A projection-iterative method for solving the periodic problem for integrodifferential equations with impulse effect (Q1119387)

The authors study a projection-iterative method for periodic solutions of nonlinear integro-differential equations of Volterra type: \[ \dot x(t)=X(t,x(t),\int^{t}_{t-\omega}\phi (t,s,x(s))ds),\quad t\neq t \] and \(\Delta x|_{t=t_ i}:=x(t_ i+0)-x(t_ i-0)=I_ i(x(t_ i- 0))\), where \(x\in {\mathbb{R}}^ n\), \(\phi \in {\mathbb{R}}^ m\), \(\omega >0\) and \(t_ i<t_{i+1}\) for \(i\in {\mathbb{Z}}\), with \(\lim_{i\to \pm \infty}t_ i=\pm \infty\). Under some Lipschitz conditions on the functions involved one proves existence, uniqueness and approximation by Fourier series of the periodic solution.