Periodic boundary value problem for second-order impulsive functional differential equations

Abstract

This paper considers the existence of extreme solutions of periodic boundary value problem for the second-order impulsive functional differential equations. We prove some new maximum principles. This allows us to introduce a new definition of lower and upper solutions and present that the method of lower and upper solutions coupled with monotone iterative technique is still valid. Meanwhile, we extend previous results.

Introduction

In the mathematical simulation in various fields of science and technology, impulsive differential equations are often used [1], [2]. This leads to necessity of justification of methods for their approximate solution, in particular for periodic solution [19], [20], [21], [22]. The existence analysis of periodic boundary value problems of ordinary or functional differential equations with impulsive effects has been the subject of many investigations in recent years, and various interesting results have been reported, we refer the readers to the articles [3], [4], [5], [6], [8], [10], [11], [12], [13], [14], [15], [16], [17], [18]. Recently, Nieto and Rodriguez-Lopez [7], [9] consider periodic boundary value problem for the following first-order functional differential equations

$$\left\{\begin{array}{c}{u}^{\prime }(t)=g(t\text{,}u(t)\text{,}u(\theta (t)))\text{,}t\ne t_k\text{,}t\in J=[0\text{,}T]\text{,}\hfill \\ u(0)=u(T).\hfill \end{array}\right.$$

In their paper, they introduce a new concept of lower and upper solutions. Motivated by Nieto and Rodriguez-Lopez [7], [9], [12], in this paper, we present the similar definition of lower and upper solutions for the second-order impulsive functional differential equations,

$$\left\{\begin{array}{c}-u^{″}(t)=f(t\text{,}u(t)\text{,}u(\theta (t)))\text{,}t\ne t_k\text{,}t\in J=[0\text{,}T]\text{,}\hfill \\ \mathrm{\Delta }u{|}_{t=t_k}=I_k(u(t_k))\text{,}\hfill \\ \mathrm{\Delta }{u}^{\prime }{|}_{t=t_k}=I_k^{\ast }(u(t_k))\text{,}k=1\text{,}2\text{,}\dots \text{,}m\text{,}\hfill \\ u(0)=u(T)\text{,}{u}^{\prime }(0)=u^{\prime }(T).\hfill \end{array}\right.$$

where $f\in C(J\times R^2\text{,}R)\text{,}0⩽\theta (t)⩽t\text{,}t\in J\text{,}0<t_1<t_2<\cdots <t_m<T\text{,}{I}_k\text{,}{I}_k^{\ast }\in C(R\text{,}R)\text{,}\mathrm{\Delta }u{|}_{t=t_k}=u(t_k^{+})-u(t_k^{-})\text{,}\mathrm{\Delta }{u}^{\prime }{|}_{t=t_k}=u^{\prime }(t_k^{+})-u^{\prime }(t_k^{-})\text{,}k=1\text{,}2\text{,}\dots \text{,}m$. In recent paper [11], the authors also investigated the PBVP (1), they claimed the following assertion:

Lemma 1 [11]

Assume that $u\in E\cap C^2(J^{-}\text{,}R)$ satisfies

$$\begin{array}{cc}\hfill & -u^{″}(t)+\mathit{Mu}(t)+\mathit{Nu}(\theta (t))⩽0\text{,}t\ne t_k\text{,}t\in J\text{,}\hfill \\ \hfill & \mathrm{\Delta }u{|}_{t=t_k}⩾L_k{u}^{\prime }(t_k)\text{,}\hfill \\ \hfill & \mathrm{\Delta }{u}^{\prime }{|}_{t=t_k}⩾L_k^{\ast }u(t_k)\text{,}k=1\text{,}2\text{,}\dots \text{,}m\text{,}\hfill \\ \hfill & u(0)=u(T)\text{,}{u}^{\prime }(0)=u^{\prime }(T)\text{,}\hfill \end{array}$$

where constants $M>0\text{,}N⩾0\text{,}{L}_k⩾0$, $L_k^{\ast }⩾0(k=1\text{,}2\text{,}\dots \text{,}m)$, and they satisfy

$$\left(\sum _{k=1}^m{L}_k^{\ast }+a(m+1)(M+N)\right)\left(\sum _{k=1}^m{L}_k+a(m+1)\right)⩽\frac{1}{2}\text{.}$$

Then, u(t) ⩽ 0 for all t ∈ J.

Unfortunately this statement is wrong. Lemma 1 means $\mathrm{\Delta }u{|}_{t=t_k}⩾L_k{u}^{\prime }(t_k)$, but in the proof of the lemma, it must satisfy $\mathrm{\Delta }u{|}_{t=t_k}=L_k{u}^{\prime }(t_k)$. Therefore the new lower and upper solutions in their paper are not valid. The aim of this paper is to introduce a new definition of lower and upper solutions.

Denote $J^{-}=J⧹\{t_i\text{,}i=1\text{,}2\text{,}\dots \text{,}m\}$. Let $\mathit{PC}(J\text{,}R)=\{u:J\to R\text{;}u(t)$is continuous everywhere except for some t_k at which $u(t_k^{+})$ and $u(t_k^{-})$ exist and $u(t_k^{-})=u(t_k)\text{,}k=1\text{,}2\text{,}\dots \text{,}m\}$; ${\mathit{PC}}^{\prime }(J\text{,}R)=\{u\in \mathit{PC}(J\text{,}R)\text{;}{u}^{\prime }(t)$ is continuous everywhere except for some t_k at which $u^{\prime }(t_k^{+})$ and $u^{\prime }(t_k^{-})$ exist and $u^{\prime }(t_k^{-})=u^{\prime }(t_k)\text{,}k=1\text{,}2\text{,}\dots \text{,}m\}$.

Let $E=\{u\in {\mathit{PC}}^{\prime }(J\text{,}R)\}$ with norm

$$\parallel u{\parallel }_{{\mathit{PC}}^{\prime }}=\max \{\parallel u(t){\parallel }_{\mathit{PC}}\text{,}\parallel u^{\prime }(t){\parallel }_{\mathit{PC}}\}\text{,}$$

where $\parallel u{\parallel }_{\mathit{PC}}={\sup }_{t\in J}|u(t)|$, then E is a Banach space. A function $u\in E\cap C^2(J^{-}\text{,}R)$is called a solution of PBVP (1) if it satisfies (1).

This paper is organized as follows. In Section 2, we present new comparison results. In Section 3, we introduce a new more general concept of upper and lower solutions relative to problem (1). These results are an important tool to develop the monotone iterative technique for (1) and to obtain two sequences approximating the extremal solutions of this problem between lower and upper solutions (see section 4).