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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the solutions of a class of linear selfadjoint differential equations


Authors: Larry R. Anderson and A. C. Lazer
Journal: Trans. Amer. Math. Soc. 152 (1970), 519-530
MSC: Primary 34.20
MathSciNet review: 0268441
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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$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1970-0268441-9
PII: S 0002-9947(1970)0268441-9
Keywords: $ {L^2}$ solutions of linear differential equations, periodic coefficients, asymptotic behavior of solutions
Article copyright: © Copyright 1970 American Mathematical Society