Resonance and Nonresonance in Terms of Average Values

We prove existence results for nonlinear two-point boundary value problems. What sets our results apart is that we impose sufficient conditions for solvability in terms of the (asymptotic) average values of the nonlinearities. Our solvability conditions allow the nonlinear term to have significant oscillations outside the given spectral gap as long as it remains within the interval on the average in some sense.