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Criterion for the Solvability of the Weighted Cauchy Problem for an Abstract Euler–Poisson–Darboux Equation

Criterion for the Solvability of the Weighted Cauchy Problem for an Abstract... In a Banach space E, we consider the abstract Euler–Poisson–Darboux equation u″(t) + kt −1 u′(t) = Au(t) on the half-line. (Here k ∈ ℝ is a parameter, and A is a closed linear operator with dense domain on E.) We obtain a necessary and sufficient condition for the solvability of the Cauchy problem u(0) = 0, lim t→0+ t k u′(t) = u 1, k < 0, for this equation. The condition is stated in terms of an estimate for the norms of the fractional power of the resolvent of A and its derivatives. We introduce the operator Bessel function with negative index and study its properties. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Criterion for the Solvability of the Weighted Cauchy Problem for an Abstract Euler–Poisson–Darboux Equation

Differential Equations , Volume 54 (5) – Jun 11, 2018

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References (21)

Publisher
Springer Journals
Copyright
Copyright © 2018 by Pleiades Publishing, Ltd.
Subject
Mathematics; Ordinary Differential Equations; Partial Differential Equations; Difference and Functional Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266118050063
Publisher site
See Article on Publisher Site

Abstract

In a Banach space E, we consider the abstract Euler–Poisson–Darboux equation u″(t) + kt −1 u′(t) = Au(t) on the half-line. (Here k ∈ ℝ is a parameter, and A is a closed linear operator with dense domain on E.) We obtain a necessary and sufficient condition for the solvability of the Cauchy problem u(0) = 0, lim t→0+ t k u′(t) = u 1, k < 0, for this equation. The condition is stated in terms of an estimate for the norms of the fractional power of the resolvent of A and its derivatives. We introduce the operator Bessel function with negative index and study its properties.

Journal

Differential EquationsSpringer Journals

Published: Jun 11, 2018

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