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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


An existence result on a Volterra equation in a Banach space

Author: Stig-Olof Londen
Journal: Trans. Amer. Math. Soc. 235 (1978), 285-304
MSC: Primary 45N05; Secondary 45D05, 47H15
MathSciNet review: 0473770
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Abstract: Let W be a real reflexive Banach space, dense in a Hilbert space H and with dual $ W'$. Let the injection $ W \to H$ be continuous and compact. We consider the nonlinear integral equation

$\displaystyle u'(t) + \int_0^t {a(t - \tau )Au(\tau )d\tau = f(t),\quad t \geqslant 0,}$ ($ 1$)

where a, f, A are given and u is the unknown. The kernel $ a(t)$ maps $ {R^ + }$ into R and f takes values in H. The nonlinear function A is a maximal monotone mapping $ W \to W'$.

Making use of the theory of maximal monotone operators we prove an existence result on (1). This result is used to obtain approximate solutions to the related nonlinear hyperbolic differential equation $ u''(t) + Au(t) = f'(t),t \geqslant 0$.

References [Enhancements On Off] (What's this?)

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  • [3] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 0348562 (50 #1060)
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Additional Information

PII: S 0002-9947(1978)0473770-7
Keywords: Abstract integral equations, Volterra equations, integrodifferential equations, monotone operators
Article copyright: © Copyright 1978 American Mathematical Society