A comparison theorem for linear difference equations
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- by P. W. Eloe PDF
- Proc. Amer. Math. Soc. 103 (1988), 451-457 Request permission
Abstract:
Let a be real, $I = \{ a,a + 1, \cdots ,b\}$ where $b - a$ is a positive integer or $I = \{ a,a + 1, \ldots \}$. Let $n$ be a positive integer and let ${I^n} = \{ a,a + 1, \ldots ,b + n\}$ if $b < \infty \;{\text {or}}\;{I^n} = I$ otherwise. Consider the $n$th order difference equation $Pu(m) = \sum \nolimits _{j = 0}^n {{\alpha _j}(m)u(m + j) = 0,\;{\alpha _n}(m) = 1,\;{\alpha _0}(m) \ne 0,\;m \in I}$. It is shown that if $0 \leq r(m) \leq q(m),\;m \in I$ and if the equations $Pu(m) = 0$ and $Pu(m) + q(m)u(m) = 0$ are disconjugate on ${I^n}$, then the equation $Pu(m) + r(m)u(m) = 0$ is disconjugate on ${I^n}$.References
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Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 451-457
- MSC: Primary 39A10
- DOI: https://doi.org/10.1090/S0002-9939-1988-0943065-5
- MathSciNet review: 943065