GLOBAL SOLUTIONS TO A CLASS OF SECOND-ORDER DIFFERENTIAL EQUATIONS.

The aim of this paper is to review some classical results and present new theorems of existence and extend- ability of solutions to the following second-order nonlinear Cauchy problem: This equation cannot be represented in ASCII code value. where f : [0, T ] · R → R is continuous, derivative independent and T > 0, u0 , u1 are real numbers. We ?rst 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 ? 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 ?xed point theorem are used to prove new existence theorems. Several examples and counterexamples illustrate the obtained results. [ABSTRACT FROM AUTHOR]