Abstract
This paper specifically focuses on a specific type of q-difference equations that incorporate multiple delays. The main objective is to explore the existence of positive periodic solutions using coincidence degree theory. Notably, the equation studied in this paper has relevance to important biological growth models constructed on quantum domains. The significance of this research lies in the fact that quantum domains are not translation invariant. By investigating periodic solutions on quantum domains, the paper introduces a new perspective and makes notable advancements in the related literature, which predominantly focuses on translation invariant domains. This research contributes to a better understanding of periodic dynamics in systems governed by q-difference equations with multiple delays, particularly in the context of biological growth models on quantum domains.
Similar content being viewed by others
Data availability
Not applicable.
References
Adıvar, M.: A new periodicity concept for time scales. Math. Slovaca 63(4), 817–828 (2013). https://doi.org/10.2478/s12175-013-0127-0
Adıvar, M., Koyuncuoğlu, H.C.: Floquet theory based on new periodicity concept for hybrid systems involving \(q\)-difference equations. Appl. Math. Comput. 273, 1208–1233 (2016). https://doi.org/10.1016/j.amc.2015.08.124
Adıvar, M., Raffoul, Y.: Stability and periodicity in dynamic delay equations. Comput. Math. Appl. 58(2), 264–272 (2009). https://doi.org/10.1016/j.camwa.2009.03.065
Alvarez, E., Castillo, S., Pinto, M.: \((\omega, c)\)-asymptotically periodic functions, first-order Cauchy problem, and Lasota-Wazewska model with unbounded oscillating production of red cells. Math. Methods Appl. Sci. 43(1), 305–319 (2020). https://doi.org/10.1002/mma.5880
Alvarez, E., Diaz, S., Lizama, C.: On the existence and uniqueness of \(({N},\lambda )\)-periodic solutions to a class of Volterra difference equations. Differ. Equ. Adv. (2019). https://doi.org/10.1186/s13662-019-2053-0
Alvarez, E., Gomez, A., Pinto, M.: \((\omega, c)\)-periodic functions and mild solutions to abstract fractional integro-differential equations. Electron. J. Qual. Theory Differ. Equ. 2018(16), 1–8 (2018). https://doi.org/10.14232/ejqtde.2018.1.16
Bohner, M., Chieochan, R.: Floquet theory for \(q\)-difference equations. Sarajev. J. Math. 8(2), 1–12 (2012). https://doi.org/10.5644/SJM.08.2.14
Bohner, M., Chieochan, R.: Positive periodic solutions for higher-order functional \(q\)-difference equations. J. Appl. Funct. Anal. 8(1), 14–22 (2013)
Bohner, M., Chieochan, R.: The Beverton-Holt \(q\)-difference equation. J. Biol. Dyn. 7(1), 86–95 (2013). https://doi.org/10.1080/17513758.2013.804599
Bohner, M., Mesquita, J.G.: Periodic averaging principle in quantum calculus. J. Math. Anal. Appl. 435(2), 1146–1159 (2016). https://doi.org/10.1016/j.jmaa.2015.10.078
Bohner, M., Mesquita, J.G.: Massera’s theorem in quantum calculus. Proc. Am. Math. Soc. 146(11), 4755–4766 (2018). https://doi.org/10.1090/proc/14116
Bohner, M., Streipert, S.: Optimal harvesting policy for the Beverton-Holt quantum difference model. Math. Morav. 20(2), 39–57 (2016). https://doi.org/10.5937/MatMor1602039B
Bohner, M., Streipert, S.: The second Cushing-Henson conjecture for the Beverton-Holt \(q\)-difference equation. Opusc. Math. 37(6), 795–819 (2017). https://doi.org/10.7494/OpMath.2017.37.6.795
Dobrogowska, A., Odzijewicz, A.: Second order \(q\)-difference equations solvable by factorization method. J. Comp. Appl. Math. 193(1), 319–346 (2006). https://doi.org/10.1016/j.cam.2005.06.009
Gaines, R.E., Mahwin, J.L.: Coincidence Degree, and Nonlinear Differential Equations, in: Lecture Notes in Mathematics. Springer, Heidelberg (2006). doi:10.1007/BFb0089537
Islam, M., Neugebauer, J.T.: Existence of periodic solutions for a quantum Volterra equation. Adv. Dyn. Syst. Appl. 11(1), 67–80 (2016)
Islam, M., Neugebauer, J.T.: Asymptotically \(p\)-periodic solutions of a quantum Volterra integral equation. Sarajev. J. Math. 14(1), 59–70 (2018). https://doi.org/10.5644/SJM.14.1.06
Jiang, D., Agarwal, R.P.: Existence of positive periodic solutions for a class of difference equations with several deviating arguments. Comput. Math. Appl. 45(6–9), 1303–1309 (2003). https://doi.org/10.1016/S0898-1221(03)00103-2
Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2012). doi:10.1007/978-1-4613-0071-7
Kaufmann, E.R., Raffoul, Y.: Periodic solutions for a neutral nonlinear dynamical equation on a time scale. J. Math. Anal. Appl. 319(1), 315–325 (2006). https://doi.org/10.1016/j.jmaa.2006.01.063
Koyuncuoğlu, H.C.: \(q\)-Floquet theory and its extensions to time scales periodic in shifts. Ph.D. Thesis, Izmir University of Economics, Izmir, Turkey (2016)
Koyuncuoğlu, H.C.: A generalization of new periodicity concept on time scales. In: Conference Proceedings of 4th International E-Conference on Mathematical Advances and Applications (ICOMAA-2021). Conference Proceedings of Science and Technology (2021)
Koyuncuoğlu, H.C.: Unified Massera type theorems for dynamic equations on time scales. Filomat 37(8), 2405–2419 (2023). https://doi.org/10.2298/FIL2308405K
Li, Y.: Existence and global attractivity of positive periodic solution for a class of delay differential equations. Sci. China Ser. A. 41, 273–284 (1998). https://doi.org/10.1007/BF02879046
Malkiewicz, P., Nieszporski, M.: Darboux transformations for \(q\)-discretizations of \({2{\rm D}}\) second order differential equations. J. Nonlinear Math. Phys. 12(2), 231–239 (2013). https://doi.org/10.2991/jnmp.2005.12.s2.17
Ostorovska, S.: The approximation by \(q\)-Bernstein polynomials in the case \(q\downarrow 1\). Arch. Math. 86, 282–288 (2006). https://doi.org/10.1007/s00013-005-1503-y
Pulita, A.: \(p\)-Adic confluence of \(q\)-difference equations. Compos. Math. 144(4), 867–919 (2008). https://doi.org/10.1112/S0010437X07003454
Zhang, J., Fan, M., Zhu, H.: Periodic solution of single population models on time scales. Math. Comput. Model. 52(3–4), 515–521 (2010). https://doi.org/10.1016/j.mcm.2010.03.048
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by Gradimir Milovanovic.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kostić, M., Koyuncuoğlu, H.C. & Raffoul, Y.N. Positive periodic solutions for certain kinds of delayed q-difference equations with biological background. Ann. Funct. Anal. 15, 5 (2024). https://doi.org/10.1007/s43034-023-00306-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43034-023-00306-9