Abstract
We introduce the class of (local)(a,k)-regularizedC-resolvent families and discuss its basic structural properties. In particular, our analysis covers subjects like regularity, perturbations, duality, spectral properties and subordination principles. We apply our results in the study of the backwards fractional diffusion-wave equation and provide several illustrative examples of differentiable(a,k)-regularizedC-resolvent families.
Highlights
Introduction and PreliminariesIn this review, we will report how a large number of known results concerning a, k regularized resolvents 1–6, C-regularized resolvents 7, and local convoluted Csemigroups and cosine functions 8, 9 can be formulated in the case of general a, k regularized C-resolvent families.The paper is organized as follows
Throughout this paper E denotes a nontrivial complex Banach space, L E denotes the space of bounded linear operators from E into E, E∗ denotes the dual space of E, and A denotes a closed linear operator acting on E
The range and the resolvent set of Aare denoted by Rang A and ρ A, respectively; D A denotes the Banach space D A equipped with the graph norm
Summary
We introduce the class of local a, k -regularized C-resolvent families and discuss its basic structural properties. Our analysis covers subjects like regularity, perturbations, duality, spectral properties and subordination principles. We apply our results in the study of the backwards fractional diffusion-wave equation and provide several illustrative examples of differentiable a, k -regularized C-resolvent families.
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