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Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales
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 |
$ C_μ $ |
$ 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} $ |
$ \mathfrak{A}(t) $ |
$ \varpi $ |
$ \mathfrak{A}(t) $ |
$ t>0 $ |
$ x_t (s)=x(t+s),\; s≤0$ |
$ Δ x(t_i)= x(t_i ^+)-x(t_i ^- ) $ |
$ \mathfrak{F} $ |
$ φ $ |
$ \mathfrak{I}_i\ (i=1,···,n) $ |
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. |
show all references
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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