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

Existence of integrals and the solution of integral equations


Author: Jon C. Helton
Journal: Trans. Amer. Math. Soc. 229 (1977), 307-327
MSC: Primary 45A05
MathSciNet review: 0445245
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Abstract: Functions are from R to N or $ R \times R$ to N, where R denotes the real numbers and N denotes a normed complete ring. If S, T and G are functions from $ R \times R$ to N, each of $ S({p^ - },p),S({p^ - },{p^ - }),T({p^ - },p)$ and $ T({p^ - },{p^ - })$ exists for $ a < p \leqslant b$, each of $ S(p,{p^ + }),S({p^ + },{p^ + }),T(p,{p^ + })$ and $ T({p^ + },{p^ + })$ exists for $ a \leqslant p < b$, G has bounded variation on [a, b] and $ \smallint _a^bG$ exists, then each of

$\displaystyle \int_a^b S \left[ {G - \int G } \right]T\quad {\text{and}}\quad \int_a^b {S\left[ {1 + G - \prod {(1 + G)} } \right]} \;T$

exists and is zero. These results can be used to solve integral equations without the existence of integrals of the form

$\displaystyle \int_a^b {\left\vert {G - \int G } \right\vert = 0} \quad {\text{and}}\quad \int_a^b {\left\vert {1 + G - \prod {(1 + G)} } \right\vert} = 0.$

This is demonstrated by solving the linear integral equation

$\displaystyle f(x) = h(x) + (LR)\int_a^x {(fG + fH)} $

and the Riccati integral equations

$\displaystyle f(x) = w(x) + (LRLR)\int_a^x {(fH + Gf + fKf)} $

without the existence of the previously mentioned integrals.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0445245-1
PII: S 0002-9947(1977)0445245-1
Keywords: Sum integral, product integral, subdivision-refinement integral, existence, interval function, normed complete ring, linear integral equation, Riccati integral equation
Article copyright: © Copyright 1977 American Mathematical Society