Initial Value Problems of Linear Equations with the Dzhrbashyan-Nersesyan Derivative in Banach Spaces

Among the many different definitions of the fractional derivative, the Riemann-Liouville and Gerasimov-Caputo derivatives are most commonly used. In this paper, we consider the equations with the Dzhrbashyan-Nersesyan fractional derivative, which generalizes the Riemann-Liouville and the Gerasimov-Caputo derivatives; it is transformed into such derivatives for two sets of parameters that are, in a certain sense, symmetric. The issues of the unique solvability of initial value problems for some classes of linear inhomogeneous equations of general form with the fractional Dzhrbashyan-Nersesyan derivative in Banach spaces are investigated. An inhomogeneous equation containing a bounded operator at the fractional derivative is considered, and the solution is presented using the Mittag-Leffler functions. The result obtained made it possible to study the initial value problems for a linear inhomogeneous equation with a degenerate operator at the fractional Dzhrbashyan-Nersesyan derivative in the case of relative p-boundedness of the operator pair from the equation. Abstract results were used to study a class of initial boundary value problems for equations with the time-fractional Dzhrbashyan-Nersesyan derivative and with polynomials in a self-adjoint elliptic differential operator with respect to spatial variables.