×

zbMATH — the first resource for mathematics

Periodic solutions of neutral nonlinear system of differential equations with functional delay. (English) Zbl 1118.34057
Summary: We study the existence of periodic solutions of the nonlinear neutral system of differential equations of the form \[ \frac{d}{dt}x(t)=A(t)x(t)+ \frac{d}{dt} Q(t,x(t-g(t)))+G(t,x(t),x(t-g(t))). \] We use the fundamental matrix solution of \[ y'=A(t)y \] and convert the given neutral differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii’s fixed-point theorem to show the existence of a periodic solution of this neutral differential equation. We also use the contraction mapping principle to show the existence of a unique periodic solution of the equation.

MSC:
34K13 Periodic solutions to functional-differential equations
34K40 Neutral functional-differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Appleby, J.A.; Gyori, I.; Reynolds, D.W., Subexponential solutions of scalar linear integro-differential equations with delay, Funct. differ. equ., 11, 11-18, (2004) · Zbl 1063.45005
[2] Dib, Y.M.; Maroun, M.R.; Raffoul, Y.N., Periodicity and stability in neutral nonlinear differential equations with functional delay, Electron. J. differential equations, 142, 1-11, (2005) · Zbl 1097.34049
[3] Hale, J.K., Ordinary differential equations, (1980), Robert E. Krieger Publishing Company New York · Zbl 0186.40901
[4] Hale, J.K.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer New York · Zbl 0787.34002
[5] Cook, K.; Krumme, D., Differential difference equations and nonlinear initial-boundary-value problems for linear hyperbolic partial differential equations, J. math. anal. appl., 24, 372-387, (1968) · Zbl 0186.16902
[6] Gopalsamy, K.; He, X.; Wen, L., On a periodic neutral logistic equation, Glasg. math. J., 33, 281-286, (1991) · Zbl 0737.34050
[7] Gopalsamy, K.; Zhang, B.G., On a neutral delay-logistic equation, Dynam. stability systems, 2, 183-195, (1988) · Zbl 0665.34066
[8] Gyori, I.; Hartung, F., Preservation of stability in a linear neutral differential equation under delay perturbations, Dynam. systems appl., 10, 225-242, (2001) · Zbl 0994.34065
[9] Gyori, I.; Ladas, G., Positive solutions of integro-differential equations with unbounded delay, J. integral equations appl., 4, 377-390, (1992) · Zbl 0761.45009
[10] Lakshmikantham, V.; Deo, S.G., Methods of variation of parameters for dynamical systems, (1998), Gordon and Breach Science Publishers Australia · Zbl 0920.34001
[11] Kun, L.Y., Periodic solutions of a periodic neutral delay equation, J. math. anal. appl., 214, 11-21, (1997) · Zbl 0894.34075
[12] M. Maroun, Y.N. Raffoul, Periodic solutions in nonlinear neutral difference equations with functional delay, J. Korean Math. Soc., in press · Zbl 1070.39020
[13] Raffoul, Y.N., Periodic solutions for neutral nonlinear differential equations with functional delay, Electron. J. differential equations, 102, 1-7, (2003) · Zbl 1054.34115
[14] Raffoul, Y.N., Periodic solutions for scalar and vector nonlinear difference equations, Panamer. math. J., 9, 97-111, (1999) · Zbl 0960.39004
[15] Rubanik, V.P., Oscillations of quasilinear systems with retardation, (1969), Nauka Moscow · Zbl 0403.34061
[16] Smart, D.R., Fixed points theorems, (1980), Cambridge Univ. Press Cambridge · Zbl 0427.47036
[17] Ding, T.R.; Iannacci, R.; Zanolin, F., On periodic solutions of sublinear Duffing equations, J. math. anal. appl., 158, 316-332, (1991) · Zbl 0727.34030
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.