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Asymptotic behaviour of an age and infection age structured model for the propagation of fungal diseases in plants
LaSalle type stationary oscillation theorems for Affine-Periodic Systems
1. | College of mathematics, Jilin Normal University, Jilin 136000, China |
2. | College of Mathematics, Jilin University, Changchun 130012, China |
3. | School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China |
4. | Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China |
5. | Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China |
The paper concerns the existence of affine-periodic solutions for affine-periodic (functional) differential systems, which is a new type of quasi-periodic solutions if they are bounded. Some more general criteria than LaSalle's one on the existence of periodic solutions are established. Some applications are also given.
References:
[1] |
T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional-Differential Equations Mathematics in Science and Engineering, 178, Academic Press, Inc. , Orlando, FL, 1985. |
[2] |
O. Chadli, Q. H. Ansari and J. C. Yao,
Mixed equilibrium problems and anti-periodic solutions for nonlinear evolution equations, J. Optim. Theory Appl., 168 (2016), 410-440.
doi: 10.1007/s10957-015-0707-y. |
[3] |
X. Chang and Y. Li,
Rotating periodic solutions of second order dissipative dynamical systems, Discrete Contin. Dyn. Syst., 36 (2016), 643-652.
doi: 10.3934/dcds.2016.36.643. |
[4] |
Y. Chen, J. J. Nieto and D. O'Regan,
Anti-periodic solutions for evolution equations associated with maximal monotone mappings, Appl. Math. Lett., 24 (2011), 302-307.
doi: 10.1016/j.aml.2010.10.010. |
[5] |
Y. Chen, D. O'Regan and R. P. Agarwal,
Anti-periodic solutions for semilinear evolution equations in Banach spaces, J. Appl. Math. Comput., 38 (2012), 63-70.
doi: 10.1007/s12190-010-0463-y. |
[6] |
W. A. Coppel, Dichotomies in Stability Theory Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. |
[7] |
J. R. Graef and L. Kong,
Periodic solutions of first order functional differential equations, Appl. Math. Lett., 24 (2011), 1981-1985.
doi: 10.1016/j.aml.2011.05.020. |
[8] |
T. Haddad and T. Haddad, Existence and uniqueness of anti-periodic solutions for nonlinear third-order differential inclusions, Electron. J. Differential Equations (2013), 10 pp. |
[9] |
J. K. Hale, Functional Differential Equations Applied Mathematical Sciences, Vol. 3, Springer-Verlag New York, New York-Heidelberg, 1971. |
[10] |
S. Heidarkhani, G. A. Afrouzi, A. Hadjian and J. Henderson, Existence of infinitely many anti-periodic solutions for second-order impulsive differential inclusions, Electron. J. Differential Equations (2013), 13 pp. |
[11] |
J. Knežević-Miljanović,
Periodic solutions of nonlinear differential equations, Adv. Dyn. Syst. Appl., 7 (2012), 89-93.
|
[12] |
T. Küpper, Y. Li and B. Zhang,
Periodic solutions for dissipative-repulsive systems, Tohoku Math. J. (2), 52 (2000), 321-329.
doi: 10.2748/tmj/1178207816. |
[13] |
J. LaSalle and S. Lefschetz, Stabillity by Lyapunov's Direct Method, with Application Mathematics in Science and Engineering, Vol. 4, New York-London 1961. |
[14] |
Y. Li and F. Huang,
Levinson's problem on affine-periodic solutions, Adv. Nonlinear Stud., 15 (2015), 241-252.
doi: 10.1515/ans-2015-0113. |
[15] |
Y. Li, Q. Zhou and X. Lu,
Periodic solutions for functional-differential inclusions with infinite delay, Sci. China Ser. A, 37 (1994), 1289-1301.
|
[16] |
Y. Li, Q. Zhou and X. Lu,
Periodic solutions and equilibrium states for functional-differential inclusions with nonconvex right-hand side, Quart. Appl. Math., 55 (1997), 57-68.
doi: 10.1090/qam/1433752. |
[17] |
J. Liang, J. Liu and T. Xiao,
Periodic solutions of delay impulsive differential equations, Nonlinear Anal., 74 (2011), 6835-6842.
