Global solutions to a class of second-order differential equations

Abstract EN

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 : where f : [0, T] × R → R is continuous, derivative independent and T > 0, u0, u1 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)ψ(u) or f (x, u) = h(x, u) + g (x, u) where ψ is nondecreasing, h is dominated by nondecreasing functions in u while g looks like a contraction in the second argument.

Methods based on the Leray-Schauder nonlinear alternative and the Krasnosel skii fixed point the Orem are used to prove new existence theorems.

Several examples and counterexamples illustrate the obtained results.