LANDESMAN { LAZER TYPE PROBLEMS AT AN EIGENVALUE 75 Lemma

The aim of this paper is to establish some a priori bounds for solutions of Landesman-Lazer problem. We show the application for the solution structure of the nonlinear diierential equation of the fourth order 1. The general theory Let X be a real Banach space with the norm k k and let D(L) X be the domain of the closed Fredholm operator L : D(L) ! X with index zero. We shall suppose that 0 is an isolated eigenvalue of odd multiplic-ity of L, hence there exists such a 0 > 0 that for 2 (? 0 ; 0), 6 = 0, (L?I) ?1 exists on X and that mapping is continuous. Let there exists a continuous positive deenite bilinear form hh ; i : X X ! R such that z 2 R(L) (range of L) ii hz ; ui = 0 for all u 2 NS(L) (nullspace of the operator L). Let P 0 : X ! X be a continuous linear projection onto NS(L). We denote the operator L P0 : D(L) \ NS(P 0) ! R(L) deened by L P0 = Lj D(L)\NS(P0) : The operator L P0 is one-to-one on D(L) \ NS(P 0) and therefore there exists an inverse operator which we denote by L ?1 P0 = K P0. We suppose that K P0 be a completely continuous. Let