We study the singular periodic boundary value problem of the form (φ(u ' ))' + h(u)u ' = g(u) + e(t), u(0) = u(T), u ' (0) = u ' (T), where φ: R→R is an increasing and odd homeomorphism such that φ(R ) = R, h ∈ C[0, ∞), e ∈ L1[0, T] and g ∈ C(0, ∞) can have a space singularity at x = 0, i.e. lim sup x→0+ |g(x)| = ∞ may hold. We prove new existence results both for the case of an attractive singularity, when lim inf x→0+ g(x) = −∞, and for the case of a strong repulsive singularity, when lim x→0+ R 1 x g(ξ)dξ = ∞. In the latter case we assume that φ(y) = φp(y) = |y| p−2 y, p > 1, is the well-known p-Laplacian. Our results extend and complete those obtained recently by Jebelean and Mawhin and by Liu Bing.
The a priori boundedness principle is proved for the Dirichlet boundary value problems for strongly singular higher-order nonlinear functional-differential equations. Several sufficient conditions of solvability of the Dirichlet problem under consideration are derived from the a priori boundedness principle. The proof of the a priori boundedness principle is based on the Agarwal-Kiguradze type theorems, which guarantee the existence of the Fredholm property for strongly singular higher-order linear differential equations with argument deviations under the two-point conjugate and right-focal boundary conditions.