Positive Solutions of Second Order Systems of Boundary Value Problems

The existence of nonnegative solutions of boundary value problems of the form x″(t) + λa(t) ƒ(x(t), y(t)) = 0y″(t) + λb(t) g(x(t), y(t)) = 0x(0) = x(1) = y(0) = y(1) = 0 is investigated where ƒ and g are functions from Rp × Rq into Rp and Rq, respectively, and a, b are functions defined on [0, 1] with values in the spaces of diagonal matrices of order p and q, respectively. All functions are assumed to be continuous and nonnegativity as well as monotonicity and growth conditions are imposed on ƒ and g; the existence results hold for a bounded interval of values of λ if ƒ and g are assumed to grow fast and have only nonzero values while if ƒ(0, 0) = 0, g(0, 0) = 0 it is shown that under reasonable conditions there exists a nontrivial nonnegative solution for every λ > 0.