We consider linear boundary-value problems for operator equations with generalized invertible operators in Banach spaces that have bases. Using the technique of generalized inverse operators applied to generalized invertible operators in Banach spaces, we establish conditions for the solvability of linear boundary-value problems for these operator equations and obtain formulas for the representation of their solutions. We consider special cases of these boundary-value problems, namely, so-called n- and d-normally solvable boundary-value problems as well as normally solvable problems for Noetherian operator equations.
Similar content being viewed by others
References
I. Ts. Gokhberg and N. Ya. Krupnik, Introduction to the Theory of One-Dimensional Singular Integral Operators [in Russian], Shtiintsa, Kishinev (1973).
M. I. Kadets and B. S. Mityagin, “Complementable subspaces in Banach spaces,” Usp. Mat. Nauk, 28, Issue 6, 77–94 (1973).
Yu. M. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1970).
M. M. Grinblyum, “Biorthogonal systems in a Banach space,” Dokl. Akad. Nauk SSSR, 47, No. 2, 79–82 (1945).
V. F. Zhuravlev, “Solvability criterion and representation of solutions of n-normal and d-normal linear operator equations in a Banach space,” Ukr. Mat. Zh., 62, No. 2, 167–182 (2010); English translation: Ukr. Math. J., 62, No. 2, 186–202 (2010).
V. F. Zhuravlev, “Solution of normally solvable operator equations in Banach spaces with basis,” Dokl. Ros. Akad. Nauk, 352, No. 3, 304–306 (1997).
A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht (2004).
A. A. Boichuk, V. F. Zhuravlev, and A. M. Samoilenko, Generalized Inverse Operators and Noetherian Boundary-Value Problems [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1995).
L. A. Lyusternik and V. I. Sobolev, A Brief Course in Functional Analysis [in Russian], Vysshaya Shkola, Moscow (1982).
V. F. Zhuravlev, “Linear boundary-value problems for differential equations in Banach spaces,” in: Abstracts of the International Scientific Conference “Differential Equations, Theory of Functions, and Their Applications” Dedicated to the 70th birthday of Academician A. M. Samoilenko (June 18–21, 2008), Melitopol’ (2008), p. 50.
O. A. Boichuk and E. V. Panasenko, “Boundary-value problems for differential equations in a Banach space,” Nelin. Kolyvannya, 12, No. 1, 16–19 (2009); English translation: Nonlin. Oscillations, 12, No. 1, 15–18 (2009).
V. S. Korolyuk and A. F. Turbin, Mathematical Foundations of Phase Lumping of Complex Systems [in Russian], Naukova Dumka, Kiev (1978).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Neliniini Kolyvannya, Vol. 13, No. 4, pp. 522–532, October–December, 2010.
Rights and permissions
About this article
Cite this article
Zhuravlev, V.F. Boundary-value problems for linear equations with a generalized invertible operator in a Banach space with basis. Nonlinear Oscill 13, 558–568 (2011). https://doi.org/10.1007/s11072-011-0131-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11072-011-0131-7