# 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 and Continuous Dynamical Systems - S, 2017, 10 (3) : 475-485. doi: 10.3934/dcdss.2017023
##### References:
 [1] H. Amann, Periodic solutions of semi-linear parabolic equations, Nonlinear Analysis, A Collection of Papers in Honor of Erich Roth, Academic Press, New York, (1978), 1-29. [2] B. de Andrade and C. Lizama, Existence of asymptotically almost periodic solutions for damped wave equations, J. Math. Anal. Appl., 382 (2011), 761-771.  doi: 10.1016/j.jmaa.2011.04.078. [3] T. Diagana, Almost periodic solutions to some second-order nonautonomous differential equations, Proc. Amer. Math. Soc., 140 (2012), 279-289.  doi: 10.1090/S0002-9939-2011-10970-5. [4] T. Diagana, Pseudo-almost periodic solutions for some classes of nonautonomous partial evolution equations, J. Franklin Inst., 348 (2011), 2082-2098.  doi: 10.1016/j.jfranklin.2011.06.001. [5] Z. J. Du and Z. S. Feng, Periodic solutions of a neutral impulsive predator-prey model with Beddington-DeAngelis functional response with delays, J. Comput. Appl. Math., 258 (2014), 87-98.  doi: 10.1016/j.cam.2013.09.008. [6] Z. S. Feng, The uniqueness of the periodic solution for a class of differential equations, Electron. J. Qual. Theory Differ. Equ., 2000 (2000), 9 pp. [7] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Vol. 1 Academic Press, New York, 1969. [8] J. Liang, J. Liu and T. J. Xiao, Periodic solutions of delay impulsive differential equations, Nonlinear Anal., 74 (2011), 6835-6842.  doi: 10.1016/j.na.2011.07.008. [9] J. Liang, J. Liu and T. J. Xiao, Periodic solutions to operational differential equations with finite delay and impulsive conditions, J. Abstr. Diff. Equ. Appl., 3 (2012), 42-47. [10] J. Liang, J. Liu and T. J. Xiao, Periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations in Banach spaces, Anal. Appl, 1 (2015).  doi: 10.1142/S0219530515500281. [11] J. Liu, Periodic solutions of infinite delay evolution equations, J. Math. Anal. Appl., 247 (2000), 627-644.  doi: 10.1006/jmaa.2000.6896. [12] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [13] B. Sadovskii, On a fixed point principle, Funct. Anal. Appl., 1 (1967), 74-76. [14] G. T. Stamov, Almost Periodic Solutions of Impulsive Differential Equations, Lecture Notes in Math. , Vol. 2047, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-27546-3. [15] G. T. Stamov and I. M. Stamova, Impulsive fractional functional differential systems and Lyapunov method for the existence of almost periodic solutions, Rep. Math. Phys., 75 (2015), 73-84.  doi: 10.1016/S0034-4877(15)60025-8. [16] N. Van Minh, G. N'Guerekata and S. Siegmund, Circular spectrum and bounded solutions of periodic evolution equations, J. Differential Equations, 246 (2009), 3089-3108.  doi: 10.1016/j.jde.2009.02.014.

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##### References:
 [1] H. Amann, Periodic solutions of semi-linear parabolic equations, Nonlinear Analysis, A Collection of Papers in Honor of Erich Roth, Academic Press, New York, (1978), 1-29. [2] B. de Andrade and C. Lizama, Existence of asymptotically almost periodic solutions for damped wave equations, J. Math. Anal. Appl., 382 (2011), 761-771.  doi: 10.1016/j.jmaa.2011.04.078. [3] T. Diagana, Almost periodic solutions to some second-order nonautonomous differential equations, Proc. Amer. Math. Soc., 140 (2012), 279-289.  doi: 10.1090/S0002-9939-2011-10970-5. [4] T. Diagana, Pseudo-almost periodic solutions for some classes of nonautonomous partial evolution equations, J. Franklin Inst., 348 (2011), 2082-2098.  doi: 10.1016/j.jfranklin.2011.06.001. [5] Z. J. Du and Z. S. Feng, Periodic solutions of a neutral impulsive predator-prey model with Beddington-DeAngelis functional response with delays, J. Comput. Appl. Math., 258 (2014), 87-98.  doi: 10.1016/j.cam.2013.09.008. [6] Z. S. Feng, The uniqueness of the periodic solution for a class of differential equations, Electron. J. Qual. Theory Differ. Equ., 2000 (2000), 9 pp. [7] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Vol. 1 Academic Press, New York, 1969. [8] J. Liang, J. Liu and T. J. Xiao, Periodic solutions of delay impulsive differential equations, Nonlinear Anal., 74 (2011), 6835-6842.  doi: 10.1016/j.na.2011.07.008. [9] J. Liang, J. Liu and T. J. Xiao, Periodic solutions to operational differential equations with finite delay and impulsive conditions, J. Abstr. Diff. Equ. Appl., 3 (2012), 42-47. [10] J. Liang, J. Liu and T. J. Xiao, Periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations in Banach spaces, Anal. Appl, 1 (2015).  doi: 10.1142/S0219530515500281. [11] J. Liu, Periodic solutions of infinite delay evolution equations, J. Math. Anal. Appl., 247 (2000), 627-644.  doi: 10.1006/jmaa.2000.6896. [12] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [13] B. Sadovskii, On a fixed point principle, Funct. Anal. Appl., 1 (1967), 74-76. [14] G. T. Stamov, Almost Periodic Solutions of Impulsive Differential Equations, Lecture Notes in Math. , Vol. 2047, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-27546-3. [15] G. T. Stamov and I. M. Stamova, Impulsive fractional functional differential systems and Lyapunov method for the existence of almost periodic solutions, Rep. Math. Phys., 75 (2015), 73-84.  doi: 10.1016/S0034-4877(15)60025-8. [16] N. Van Minh, G. N'Guerekata and S. Siegmund, Circular spectrum and bounded solutions of periodic evolution equations, J. Differential Equations, 246 (2009), 3089-3108.  doi: 10.1016/j.jde.2009.02.014.
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