In the Sobolev-type space with exponential weight, we obtain sufficient conditions for the well-posed and unique solvability on the entire axis of a fourth-order operator-differential equation whose main part has a multiple characteristic. We establish estimates for the norms of the operators of intermediate derivatives related to the conditions of solvability. In addition, we deduce the relationship between the exponent of the weight and the lower bound of the spectrum of the main operator appearing in the principal part of the equation. The obtained results are illustrated by an example of a problem for partial differential equations.
This is a preview of subscription content, log in via an institution to check access.
References
S. G. Krein, Linear Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1967).
V. I. Gorbachuk and M. L. Gorbachuk, Boundary-Value Problems for Differential-Operator Equations [in Russian], Naukova Dumka, Kiev (1984).
S. Ya. Yakubov, Linear Differential-Operator Equations and Their Applications [in Russian], Élm, Baku (1985).
A. R. Aliev, “On a boundary-value problem for one class of differential equations of the fourth order with operator coefficients,” Azerb. J. Math., 1, No. 1, 145–156 (2011).
A. A. Shkalikov, “Elliptic equations in a Hilbert space and related spectral problems,” in: Proc. of the I. G. Petrovskii Seminar [in Russian], 14 (1989), pp. 140–224.
J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications [Russian translation], Mir, Moscow (1971).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 5, pp. 699–707, May, 2014.
Rights and permissions
About this article
Cite this article
Aliev, A.R. On the Solvability of a Fourth-Order Operator-Differential Equation with Multiple Characteristic. Ukr Math J 66, 781–791 (2014). https://doi.org/10.1007/s11253-014-0972-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-014-0972-1