First Order System with Linear Boundary Conditions

Abstract

The chapter generalizes and extends the previous results contained in Chaps. 6–9. We study the solvability of a boundary value problem with state-dependent impulses of the form

where the impulse instants $t \in (a,b)$ are a priori unknown and depend on z. Here, $n,p\in \mathbb {N}$, $\mathbf {c} \in \mathbb {R}^n$, the vector function $\mathbf {f}$ satisfies the Carathéodory conditions on $[a,b]\times \mathbb {R}^n$. The impulse functions $\mathbf {J}_i$, $i=1,\ldots ,p$, are continuous on $[a,b]\times \mathbb {R}^n$, and the barrier functions $\gamma _i$, $i = 1,\ldots ,p$, are continuous on $\mathbb {R}^n$. The operator $\ell$ is an arbitrary linear and bounded operator on the space of left-continuous regulated on [a, b] vector valued functions. Arbitrary linear local and nonlocal boundary conditions are covered by $\ell$. Under the assumptions of boundedness of data functions $\mathbf {f}$ and $\mathbf {J}_i$ and transversality conditions, the existence result for a solution of the investigated boundary value problem having exactly one cross through each barrier $t = \gamma _i(\mathbf {x})$ is guaranteed.