The Defect of a Cauchy Type Problem for Linear Equations with Several Riemann–Liouville Derivatives

We consider the existence and uniqueness of solutions to initial value problems for general linear nonhomogeneous equations with several Riemann–Liouville fractional derivatives in Banach spaces. Considering the equation solved for the highest fractional derivative $ D^{\alpha}_{t} $ , we introduce the concept of the defect $ m^{*} $ of a Cauchy type problem which determines the number of the zero initial conditions $ D^{\alpha-m+k}_{t}z(0)=0 $ , $ k=0,1,\dots,m^{*}-1 $ , necessary for the existence of the finite limits $ D^{\alpha-m+k}_{t}z(t) $ as $ t\to 0+ $ for all $ k=0,1,\dots,m-1 $ . We show that the defect $ m^{*} $ is uniquely determined by the set of orders of the Riemann–Liouville fractional derivatives in the equation. Also we prove the unique solvability of the incomplete Cauchy problem $ D^{\alpha-m+k}_{t}z(0)=z_{k} $ , $ k=m^{*},m^{*}+1,\dots,m-1 $ , for the equation with bounded operator coefficients solved for the highest Riemann–Liouville derivative. The obtained result allowed us to investigate initial problems for a linear nonhomogeneous equation with a degenerate operator at the highest fractional derivative, provided that the operator at the second highest order derivative is 0-bounded with respect to this operator, while the cases are distinguished that the fractional part of the order of the second derivative coincides or does not coincide with the fractional part of the order of the highest derivative. The results for equations in Banach spaces are used for the study of initial boundary value problems for a class of equations with several Riemann–Liouville time derivatives and polynomials in a selfadjoint elliptic differential operator of spatial variables.

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