In the paper, the existence of affine-periodic solutions for impulsive differential systems of the form \begin{align*} &x'=f(t,x);\qquad\quad t\ne\tau_i(x),\\ &\Delta x(t)=I_i(x);\qquad t=\tau_i(x)\quad (i\in{\mathbb Z}) \end{align*} are discussed which is a new periodic concept which has been founded in recent years. It was first introduced by Zhang et al., which describes those natural phenomena that exhibit certain symmetry in space rather than periodicity in time. The existence of affine-periodic solutions for such equations, for instance, dissipative systems and nonlinear systems on time scales etc, was discussed in (see [Y. Li and F. Huang, Adv. Nonlinear Stud. 15, No. 1, 241–252 (2015; Zbl 1351.34047); C. Wang and Y. Li, Adv. Difference Equ. 2015, Paper No. 286, 16 p. (2015; Zbl 1351.34110); C. Wang et al., Rocky Mt. J. Math. 46, No. 5, 1717–1737 (2016; Zbl 1370.34067); Acta Math. Appl. Sin., Engl. Ser. 31, No. 2, 307–312 (2015; Zbl 1328.34035)]). Further, to know about the recent work, it is referred to [G. Liu et al., “Existence and multiplicity of rotating periodic solutions for resonant hamiltonian systems”, J. Differ. Equ. 265, No. 4, 1324–1352 (2018; doi:10.1016/j.jde.2018.04.001); “Rotating periodic solutions for super-linear second order hamiltonian systems”, Appl. Math. Lett. Volume 79, 73–79 (2018; doi:10.1016/j.aml.2017.11.024); T. Shen and W. Liu, Appl. Math. Lett. 88, 164–170 (2019; Zbl 1406.37047); J. Xing et al., Appl. Math. Lett. 89, 91–96 (2019; Zbl 1472.37066)].
Several existence theorems are proved for dissipative impulsive (functional) differential systems. Some applications are also given by combining Lyapunov’s methods.