Existence of solutions of second order Volterra integrodifferential equations in Banach spaces (Q2709191)

Let \(X\) be a Banach space, let \(A\) be the infinitesimal generator of the strongly continuous cosine family of bounded linear operators on \(X\), let \(g\) be a nonlinear mapping from \(R\times R\times X\) to \(X\), and let \(f\) be a nonlinear mapping from \(R\times X\times X\) to \(X\). Under suitable assumptions, the authors prove that the nonlinear second order Volterra integrodifferential equation \(x''(t)=Ax(t)+f(t,x(t)\), \(\int_0^tg(t,s,x(s)) ds)\), \(t\geq 0\), with the initial conditions \(x(0)=x_0\in X\) and \(x'(0)=y_0\in X\), has at least one mild solution.