# American Institute of Mathematical Sciences

June  2017, 10(3): 475-485. doi: 10.3934/dcdss.2017023

## 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

Received  March 2016 Revised  October 2016 Published  February 2017

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

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.
Citation: Jin Liang, James H. Liu, Ti-Jun Xiao. Condensing operators and periodic solutions of infinite delay impulsive evolution equations. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 475-485. doi: 10.3934/dcdss.2017023
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