Solvability of initial problems for one class of dynamical equations in quasi-Sobolev spaces

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 spases. 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.