further assume that

∫_{0}^{T} a(t)dt + \frac{1}{2} ∫_{0}^{T} c(t)dt < 2(2 − c^{∗}), (3.15)

where c^{∗} = max_{t∈J} c(t) < 2. Then BVP (1.1) has at least one solution.

Proof. Set α_{k} = γ_{k} = 0, k = 1, 2, ⋯, m. Then (H_{3}) reduces to (H_{4}), and (3.1) reduces to (3.15). So, by Theorem 3.1, we know that Corollary 3.2 holds.

Theorem 3.3. Suppose that there exist constants K_{1}, K_{2}, and L_{k}, l_{k} (k = 1, 2, ⋯, m) such that

|f(t, u, v) − f(t, x, y)| ≤ K_{1}|u − x| + K_{2}|v − y|, ∀u, v, x, y ∈ R,

and

|I_{k}(u) − I_{k}(v)| ≤ L_{k}|u − v|, |I_{k}^{∗}(u) − I_{k}^{∗}(v)| ≤ l_{k}|u − v|, ∀u, v ∈ R.

Moreover suppose that

(K_{1} + K_{2} + p + q)T max{G_{1}, G_{2}} + ∑_{k=1}^{m}(max{G_{1}, G_{2}}L_{k} + max{G_{2}, G_{3}}l_{k}) < 1, (3.16)

where G_{1}, G_{2} and G_{3} are as in (2.18), (2.20) and (2.21), respectively, then BVP (1.1) has a unique solution.

Proof. From (2.15), we have

|Au(t) − Av(t)| = |∫_{0}^{T} G(t, s)[(f(s, u(s), u′(s)) − f(s, v(s), v′(s)) + p(u′(s) − v′(s))

+q(u(s) − v(s))]ds + ∑_{k=1}^{m}[G(t, t_{k})(I_{k}^{∗}(v(t_{k})) − I_{k}^{∗}(u(t_{k}))) + W (t, t_{k})(I_{k}(u(t_{k})) − I_{k}(v(t_{k})))]|

≤ ∫_{0}^{T} |G(t, s)|[|f(s, u(s), u′(s)) − f(s, v(s), v′(s)| + p|u′(s) − v′(s)|

+q|u(s) − v(s)|]ds + ∑_{k=1}^{m}[|G(t, t_{k})||I_{k}^{∗}(v(t_{k})) − I_{k}^{∗}(u(t_{k}))| + |W (t, t_{k})||I_{k}(u(t_{k})) − I_{k}(v(t_{k}))|]

≤ ∫_{0}^{T} |G(t, s)|[K_{1}|u(s) − v(s)| + K_{2}|u′(s) − v′(s)| + p|u′(s) − v′(s)|

+q|u(s) − v(s)|]ds + ∑_{k=1}^{m}[|G(t, t_{k})|L_{k}|u(t_{k})) − v(t_{k}))| + |W (t, t_{k})|l_{k}|u(t_{k})) − v(t_{k}))|]

≤ (K_{1} + K_{2} + p + q)∥u − v∥_{P C^{1}} ∫_{0}^{T} |G(t, s)|ds + ∑_{k=1}^{m}[|G(t, t_{k})|L_{k} + |W (t, t_{k})|l_{k}]∥u − v∥_{P C^{1}}

≤ [(K_{1} + K_{2} + p + q)T G_{1} + ∑_{k=1}^{m}(G_{1}L_{k} + G_{2}l_{k})]∥u − v∥_{P C^{1}}.

|(Au)′(t) − (Av)′(t)| = |∫_{0}^{T} \frac{∂}{∂t}G(t, s)[(f(s, u(s), u′(s)) − f(s, v(s), v′(s)) + p(u′(s) − v′(s))

+q(u(s) − v(s))]ds + ∑_{k=1}^{m}[\frac{∂}{∂t}G(t, t_{k})(I_{k}^{∗}(v(t_{k})) − I_{k}^{∗}(u(t_{k}))) + \frac{∂}{∂t}W (t, t_{k})(I_{k}(u(t_{k})) − I_{k}(v(t_{k})))]|

