Abstract
We prove that the operator d/dt + A constructed on the basis of a sectorial operator A with spectrum in the right half-plane of ℂ is continuously invertible in the Sobolev spaces W 1p (ℝ, D α), α ≥ 0. Here, D α is the domain of definition of the operator A α and the norm in D α is the norm of the graph of A α.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Henry, Geometric Theory of Semilinear Parabolic Equations [Russian translation], Mir, Moscow (1985).
E. Hille and R. Phillips, Functional Analysis and Semi-Groups [Russian translation], Inostrannaya Literatura, Moscow (1962).
B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations [in Russian], Moscow University, Moscow (1978).
A. G. Baskakov, “Semigroups of difference operators in the spectral analysis of linear differential operators,” Funct. Anal. Prilozhen., 30, No. 3, 1–11 (1996).
A. G. Baskakov and A. I. Pastukhov, “Spectral analysis of operators of weighted translation with unbounded operator coefficients,” Sib. Mat. Zh., 42, No. 6, 1231–1243 (2001).
M. F. Horodnii, “l p -solutions of one difference equation in a Banach space,” Ukr. Mat. Zh., 55, No. 3, 425–430 (2003).
J. A. Goldstein, Semigroups of Linear Operators and Applications [Russian translation], Vyshcha Shkola, Kiev (1989).
M. F. Horodnii, “L p -solutions of a differential equation with bounded operator coefficient,” Nauk. Zap. NaUkma, Fiz.-Mat. Nauki, 21, 32–35 (2003).
M. F. Horodnii and A. V. Chaikovs’kyi, “On approximation of a bounded solution of a differential equation with unbounded operator coefficient,” Dopov. Nats. Akad.Nauk Ukr., No. 6, 10–14 (2002).
E. M. Stein, Singular Integrals and Differentiability Properties of Functions [Russian translation], Mir, Moscow (1973).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1020–1025, August, 2007.
Rights and permissions
About this article
Cite this article
Horodnii, M.F. On the invertibility of the operator d/dt + A in certain functional spaces. Ukr Math J 59, 1130–1136 (2007). https://doi.org/10.1007/s11253-007-0074-4
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11253-007-0074-4