Dirichlet Problem with One Impulse Condition

Abstract

The chapter deals with the second order Dirichlet boundary value problem with one state-dependent impulse condition $$\begin{aligned}&z''(t) = f(t,z(t)) \quad \text {for a.e. }t \in [0,T]\subset \mathbb {R}, \\&z'(t+) - z'(t-) = M(z(t)), \quad t = \gamma (z(t)), \\&\qquad \quad z(0) = 0, \quad z(T) = 0. \end{aligned}$$ Proofs of the main results make use of a new approach to boundary value problems with state-dependent impulses which is based on a transformation to a fixed point problem of an appropriate operator in the space \({\mathbb C}^1([0,T])\times {\mathbb C}^1([0,T])\). Sufficient conditions for the existence of solutions to the problem are given. The presented approach is extended to more impulses and to other boundary conditions in the next chapters.