Dirichlet BVP for the second order nonlinear ordinary differential equations at resonance

Landesman-Lazer’s type efficient sufficient conditions are established for the solvability of the Dirichlet problem , for where ;R) and f is the L([a,b]; R) Caratheodory function, in the case where the linear problem has nontrivial solutions. The results obtained in the paper are optimal in the sense that if , i.e., when nonlinear equation turns to the linear equation, from our results follows the first part of Fredholm’s theorem