Canonical Type First Order Boundedly Solvable Differential Operators

Abstract

The general information on the degenerate differential equations in Banach spaces can be found in book of A. Favini and A. Yagi [1]. The fundamental interest to such equations are motivated by applications in different fields of life sciences.

The solvability of the considered problems may be seen as boundedly solvability of linear differential operators in corresponding functional Banach spaces. Note that the theory of boundedly solvable extensions of a linear densely defined closed operator in Hilbert spaces was presented in the important works of M. I. Vishik in [2], [3]. Generalization of these results to the nonlinear and complete additive Hausdorff topological spaces in abstract terms of abstract boundary conditions have been done by B. K. Kokebaev, M. O. Otelbaev and A. N. Synybekov in [4]-[6]. Another approach to the description of regular extensions for some classes of linear differential operators in Hilbert spaces of vector-functions at finite interval has been offered by A. A. Dezin [7] and N. I. Pivtorak [8].

Remember that a linear closed densely defined operator on any Hilbert space is called boundedly solvable, if it is one-to-one and onto and its inverse is bounded.

The main goal of this work is to describe of all boundedly solvable extensions of the minimal operator generated by first-order linear canonical type differential-operator expression in the weighted Hilbert space of vector-functions at finite interval in terms of boundary conditions by using the methods of operator theory. Later on, the structure of spectrum of this type extension will be investigated. 

Keywords

• Boundedly solvable operator,Differential-operator expression,Spectrum

References

1. Dezin, A., General Problems in the Theory of Boundary Value Problems, Nauka, Moskov, 1980.
2. Favini, A., Yagi, A., Degenerate Di_erential Equations in Banach Spaces, Marcel Dekker, Inc., New-York-Basel-Hong Kong, 1999.
3. Goldstein, J. A., Semigroups of Linear Operators and Applications, Oxford University Press, New York and Oxford, 1985.
4. Hörmander, L., On the theory of general partial di_erential operators, Acta. Math. 94 (1955), 162-166.
5. Kokebaev, B. K., Otelbaev, M., Shynybekov, A. N., On the theory of contraction and extension of operators I, Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat. 5 (1982), 24-26 (in Russian).
6. Kokebaev, B. K., Otelbaev, M., Shynybekov, A. N., On the theory of contraction and extension of operators II, Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat. 110 (1983), 24-26 (in Russian).
7. Kokebaev, B. K., Otelbaev, M., Shynybekov, A. N., On questions of extension and restriction of operator, Sov. Math. Dokl. 28 (1983), 259-262.
8. Pivtorak, N. I., Solvable boundary value problems for an operator-differential equations for parabolic type, Akad. Nauk. Ukr. SSR Inst. Mat. Kiev 9 (1985), 104-107 (in Russian).

9. Vishik, M. I., On linear boundary problems for di_erential equations, Doklady Akad. Nauk SSSR (N.S) 65 (1949), 785-788.
10. Vishik, M. I., On general boundary problems for elliptic di_erential equations, Amer. Math. Soc. Transl. II 24 (1963), 107-172.