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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Solutions of $f(x)=f(a)+(RL)\int _{a}^{x} (fH+fG)$ for rings
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by Burrell W. Helton
Proc. Amer. Math. Soc. 25 (1970), 735-742
DOI: https://doi.org/10.1090/S0002-9939-1970-0259521-8

Abstract:

We show that there is a solution $f$ of the equation \[ f(x) = f(a) + (RL)\int _a^x {(fH + fG)} \] such that $f(p) = 0$ and $f(q) \ne 0$ for some pair $p,q \in [a,b]$ iff there is a number $t \in [a,b]$ such that one of $1 - H({t^ - },t),1 - H(t,{t^ + }),1 + G({t^ - },t)$ or $1 + G(t,{t^ + })$ is zero or a right divisor of zero, where $f,G$ and $H$ are functions of bounded variation with ranges in a normed ring $N$. Furthermore, if $N$ is a field, then for each discontinuity of $H$ on $[a,b]$ there exists $\lambda \in N$ and a finite set of linearly independent nonzero solutions on $[a,b]$ of the equation $f(x) = f(a) + (RL)\int _a^x {(fH + fG)\lambda }$ such that if $f$ is a solution and has bounded variation on $[a,b]$, then $f$ is a linear combination of this set of solutions. Product integrals are used extensively in the proofs.
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Bibliographic Information
  • © Copyright 1970 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 25 (1970), 735-742
  • MSC: Primary 45.10
  • DOI: https://doi.org/10.1090/S0002-9939-1970-0259521-8
  • MathSciNet review: 0259521