The second-order Cauchy problem in a scale of Banach spaces with vector-valued measures of noncompactness and an application to Kirchhoff equations

Abstract

In the paper, by using Darbo-Sadovskii fixed point theorem for condensing operators on Fréchet spaces with respect to vector-valued measure of noncompactness, we prove the existence results for the second-order Cauchy problem u (t) = f (t, u(t)), t ∈ (0, T ), u(0) = u 0 , u (0) = u 1 , in a scale of Banach spaces. The result is applied to the Kirchhoff equations in Gevrey class.