≤ ∫_{0}^{T} |\frac{∂}{∂t}G(t, s)| [K_{1}|u(s) − v(s)| + K_{2}|u′(s) − v′(s)| + p|u′(s) − v′(s)|

+q|u(s) − v(s)|]ds + ∑_{k=1}^{m} [|\frac{∂}{∂t}G(t, t_{k})| L_{k}|u(t_{k})) − v(t_{k}))| + |\frac{∂}{∂t}W (t, t_{k})| l_{k}|u(t_{k})) − v(t_{k}))|]

≤ (K_{1} + K_{2} + p + q)∥u − v∥_{P C^{1}} ∫_{0}^{T} |\frac{∂}{∂t}G(t, s)| ds + ∑_{k=1}^{m} [|\frac{∂}{∂t}G(t, t_{k})| L_{k}

+ |\frac{∂}{∂t}W (t, t_{k})| l_{k}] ∥u − v∥_{P C^{1}}

≤ [(K_{1} + K_{2} + p + q)T G_{2} + ∑_{k=1}^{m}(G_{2}L_{k} + G_{3}l_{k})]∥u − v∥_{P C^{1}}.

Hence,

∥Au − Av∥_{P C} ≤ [(K_{1} + K_{2} + p + q)T G_{1} + ∑_{k=1}^{m}(G_{1}L_{k} + G_{2}l_{k})]∥u − v∥_{P C^{1}},

∥(Au)′ − (Av)′∥_{P C} ≤ [(K_{1} + K_{2} + p + q)T G_{2} + ∑_{k=1}^{m}(G_{2}L_{k} + G_{3}l_{k})]∥u − v∥_{P C^{1}}.

Thus, we obtain

∥Au − Av∥_{P C^{1}} ≤ [(K_{1} + K_{2} + p + q)T max{G_{1}, G_{2}}

+ ∑_{k=1}^{m}(max{G_{1}, G_{2}}L_{k} + max{G_{2}, G_{3}}l_{k})]∥u − v∥_{P C^{1}}.

In view of (3.16) and Banach fixed point theorem, A has a unique fixed point. The proof is complete. □

4. Example In this section, we give an example to illustrate the effectiveness of our results.

Example 4.1. Consider the problem

\{ \begin{matrix}u′′(t) + u(t) sin^{2} t − e^{t}u^{1/2}(t) + \frac{1}{2}u′(t) cos^{2} t + t^{2} + sin 2t = 0, t ∈ [0, \frac{π}{2} ] \ \{\frac{1}{3}, \frac{2}{3}\}, \\ △u (\frac{1}{3}) = sin (u (\frac{1}{3})) , △u (\frac{2}{3}) = cos (u (\frac{2}{3})) , \\ △u′ (\frac{1}{3}) = 1 + \frac{1}{2} cos^{2} (u (\frac{1}{3})) , △u′ (\frac{2}{3}) = \frac{1}{3} sin (u (\frac{2}{3})) , \\ u(0) + u (\frac{π}{2} ) = 0, u′(0) + u′ (\frac{π}{2} ) = 0, \end{matrix} (4.1)

Let f(t, u, v) = u(t) sin^{2} t − e^{t}u^{1/2}(t) + \frac{v}{2} cos^{2} t + t^{2} + sin 2t, I_{1}(u) = sin u, I_{2}(u) = cos u, I_{1}^{∗}(u) = 1 + \frac{1}{2} cos^{2} u, I_{2}^{∗}(u) = \frac{1}{3} sin u, T = \frac{π}{2} , J = [0, \frac{π}{2} ]. It is easy to show that

|I_{1}(u)| ≤ 1, |I_{2}(u)| ≤ 1, |I_{1}^{∗}(u)| ≤ \frac{3}{2}, |I_{2}^{∗}(u)| ≤ \frac{1}{3}, and

|f(t, u, v)| ≤ a(t)|u| + b(t)|u|^{1/2} + c(t)|v| + h(t),

where

a(t) = sin^{2} t, b(t) = e^{t}, c(t) = \frac{1}{2} cos^{2} t, h(t) = t^{2} + sin 2t. (4.2)

