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Solvability of a Nonlocal Problem for an Evolution Equation with a Superstable Semigroup

Solvability of a Nonlocal Problem for an Evolution Equation with a Superstable Semigroup We study a linear nonlocal problem for an evolution equation in a Banach space. Thestandard semigroup approach is used but integral averaging over time is used instead of thetraditional initial condition. It is assumed that the evolution semigroup associated with theabstract differential equation is superstable (quasinilpotent), i.e., has an infinite negativeexponential type. A theorem about unique solvability of the posed nonlocal problem is proved. Itis shown that the solution can be represented by a convergent Neumann series. Some corollariesare noted. The case in which the semigroup is nilpotent is treated separately. The class ofexamples of superstable semigroups that are of interest in mathematical physics is outlined. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Solvability of a Nonlocal Problem for an Evolution Equation with a Superstable Semigroup

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References (28)

Publisher
Springer Journals
Copyright
Copyright © Pleiades Publishing, Ltd. 2020
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266120040072
Publisher site
See Article on Publisher Site

Abstract

We study a linear nonlocal problem for an evolution equation in a Banach space. Thestandard semigroup approach is used but integral averaging over time is used instead of thetraditional initial condition. It is assumed that the evolution semigroup associated with theabstract differential equation is superstable (quasinilpotent), i.e., has an infinite negativeexponential type. A theorem about unique solvability of the posed nonlocal problem is proved. Itis shown that the solution can be represented by a convergent Neumann series. Some corollariesare noted. The case in which the semigroup is nilpotent is treated separately. The class ofexamples of superstable semigroups that are of interest in mathematical physics is outlined.

Journal

Differential EquationsSpringer Journals

Published: Apr 1, 2020

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