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
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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.References
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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