Ž . JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 202, 126 1996 ARTICLE NO. 0300 Strongly Nonlinear Second-Order ODEs with Rapidly Growing Terms* M. Garcıa-Huidobro ´ Departamento de Matematicas, Uni  ersidad Catolica de Chile, Casilla 306, Correo 22, ´ ´ Santiago, Chile R. Manasevich ´ Departamento de Ingenierıa Matematica, Uni  ersidad de Chile, Casilla 170, Correo 3, ´ ´ Santiago, Chile and F. Zanolin Dipartimento di Matematica e Informatica, Uni  ersita di Udine,  ia delle Scienze 208 ` ( ) loc. Rizzi , 33100 Udine, Italy Submitted by Jean Mawhin Received July 3, 1995 1. INTRODUCTION   Let g :    be a continuous function and let q : a, b        Ž .  Ž. satisfy the Caratheodory conditions and be such that qt , x, y  et for   Ž . 2 Ž. almost every t  a, b and all x, y   , where e  is a Lebesgue integrable function. The study of existence and multiplicity of solutions to various boundary value problems associated to the nonlinear second-order differential equation u  g u  qt , u , u' , t  a , b 1.1 Ž . Ž . Ž . Ž . *Work performed in the frame of the EC project ‘‘Some nonlinear boundary value problems for differential equations,’’ Grant CI 1*-CT93-0323. The work of M. Garcıa- ´ Huidobro and R. Manasevich has been also partially supported by FONDECYT 1940409-94. ´ 1 0022-247X96 $18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.