Singular Periodic Problem for Nonlinear Ordinary Differential Equations with Φ-laplacian

We investigate the singular periodic boundary-value problem with φ-Laplacian, (φ(u′))′ = f(t, u, u′), u(0) = u(T ), u′(0) = u′(T ), where φ is an increasing homeomorphism, φ(R) = R, φ(0) = 0. We assume that f satisfies the Carathéodory conditions on each set [a, b] × R2, [a, b] ⊂ (0, T ) and f does not satisfy the Carathéodory conditions on [0, T ]×R2, 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 C1[0, T ]. We also find conditions which guarantee the existence of a sign-changing solution to the problem.