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On periodic solutions of systems of linear functional-differential equations
Archivum mathematicum, Tome 33 (1997) no. 3, pp. 197-212 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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This paper deals with the system of functional-differential equations \[ \frac{dx(t)}{dt}=p(x)(t)+q(t), \] where $p:C_\omega ({R}^n)\rightarrow L_\omega ({R}^n)$ is a linear bounded operator, $q\in L_\omega ({R}^n)$, $\omega >0$ and $C_\omega ({R}^n)$ and $L_\omega ({R}^n)$ are spaces of $n$-dimensional $\omega $-periodic vector functions with continuous and integrable on $[0,\omega ]$ components, respectively. Conditions which guarantee the existence of a unique $\omega $-periodic solution and continuous dependence of that solution on the right hand side of the system considered are established.
This paper deals with the system of functional-differential equations \[ \frac{dx(t)}{dt}=p(x)(t)+q(t), \] where $p:C_\omega ({R}^n)\rightarrow L_\omega ({R}^n)$ is a linear bounded operator, $q\in L_\omega ({R}^n)$, $\omega >0$ and $C_\omega ({R}^n)$ and $L_\omega ({R}^n)$ are spaces of $n$-dimensional $\omega $-periodic vector functions with continuous and integrable on $[0,\omega ]$ components, respectively. Conditions which guarantee the existence of a unique $\omega $-periodic solution and continuous dependence of that solution on the right hand side of the system considered are established.
Classification : 34C25, 34K05, 34K13, 34K15
Keywords: linear functional-differential system; differential system with deviated argument; $\omega$-periodic solution 
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Kiguradze, Ivan; Půža, Bedřich. On periodic solutions of systems of linear functional-differential equations. Archivum mathematicum, Tome 33 (1997) no. 3, pp. 197-212. http://geodesic.mathdoc.fr/item/ARM_1997_33_3_a2/

[1] N.V. Azbelev, V.P. Maksimov and L.F. Rakhmatullina: Introduction to the Theory of Functional Differential Equations. (Russian) Nauka, Moscow, 1991. | MR

[2] E.A. Coddington and N. Levinson: Theory of Ordinary Differential Equations. McGraw-Hill Book Company, Inc., New York Toronto London, 1955. | MR

[3] J. Cronin: Periodic solutions of some nonlinear differential equations. J. Differential Equations 3 (1967), 31-46. | MR | Zbl

[4] J.K. Hale: Periodic and almost periodic solutions of functional-differential equations. Arch. Rational Mech. Anal. 15 (1964), 289-304. | MR | Zbl

[5] P. Hartman: Ordinary Differential Equations. John Wiley, New York, 1964. | MR | Zbl

[6] A. Halanay: Optimal control of periodic solutions. Rev. Roumaine Math. Pures Appl. 19 (1974), 3-16. | MR | Zbl

[7] G.S. Jones: Asymptotic fixed point theorems and periodic systems of functional-differential equations. Contr. Diff. Eqn 2 (1963), 385-405. | MR | Zbl

[8] I.T. Kiguradze: On periodic solutions of systems of non-autonomous ordinary differential equations. (Russian) Mat. zametki 39 (1986), No.4, 562-575. | MR

[9] I.T. Kiguradze: Boundary Value Problems for Systems of Ordinary Differential Equations. (Russian) Modern problems in mathematics. The latest achievements (Itogi Nauki i Tekhniki. VINITI Acad. Sci. USSR), Moscow, 1987, V.30, 3-103. | MR | Zbl

[10] I.T. Kiguradze and B. Půža: On some boundary value problems for ordinary differential equation systems. (Russian) Diff. Uravnenija 12 (1976), No.12, 2139-2148. | MR

[11] I.T. Kiguradze and B. Půža: On boundary value problems for systems of linear functional differential equations. Czech. Math. J. To appear.

[12] M.A. Krasnosel’skij: An alternative priciple for establishing the existence of periodic solutions of differential equations with a lagging argument. Soviet Math. Dokl. 4 (1963), 1412-1415.

[13] M.A. Krasnosel’skij: The theory of periodic solutions of non-autonomous differential equations. (Russian) Math. Surveys 21 (1966), 53-74.

[14] M.A. Krasnosel’skij: The Operator of Translation along the Trajectories of Differential Equations. (Russian) Nauka, Moscow, 1966.

[15] M.A. Krasnosel’skij and A.I. Perov: On a existence principle for bounded, periodic and almost periodic solutions of systems of ordinary differential equations. (Russian) Dokl. Akad. Nauk SSSR 123 (1958), 235-238. | MR

[16] J. Kurzweil: Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslovak Math. J. 7 (1957), 418-449. | MR | Zbl

[17] J. Kurzweil and Z. Vorel: Continuous dependence of solutions of differential equations on a parameter. (Russian) Czechoslovak Math. J. 7 (1957), 568-583. | MR

[18] J. Mawhin: Periodic Solutions of Nonlinear Functional Differential Equations. J. Diff. Equ. 10 (1971), 240-261. | MR | Zbl

[19] Z. Opial: Continuous parameter dependence in linear systems of differential equations. J. Diff. Eqs. 3 (1967), 571-579. | MR | Zbl

[20] T.A. Osechkina: Kriterion of unique solvability of periodic boundary value problem for functional differential equation. (Russian) Izv. VUZ. Matematika No. 10 (1994), 48-52. | MR

[21] K. Schmitt: Periodic solutions of nonlinear differential systems. J. Math. Anal. and Appl. 40 (1972), No.1, 174-182. | MR | Zbl

[22] æ. Schwabik, M. Tvrdì and O. Vejvoda: Differential and Integral Equations: Boundary Value Problems and Adjoints. Academia, Praha, 1979. | MR

[23] S. Sedziwy: Periodic solutions of a system of nonlinear differential equations. Proc. Amer. Math. Soc. 48 (1975), No.8, 328-336. | MR | Zbl

[24] Z. Vorel: Continuous dependence of parameters. Nonlinear Anal. Theory Meth. and Appl. 5 (1981), No.4, 373-380. | MR

[25] T. Yoshizawa: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. Springer-Verlag, New York Heidelberg Berlin, 1975. | MR | Zbl