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Condensing operators and periodic solutions of infinite delay impulsive evolution equations

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    * Corresponding author 
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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  • 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.

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


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