Periodic solutions of systems with asymptotically even nonlinearities. (2000) - Kloeden, Peter E.; Krasnosel'skii, Alexander M.
Abstract : New conditions of solvability based on a general theorem on the calculation of the index at infinity for vector fields that have degenerate principal linear part as well as degenerate next order terms are obtained for the 2 π -periodic problem for the scalar equation x ″ + n 2 x = g ( | x | ) + f ( t, x ) + b ( t ) with bounded g ( u ) and f ( t, x ) → 0 as | x | → 0 . The result is also applied to the solvability of a two-point boundary value problem and to resonant problems for equations arising in control theory.