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Hartman and Nirenberg type results for systems of delay differential equations under $ (\omega,Q) $-periodic conditions
1. | Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina and IMAS-CONICET, Ciudad Universitaria, Pab. I (1428), Buenos Aires, Argentina |
2. | Instituto de Ciencias, Universidad Nacional de General Sarmiento, Juan María Gutiérrez 1150, (1613) Los Polvorines, Buenos Aires, Argentina |
3. | Universidad de Chile - Departamento de Matemática, Facultad de Ciencias, Casilla 653, Santiago, Chile |
We consider the $ (\omega,Q) $-periodic problem for a system of delay differential equations, where $ Q $ is an invertible matrix. Existence and multiplicity of solutions is proven under different conditions that extend well-known results for the periodic case $ Q = I $ and anti-periodic case $ Q = -I $. In particular, the results apply to biological models with mixed terms of Nicholson, Lasota or Mackey type, and also vectorial versions of Nicholson or Mackey-Glass models.
References:
[1] |
S. Abbas, S. Dhama, M. Pinto and D. Sepúlveda, Pseudo compact almost automorphic solutions for a family of delayed population model of Nicholson type, Journal of Mathematical Analysis and Applications, 495 (2021), 124722, 22 pp.
doi: 10.1016/j.jmaa.2020.124722. |
[2] |
M. Agaoglu, M. Feckan and A. Panagiotidon,
Existence and uniqueness of $(\omega, c)$-periodic solutions of semilinear evolution equations, Int. J. Dyn Systems and Diff. Eqs., 10 (2020), 149-166.
doi: 10.1504/IJDSDE.2020.106027. |
[3] |
E. Alvarez, S. Castillo and M. Pinto, $(\omega, c)$-Pseudo periodic functions, first order Cauchy Problem and Lasota-Wazewska model, Bound. Value Prob., 2019 (2019), Paper No. 106, 20 pp.
doi: 10.1186/s13661-019-1217-x. |
[4] |
E. Alvarez, S. Castillo and M. Pinto,
Asymptotically $(\omega, c)$-periodic first-order Cauchy problem and Lasota-Wazewska model with unbounded oscillating production of red cells, Math. Meth. Appl. Sci., 43 (2020), 305-319.
doi: 10.1002/mma.5880. |
[5] |
E. Alvarez, S. Díaz and C. Lizama, On the existence and uniqueness of $(N, \lambda)$-periodic solutions to a class of Volterra difference equations, Advances in Difference Equations, 2019 (2019), Art 105, 12pp.
doi: 10.1186/s13662-019-2053-0. |
[6] |
E. Alvarez, A. Gómez and M. Pinto, $(\omega, c)$-periodic functions and mild solutions to abstract fractional integro differential equations, Electr.J. Qual. Th. Differ. Equ., 2018 (2018), Paper No. 16, 8 pp.
doi: 10.14232/ejqtde.2018.1.16. |
[7] |
P. Amster, M. P. Kuna and G. Robledo,
Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium, Communications on Pure and Applied Analysis, 18 (2019), 1695-1709.
doi: 10.3934/cpaa.2019080. |
[8] |
X. Chang and Y. Liu,
Rotating periodic solutions of second order dissipative dynamical systems, Disc. Cont. Dyn. Systems A, 36 (2016), 643-652.
doi: 10.3934/dcds.2016.36.643. |
[9] |
T. Faria,
Periodic solutions for a non-monotone family of delayed differential equations with applications to Nicholson systems, Journal of Differential Equations, 263 (2017), 509-533.
doi: 10.1016/j.jde.2017.02.042. |
[10] |
R. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, 1977. |
[11] |
P. Hartman,
On boundary value problems for systems of ordinary nonlinear second order differential equations, Transactions of the American Mathematical Society, 96 (1960), 493-509.
doi: 10.1090/S0002-9947-1960-0124553-5. |
[12] |
M. Khalladi, M. Kostic, M. Pinto, A. Rahmani and D. Velinov,
$c$-Almost periodic functions and applications, Nonautonomous Dynamical Systems, 7 (2020), 176-193.
