  We study the existence of $T$-periodic solutions $(T > 0)$ for the first
order differential equations being at resonance at infinity, where the right
hand side is the perturbations of a sectorial operator. Our aim is to prove an
index formula expressing the topological degree of the associated translation
along trajectories operator on appropriately large ball, in terms of special
geometrical assumptions imposed on the nonlinearity. We also prove that the
geometrical assumptions are generalization of well known Landesman-Lazer and
strong resonance conditions. Obtained index formula is used to derive the
criteria determining the existence of $T$-periodic solutions for the heat
equation being at resonance at infinity.
