Canonical Type First Order Boundedly Solvable Differential Operators

Year 2018, Volume: 10, 50 - 55, 29.12.2018

Zameddin Ismaılov , Pembe Ipek Al

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

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