doi: 10.1016/j.na.2011.07.008. |
[18] |
A. Lipowski, B. Przeradzki and K. Szymańska-Debowska,
Periodic solutions to differential equations with a generalized p-Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2593-2601.
doi: 10.3934/dcdsb.2014.19.2593. |
[19] |
J. Mawhin,
Periodic solutions of differential and difference systems with pendulum-type nonlinearities: Variational approaches, Differential and difference equations with applications, Springer Proc. Math. Stat., Springer, New York, 47 (2013), 83-98.
doi: 10.1007/978-1-4614-7333-6_7. |
[20] |
X. Meng and Y. Li,
Affine-periodic solutions for discrete dynamical systems, J. Appl. Anal. Comput., 5 (2015), 781-792.
|
[21] |
S. Padhi, John R. Graef and P. D. N. Srinivasu, Periodic Solutions of First-Order Functional Differential Equations in Population Dynamics Springer, New Delhi, 2014.
doi: 10.1007/978-81-322-1895-1. |
[22] |
P. Sa Ngiamsunthorn, Existence of periodic solutions for differential equations with multiple delays under dichotomy condition Adv. Difference Equ. 2015 (2015), 11 pp.
doi: 10.1186/s13662-015-0598-0. |
[23] |
C. Wang and Y. Li, Affine-periodic solutions for nonlinear dynamic equations on time scales Adv. Difference Equ. 2015 (2015), 16 pp.
doi: 10.1186/s13662-015-0634-0. |
[24] |
C. Wang, X. Yang and Y. Li,
Affine-periodic solutions for nonlinear differential equations, Rocky Mountain J. Math., 46 (2016), 1717-1737.
doi: 10.1216/RMJ-2016-46-5-1717. |
[25] |
J. R. Ward,
Periodic solutions of first order systems, Discrete Contin. Dyn. Syst., 33 (2013), 381-389.
doi: 10.3934/dcds.2013.33.381. |
[26] |
R. Wu, F. Cong and Y. Li,
Anti-periodic solutions for second order differential equations, Appl. Math. Lett., 24 (2011), 860-863.
doi: 10.1016/j.aml.2010.12.031. |
[27] |
T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions Applied Mathematical Sciences, Vol. 14. Springer-Verlag, New York-Heidelberg, 1975. |
[28] |
Y. Zhang, X. Yang and Y. Li, Affine-periodic solutions for dissipative systems, Abstr. Appl. Anal. (2013), Art. ID 157140, 4 pp. |
show all references
References:
[1] |
T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional-Differential Equations Mathematics in Science and Engineering, 178, Academic Press, Inc. , Orlando, FL, 1985. |
[2] |
O. Chadli, Q. H. Ansari and J. C. Yao,
Mixed equilibrium problems and anti-periodic solutions for nonlinear evolution equations, J. Optim. Theory Appl., 168 (2016), 410-440.
doi: 10.1007/s10957-015-0707-y. |
[3] |
X. Chang and Y. Li,
Rotating periodic solutions of second order dissipative dynamical systems, Discrete Contin. Dyn. Syst., 36 (2016), 643-652.
doi: 10.3934/dcds.2016.36.643. |
[4] |
Y. Chen, J. J. Nieto and D. O'Regan,
Anti-periodic solutions for evolution equations associated with maximal monotone mappings, Appl. Math. Lett., 24 (2011), 302-307.
doi: 10.1016/j.aml.2010.10.010. |
[5] |
Y. Chen, D. O'Regan and R. P. Agarwal,
Anti-periodic solutions for semilinear evolution equations in Banach spaces, J. Appl. Math. Comput., 38 (2012), 63-70.
doi: 10.1007/s12190-010-0463-y. |
[6] |
W. A. Coppel, Dichotomies in Stability Theory Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. |
[7] |
J. R. Graef and L. Kong,
Periodic solutions of first order functional differential equations, Appl. Math. Lett., 24 (2011), 1981-1985.
