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where E: D(E) -+ Y, B(E) C X, N: X + Y, A: X+ Y are operators, E linear, not necessarily bounded, N and A continuous, not necessarily linear, X, Y Banach spaces over the reals, 01 a real parameter. In any application E may be a linear differential operator in some domain G C E*, v 3 1, with linear homogeneous boundary conditions. We consider the case in which Ex = 0 has a nontrivial set X0 = ker E of solutions; in other words, the equation Ex + hx = 0 has X = 0 as an eigenvalue. We assume, however, that X,, is finite dimensional, thus, 1 0, C > 0 such that, for every real ol with 1 ti / < tiO , the equation Ex + ol Ax = Nx has at least a solution x E X with 11 x /I < C (existence of solutions across a point of resonance). In other words, the parameter 01 is allowed to go through the point of resonance OL = 0, and yet uniformly bounded solutions x of (1) can be guaranteed. This phenomenon has physical significance. For the case of periodic solutions