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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.

34K19Invariant manifolds (functional-differential equations)
34K12Growth, boundedness, comparison of solutions of functional-differential equations
34K13Periodic solutions of functional differential equations
Full Text: DOI
[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) · 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) · 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) · Zbl 0464.58001
[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) · Zbl 0376.34001
[11] Halanay, A.; Wexler, D.: Qualitative theory of impulsive systems. Edit. acad. RPR, bucuresti (1968) · Zbl 0176.05202
[12] Hale, J.; Lunel, S. M. V.: Introduction to functional differential equations. (1993) · Zbl 0787.34002
[13] Hartman, P.: Ordinary differential equations. (1964) · Zbl 0125.32102
[14] Kelley, A.: The stable, center-stable, center, center-unstable, unstable manifolds. An appendix in transversal mappings and flows (1967) · 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) · Zbl 0719.34002
[17] Lyapunov, A. M.: Probléme général de la stabilité du mouvement. (1949)
[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) · Zbl 0463.34002
[24] Poincaré, H.: LES méthodes nouvelles de la mécanique céleste. 1--2 (1892)
[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) · 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) · 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