Existence Theorems across a Point of Resonance

where E: D(E) —• S, D(E) C S, is a linear not necessarily bounded operator in a real Hilbert space S with 1 < dim W < °°, W = ketE, and N: S —> S is a continuous nonlinear operator. In terms of the alternative method (see Cesari [1] , [2]), let P: S —> S be the orthogonal projector with range PS = S0 = W = ker E, let St = (I -P)S, and let S1 be also the range of E. Let H: Sx -—• 5'1 denote a linear, bounded, compact operator (a partial inverse of E) such that the usual relations of the alternative method (selfadjoint case) hold: (hx) H(I P)E = I P; (h2) EP = PE\ (h3) £»(ƒ F) = / P. Let ( , ) and || || denote inner product and norm in 5, and let I = ||//(/ P)||.