Convex sets and second order systems with nonlocal boundary conditions at resonance

The solvability of the resonant nonlocal boundary value problem (Formula presented.) with f: [0, 1] × Rn × Rn → Rn continuous, g = diag(g1,…, gn), gi: [0, 1] → R of bounded variation, (Formula presented.), is studied using the Leray-Schauder continuation theorem. The a priori estimates follow from the existence of an open bounded convex subset C ⊂ Rn, such that, for each t ∈ [0, 1] and x ∈ C, the vector fields f(t, x, ·) satisfy suitable geometrical conditions on ∂C. The special cases where C is a ball or a parallelotope are considered.