Geodesic
Bounded and periodic solutions of nonlinear functional differential equations
Sbornik. Mathematics, Tome 203 (2012) no. 5, pp. 743-767 Cet article a éte moissonné depuis la source Math-Net.Ru

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Conditions for the existence of bounded and periodic solutions of the nonlinear functional differential equation $$ \frac{d^mx(t)}{dt^m}+(Fx)(t)=h(t), \qquad t\in \mathbb{R}, $$ are presented, involving local linear approximations to the operator $F$. Bibliography: 23 titles.
Keywords: bounded and periodic solutions, nonlinear functional differential equations, invertibility of linear operators. 
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V. E. Slyusarchuk. Bounded and periodic solutions of nonlinear functional differential equations. Sbornik. Mathematics, Tome 203 (2012) no. 5, pp. 743-767. http://geodesic.mathdoc.fr/item/SM_2012_203_5_a4/

[1] V. E. Slyusarchuk, “The method of local linear approximation in the theory of nonlinear functional-differential equations”, Sb. Math., 201:8 (2010), 1193–1215 | DOI | Zbl

[2] V. Yu. Slyusarchuk, “Conditions of the existence of bounded solutions of non-linear difference equations”, Nauk. Visn. Chernivets'kogo Univ., Mat., 454 (2009), 88–94 | Zbl

[3] V. Yu. Slyusarchuk, “Method of local linear approximation in the theory of bounded solutions of nonlinear difference equations”, Nonlinear Oscil. (N. Y.), 12:3 (2009), 380–391 | DOI | MR

[4] V. Yu. Slyusarchuk, “Method of local linear approximation in the theory of bounded solutions of nonlinear differential equations”, Ukrainian Math. J., 61:11 (2009), 1809–1829 | DOI | MR | Zbl

[5] É. Mukhamadiev, “On the inversion of functional operators in a space of functions bounded on the axes”, Math. Notes, 11:3 (1972), 169–172 | DOI | Zbl

[6] É. Mukhamadiev, “Investigations in the theory of bounded and periodic solutions of differential equations”, Math. Notes, 30:3 (1981), 713–722 | DOI | MR | Zbl | Zbl

[7] V. E. Sljusarčuk, “Invertibility of almost periodic $c$-continuous functional operators”, Math. USSR-Sb., 44:4 (1983), 431–446 | DOI | MR | Zbl | Zbl

[8] V. E. Slyusarchuk, “Integralnoe predstavlenie $c$-nepreryvnykh lineinykh operatorov”, Dokl. AN USSR. Ser. A, 8 (1981), 35–38 | MR | Zbl

[9] V. E. Slyusarchuk, “Invertibility of nonautonomous functional-differential operators”, Math. USSR-Sb., 58:1 (1987), 83–100 | DOI | MR | Zbl

[10] V. E. Slyusarchuk, “Necessary and sufficient conditions for invertibility of nonautonomous functional-differential operators”, Math. Notes, 42:2 (1987), 648–651 | DOI | MR | Zbl | Zbl

[11] V. E. Slyusarchuk, “Necessary and sufficient conditions for invertibility of uniformly c-continuous functional-differential operators”, Ukrainian Math. J., 41:2 (1989), 180–183 | DOI | MR | Zbl

[12] Chan Khyu Bong, Pochti periodicheskie i ogranichennye resheniya lineinykh funktsionalno-differentsialnykh uravnenii, Dis. ... dokt. fiz.-matem. nauk, In-t matem. AN Ukrainy, Kiev, 1993

[13] V. E. Slyusarchuk, “Slabo nelineinye vozmuscheniya impulsnykh sistem”, Matem. fizika i nelineinaya mekh., 15:49 (1991), 32–35 | MR

[14] M. A. Krasnosel'skii, V. Sh. Burd, Yu. S. Kolesov, Nonlinear almost periodic oscillations, Halsted Press, New York–Toronto, ON; Israel Program for Scientific Transl., Jerusalem–London, 1973 | MR | MR | Zbl | Zbl

[15] A. A. Andronov, A. A. Vitt, S. E. Khaikin, Theory of oscillators, Pergamon Press, Oxford, 1966 | MR | MR | Zbl | Zbl

[16] N. N. Bogoliubov, Yu. A. Mitropolsky, Asymptotic methods in the theory of non-linear oscillations, Hindustan Publishing Corp., Delhi; Gordon and Breach, New York, 1961 | MR | MR | Zbl | Zbl

[17] R. Reissig, G. Sansone, R. Conti, Qualitative Theorie nichtlinearer Differentialgleichungen, Edizioni Cremonese, Rome, 1963 | MR | MR | Zbl

[18] E. F. Mishchenko, N. Kh. Rozov, Differential equations with small parameters and relaxation oscillations, Math. Concepts Methods Sci. Engrg., 13, Plenum Press, New York, 1980 | MR | MR | Zbl | Zbl

[19] M. A. Krasnoselskii, P. P. Zabreiko, Geometricheskie metody nelineinogo analiza, Nauka, M., 1975 | MR | Zbl

[20] L. Nirenberg, Topic in nonlinear functional analysis, Courant Institute of Math. Sciences, New York Univ., New York, 1974 | MR | MR | Zbl | Zbl

[21] Ju. L. Dalec'kiǐ, M. G. Kreǐn, Stability of solutions of differential equations in Banach space, Amer. Math. Soc., Providence, RI, 1974 | MR | MR | Zbl | Zbl

[22] G. M. Fikhtengol'ts, Differential and integral calculus, v. I, Deutscher, Berlin, 1977 | MR | Zbl | Zbl

[23] A. N. Kolmogorov, S. V. Fomin, Introductory real analysis, Prentice-Hall, Englewood Cliffs, NJ, 1970 | MR | MR | Zbl | Zbl