doi: 10.3934/dcdsb.2021171
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

Received  January 2021 Revised  April 2021 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.

Citation: Pablo Amster, Alberto Déboli, Manuel Pinto. Hartman and Nirenberg type results for systems of delay differential equations under $ (\omega,Q) $-periodic conditions. Discrete & Continuous Dynamical Systems - B, 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.  Google Scholar

[2]

M. AgaogluM. 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.  Google Scholar

[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.  Google Scholar

[4]

E. AlvarezS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

P. AmsterM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, 1977.  Google Scholar

[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.  Google Scholar

[12]

M. KhalladiM. KosticM. PintoA. Rahmani and D. Velinov, $c$-Almost periodic functions and applications, Nonautonomous Dynamical Systems, 7 (2020), 176-193.  doi: 10.1515/msds-2020-0111.  Google Scholar

[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.  Google Scholar

[14]

E. Landesman and A. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1969/70), 609-623.   Google Scholar

[15]

G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, 1978.  Google Scholar

[16]

M. LiJ. Wang and M. Feckan, $(\omega, c)$-periodic solutions for impulsive differential systems, Communicat. Math. Analysis, 21 (2018), 35-64.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

L. Nirenberg, Generalized degree and nonlinear problems, Contributions to Nonlinear Functional Analysis Academic Press, 1971, 1–9.  Google Scholar

[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.  Google Scholar

[21]

M. Pinto, Ergodicity and Oscillations, Conference, Universidad Católica del Norte, Antofagasta, Chile, 2014. Google Scholar

[22]

H. Schaefer, Über die methode der a priori-Schranken, Math. Ann., 129 (1955), 415-416.  doi: 10.1007/BF01362380.  Google Scholar

[23]

C. WangX. 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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

M. AgaogluM. 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.  Google Scholar

[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.  Google Scholar

[4]

E. AlvarezS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

P. AmsterM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, 1977.  Google Scholar

[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.  Google Scholar

[12]

M. KhalladiM. KosticM. PintoA. Rahmani and D. Velinov, $c$-Almost periodic functions and applications, Nonautonomous Dynamical Systems, 7 (2020), 176-193.  doi: 10.1515/msds-2020-0111.  Google Scholar

[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.  Google Scholar

[14]

E. Landesman and A. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1969/70), 609-623.   Google Scholar

[15]

G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, 1978.  Google Scholar

[16]

M. LiJ. Wang and M. Feckan, $(\omega, c)$-periodic solutions for impulsive differential systems, Communicat. Math. Analysis, 21 (2018), 35-64.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

L. Nirenberg, Generalized degree and nonlinear problems, Contributions to Nonlinear Functional Analysis Academic Press, 1971, 1–9.  Google Scholar

[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.  Google Scholar

[21]

M. Pinto, Ergodicity and Oscillations, Conference, Universidad Católica del Norte, Antofagasta, Chile, 2014. Google Scholar

[22]

H. Schaefer, Über die methode der a priori-Schranken, Math. Ann., 129 (1955), 415-416.  doi: 10.1007/BF01362380.  Google Scholar

[23]

C. WangX. 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.  Google Scholar

[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.  Google Scholar

[1]

Yong Li, Hongren Wang, Xue Yang. Fink type conjecture on affine-periodic solutions and Levinson's conjecture to Newtonian systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2607-2623. doi: 10.3934/dcdsb.2018123

[2]

Hongren Wang, Xue Yang, Yong Li, Xiaoyue Li. LaSalle type stationary oscillation theorems for Affine-Periodic Systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2907-2921. doi: 10.3934/dcdsb.2017156

[3]

Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861

[4]

Pablo Amster, Mónica Clapp. Periodic solutions of resonant systems with rapidly rotating nonlinearities. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 373-383. doi: 10.3934/dcds.2011.31.373

[5]

Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765

[6]

Jehad O. Alzabut. A necessary and sufficient condition for the existence of periodic solutions of linear impulsive differential equations with distributed delay. Conference Publications, 2007, 2007 (Special) : 35-43. doi: 10.3934/proc.2007.2007.35

[7]

Xiaowei Tang, Xilin Fu. New comparison principle with Razumikhin condition for impulsive infinite delay differential systems. Conference Publications, 2009, 2009 (Special) : 739-743. doi: 10.3934/proc.2009.2009.739

[8]

P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220

[9]

Nguyen Thi Van Anh. On periodic solutions to a class of delay differential variational inequalities. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021045

[10]

Weigao Ge, Li Zhang. Multiple periodic solutions of delay differential systems with $2k-1$ lags via variational approach. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4925-4943. doi: 10.3934/dcds.2016013

[11]

Klaudiusz Wójcik, Piotr Zgliczyński. Topological horseshoes and delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 827-852. doi: 10.3934/dcds.2005.12.827

[12]

Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301

[13]

Alexander Krasnosel'skii. Resonant forced oscillations in systems with periodic nonlinearities. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 239-254. doi: 10.3934/dcds.2013.33.239

[14]

Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure & Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541

[15]

Fritz Colonius, Alexandre J. Santana. Topological conjugacy for affine-linear flows and control systems. Communications on Pure & Applied Analysis, 2011, 10 (3) : 847-857. doi: 10.3934/cpaa.2011.10.847

[16]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031

[17]

Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105

[18]

Zhiming Guo, Xiaomin Zhang. Multiplicity results for periodic solutions to a class of second order delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1529-1542. doi: 10.3934/cpaa.2010.9.1529

[19]

Vera Ignatenko. Homoclinic and stable periodic solutions for differential delay equations from physiology. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3637-3661. doi: 10.3934/dcds.2018157

[20]

Diego Averna, Nikolaos S. Papageorgiou, Elisabetta Tornatore. Multiple solutions for nonlinear nonhomogeneous resonant coercive problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 155-178. doi: 10.3934/dcdss.2018010

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (52)
  • HTML views (132)
  • Cited by (0)

Other articles
by authors

[Back to Top]