Analytic Resolving Families for Equations with Distributed Riemann–Liouville Derivatives

We study a class of equations in Banach spaces with a Riemann–Liouville-type integro-differential operator with an operator-valued convolution kernel. The properties of \(k\)-resolving operators of such equations are studied and the class \(\mathcal A_{m,K,\chi}\) of linear closed operators is defined such that the belonging to this class is necessary and, in the case of commutation of the operator with the convolution kernel, is sufficient for the existence of analytic in the sector \(k\)-resolving families of operators of the equation under study. Under certain additional conditions on the convolution kernel, we prove theorems on the unique solvability of the nonhomogeneous linear equation of the class under consideration if the nonhomogeneity is continuous in the norm of the graph of the operator from the equation or Hölder continuous. We obtain the theorem on sufficient conditions on an additive perturbation of an operator of the class \(\mathcal A_{m,K,\chi}\) in order that the perturbed operator also belong to such a class. Abstract results are used in the study of initial-boundary value problems for a system of partial differential equations with several fractional Riemann–Liouville derivatives of different orders with respect to time and for an equation with a fractional Prabhakar derivative with respect to time.

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Analytic Resolving Families for Equations with Distributed Riemann–Liouville Derivatives

Cited by 6 publications

References 17 publications

Transmutation operators intertwining first-order and distributed-order derivatives

Transmutation operators intertwining first-order and distributed-order derivatives

Linear Equations with Distributed Riemann–Liouville Derivatives Given by Stieltjes Integrals and Their Analytic Resolving Families of Operators

Integro-differential equations in Banach spaces and analytic resolving families of operators

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