HOMOCLINIC AND PERIODIC SOLUTIONS FOR A CLASS OF SECOND ORDER DIFFERENTIAL EQUATIONS Maria do Rosario Grossinho 1 Departamento de MatemÆtica, ISEG, Universidade TØcnica de Lisboa, Rua do Quelhas, 6, 1200 Lisboa, Portugal and CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1699 Lisboa Codex, Portugal; Feliz Minhos C.I.M.A.-U.E., Universidade de vora, vora, Portugal; Stepan Tersian 2 Center of Applied Mathematics and Informatics, University of Rousse, 8, Studentska, 7017 Rousse, Bulgaria. 1. INTRODUCTION In this paper we are concerned with the existence of positive homoclinic solutions of the second order equation u 00  a (x) u + b (x) u 2 + c (x) u 3 =0; x 2 R: (I) where the coe¢ cient functions a (x) ;b (x) and c (x) are continuous and there are positive constants a; A; b; B; c and C verifying 0 < a  a (x)  A; 0 <c  c (x)  C; (1) 0  b  b (x)  B: (2) We study also the existence of periodic solutions of the equation u 00 + a (x) u  b (x) u 2 + c (x) u 3 =0; (II) where a (x) ;b (x) ;c (x) are continuous 2periodic functions satisfying (1) and instead of (2) jb (x) j B: (3) 1 Supported by FCT, PRAXIS XXI, FEDER, under projects PRAXIS/PCEX/P/MAT/36/96 and PRAXIS/2/2.1/MAT/125/94. 2 Supported by NATO fellowship, INVOTAN, and National Research Foundation in Bulgaria, Grant MM519/95. 1