×

Three positive periodic solutions to nonlinear neutral functional differential equations with parameters on variable time scales. (English) Zbl 1235.34243

Summary: Using two successive reductions: B-equivalence of the system on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation and by Leggett-Williams fixed point theorem, we investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equation on variable time scales with a transition condition between two consecutive parts of the scale \((d/dt)(x(t) + c(t)x(t - \alpha)) = a(t)g(x(t))x(t) - \sum^n_{j=1} \lambda_j f_j (t, x(t - v_j(t))), (t, x) \in \mathbb T_0 (x), \Delta t|_{(t, x) \in \varsigma_{2i}} = \Pi^1_i (t, x) - t, \Delta x|_{(t, x) \in \varsigma_{2i}} = \Pi^2_i (t, x) - x\), where \(\Pi^1_i (t, x) = t_{2i+1} + \tau_{2i+1} (\Pi^2_i(t, x))\) and \(\Pi^2_i (t, x) = B_i x + J_i (x) + x, i = 1, 2, \dots\). \(\lambda_j(j = 1, 2, \dots, n)\) are parameters, \(\mathbb T_0 (x)\) is a variable time scale with \((\omega, p)\)-property, \(c(t), a(t), v_j(t)\), and \(f_j(t, x) (j = 1, 2, \dots, n)\) are \(\omega\)-periodic functions of \(t, B_{i+p} = B_i, J_{i+p}(x) = J_i(x)\) uniformly with respect to \(i \in \mathbb Z\).

MSC:

34N05 Dynamic equations on time scales or measure chains
34A05 Explicit solutions, first integrals of ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäuser Boston, Boston, Mass, USA, 2001. · Zbl 1107.34304
[2] V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan, Dynamic Systems on Measure Chains, vol. 370 of Mathematics and its Applications, Kluwer Academic, Dodrecht, The Netherlands, 1996. · Zbl 1060.65591
[3] V. Lakshmikantham and A. S. Vatsala, “Hybrid systems on time scales. Dynamic equations on time scales,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 227-235, 2002. · Zbl 1032.34050
[4] S. Sivasundaram, “Stability of dynamic systems on the time scales,” Nonlinear Dynamics and Systems Theory, vol. 2, no. 2, pp. 185-202, 2002. · Zbl 1027.34059
[5] M. U. Akhmet and M. Turan, “The differential equation on time scales through impulsive differential equations,” Nonlinear Analysis, vol. 65, no. 11, pp. 2043-2060, 2006. · Zbl 1110.34006
[6] M. U. Akhmet and M. Turan, “Differential equations on variable time scales,” Nonlinear Analysis, vol. 70, no. 3, pp. 1175-1192, 2009. · Zbl 1170.34010
[7] M. U. Akhmet, “Perturbations and Hopf bifurcation of the planar discontinuous dynamical system,” Nonlinear Analysis, vol. 60, no. 1, pp. 163-178, 2005. · Zbl 1066.34008
[8] E. Akalin and M. U. Akhmet, “The principles of B-smooth discontinuous flows,” Computers & Mathematics with Applications, vol. 49, no. 7-8, pp. 981-995, 2005. · Zbl 1093.37004
[9] M. U. Akhmetov and N. A. Perestyuk, “The comparison method for differential equations with impulse action,” Differential Equations, vol. 26, no. 9, pp. 1079-1086, 1990. · Zbl 0712.34013
[10] Y. M. Dib, M. R. Maroun, and Y. N. Raffoul, “Periodicity and stability in neutral nonlinear differential equations with functional delay,” Electronic Journal of Differential Equations, vol. 142, pp. 1-11, 2005. · Zbl 1097.34049
[11] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99, Springer, New York, NY, USA, 1993. · Zbl 0787.34002
[12] K. Gopalsamy, X. Z. He, and L. Z. Wen, “On a periodic neutral logistic equation,” Glasgow Mathematical Journal, vol. 33, no. 3, pp. 281-286, 1991. · Zbl 0737.34050
[13] K. Gopalsamy and B. G. Zhang, “On a neutral delay logistic equation,” Dynamics and Stability of Systems, vol. 2, no. 3-4, pp. 183-195, 1987. · Zbl 0665.34066
[14] I. Gy\Hori and F. Hartung, “Preservation of stability in a linear neutral differential equation under delay perturbations,” Dynamic Systems and Applications, vol. 10, no. 2, pp. 225-242, 2001. · Zbl 0994.34065
[15] Y. Li, “Positive periodic solutions of periodic neutral Lotka-Volterra system with state dependent delays,” Journal of Mathematical Analysis and Applications, vol. 330, no. 2, pp. 1347-1362, 2007. · Zbl 1118.34059
[16] M. N. Islam and Y. N. Raffoul, “Periodic solutions of neutral nonlinear system of differential equations with functional delay,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 1175-1186, 2007. · Zbl 1118.34057
[17] Y. N. Raffoul, “Periodic solutions for neutral nonlinear differential equations with functional delay,” Electronic Journal of Differential Equations, vol. 102, pp. 1-7, 2003. · Zbl 1054.34115
[18] Y. N. Raffoul, “Periodic solutions for scalar and vector nonlinear difference equations,” Panamerican Mathematical Journal, vol. 9, no. 1, pp. 97-111, 1999. · Zbl 0960.39004
[19] T. R. Ding, R. Iannacci, and F. Zanolin, “On periodic solutions of sublinear Duffing equations,” Journal of Mathematical Analysis and Applications, vol. 158, no. 2, pp. 316-332, 1991. · Zbl 0727.34030
[20] C. Wang, Y. Li, and Y. Fei, “Three positive periodic solutions to nonlinear neutral functional differential equations with impulses and parameters on time scales,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1451-1462, 2010. · Zbl 1210.34097
[21] R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces,” Indiana University Mathematics Journal, vol. 28, no. 4, pp. 673-688, 1979. · Zbl 0421.47033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.