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Solutions of $ f(x)=f(a)+(RL)\int \sb{a}\sp{x}\,(fH+fG)$ for rings


Author: Burrell W. Helton
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
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Abstract: We show that there is a solution $ f$ of the equation

$\displaystyle 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.

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

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0259521-8
Keywords: Integral equations, product integrals, normed ring, nonzero solutions, linearly independent solutions, divisors of zero, bounded variation
Article copyright: © Copyright 1970 American Mathematical Society

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