Periodic solutions for nonlinear evolution equations at resonance

We are concerned with periodic problems for nonlinear evolution equations at resonance of the form u ˙ ( t ) = − A u ( t ) + F ( t , u ( t ) ) , where a densely defined linear operator A : D ( A ) → X on a Banach space X is such that −A generates a compact C 0 semigroup and F : [ 0 , + ∞ ) × X → X is a nonlinear perturbation. Imposing appropriate Landesman–Lazer type conditions on the nonlinear term F, we prove a formula expressing the fixed point index of the associated translation along trajectories operator, in the terms of a time averaging of F restricted to Ker A. By the formula, we show that the translation operator has a nonzero fixed point index and, in consequence, we conclude that the equation admits a periodic solution.