doi: 10.1515/msds-2020-0111. |
[13] |
M. Khalladi, M. Kostic, M. Pinto, A. Rahmani and D. Velinov, On Semi $c$-periodic functions, J. Math., 2021 (2021), Art. ID 6620625, 5 pp.
doi: 10.1155/2021/6620625. |
[14] |
E. Landesman and A. Lazer,
Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1969/70), 609-623.
|
[15] |
G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, 1978. |
[16] |
M. Li, J. Wang and M. Feckan,
$(\omega, c)$-periodic solutions for impulsive differential systems, Communicat. Math. Analysis, 21 (2018), 35-64.
|
[17] |
K. Liu, M. Feckan, D. O'Regan and J. Wang, $(\omega, c)$-periodic solutions for time-varying non-instantaneous impulsive differential systems, Applicable Analysis, 2021, 1895123.
doi: 10.1080/00036811.2021.1895123. |
[18] |
K. Liu, J. Wang, D. O'Regan and M. Feckan, A new class of $(\omega, c)$-periodic non-instantaneous impulsive differential Equations, Mediterr. J. Math., 17 (2020), Paper No. 155, 22 pp.
doi: 10.1007/s00009-020-01574-8. |
[19] |
L. Nirenberg, Generalized degree and nonlinear problems, Contributions to Nonlinear Functional Analysis Academic Press, 1971, 1–9. |
[20] |
R. Ortega and L. Sánchez,
Periodic solutions of forced oscillators with several degrees of freedom, Bulletin of the London Mathematical Society, 34 (2002), 308-318.
doi: 10.1112/S0024609301008748. |
[21] |
M. Pinto, Ergodicity and Oscillations, Conference, Universidad Católica del Norte, Antofagasta, Chile, 2014. |
[22] |
H. Schaefer,
Über die methode der a priori-Schranken, Math. Ann., 129 (1955), 415-416.
doi: 10.1007/BF01362380. |
[23] |
C. Wang, X. Yang and Y. Li,
Affine-periodic solutions for nonlinear differential equations, Rocky Mountain Journal of Mathematics, 46 (2016), 1717-1737.
doi: 10.1216/RMJ-2016-46-5-1717. |
[24] |
Y. Zhang, X. Yang and Y. Li, Affine-periodic solutions for dissipative systems, Abstract and Applied Analysis, 2013 (2013), Art. ID 157140, 4pp.
doi: 10.1155/2013/157140. |
show all references
References:
[1] |
S. Abbas, S. Dhama, M. Pinto and D. Sepúlveda, Pseudo compact almost automorphic solutions for a family of delayed population model of Nicholson type, Journal of Mathematical Analysis and Applications, 495 (2021), 124722, 22 pp.
doi: 10.1016/j.jmaa.2020.124722. |
[2] |
M. Agaoglu, M. Feckan and A. Panagiotidon,
Existence and uniqueness of $(\omega, c)$-periodic solutions of semilinear evolution equations, Int. J. Dyn Systems and Diff. Eqs., 10 (2020), 149-166.
doi: 10.1504/IJDSDE.2020.106027. |
[3] |
E. Alvarez, S. Castillo and M. Pinto, $(\omega, c)$-Pseudo periodic functions, first order Cauchy Problem and Lasota-Wazewska model, Bound. Value Prob., 2019 (2019), Paper No. 106, 20 pp.
doi: 10.1186/s13661-019-1217-x. |
[4] |
E. Alvarez, S. Castillo and M. Pinto,
Asymptotically $(\omega, c)$-periodic first-order Cauchy problem and Lasota-Wazewska model with unbounded oscillating production of red cells, Math. Meth. Appl. Sci., 43 (2020), 305-319.
doi: 10.1002/mma.5880. |
[5] |
E. Alvarez, S. Díaz and C. Lizama, On the existence and uniqueness of $(N, \lambda)$-periodic solutions to a class of Volterra difference equations, Advances in Difference Equations, 2019 (2019), Art 105, 12pp.