Thus, (H_{2}) and (H_{4}) hold. Let p = 0 and q = \frac{1}{4}, we have by (3.1) and (4.2) that

a_{1} = \frac{12π}{16 − 3π} , a_{2} = \frac{24(e^{π/2} − 1)}{16 − 3π}, a_{3} = \frac{π^{3} + 68}{16 − 3π} ,

and

3 ∫_{0}^{T} c(t)dt + 6pT = \frac{3}{2} ∫_{0}^{π/2} cos^{2} tdt = \frac{3π}{8} < 2,

which implies that (H_{1}) holds. Moreover, we see that

b_{1} = ∫_{0}^{π/2} a(t)dt + \frac{1}{2} ∫_{0}^{π/2} c(t)dt = \frac{5}{16}π < 3 = 2(2 − c^{∗}),

where c^{∗} = max_{t∈J} c(t) = \frac{1}{2}. Thus, (3.15) holds. So, all the conditions of Corollary 3.2 are satisfied. By Corollary 3.2, anti-period boundary value problem (4.1) has at least one solution.

Acknowledgements

The author is very grateful to the referees for careful reading of the original manuscript and for valuable suggestions on improving this paper. Project supported by the National Natural Science Foundation of China (10771212) and the Natural Science Foundation of Jiangsu Education Office (06KJB110010).

References

[1] A. R. Aftabizadeh, S. Aizicovici, N.H. Pavel, On a class of second-order anti-periodic boundary value problems, J. Math. Anal. Appl. 171 (1992) 301-320.

[2] D. Franco, J.J. Nieto, D. O’Regan, Anti-periodic boundary value problem for non-linear first order ordinary differential equations, Math. Inequal. Appl. 6 (2003) 477-485.

[3] T. Jankowski, Ordinary differential equations with anti-periodic and nonlinear boundary value conditions of anti-periodic type, Computer Math. Applic. 47 (2004) 1429-1436.

[4] Y. Yin, Remarks on first order differential equations with anti-periodic and nonlinear boundary value conditions, Nonlinear Times Digest. 2 (1995) 83-94.

[5] Y. Yin, Monotone iterative technique and quasilinearization for some anti-periodic problem, Nonlinear World. 3 (1996) 253-266.

[6] C. Ahn, C. Rim, Boundary flows in general coset theories, J. Phys. A. 32 (1999) 2509-2525.

[7] A. Abdurrahman, F. Anto, J. Bordes, Half-string oscillator approach to string field theory (ghost sector 2), Nuclear Phys. B. 397 (1993) 260-282.

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

EJQTDE, 2009 No. 11, p. 14 [9] A.M. Samoilenko, N.A. Perestyuk. Impulsive Differential Equations, World Scientific,

Singapore, 1995.

[10] S.T. Zavalishchin, A.N. Sesekin, Dynamic Impulse Systems. Theory and applications, Kluwer Academic Publishers Group, Dordrecht, 1997.

[11] X. Liu (Ed.), Advances in Impulsive Differential Equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 9 (2002) 313-462.

[12] T. Jankowski, Existence of solutions for second order impulsive differential equations with deviating arguments, Nonlinear Anal. 67 (2007) 1764-1774.

[13] D. Franco, J.J. Nieto, First-order impulsive ordinary differential equations with antiperiodic and nonlinear boundary value conditions, Nonlinear Anal. 42 (2000) 163-173.

[14] Z. Luo, J. Shen, J. Nieto, Anti-periodic boundary value problem for first-order impulsive ordinary differential equation, Computer Math. Applic. 49 (2005) 253-261.

[15] W. Wang, J. Shen, Existence of solution for anti-periodic boundary value problems, Nonlinear Anal. 79 (2009) 598-605.

[16] N. G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, no. 73, Cambridge University Press, Cambridge, UK, 1978.

(Received December 16, 2008)