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Solvability of initial problems for one class of dynamical equations in quasi-Sobolev spaces. (English) Zbl 1359.47039

Summary: The equations, which are not solved with respect to the highest derivative, are now actively studied. Such equations are also called the Sobolev type equations. Note that these equations in Banach spaces are studied quite well. Quasi-Sobolev spaces are quasi normalized complete spaces of sequences. Recently these spaces began to be studied. The interest to such spaces and its equations is connected with a desire to fill up the theory more than with practical applications.
The paper is devoted to the study of solvability of the Cauchy problem and the Showalter-Sidorov problem for a class of equations considered in the quasi-Sobolev spaces. To this end we use properties of the equation operators, namely the relative boundedness of the operators. To illustrate abstract results we consider an analogue of the Hoff equation in the quasi-Sobolev spaces.

MSC:

47D06 One-parameter semigroups and linear evolution equations
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46B45 Banach sequence spaces
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