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- Title
Representations of abstract resolvent families on time scales via Laplace Transform.
- Authors
Grau, Rogelio; Pereira, Aldo
- Abstract
In this work we introduce a general formulation of resolvent family, also named resolvent operator, to describe explicit formulas for the solutions of dynamic equations on time scales of order 0 < α ≤ 1 . The treatment developed here is based on a formulation of Laplace Transform on time scales that includes continuous and discrete cases, to obtain concise expressions for such explicit formulas. Moreover, this formulation of Laplace Transform allows to obtain discrete counterparts of some important properties in the context of fractional calculus. As main results in this work, we study the relationship between an abstract resolvent family and its infinitesimal generator, along with the main properties of resolvent families, and the existence of solutions for abstract dynamic equations on time scales. In addition, we introduce formulas for resolvent families as solutions of several types of dynamic equations of order 0 < α ≤ 1 on continuous, discrete and quantum time scales.
- Subjects
RESOLVENTS (Mathematics); FRACTIONAL calculus; FAMILIES; CAUCHY problem
- Publication
Fractional Calculus & Applied Analysis, 2024, Vol 27, Issue 1, p218
- ISSN
1311-0454
- Publication type
Article
- DOI
10.1007/s13540-023-00227-3