On three solutions of the second order periodic boundary-value problem

Abstract

We consider the periodic boundary-value problem x'' + a(t)x' + b(t)x = f(t, x, x'), x(') =x(2π), x'(0) = x' (2π), where a, b are Lebesgue integrable functions and f fulfils the Caratheodory conditions. We extend results about the Leray – Schauder topological degree and ´ present conditions implying nonzero values of the degree on sets defined by lower and upper functions. Using such results we prove the existence of at least three different solutions to the above problem.