Condensing operators and periodic solutions of infinite delay impulsive evolution equations

Jin Liang; James H. Liu; Ti-Jun Xiao

doi:10.3934/dcdss.2017023

Article Contents

2017, Volume 10, Issue 3: 475-485. Doi: 10.3934/dcdss.2017023

This issue Previous Article Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales Next Article The mixed-mode oscillations in Av-Ron-Parnas-Segel model

Condensing operators and periodic solutions of infinite delay impulsive evolution equations

1.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
2.
Department of Mathematics, James Madison University, Harrisonburg, VA 22807, USA
3.
School of Mathematical Sciences, Fudan University, Shanghai 200433, China

^* Corresponding author
^* Corresponding author

Received: February 29, 2016

Revised: September 30, 2016

Published: May 2017

The first author is supported by NSF of China Grant No. 11571229. The third author is supported by NSF of China Grant No. 11371095.

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Abstract

By showing the existence of the fixed point of the condensing operators in the phasespace $ C_μ $ for the Cauchy problem for impulsive evolution equations with infinite delay in a Banach space $ X $:

$\begin{align} &{{x}^{\prime }}(t)+\mathfrak{A}(t)x(t)=\mathfrak{F}(t,x(t),{{x}_{t}}),\ \ t>0,\ t\ne {{t}_{i}}, \\ &x(s)=\varphi (s),\ s\le 0, \\ &\Delta x({{t}_{i}})={{\Im }_{i}}(x({{t}_{i}})),\ \ i=1,2,\cdots ,\ \ 0<{{t}_{1}}<{{t}_{2}}<\cdots <\infty , \\ \end{align} $

where $ \mathfrak{A}(t) $ is $ \varpi $-periodic, the operator $ \mathfrak{A}(t) $ is unbounded for each $ t>0 $, $ x_t (s)=x(t+s),\; s≤0$, $ Δ x(t_i)= x(t_i ^+)-x(t_i ^- ) $, $ \mathfrak{F} $, $ φ $ and $ \mathfrak{I}_i\ (i=1,···,n) $ are given functions, we derive periodic solutions from bounded solutions. The new periodic solution existence results obtained here extend earlier results in this area for evolution equations without impulsive conditions or without infinite delay.

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Mathematics Subject Classification: Primary: 34G20, 37C25; Secondary: 47H08.

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References

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