Abstract
AbstractA class of semilinear impulsive periodic system on Banach spaces is considered. First, we introduce the "Equation missing"-periodic PC-mild solution of semilinear impulsive periodic system. By virtue of Gronwall lemma with impulse, the estimate on the PC-mild solutions is derived. The continuity and compactness of the new constructed Poincaré operator determined by impulsive evolution operator corresponding to homogenous linear impulsive periodic system are shown. This allows us to apply Horn's fixed-point theorem to prove the existence of "Equation missing"-periodic PC-mild solutions when PC-mild solutions are ultimate bounded. This extends the study on periodic solutions of periodic system without impulse to periodic system with impulse on general Banach spaces. At last, an example is given for demonstration.
Highlights
It is well known that impulsive periodic motion is a very important and special phenomenon in natural science and in social science such as climate, food supplement, insecticide population, and sustainable development
There are some papers on periodic solution of periodic systems on infinite dimensional spaces see 8–13 and some results about the impulsive systems on infinite dimensional spaces see 14–18
Particulary, Professor Jean Mawhin investigated the periodic solutions of all kinds of systems on in finite dimensional spaces extensively see 2, 19–23
Summary
It is well known that impulsive periodic motion is a very important and special phenomenon in natural science and in social science such as climate, food supplement, insecticide population, and sustainable development. There are many results, such as existence, the relationship between bounded solutions and periodic solutions, stability, food limited, and robustness, about impulsive periodic system on finite dimensional spaces see 1–7. We have been established periodic solution theory under the existence of a bounded solution for the linear impulsive periodic system on infinite dimensional spaces. This paper is mainly concerned with the existence of periodic solution for semilinear impulsive periodic system on infinite dimensional Banach space X. We use Horn’s fixed-point theorem to obtain the existence of periodic solution for semilinear impulsive periodic system 1.1. After the continuity and compactness of Poincareoperator P are shown, the existence of T0periodic P C-mild solutions for semilinear impulsive periodic system is established by virtue of Horn’s fixed-point theorem when P C-mild solutions are ultimate bounded. An example is given to demonstrate the applicability of our result
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