Abstract
Incomplete Cauchy-type problems are considered for linear multi-term equations solved with respect to the highest derivative in Banach spaces with fractional Riemann–Liouville derivatives and with linear closed operators at them. Some new existence and uniqueness theorems for solutions are presented explicitly and the analyticity of the solutions of the homogeneous equations are also shown. The asymmetry of the Cauchy-type problem under study is expressed in the presence of a so-called defect, which shows the number of lower-order initial conditions that should not be set when setting the problem. As applications, our abstract results are used in the study of a class of initial-boundary value problems for multi-term equations with Riemann–Liouville derivatives in time and with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables.
Highlights
IntroductionPublisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
A linear multi-term fractional equationReceived: 4 December 2021Accepted: 23 December 2021Published: 5 January 2022Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Licensee MDPI, Basel, Switzerland. m −1 Dtα z(t) = ∑ α−m+ j A j Dt n z(t) +
For a linear equation with several Riemann–Liouville derivatives and with constant coefficients, the authors of this work showed that the set of orders of derivatives from the equation determines the defect of the Cauchy-type problem, which determines the number of lower-order initial conditions that must be excluded from the statement of the problem for its solvability
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The unique solvability of the Cauchy-type problem for linear and nonlinear equations with the Riemann–Liouville derivative and with an operator from Aα (θ0 , a0 ) in the right-hand side was studied in [11,12,13,14,15]. In contrast to the results of work [8] on the problem (1) and (3) with bounded operators, in this case the equations are considered, in which the degrees of polynomials at lower fractional derivatives in time can exceed the degree of the polynomial at the highest time-derivative
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