Wang, Jinrong; Xiang, X.; Wei, W.; Chen, Qian Bounded and periodic solutions of semilinear impulsive periodic system on Banach spaces. (English) Zbl 1155.35306 Fixed Point Theory Appl. 2008, Article ID 401947, 15 p. (2008). Summary: A class of semilinear impulsive periodic systems on Banach spaces is considered. First, we introduce the \(T_{0}\)-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 homogeneous linear impulsive periodic system are shown. This allows us to apply Horn’s fixed-point theorem to prove the existence of \(T_{0}\)-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. Cited in 5 Documents MSC: 35B10 Periodic solutions to PDEs 35R12 Impulsive partial differential equations 35K90 Abstract parabolic equations 47D06 One-parameter semigroups and linear evolution equations Keywords:Gronwall lemma with impulse; Horn’s fixed-point theorem PDFBibTeX XMLCite \textit{J. Wang} et al., Fixed Point Theory Appl. 2008, Article ID 401947, 15 p. (2008; Zbl 1155.35306) Full Text: DOI EuDML References: [2] doi:10.1112/blms/18.2.173 · Zbl 0586.34038 · doi:10.1112/blms/18.2.173 [6] doi:10.1155/2007/81756 · Zbl 1146.37370 · doi:10.1155/2007/81756 [7] doi:10.1155/DDNS/2006/31614 · doi:10.1155/DDNS/2006/31614 [8] doi:10.1016/0096-3003(94)90171-6 · Zbl 0819.34041 · doi:10.1016/0096-3003(94)90171-6 [10] doi:10.1016/S0362-546X(97)00606-8 · Zbl 0934.34066 · doi:10.1016/S0362-546X(97)00606-8 [11] doi:10.1016/S0022-247X(03)00512-2 · Zbl 1045.34052 · doi:10.1016/S0022-247X(03)00512-2 [12] doi:10.1016/S0022-247X(02)00378-5 · Zbl 1029.34045 · doi:10.1016/S0022-247X(02)00378-5 [13] doi:10.1016/0362-546X(92)90195-K · Zbl 0765.34057 · doi:10.1016/0362-546X(92)90195-K [15] doi:10.1016/S0362-546X(03)00117-2 · Zbl 1030.34056 · doi:10.1016/S0362-546X(03)00117-2 [19] doi:10.1016/0022-0396(71)90049-0 · Zbl 0223.34055 · doi:10.1016/0022-0396(71)90049-0 [20] doi:10.1007/BF01371406 · Zbl 0537.34037 · doi:10.1007/BF01371406 [21] doi:10.1016/0960-0779(94)00169-Q · Zbl 1079.35532 · doi:10.1016/0960-0779(94)00169-Q [22] doi:10.1023/B:JODY.0000009739.00640.44 · Zbl 1055.34035 · doi:10.1023/B:JODY.0000009739.00640.44 [23] doi:10.1007/s10231-004-0139-z · Zbl 1162.34324 · doi:10.1007/s10231-004-0139-z [24] doi:10.1155/2007/26196 · Zbl 1148.39015 · doi:10.1155/2007/26196 [25] doi:10.1155/2008/524945 · Zbl 1154.34030 · doi:10.1155/2008/524945 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.