Boundary Value Problems
Volume 2008 (2008), Article ID 236386, 9 pages
doi:10.1155/2008/236386

Nonlinear systems of second-order ODEs

Abstract

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,c2≥0, h1,h2∈L1(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.