Abstract
We consider a planar system $$z'=f(t,z)$$ under non-resonance or double resonance conditions and obtain the existence of $$2\uppi $$ -periodic solutions by combining a rotation number approach together with Poincare-Bohl theorem. Firstly, we allow that the angular velocity of solutions of $$z'=f(t,z)$$ is controlled by the angular velocity of solutions of two positively homogeneous system $$z'=L_i(t,z),i=1,2$$ , whose rotation numbers satisfy $$\rho (L_1)>n$$ and $$\rho (L_2)<n+1$$ , namely, nonresonance occurs in the sense of the rotation number. Secondly, we prove the existence of $$2\uppi $$ -periodic solutions when the nonlinearity is allowed to interact with two positively homogeneous system $$z'=L_i(t,z),i=1,2$$ , with $$\rho (L_1)\ge n$$ and $$\rho (L_2)\le n+1$$ , which gives rise to double resonance, and some kind of Landesman–Lazer conditions are assumed at both sides.
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