Periodic and asymptotically periodic solutions in nonlinear coupled Volterra integro-dynamic systems with infinite delay on time scales

Let T be a periodic time scale. The purpose of this paper is to use Schauder's fixed point theorem to prove the existence of periodic and asymptotically periodic solutions of nonlinear coupled Volterra integro-dynamic systems with infinite delay on time scales. The results obtained here extend the work of Raffoul r1.

Keywords:

Fixed points, periodic solutions, asymptotically periodic solutions Volterra integro-dynamic systems, time scales,

PDF

___

[1] M. Adivar, H.C. Koyuncuoglu, Y.N. Raffoul, Classification of positive solutions of nonlinear systems of Volterra integrodynamic equations on time scales, Commun. Appl. Anal. 16(3) (2012) 359-375.
[2] M. Adivar, Y. N. Raffoul, Existence of periodic solutions in totally nonlinear delay dynamic equations, Electronic Journal of Qualitative Theory of Differential Equations 2009(1) (2009) 1-20.
[3] M. Adivar, Y.N. Raffoul, Existence results for periodic solutions of integro-dynamic equations on time scales, Annali di Matematica 188 (2009) 543-559.
[4] E. Akin, O. Ozturk, On Volterra integro dynamical systems on time scales, Communications in Applied Analysis 23(1) (2019) 21-30.
[5] A. Ardjouni, A. Djoudi, Existence of positive periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale, Malaya Journal of Matematik 2(1) (2013) 60-67.
[6] A. Ardjouni, A. Djoudi, Existence of periodic solutions for nonlinear neutral dynamic equations with functional delay on a time scale, Acta Univ. Palacki. Olomnc., Fac. rer. nat., Mathematica 52(1) (2013) 5-19.
[7] A. Ardjouni, A. Djoudi, Existence of periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale, Commun Nonlinear Sci Numer Simulat 17 (2012) 3061-3069.
[8] A. Ardjouni, A. Djoudi, Periodic solutions in totally nonlinear dynamic equations with functional delay on a time scale, Rend. Sem. Mat. Univ. Politec. Torino Vol. 68(4)(2010) 349-359.
[9] M. Bohner, A. Peterson, Dynamic equations on time scales, An introduction with applications, Birkhäuser, Boston, (2001).
[10] M. Bohner, A. Peterson, Advances in dynamic equations on time scales, Birkhäuser, Boston, (2003).
[11] F. Bouchelaghem, A. Ardjouni, A. Djoudi, Existence and stability of positive periodic solutions for delay nonlinear dynamic equations, Nonlinear Studies 25(1) (2018) 191-202.
[12] F. Bouchelaghem, A. Ardjouni, A. Djoudi, Existence of positive solutions of delay dynamic equations, Positivity 21(4) (2017) 1483-1493.
[13] F. Bouchelaghem, A. Ardjouni, A. Djoudi, Existence of positive periodic solutions for delay dynamic equations, Proyecciones (Antofagasta) 36(3) (2017) 449-460.
[14] J.A. Cid, G. Propst, M. Tvrdy, On the pumping effect in a pipe/tank flow configuration with friction, Physica D: Nonlinear Phenomena 273/274 (2014) 28-33.
[15] I. Culakova, L. Hanustiakova, R. Olach, Existence for positive solutions of second-ordre neutral nonlinear differential equations, Applied Mathematics Letters 22 (2009) 1007-1010.
[16] B. Dorociakova, M. Michalkova, R. Olach, M. Saga, Existence and stability of periodic solution related to valveless pumping, Mathematical Problems in Engineering 2018 (2018) 1-8.
[17] M. Gouasmia, A. Ardjouni, A. Djoudi, Periodic and nonnegative periodic solutions of nonlinear neutral dynamic equations on a time scale, International Journal of Analysis and Applications 16(2) (2018) 162-177.
[18] S. Hilger, Ein Masskettenkalkul mit Anwendung auf Zentrumsmanningfaltigkeiten, PhD thesis, Universitat Wurzburg, (1988).
[19] E.R. Kaufmann, Y.N. Raffoul, Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J. Math. Anal. Appl. 319 (2006) 315-325.
[20] V. Lakshmikantham, S. Sivasundaram, B. Kaymarkcalan, Dynamic systems on measure chains, Kluwer Academic Publishers, Dordrecht, (1996).
[21] Z. Li, C. Wang, R.P. Agarwal, D. O'Regan, Commutativity of quaternion-matrix-valued functions and quaternion matrix dynamic equations on time scales, Studies in Applied Mathematics (2020), https://doi.org/10.1111/sapm.12344.
[22] Y.N. Raffoul, Analysis of periodic and asymptotically periodic solutions in nonlinear coupled Volterra integro-differential systems, Turk. J. Math. 42 (2018), 108-120.
[23] D.R. Smart, Fixed points theorems, Cambridge Univ. Press, Cambridge, UK, (1980).
[24] C. Wang, R.P. Agarwal, A classication of time scales and analysis of the general delays on time scales with applications, Mathematical Methods in the Applied Sciences 39(6) (2016) 1568-1590.
[25] C. Wang, R.P. Agarwal, D. O' Regan, R. Sakthivel, Theory of translation closedness for time scales, Developments in Mathematics, Vol. 62, Springer, (2020).
[26] C. Wang, R.P. Agarwal, S. Rathinasamy, Almost periodic oscillations for delay impulsive stochastic Nicholsons blowflies timescale model, Computational and Applied Mathematics 37 (2018) 3005-3026.