The Dirichlet Problem for the Fourth Order Nonlinear Ordinary Differential Equations at Resonance

摘要

Landesman-Lazer's type efficient sufficient conditions are established for the solvability of the two-point boundary value problem u((4))(t) = p(t)u(t) + f(t, u(t)) + h(t) for a = t = b, u((i))(a) = 0, u((i))(b) = 0, (i = 0, 1), where h, p is an element of L([a, b]; R) and f is an element of K([a, b] x R; R), in the case where the linear problem w((4))(t) = p(t)w(t), w((i))(a) = 0, w((i))(b) = 0, (i = 0, 1) has nontrivial solutions. The results obtained in the paper are optimal in the sense that if f = 0, i.e. when nonlinear equation turns to the linear equation, from our results follows the first part of Fredholm's theorem.