Abstract
We consider a linear differential equation unresolved with respect to the derivative. We assume that the spectrum of the corresponding pencil is contained in two sectors. We study the unique existence of a bounded solution with any bounded free term.
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References
O. Perron, “Die Stabilitätsfrage bie Differentialgleichungen,” Math. Z. 32(5), 703–728 (1930).
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space (Nauka, Moscow, 1970) [in Russian].
M. A. Krasnosel’skii, V. Sh. Burd, and Yu. S. Kolesov, Nonlinear Almost Periodic Oscillations (Nauka, Moscow, 1970; J. Wiley, New York, 1973).
A. G. Baskakov, “Theory of Representations of Banach Algebras and Abelian Groups and Semigroups in the Spectral Analysis of Linear Operators,” Sovremenn.Matem. Fundament. Napravleniya 9, 3–151 (MAI, Moscow, 2004).
A. G. Baskakov and K. I. Chernyshov, “On Distribution Semigroups with Singularity at Zero and Bounded Solutions of Linear Differential Inclusions,” Matem. Zametki 79(1), 19–33 (2006).
A. G. Baskakov, “Spectral Analysis of Differential Operators with Unbounded Operator Coefficients, Difference Relations, and Semigroups of Difference Relations,” Izv. Ross. Akad. Nauk, Ser. Matem. 73(2), 3–68 (2009).
M. S. Bichegkuev, “To the Theory of Infinitely Differentiable Semigroups of Operators,” Algebra i Analiz 22(2), 1–13 (2010).
D. Henry, Geometric Theory of Semilinear Parabolic Equations (Springer-Verlag, Berlin, 1981; Mir, Moscow, 1985).
H. Bart, I. Gohberg, and M. A. Kaashoek, “Wiener-Hopf Factorization, Inverse Fourier Transforms and Exponentially Dichotomous Operators,” J. Funct. Anal. 68(1), 1–42 (1986).
C. V. M. Van der Mee, Exponentially Dichotomous Operators and Applications (Birkhäuser, Berlin, 2008).
I. V. Kurbatova, “Banach Algebra Associated with Linear Operator Pencils,” Matem. Zametki 86(3), 394–401 (2009).
A.G. Baskakov, “Spectral Properties of the Differential Operator \(\tfrac{d} {{dt}} - A_0\) with the Unbounded Operator A 0,” Differents. Uravneniya 27(12), 2162–2164 (1991).
V. E. Fedorov and M. A. Sagadeeva, “The Bounded on the Axis Solutions of Linear Equations of the Sobolev Type with Relatively Sectorial Operators,” Izv. Vysh. Uchebn. Zaved. Mat., No. 4, 81–84 (2005) [Russian Mathematics (Iz. VUZ) 49 (4), 77–80 (2005)].
I. V. Kurbatova, “A Generalized Impulse Characteristic,” Vestn. Voronezhsk. Gos. Univ. Fiz.-Mat. Nauki, No. 1, 148–152 (2007).
A. V. Pechkurov, “Operator Pencils, Bisemigroups, and Problems on Bounded Solutions,” Spectral and Evolution Problems, Tavrich. Nats. Univ. im. V. I. Vernadskogo (Simferopol, 2011), Vol. 21, pp. 75–86.
M. A. Lavrent’ev and B. V. Shabat, Methods of Theory of Functions of Complex Variables (Nauka, Moscow, 1965) [in Russian].
L. Schwartz, “Distributions á Valeurs Vectorielles,” Ann. Inst. Fourier 7, 1–141 (1957).
L. Schwartz, “Distributions á Valeurs Vectorielles. II,” Ann. Inst. Fourier 8, 1–209 (1957).
A. V. Pechkurov, “Invertibility in the Schwartz Space of an Operator Generated by a Pencil of Moderate Growth,” Vestn. Voronezhsk. Gos. Univ. Fiz.-Matem. Nauki, No. 2, 111–118 (2011).
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Original Russian Text © A.V. Pechkurov, 2012, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2012, No. 3, pp. 31–41.
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Pechkurov, A.V. Bisectorial operator pencils and the problem of bounded solutions. Russ Math. 56, 26–35 (2012). https://doi.org/10.3103/S1066369X1203005X
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DOI: https://doi.org/10.3103/S1066369X1203005X