Averaging principle and periodic solutions for nonlinear evolution equations at resonance

摘要

We study the existence of $T$-periodic solutions $(T > 0)$ for the firstorder differential equations being at resonance at infinity, where the righthand side is the perturbations of a sectorial operator. Our aim is to prove anindex formula expressing the topological degree of the associated translationalong trajectories operator on appropriately large ball, in terms of specialgeometrical assumptions imposed on the nonlinearity. We also prove that thegeometrical assumptions are generalization of well known Landesman-Lazer andstrong resonance conditions. Obtained index formula is used to derive thecriteria determining the existence of $T$-periodic solutions for the heatequation being at resonance at infinity.