Positive solutions of nonlocal boundary value problems involving integral conditions

Abstract

We establish the existence of multiple positive solutions of nonlinear equations of the form -u″(t) = g(t)f(t, u(t)), t ∈ (0, 1), where g, f are non-negative functions, subject to various nonlocal boundary conditions. The common feature is that each can be written as an integral equation, in the space C[0, 1], of the form -u(t) = γ(t)α [u] + ∫ 01 κ(t, s)g(s)f(s, u(s))ds where α[u] is a linear functional given by a Stieltjes integral but is not assumed to be positive for all positive u. Our new results cover many non-local boundary conditions previously studied on a case by case basis for particular positive functionals only, for example, many m-point BVPs are special cases. Even for positive functionals our methods give improvements on previous work. Also we allow weaker assumptions on the nonlinear term than were previously imposed. © 2008 Birkhaueser.