Some Local Existence Results on an Integral Equation in a Banach Space.

摘要

A local existence result is proved for the nonlinear integral equation u'(t) + the integral from 0 to t of a(t-tau)Au(tau)d tau = f(t), t > or = 0, where the kernel a of the equation is a real-valued function, A is a nonlinear monotone mapping and u takes values in a Banach space. The author's aim is to obtain results on 'bad' nonlinearities and to obtain these he finds it necessary to make rather strong assumptions on the kernel. The results are used to obtain 'approximating' solutions to the related nonlinear hyperbolic differential equation (u double prime)(t) + Au(t) = f(t), t > or = 0.