We establish necessary and sufficient conditions for autonomous nonlinear differential operators defined in the space of functions bounded and continuous on the axis to be C1-diffeomorphisms.
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References
S. Lang, Introduction to Differentiable Manifolds, Wiley, New York (1962).
J. W. Milnor, Topology from the Differentiable Viewpoint, University Press of Virginia, Charlottesville (1965); A. H. Wallace, Differential Topology, Benjamin, New York (1968).
M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer, New York (1973).
V. A. Zorich, Mathematical Analysis [in Russian], Vol. 2, Nauka, Moscow (1984).
V. Yu. Slyusarchuk, “Necessary and sufficient conditions for the invertibility of nonlinear differentiable maps,” Ukr. Mat. Zh., 68, No. 4, 563–576 (2016); English translation: Ukr. Math. J., 68, No. 4, 638–652 (2016).
V. Yu. Slyusarchuk, “Invertibility of the theorem on inverse function for differentiable functions,” Bukov. Mat. Zh., 2, No. 4, 112–113 (2014).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Ukrainian], Vyshcha Shkola, Kyiv (1974).
J. L. Massera and J. J. Schaffer, Linear Differential Equations and Function Spaces, Academic Press, New York (1966).
M. A. Krasnosel’skii, V. Sh. Burd, and Yu. S. Kolesov, Nonlinear Almost Periodic Oscillations [in Russian], Nauka, Moscow (1970).
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1970).
Yu. A. Mitropol’skii, A. M. Samoilenko, and V. L. Kulik, Investigation of the Dichotomy of Linear Systems of Differential Equations with the Help of Lyapunov Functions [in Russian], Naukova Dumka, Kiev (1990).
V. E. Slyusarchuk, “Exponential dichotomy for solutions of discrete systems,” Ukr. Mat. Zh., 35, No. 1, 109–115 (1983); English translation: Ukr. Math. J., 35, No. 1, 98–103 (1983).
V. Yu. Slyusarchuk, “Method of local linear approximation in the theory of nonlinear equations,” Nauk. Visn. Cherniv. Univ., 2, No. 2–3, 157–163 (2012).
S. Bochner, “Beiträge zur Theorie der Fastperiodischen. I Teil. Funktionen einer Variablen,” Math. Ann., 96 119–147 (1927); S. Bochner, “Beiträge zur Theorie der Fastperiodischen. II Teil. Funktionen mehrerer Variablen,” Math. Ann., 96, 383–409 (1927).
É. Mukhamadiev, “On the invertibility of functional operators in the space of functions bounded on the axis,” Mat. Zametki, 11, No. 3, 269–274 (1972).
V. E. Slyusarchuk, “Invertibility of almost periodic c-continuous functional operators,” Mat. Sb., 116(158), No. 4(12), 483–501 (1981).
L. A. Lyusternik and V. I. Sobolev, A Brief Course in Functional Analysis [in Russian], Vysshaya Shkola, Moscow (1982).
G. M. Fikhtengol’ts, A Course in Differential and Integral Calculus [in Russian], Vol. 2, Nauka, Moscow (1966).
V. E. Slyusarchuk, “Bounded and periodic solutions of nonlinear functional-differential equations,” Mat. Sb., 203, No. 5, 135–160 (2012).
V. E. Slyusarchuk, “Necessary and sufficient conditions for existence and uniqueness of bounded and almost-periodic solutions of nonlinear differential equations,” Acta Appl. Math., 65, No. 1–3, 333–341 (2001).
V. A. Zorich, “M. A. Lavrent’ev’s theorem on quasiconformal mappings of the space,” Mat. Sb., 74 (116), No. 3, 417–433 (1967).
M. Radulescu and S. Radulescu, “Global inversion theorems and applications to differential equations,” Nonlin. Anal., 4, No. 4, 951–965 (1980).
V. A. Trenogin, “Global invertibility of nonlinear operators and the method of continuation with respect to a parameter,” Dokl. Ros. Akad. Nauk, 350, No. 4, 455–457 (1996).
V. Yu. Slyusarchuk, Method of Local Linear Approximation in the Theory of Nonlinear Equations [in Ukrainian], National University of Water Management and Utilization of Natural Resources, Rivne (2011).
V. E. Slyusarchuk, “Invertibility of nonautonomous functional-differential operators,” Mat. Sb., 130(172), No. 1(5), 86–104 (1986).
V. E. Slyusarchuk, “Necessary and sufficient conditions for the invertibility of nonautonomous functional-differential operators,” Mat. Zametki, 42, No. 2, 262–267 (1987).
V. E. Slyusarchuk, “Necessary and sufficient conditions for invertibility of uniformly c-continuous functional-differential operators,” Ukr. Mat. Zh., 41, No. 2, 201–205 (1989); English translation: Ukr. Math. J., 41, No. 2, 180–183 (1989).
Yu. V. Trubnikov and A. I. Perov, Differential Equations with Monotone Nonlinearities [in Russian], Nauka i Tekhnika, Minsk (1986).
L. Amerio, “Soluzioni quasiperiodiche, o limital, di sistemi differenziali non lineari quasi-periodici, o limitati,” Ann. Mat. Pura Appl., 39, 97–119 (1955).
R. Reissig, G. Sansone, and R. Conti, Qualitative Theorie Nichtlinear Differentialgleichungen, Edizioni Cremonese, Roma (1963).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 11, pp. 1507–1515, November, 2021. Ukrainian DOI: https://doi.org/10.37863/umzh.v73i11.2323.
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Slyusarchuk, V.Y. Invertibility Conditions for Nonlinear Autonomous Differential Operators in the Space of Functions Bounded on the Axis. Ukr Math J 73, 1741–1751 (2022). https://doi.org/10.1007/s11253-022-02027-z
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DOI: https://doi.org/10.1007/s11253-022-02027-z