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June  2022, 27(6): 3019-3037. doi: 10.3934/dcdsb.2021171

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

Pablo Amster 1,, , Alberto Déboli 2, and  Manuel Pinto 3,

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

Received  January 2021 Revised  April 2021 Published  June 2022 Early access  June 2021

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

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: Delay differential systems, Affine-periodic solutions, Hartman condition, Resonant problems, Topological degree.
Mathematics Subject Classification: Primary: 34K10; Secondary: 37C27.
Citation: Pablo Amster, Alberto Déboli, Manuel Pinto. Hartman and Nirenberg type results for systems of delay differential equations under $ (\omega,Q) $-periodic conditions. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3019-3037. doi: 10.3934/dcdsb.2021171
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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