Lyapunov-Bohl exponent and Green's function for some class of integro-differential equations
The aim of this paper is to study the solutions of the following nonlinear integro-differential equation of Barbashin type: \[ {{\partial x (t,s)} \over {\partial t}}= c(t, s) x(t, s)+ \int^b_a k(t, s, \sigma, x(t, \sigma)) d\sigma, \tag{1} \] and the linear integro-differential equation \[ {{\partial x (t, s)} \over {\partial t}}= c(t, s) x(t, s)+ \int^b_a k(t, s, \sigma) x(t, \sigma) d\sigma. \tag{2} \] The equation (1) can be written as a differential equation \[ {{dx} \over {dt}}= c(t) x+ k(t,x), \tag{3} \] with multiplication operator \((c(t) x) (s)= c(t, s) x(s)\) and integral operator \[ k(t, x) (s)= \int^b_a k(t, s, \sigma, x(\sigma)) d\sigma. \] The author discusses the upper estimates for Lyapunov-Bohl exponent of (1) and (2), and gives necessary and sufficient conditions for the exponential dichotomy of the differential equation (3).
nonlinear integro-differential equation of Barbashin type
scientific article; zbMATH DE number 856798