doi: 10.1186/s13662-019-2053-0. |
[6] |
E. Alvarez, A. Gómez and M. Pinto, $(\omega, c)$-periodic functions and mild solutions to abstract fractional integro differential equations, Electr.J. Qual. Th. Differ. Equ., 2018 (2018), Paper No. 16, 8 pp.
doi: 10.14232/ejqtde.2018.1.16. |
[7] |
P. Amster, M. P. Kuna and G. Robledo,
Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium, Communications on Pure and Applied Analysis, 18 (2019), 1695-1709.
doi: 10.3934/cpaa.2019080. |
[8] |
X. Chang and Y. Liu,
Rotating periodic solutions of second order dissipative dynamical systems, Disc. Cont. Dyn. Systems A, 36 (2016), 643-652.
doi: 10.3934/dcds.2016.36.643. |
[9] |
T. Faria,
Periodic solutions for a non-monotone family of delayed differential equations with applications to Nicholson systems, Journal of Differential Equations, 263 (2017), 509-533.
doi: 10.1016/j.jde.2017.02.042. |
[10] |
R. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, 1977. |
[11] |
P. Hartman,
On boundary value problems for systems of ordinary nonlinear second order differential equations, Transactions of the American Mathematical Society, 96 (1960), 493-509.
doi: 10.1090/S0002-9947-1960-0124553-5. |
[12] |
M. Khalladi, M. Kostic, M. Pinto, A. Rahmani and D. Velinov,
$c$-Almost periodic functions and applications, Nonautonomous Dynamical Systems, 7 (2020), 176-193.
doi: 10.1515/msds-2020-0111. |
[13] |
M. Khalladi, M. Kostic, M. Pinto, A. Rahmani and D. Velinov, On Semi $c$-periodic functions, J. Math., 2021 (2021), Art. ID 6620625, 5 pp.
doi: 10.1155/2021/6620625. |
[14] |
E. Landesman and A. Lazer,
Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1969/70), 609-623.
|
[15] |
G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, 1978. |
[16] |
M. Li, J. Wang and M. Feckan,
$(\omega, c)$-periodic solutions for impulsive differential systems, Communicat. Math. Analysis, 21 (2018), 35-64.
|
[17] |
K. Liu, M. Feckan, D. O'Regan and J. Wang, $(\omega, c)$-periodic solutions for time-varying non-instantaneous impulsive differential systems, Applicable Analysis, 2021, 1895123.
doi: 10.1080/00036811.2021.1895123. |
[18] |
K. Liu, J. Wang, D. O'Regan and M. Feckan, A new class of $(\omega, c)$-periodic non-instantaneous impulsive differential Equations, Mediterr. J. Math., 17 (2020), Paper No. 155, 22 pp.
doi: 10.1007/s00009-020-01574-8. |
[19] |
L. Nirenberg, Generalized degree and nonlinear problems, Contributions to Nonlinear Functional Analysis Academic Press, 1971, 1–9. |
[20] |
R. Ortega and L. Sánchez,
Periodic solutions of forced oscillators with several degrees of freedom, Bulletin of the London Mathematical Society, 34 (2002), 308-318.
doi: 10.1112/S0024609301008748. |
[21] |
M. Pinto, Ergodicity and Oscillations, Conference, Universidad Católica del Norte, Antofagasta, Chile, 2014. |
[22] |
H. Schaefer,
Über die methode der a priori-Schranken, Math. Ann., 129 (1955), 415-416.
doi: 10.1007/BF01362380. |
[23] |
C. Wang, X. Yang and Y. Li,
Affine-periodic solutions for nonlinear differential equations, Rocky Mountain Journal of Mathematics, 46 (2016), 1717-1737.
doi: 10.1216/RMJ-2016-46-5-1717. |
[24] |
Y. Zhang, X. Yang and Y. Li, Affine-periodic solutions for dissipative systems, Abstract and Applied Analysis, 2013 (2013), Art. ID 157140, 4pp.
doi: 10.1155/2013/157140. |
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