On linear integro-differential equations of Barbashin type in spaces of continuous and measurable functions (Q1106454)

This paper surveys several important properties of linear integro- differential equations \[ (1)\quad du/dt=A(t)u \] of Barbashin type \[ (2)\quad A(t)x(s)=c(t,s)x(s)+\int^{b}_{a}k(t,s,\sigma)x(\sigma)d\sigma, \] especially those related to the geometric structure of the underlying function space. In contrast to Barbashin's classical results, also discontinuous data (e.g. kernel functions) are allowed. After discussing several classes of suitable kernels, the resolvent operator (Cauchy function) generated by the operator (2) is described. Moreover, stability results for equation (1) are proved. Finally, representation formulas for the corresponding Green's function are given; a perturbed version of such formulas applies to averaging procedures of Bogolyubov-Krylov type.