SOLVABILITY OF SEMILINEAR ABSTRACT EQUATIONS AT RESONANCE

We establish a generalization of the Cesari-Kannan existence result fo r problems of the type Lx = N(x), x is an element of X where X is a se parable Hilbert functional space, L is a selfadjoint linear differenti al operator with nontrivial finite dimensional kernel and N:X --> X is a bounded continuous nonlinear operator. This generalization leads to new results when the dimension of the kernel of L is greater than one . Applications to systems of second order ordinary differential equati ons are given.