SINGULAR PERIODIC PROBLEM FOR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH φ-LAPLACIAN

We investigate the singular periodic boundary-value problem with phi-Laplacian, (phi(u'))' = f (t, u, u'), u(0) = u(T), u'(0) = u'(T), where phi is an increasing homeomorphism, phi(R) = R, phi(0) = 0. We assume that f satisfies the Caratheodory conditions on each set [a, b] x R-2, [a, b] subset of (0, T) and f does not satisfy the Caratheodory conditions on [0,T] x R-2, which means that f has time singularities at t = 0, t = T. We provide sufficient conditions for the existence of solutions to the above problem belonging to C-1[0,T]. We also find conditions which guarantee the existence of a sign-changing solution to the problem.