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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Solution of integral equations by product integration


Author: Jon C. Helton
Journal: Proc. Amer. Math. Soc. 49 (1975), 401-406
MSC: Primary 45N05
MathSciNet review: 0405048
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Abstract: Suppose $ f,h$ and $ G$ are functions with values in a normed complete ring. With suitable restrictions on these functions, it is established that

$\displaystyle f(x) = h(x) + \int_a^x {f(u)G(u,v)} $

for $ a \leq x \leq b$ only if $ \int_a^x {h(u)G{{(u,v)}_v}{\Pi ^x}} (1 + G)$ exists and is $ f(x) - h(x)$ for $ a \leq x \leq b$, and that

$\displaystyle f(x) = h(x) + \int_a^x {G(u,v)f(u)} $

for $ a \leq x \leq b$ only if $ \int_a^x {_x{\Pi ^v}} (1 + \mathcal{G})G(u,v)h(u)$ exists and is $ f(x) - h(x)$ for $ a \leq x \leq b$, where $ \mathcal{G}(s,r) = G(r,s)$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0405048-4
PII: S 0002-9939(1975)0405048-4
Keywords: Sum integral, product integral, subdivision-refinement integral, integral equation, interval function, normed complete ring
Article copyright: © Copyright 1975 American Mathematical Society