We propose an approach to the investigation of generalized solutions of linear operators that satisfy weakened a priori inequalities. This approach generalizes several well-known definitions of generalized solutions of operator equations. We prove existence and uniqueness theorems for a generalized solution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
O. A. Ladyzhenskaya, Boundary-Value Problems in Mathematical Physics [in Russian], Nauka, Moscow (1973).
O. A. Ladyzhenskaya, Mathematical Problems of Dynamics of Viscous Incompressible Liquids [in Russian], Nauka, Moscow (1970).
Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).
Yu. I. Petunin, “On the concept of generalized solution of operator equations in Banach spaces,” Ukr. Mat. Zh., 48, No. 9, 1286–1290 (1996).
D. A. Klyushin, A. A. Kushchan, S. I. Lyashko, D. A. Nomirovskii, and Yu. I. Petunin, “Generalized solutions of some operator equations in Banach spaces,” Zh. Obchysl. Prykl. Mat., Issue 86, No. 1, 29–50 (2001).
D. A. Klyushin and Yu. I. Petunin, “Concept of generalized solution of nonlinear operator equations in metric spaces,” Zh. Obchysl. Prykl. Mat., Issue 87, No. 1, 11–23 (2002).
D. A. Nomirovskii, “On generalized solvability of linear systems,” Dopov. Nats. Akad. Nauk Ukr., No. 10, 26–33 (2004).
D. A. Nomirovs’kyi, “On the problem of uniqueness of generalized solutions of operator equations,” Visn. Kyiv. Univ., Ser. Fiz.-Mat. Nauk., No. 4, 223–227 (2004).
S. I. Lyashko, D. A. Nomirovskii, Yu. I. Petunin, and V. V. Semenov, Hilbert’s Twentieth Problem: Generalized Solutions of Operator Equations [in Russian], Vil’yams, Moscow (2009).
S. I. Lyashko, Generalized Optimal Control of Linear Systems with Distributed Parameters, Kluwer, Boston (2002).
S. I. Lyashko and D. A. Nomirovskii, “Generalized solvability and optimization of parabolic systems in domains with thin weakly penetrable inclusions,” Kiber. Sist. Anal., No. 5, 131–142 (2003).
A. P. Robertson and W. J. Robertson, Topological Vector Spaces, Cambridge University Press, London (1964).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 8, pp. 1011–1021, August, 2010.
Rights and permissions
About this article
Cite this article
Anikushyn, A.V., Nomirovs’kyi, D.A. Generalized solutions for linear operators with weakened a priori inequalities. Ukr Math J 62, 1175–1186 (2011). https://doi.org/10.1007/s11253-011-0435-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-011-0435-x