Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
periodic solution; iterative differential equation; fixed point theorem; Green's function
periodic solution; iterative differential equation; fixed point theorem; Green's function
Summary:
The goal of the present paper is to establish some new results on the existence, uniqueness and stability of periodic solutions for a class of third order functional differential equations with state and time-varying delays. By Krasnoselskii's fixed point theorem, we prove the existence of periodic solutions and under certain sufficient conditions, the Banach contraction principle ensures the uniqueness of this solution. The results obtained in this paper are illustrated by an example.
References:
[1] Babbage C.: An essay towards the calculus of functions. Philosophical Transactions of The Royal Society of London 105 (1815), 389–423.
[2] Berinde V.: Existence and approximation of solutions of some first order iterative differential equations. Miskolc Math. Notes 11 (2010), no. 1, 13–26. DOI 10.18514/MMN.2010.256 | MR 2743858
[3] Cooke K. L.: Functional-differential equations: Some models and perturbation problems. Differential Equations and Dynamical Systems, Proc. Internat. Sympos., Mayaguez, 1965, Academic Press, New York, 1967, pages 167–183. MR 0222409
[4] Driver R. D.: Delay-differential Equations and an Application to a Two-body Problem of Classical Electrodynamics. Thesis Ph.D., University of Minnesota, 1960. MR 2613114
[5] Eder E.: The functional-differential equation $x^{\prime }( t) =x( x( t) ) $. J. Differential Equations 54 (1984), no. 3, 390–400. DOI 10.1016/0022-0396(84)90150-5 | MR 0760378
[6] Fečkan M.: On a certain type of functional-differential equations. Math. Slovaca 43 (1993), no. 1, 39–43. MR 1216267
[7] Ge W., Mo Y.: Existence of solutions to differential-iterative equation. J. Beijing Inst. Tech. 6 (1997), no. 3, 192–200. MR 1604507
[8] Lauran M.: Existence results for some differential equations with deviating argument. Filomat 25 (2011), no. 2, 21–31. DOI 10.2298/FIL1102021L | MR 2920248
[9] Li Y., Kuang Y.: Periodic solutions in periodic state-dependent delay equations and population models. Proc. Amer. Math. Soc. 130 (2002), no. 5, 1345–1353. DOI 10.1090/S0002-9939-01-06444-9 | MR 1879956
[10] Pelczyr A.: On some iterative differential equations. I. Zeszyty Nauk. Uniw. Jagiello. Prace Matemat. No. 12 (1968), 53–56. MR 0223627
[11] Ren J., Siegmun S., Chen Y.: Positive periodic solutions for third-order nonlinear differential equations. Electron. J. Differential Equations (2011), No. 66, 19 pages. MR 2801251
[12] Smart D. R.: Fixed Point Theorems. Cambridge Tracts in Mathematics, 66, Cambridge University Press, London, 1974. MR 0467717 | Zbl 0427.47036
[13] Wang K.: On the equation $x^{\prime }( t) =f( x( x( t) ) ) $. Funkcial. Ekvac. 33 (1990), no. 3, 405–425. MR 1086769
[14] Zhao H. Y, Liu J.: Periodic solutions of an iterative functional differential equation with variable coefficients. Math. Methods Appl. Sci. 40 (2017), no. 1, 286–292. DOI 10.1002/mma.3991 | MR 3583054
[15] Zhao H. Y., Fečkan M.: Periodic solutions for a class of differential equations with delays depending on state. Math. Commun. 23 (2018), no. 1, 29–42. MR 3742187
Search
Partner of
