Hartman and Nirenberg type results for systems of delay differential equations under $ (\omega,Q) $-periodic conditions

Pablo Amster; Alberto Déboli; Manuel Pinto

doi:10.3934/dcdsb.2021171

Article Contents

2022, Volume 27, Issue 6: 3019-3037. Doi: 10.3934/dcdsb.2021171

This issue Previous Article The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model Next Article Stage-structured models for interactive wild and periodically and impulsively released sterile mosquitoes

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

^* Corresponding author: Pablo Amster
^* Corresponding author: Pablo Amster

Received: December 31, 2020

Revised: March 31, 2021

Published: June 2022

The first two authors were partially supported by projects 20020190100039BA UBACyT and PIP 11220130100006CO CONICET. The third author was supported by project Fondecyt 1170466

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 34K10; Secondary: 37C27.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.