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Optimization and α-Disfocality for Ordinary Differential Equations

Published online by Cambridge University Press:  20 November 2018

M. Essén
M. Essén*
Affiliation:
University of Uppsala, Uppsala, Sweden
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For f ∊ L−1(0, T), we define the distribution function

where T is a fixed positive number and |·| denotes Lebesgue measure. Let Φ:[0, T] → [0, m] be a nonincreasing, right continuous function. In an earlier paper [3], we discussed the equation

(0.1)

when the coefficient q was allowed to vary in the class

We were in particular interested in finding the supremum and infimum of y(T) when q was in or in the convex hull Ω(Φ) of (see below).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 2 , 01 April 1985 , pp. 310 - 323
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Essén, M., The cos πλ theorem, Springer Lecture Notes in Mathematics 467 (Springer-Verlag, 1975).CrossRefGoogle Scholar
2. Essén, M., A theorem on convex sequences, UUDM Report 1981:12. Analysis 2 (1982), 231–252.Google Scholar
3. Essén, M., Optimization and rearrangements of the coefficient in the operator d2 / dt 2 — p(t)2 on a finite interval, UUDM Report 1983:3. To appear, Journal of Mathematical Analysis and Applications.Google Scholar
4. Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities (Cambridge University Press, 1934).Google Scholar
5. Hartman, P., Ordinary differential equations (J. Wiley, 1964).Google Scholar
6. Ryff, J., Orbits of L1-functions under doubly stochastic transformations, Trans, of the Amer. Math. Soc. 117 (1965), 92–100.Google Scholar
7. Ryff, J., Majorized functions and measures, Indag. Math. 30 (1968), 431–437.Google Scholar
8. Swanson, C. A., Comparison and oscillation theory of linear differential equations (Academic Press, 1968).Google Scholar
9. Tipler, F. J., General relativity and conjugate differential equations, Journal of Differential Equations 50 (1978), 165–174.Google Scholar