Abstract
The aim of this paper is to review some classical results and present new theorems of existence and extendability of solutions to the following second-order nonlinear Cauchy problem:
(P) {u ''(x) = f(x, u(x)), 0<x<T u(0) = u(0) u'(0) = u(1)
where f: [0, T] x R -> R is continuous, derivative independent and T > 0, u(0), u(1) are real numbers. We first consider the model case of autonomous equations and some classical results are surveyed. Particular attention is then paid to the cases where f has either the form f(x, u) = q(x)psi(u) or f(x,u) = h(x, u) + g(x, u) where psi is nondecreasing, h is dominated by nondecreasing functions in u while g looks like a constraction in the second argument. Methods based on the Leray-Schauder nonlinear alternative and the Krasnosel'skii fixed point theorem are used to prove new existense theorems. Several examples and counterexamples illustrate the obtained results.