Boundary Value Problems
Volume 2008 (2008), Article ID 236386, 9 pages
Research Article

Nonlinear Systems of Second-Order ODEs

Patricio Cerda and Pedro Ubilla

Departamento de Matemática y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

Received 2 February 2007; Accepted 16 November 2007

Academic Editor: Jean Mawhin

Copyright © 2008 Patricio Cerda and Pedro Ubilla. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study existence of positive solutions of the nonlinear system (p1(t,u,v)u)=h1(t)f1(t,u,v) in (0,1); (p2(t,u,v)v)=h2(t)f2(t,u,v) in (0,1); u(0)=u(1)=v(0)=v(1)=0, where p1(t,u,v)=1/(a1(t)+c1g1(u,v)) and p2(t,u,v)=1/(a2(t)+c2g2(u,v)). Here, it is assumed that g1, g2 are nonnegative continuous functions, a1(t), a2(t) are positive continuous functions, c1,c20, h1,h2L1(0,1), and that the nonlinearities f1,f2 satisfy superlinear hypotheses at zero and +. The existence of solutions will be obtained using a combination among the method of truncation, a priori bounded and Krasnosel'skii well-known result on fixed point indices in cones. The main contribution here is that we provide a treatment to the above system considering differential operators with nonlinear coefficients. Observe that these coefficients may not necessarily be bounded from below by a positive bound which is independent of u and v.