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Integral manifolds of differential equations with piecewise constant argument of generalized type. (English) Zbl 1122.34054

Equations with piecewise constant delayed argument are studied, like
\[ \dot x(t) = f(t, x(t), x([t])). \]
One main assumption is that there is a linear part \(\dot x(t) = A(t) x(t)\) of the equation which has an exponential dichotomy. Manifolds of solutions converging to zero in forward/backward time are constructed using a Perron-type approach. Existence and uniqueness of bounded/periodic solutions is obtained (as a consequence of the exponential dichotomy). The author uses successive approximations instead of the contraction theorem.

MSC:

34K19 Invariant manifolds of functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
34K13 Periodic solutions to functional-differential equations
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[1] Aftabizadeh, A. R.; Wiener, J.; Xu, J.-M., Oscillatory and periodic solutions of delay differential equations with piecewise constant argument, Proc. Amer. Math. Soc., 99, 673-679 (1987) · Zbl 0631.34078
[2] Akhmetov, M. U.; Perestyuk, N. A., Differential properties of solutions and integral surfaces of nonlinear impulse systems, Differ. Equ., 28, 445-453 (1992) · Zbl 0799.34007
[3] Akhmetov, M. U.; Perestyuk, N. A., Integral sets of quasilinear impulse systems, Ukrainian Math. J., 44, 1-17 (1992) · Zbl 0786.34005
[4] M.U. Akhmet, Almost periodic solutions of differential equations with piecewise constant argument of generalized type (submitted for publication); M.U. Akhmet, Almost periodic solutions of differential equations with piecewise constant argument of generalized type (submitted for publication) · Zbl 1166.34039
[5] Alonso, A.; Hong, J.; Obaya, R., Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences, Appl. Math. Lett., 13, 131-137 (2000) · Zbl 0978.34039
[6] Bogolyubov, N. N., On some statistical methods in mathematical physics, Acad. Nauk R.S.R. (1945), (in Russian) · Zbl 0063.00496
[7] Bogolyubov, N. N.; Mitropol’sky, Yu. A., The method of integral manifolds in nonlinear mechanics, Contrib. Differ. Equ., 2, 123-196 (1963)
[8] Carr, J., Applications of Center Manifold Theory (1981), Springer-Verlag: Springer-Verlag New York
[9] Cooke, K. L.; Wiener, J., Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl., 99, 265-297 (1984) · Zbl 0557.34059
[10] Coppel, W. A., Dichotomies in Stability Theory, (Lecture Notes in Mathematics (1978), Springer-Verlag: Springer-Verlag New York) · Zbl 0376.34001
[11] Halanay, A.; Wexler, D., Qualitative theory of impulsive systems, Edit. Acad. RPR, Bucuresti (1968), (in Romanian) · Zbl 0176.05202
[12] Hale, J.; Lunel, S. M.V., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0787.34002
[13] Hartman, P., Ordinary Differential Equations (1964), Wiley: Wiley New York · Zbl 0125.32102
[14] Kelley, A., The stable, center-stable, center, center-unstable, unstable manifolds, (Abraham, R.; Robbin, J., An Appendix in Transversal Mappings and Flows (1967), Benjamin: Benjamin New York) · Zbl 0173.11001
[15] Küpper, T.; Yuan, R., On quasi-periodic solutions of differential equations with piecewise constant argument, J. Math. Anal. Appl., 267, 173-193 (2002) · Zbl 1008.34063
[16] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific New York · Zbl 0719.34002
[17] Lyapunov, A. M., Probléme général de la stabilité du mouvement (1949), Princeton University Press: Princeton University Press Princeton, NJ
[18] Palmer, K. J., A generalization of Hartman’s linearisation theorem, J. Math. Anal. Appl., 41, 753-758 (1973) · Zbl 0272.34056
[19] Palmer, K. J., Linearisation near an integral manifold, J. Math. Anal. Appl., 51, 243-255 (1975) · Zbl 0311.34056
[20] Papaschinopoulos, G., Some results concerning a class of differential equations with piecewise constant argument, Math. Nachr., 166, 193-206 (1994) · Zbl 0830.34062
[21] Papaschinopoulos, G., Linearisation near the integral manifold for a system of differential equations with piecewise constant argument, J. Math. Anal. Appl., 215, 317-333 (1997) · Zbl 0892.34045
[22] Pliss, V. A., A reduction principle in the theory of the stability of motion, Izv. Akad. Nauk SSSR, Ser. Mat., 28, 1297-1324 (1964) · Zbl 0131.31505
[23] Pliss, V. A., Integral sets of periodic systems of differential equations, Izdat. Nauka, Moscow (1977), (in Russian) · Zbl 0463.34002
[24] Poincaré, H., (Les méthodes nouvelles de la mécanique céleste, vols. 1-2 (1892), Gauthier-Villars: Gauthier-Villars Paris)
[25] Pugh, C.; Shub, M., Linearisation of normally hyperbolic diffeomorphisms and flows, Invent. Math., 10, 187-190 (1970) · Zbl 0206.25802
[26] Samoilenko, A. M.; Perestyuk, N. A., Impulsive Differential Equations (1995), World Scientific: World Scientific Singapore · Zbl 0837.34003
[27] Seifert, G., Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, J. Differential Equations, 164, 451-458 (2000) · Zbl 1009.34064
[28] Stokes, A., Local coordinates around a limit cycle of a functional differential equation with applications, J. Differential Equations, 24, 153-172 (1977) · Zbl 0342.34055
[29] Wiener, J., Generalized Solutions of Functional Differential Equations (1993), World Scientific: World Scientific Singapore · Zbl 0874.34054
[30] Wiener, J.; Lakshmikantham, V., A damped oscillator with piecewise constant time delay, Nonlinear Stud., 7, 78-84 (2000) · Zbl 1016.34069
[31] Muroya, Y., Persistence, contractivity and global stability in logistic equations with piecewise constant delays, J. Math. Anal. Appl., 270, 602-635 (2002) · Zbl 1012.34076
[32] Rong, Yuan, The existence of almost periodic solutions of retarded differential equations with piecewise argument, Nonlinear Anal., 48, 1013-1032 (2002) · Zbl 1015.34058
[33] Rong, Yuan, On the spectrum of almost periodic solution of second order scalar functional differential equations with piecewise constant argument, J. Math. Anal. Appl., 303, 103-118 (2005) · Zbl 1073.34085
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