doi: 10.1016/j.aml.2011.05.020. |
[8] |
T. Haddad and T. Haddad, Existence and uniqueness of anti-periodic solutions for nonlinear third-order differential inclusions, Electron. J. Differential Equations (2013), 10 pp. |
[9] |
J. K. Hale, Functional Differential Equations Applied Mathematical Sciences, Vol. 3, Springer-Verlag New York, New York-Heidelberg, 1971. |
[10] |
S. Heidarkhani, G. A. Afrouzi, A. Hadjian and J. Henderson, Existence of infinitely many anti-periodic solutions for second-order impulsive differential inclusions, Electron. J. Differential Equations (2013), 13 pp. |
[11] |
J. Knežević-Miljanović,
Periodic solutions of nonlinear differential equations, Adv. Dyn. Syst. Appl., 7 (2012), 89-93.
|
[12] |
T. Küpper, Y. Li and B. Zhang,
Periodic solutions for dissipative-repulsive systems, Tohoku Math. J. (2), 52 (2000), 321-329.
doi: 10.2748/tmj/1178207816. |
[13] |
J. LaSalle and S. Lefschetz, Stabillity by Lyapunov's Direct Method, with Application Mathematics in Science and Engineering, Vol. 4, New York-London 1961. |
[14] |
Y. Li and F. Huang,
Levinson's problem on affine-periodic solutions, Adv. Nonlinear Stud., 15 (2015), 241-252.
doi: 10.1515/ans-2015-0113. |
[15] |
Y. Li, Q. Zhou and X. Lu,
Periodic solutions for functional-differential inclusions with infinite delay, Sci. China Ser. A, 37 (1994), 1289-1301.
|
[16] |
Y. Li, Q. Zhou and X. Lu,
Periodic solutions and equilibrium states for functional-differential inclusions with nonconvex right-hand side, Quart. Appl. Math., 55 (1997), 57-68.
doi: 10.1090/qam/1433752. |
[17] |
J. Liang, J. Liu and T. Xiao,
Periodic solutions of delay impulsive differential equations, Nonlinear Anal., 74 (2011), 6835-6842.
doi: 10.1016/j.na.2011.07.008. |
[18] |
A. Lipowski, B. Przeradzki and K. Szymańska-Debowska,
Periodic solutions to differential equations with a generalized p-Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2593-2601.
doi: 10.3934/dcdsb.2014.19.2593. |
[19] |
J. Mawhin,
Periodic solutions of differential and difference systems with pendulum-type nonlinearities: Variational approaches, Differential and difference equations with applications, Springer Proc. Math. Stat., Springer, New York, 47 (2013), 83-98.
doi: 10.1007/978-1-4614-7333-6_7. |
[20] |
X. Meng and Y. Li,
Affine-periodic solutions for discrete dynamical systems, J. Appl. Anal. Comput., 5 (2015), 781-792.
|
[21] |
S. Padhi, John R. Graef and P. D. N. Srinivasu, Periodic Solutions of First-Order Functional Differential Equations in Population Dynamics Springer, New Delhi, 2014.
doi: 10.1007/978-81-322-1895-1. |
[22] |
P. Sa Ngiamsunthorn, Existence of periodic solutions for differential equations with multiple delays under dichotomy condition Adv. Difference Equ. 2015 (2015), 11 pp.
doi: 10.1186/s13662-015-0598-0. |
[23] |
C. Wang and Y. Li, Affine-periodic solutions for nonlinear dynamic equations on time scales Adv. Difference Equ. 2015 (2015), 16 pp.
doi: 10.1186/s13662-015-0634-0. |
[24] |
C. Wang, X. Yang and Y. Li,
Affine-periodic solutions for nonlinear differential equations, Rocky Mountain J. Math., 46 (2016), 1717-1737.
doi: 10.1216/RMJ-2016-46-5-1717. |
[25] |
J. R. Ward,
Periodic solutions of first order systems, Discrete Contin. Dyn. Syst., 33 (2013), 381-389.
doi: 10.3934/dcds.2013.33.381. |
[26] |
R. Wu, F. Cong and Y. Li,
Anti-periodic solutions for second order differential equations, Appl. Math. Lett., 24 (2011), 860-863.
doi: 10.1016/j.aml.2010.12.031. |
[27] |
T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions Applied Mathematical Sciences, Vol. 14. Springer-Verlag, New York-Heidelberg, 1975. |
[28] |
Y. Zhang, X. Yang and Y. Li, Affine-periodic solutions for dissipative systems, Abstr. Appl. Anal. (2013), Art. ID 157140, 4 pp. |
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