Existence of solutions of a class of abstract second order nonlinear integrodifferential equations
The following nonlinear second order integrodifferential equation is investigated \[ Kx''(t)= Ax(t)+f\left(t,x(t), \int^t_0 g\bigl(t,x (s), x'(s)\bigr)ds, x'(t)\right), \;t\in J=[0,T] \tag{1} \] together with the initial conditions \(x(0)=x_0\), \(x'(0)=y_0\). Here \(K\) and \(A\) are linear operators with domains contained in a Banach space \(X\) and with values in a Banach space \(Y\) and the functions \(g:J\times J\times X\times X\to X\) and \(f:J \times X\times X\times X\to Y\) being continuous and sublinear. Moreover, the operator \(K\) is continuously invertible with \(AK^{-1}\) generating a strongly continuous cosine family of bounded linear operators on \(X\). Under some extra assumptions an existence theorem for the problem (1) is proved with help of the Schaefer fixed point theorem.
nonlinear second order integrodifferential equation
scientific article; zbMATH DE number 1844035