Higher Order Equation with General Linear Boundary Conditions

Abstract

The chapter investigates the nth order ($n\ge 2$) differential equation

$$\begin{aligned} \begin{aligned} \sum _{j=0}^{n} a_j(t) z^{(j)}(t) = h(t,z(t),\ldots ,z^{(n-1)}(t)) \quad \text {for a.e. }t\in [a,b]\subset \mathbb {R}\end{aligned} \end{aligned}$$

with the state-dependent impulses

$$\begin{aligned}\begin{aligned} z^{(j)}(t+) - z^{(j)}(t-) = J_{ij}(z(t-),z'(t-),\ldots ,z^{(n-1)}(t-)), \quad j=0,\ldots ,n-1, \end{aligned} \end{aligned}$$

where the impulse points $t\in (a,b)$ are determined as solutions of the equations

$$\begin{aligned} \begin{aligned} t = \gamma _i(z(t-),z'(t-),\ldots ,z^{(n-2)}(t-)),\quad i = 1,\ldots ,p \end{aligned} \end{aligned}$$

Here $n,p\in \mathbb {N}$, the functions $a_j/a_n$, $j=0,\dots ,n-1$, are Lebesgue integrable on [a, b] and $h/a_n$ satisfies the Carathéodory conditions on $[a,b]\times \mathbb {R}^n$. The impulse functions $J_{ij}$, $i=1,\ldots ,p$, $j=0,\ldots ,n-1$, and the barrier functions $\gamma _i$, $i = 1,\dots ,p$, are continuous on $\mathbb {R}^n$ and $\mathbb {R}^{n-1}$, respectively. This impulsive differential equation is subject to the general linear boundary conditions

$$\begin{aligned} \begin{aligned} \ell _j(z,z',\ldots ,z^{(n-1)}) = c_j, \quad j = 1,\ldots ,n, \end{aligned} \end{aligned}$$

where $c_1,\ldots , c_n \in \mathbb {R}$ and the functionals $\ell _j$, $j=1,\dots ,n$, are linear and bounded on the space of left-continuous regulated (i.e. having finite one-sided limits at each point) on [a, b] vector-valued functions. Provided the data functions h and $J_{ij}$ are bounded, transversality conditions guaranteeing that each possible solution of the problem in a given region crosses each barrier $\gamma _i$ at a unique impulse point $t = \tau _i$ are presented, and consequently the existence of a solution to the problem is proved.