DIRICHLET BVP FOR THE SECOND ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS AT RESONANCE

Abstract

Landesman-Lazer’s type efficient sufficient conditions are established forthe solvability of the Dirichlet problem u′′(t) = p(t)u(t) + f(t, u(t)) + h(t),for a ≤ t ≤ b, u(a) = 0, u(b) = 0, where h, p ϵ L([a, b];R) and f is the L([a, b];R) Caratheodory function, in the casewhere the linear problem u′′(t) = p(t)u(t), u(a) = 0,u(b) = 0 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 partof Fredholm’s theorem.