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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Product integral techniques for abstract hyperbolic partial differential equations

Author: J. W. Spellmann
Journal: Trans. Amer. Math. Soc. 209 (1975), 353-365
MSC: Primary 47D05; Secondary 35R20
MathSciNet review: 0377590
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Abstract: Explicit and implicit product integral techniques are used to represent a solution $ U$ to the abstract system: $ {U_{12}}(x,y) = AU(x,y);U(x,0) = p = U(0,y)$. The coefficient $ A$ is a closed linear transformation defined on a dense subspace $ D(A)$ of the Banach space $ X$ and the point $ p$ in $ D(A)$ satisfies the condition that $ \vert\vert{A^i}p\vert\vert < {S^i}{(i!)^{3/2}}$ for all integers $ i \geqslant 0$ and some $ S > 0$. The implicit technique is developed under the additional assumption that $ A$ generates a strongly continuous semigroup of bounded linear transformations on $ X$. Both methods provide representations for the $ {J_0}$ Bessel function.

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Keywords: Product integration, linear semigroups of operators, abstract partial differential equations, $ {J_0}$ Bessel function
Article copyright: © Copyright 1975 American Mathematical Society