Abstract
We obtain necessary and sufficient conditions for the Lipschitzian invertibility of the differential mapping d/dt − f, where \(\user1{f}\user2{:}\mathbb{R} \to \mathbb{R}\) is a continuous mapping, in the spaces \(\user1{L}_\user1{p} \user2{(}\mathbb{R}\user2{,}\mathbb{R}\user2{)},{\text{ }}1 \leqslant \user1{p} \leqslant \infty \).
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REFERENCES
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space [in Russian], Nauka, Moscow, 1970.
M. A. Krasnosel'skii, V. Sh. Burd, and Yu. S. Kolesov, Nonlinear Almost-Periodic Ocillations [in Russian], Nauka, Moscow, 1970.
J. Massera and J. Shäffer, Linear Differential Equations and Function Spaces, New York, 1966.
Ph. Hartman, Ordinary Differential Equations, New York, 1964.
Yu. A. Mitropol'skii, A. M. Samoilenko, and V. L. Kulik, Study of the Dichotomy of Linear Systems of Differential Equations Using Lyapunov Functions [in Russian], Naukova Dumka, Kiev, 1990.
V. G. Kurbatov, Linear Difference-Differential Equations [in Russian], Voronezh Univ., Voronezh, 1990.
É. Mukhamadiev, “On the invertibility of functional operators in the space of functions bounded on the axis,” Mat. Zametki [Math. Notes], 11 (1972), no. 3, 269–274.
V. E. Slyusarchuk, “The invertibility of almost-periodic c-continuous functional operators,” Mat. Sb. [Math. USSR-Sb.], 116 (158) (1981), no. 4 (12), 483–501.
V. E. Slyusarchuk, “The invertibility of nonautonomous functional-differential operators,” Mat. Sb. [Math. USSR-Sb.], 130 (172) (1986), no. 1 (5), 86–104.
V. E. Slyusarchuk, “Necessary and sufficient conditions for the invertibility of nonautonomous functional-differential operators,” Mat. Zametki [Math. Notes], 42 (1987), no. 2, 262–267.
V. E. Slyusarchuk, “Weakly nonlinear perturbations of normally solvable functional-differential and discrete equations,” Ukrain. Mat. Zh. [Ukrainian Math. J.], 39 (1987), no. 5, 660–662.
V. E. Slyusarchuk, “Necessary and sufficient conditions for the invertibility of uniformly c-continuous functional-differential operators,” Ukrain. Mat. Zh. [Ukrainian Math. J.], 41 (1989), no. 2, 201–205.
A. G. Baskakov, “On conditions for the invertibility of linear differential and difference operators” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 333 (1993), no. 3, 282–284.
A. G. Baskakov, “Semigroups of difference operators in the spectral analysis of linear differential operators,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 30 (1996), no. 3, 1–11.
V. M. Tyurin, “On the invertibility of linear differential operators in function spaces,” Sibirsk. Mat. Zh. [Siberian Math. J.], 32 (1991), no. 3, 160–165.
Yu. V. Trubnikov and A. I. Perov, Differential Equations with Monotone Nonlinearities [in Russian], Nauka and Tekhnika, Minsk, 1986.
V. E. Slyusarchuk, “Necessary and sufficient conditions for the existence and uniqueness of bounded solutions of nonlinear differential equations,” Neliniini Kolivannya, 2 (1999), no. 4, 523–537.
A. B. Antonevich and Ya. V. Radyno, Functional Analysis and Integral Equations [in Russian], Univers. Izd., Minsk, 1984.
I. P. Natanson, The Theory of Functions of a Real Variable [in Russian], Nauka, Moscow, 1974.
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Slyusarchuk, V.E. Necessary and Sufficient Conditions for the Lipschitzian Invertibility of the Nonlinear Differential Mapping d/dt − f in the Spaces LP(ℝ,ℝ), 1≤p≤ ∞. Mathematical Notes 73, 843–854 (2003). https://doi.org/10.1023/A:1024058015313
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DOI: https://doi.org/10.1023/A:1024058015313