A New Continuation Method for the Study of Nonlinear Equations at Resonance

A new continuation theorem for the existence of solutions to an equation Lu = N(u), where N is a nonlinear continuous operator and L a linear Fredholm noninvertible one, is proved. The continuation which makes N collapse is replaced by a deformation of L to an invertible linear operator. This implies results concerning sublinear N, N having a linear growth at infinity and superlinear N. These generalize the classical theorems on the solvability of semilinear elliptic BVP′s at resonance. The periodic solutions of Liénard equations are studied.