#### Jin Liang, James H. Liu, Minh Van Nguyen, Ti-Jun Xiao, Periodic solutions of impulsive differential equations with infinite delay in Banach spaces, Vol. 2019 (2019), Article ID 18, pp. 1-10

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DOI: 10.23952/jnfa.2019.18

Received December 24, 2018; Accepted April 26, 2019; Published May 10, 2019

Abstract. In this paper, we investigate the existence of a periodic solution to the Cauchy problem for a class of impulsive differential equations with infinite delay in Banach spaces under general fading memory phase spaces satisfying a basic axiom. It is shown that the related Poincaré operator given by ${\mathbb P}(\phi)=u_T (\phi)$ (i.e., $T$ units along the unique solution $u(\phi)$ determined by the initial function $\phi$) is a condensing operator by virtue of the Kuratowski measure of non-compactness. Based on this result, we derive the existence of a periodic solution to the Cauchy problem with help of the boundedness of the solutions and three-set fixed point theorems. The main result in this paper extends the previous results for equations without impulsive conditions or in special $C_g$ phase spaces. Finally, we give a remark on future developments of the related impulsive problems for fractional differential equations with infinite delay in Banach spaces.