## On the solutions of a class of linear selfadjoint differential equations

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- by Larry R. Anderson and A. C. Lazer
- Trans. Amer. Math. Soc.
**152**(1970), 519-530 - DOI: https://doi.org/10.1090/S0002-9947-1970-0268441-9
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## Abstract:

Let $L$ be a linear selfadjoint ordinary differential operator with coefficients which are real and sufficiently regular on $( - \infty ,\infty )$. Let ${A^ + }({A^ - })$ denote the subspace of the solution space of $Ly = 0$ such that $y \in {A^ + }(y \in {A^ - })$ iff ${D^k}y \in {L^2}[0,\infty )({D^k}y \in {L^2}( - \infty ,0])$ for $k = 0,1, \ldots ,m$ where $2m$ is the order of $L$. A sufficient condition is given for the solution space of $Ly = 0$ to be the direct sum of ${A^ + }$ and ${A^ - }$. This condition which concerns the coefficients of $L$ reduces to a necessary and sufficient condition when these coefficients are constant. In the case of periodic coefficients this condition implies the existence of an exponential dichotomy of the solution space of $Ly = 0$.## References

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*Finite-dimensional vector spaces*, The University Series in Undergraduate Mathematics, D. Van Nostrand Co., Inc., Princeton-Toronto-New York-London, 1958. 2nd ed. MR**0089819** - Marko Švec,
*Sur le comportement asymptotique des intégrales de l’équation différentielle $y^{(4)}+Q(x)y=0$*, Czechoslovak Math. J.**8(83)**(1958), 230–245 (French, with Russian summary). MR**101355**

## Bibliographic Information

- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**152**(1970), 519-530 - MSC: Primary 34.20
- DOI: https://doi.org/10.1090/S0002-9947-1970-0268441-9
- MathSciNet review: 0268441