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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the growth of solutions in the oscillatory case


Author: Robert M. Kauffman
Journal: Proc. Amer. Math. Soc. 51 (1975), 49-54
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9939-1975-0374559-2
MathSciNet review: 0374559
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Abstract: Suppose that $A$ is a bounded continuously differentiable function from $[0,\infty )$ to the real $n \times n$ Hermitian matrices such that, for every $\varepsilon > 0$ and every $\lambda > 0$, there is an $a$ (depending on $\varepsilon$ and $\lambda$) such that ${D^2} - A - \varepsilon E$ and ${D^2} + A’/\lambda - \varepsilon E$ are disconjugate on $[a,\infty )$, where $E$ is the $n \times n$ identity matrix. It follows from the result of this paper that no solution of $({D^2} + A)f = 0$ can either grow or decay exponentially.


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

  • Richard Bellman, Stability theory of differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. MR 0061235
  • W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. MR 0460785
  • N. Dunford and J. T. Schwartz, Linear operators. II: Spectral theory. Self-adjoint operators in Hilbert space, Wiley, New York, 1963. MR 32 #6181.
  • Seymour Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0200692

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Keywords: Exponential growth, Fredholm operator, disconjugacy, essential spectrum
Article copyright: © Copyright 1975 American Mathematical Society