Volterra and Urysohn integral equations in Banach spaces (Q1277067)

The author studies the existence of solutions of Volterra integral equations of the form \[ y(t) = h(t) + \int_0^t K(t,s,y(s)) ds, \quad 0\leq t < T, \] as well as the corresponding Urysohn integral equation \(y(t) = h(t) + \int_0^T K(t,s,y(s)) ds\) in a Banach space. First an existence principle, which relies on a nonlinear alternative of Leray-Schauder type and measures of noncompactness, is established and this is used in the proofs of the existence results presented later. An application to the abstract Dirichlet problem \(y''(t) + f(t,y(t)) =0\), \(y(0)=y(1)=0\), involving the notion of ``solution tube'' is given as well.