Hilbert space

The initial value problem for a differential equation with a fractional derivative and a positive definite operator coefficient in a Hilbert space is considered.  The exact solution involves the solving operator (expressed as an infinite series incorporating the Cayley transform of the operator coefficient, and certain polynomials of the independent variable, which is known as the Laguerre–Cayley polynomials) and the convolution integral of the solving operator with the right-hand side of the equation.  The approximate solution is expressed through the partial sum of th

The present paper is devoted to an infinite system of differential equations.  This system consists of ternary differential equations  corresponding to $3\times3$ Jordan blocks.  The system is considered in the Hilbert space $l_2$.  A theorem about the existence and uniqueness of solution of the system is proved.