Existence Theorems for Second Order Multi-Point Boundary Value Problems

We are interested in the existence of nontrivial solutions for the second order nonlinear differential equation (E): y ''(t) = f(t, y(t)) = 0, 0 < t < 1 subject to multi-point boundary conditions at t = 1 and either Dirichlet or Neumann conditions at t = 0. Assume that f(t, y) satisfies vertical bar f(t, y)vertical bar <= k(t)vertical bar y vertical bar + h(t) for non-negative functions k, h is an element of L-1(0, 1) for all (t, y) is an element of (0, 1) x R and f(t, 0) not equivalent to 0 for t is an element of (0, 1). We show without any additional assumption on h(t) that if parallel to k parallel to(1) is sufficiently small where parallel to.parallel to(1) denotes the norm of L-1(0, 1) then there exists at least one non-trivial solution for such boundary value problems. Our results reduce to that of Sun and Liu [11] and Sun [10] for the three point problem with Neumann boundary condition at t = 0.