Further higher monotonicity properties of Sturm-Liouville functions

Zuzana Došlá; Miloš Háčik; Martin E. Muldoon

Archivum Mathematicum (1993)

  • Volume: 029, Issue: 1-2, page 83-96
  • ISSN: 0044-8753

Abstract

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Suppose that the function q ( t ) in the differential equation (1) y ' ' + q ( t ) y = 0 is decreasing on ( b , ∞ ) where b ≥ 0 . We give conditions on q which ensure that (1) has a pair of solutions y 1 ( t ) , y 2 ( t ) such that the n -th derivative ( n ≥ 1 ) of the function p ( t ) = y 1 2 ( t ) + y 2 2 ( t ) has the sign ( - 1 ) n + 1 for sufficiently large t and that the higher differences of a sequence related to the zeros of solutions of (1) are ultimately regular in sign.

How to cite

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Došlá, Zuzana, Háčik, Miloš, and Muldoon, Martin E.. "Further higher monotonicity properties of Sturm-Liouville functions." Archivum Mathematicum 029.1-2 (1993): 83-96. <http://eudml.org/doc/247440>.

@article{Došlá1993,
abstract = {Suppose that the function $q(t)$ in the differential equation (1) $y^\{\prime \prime \}+q(t)y=0 $ is decreasing on $(b,\infty )$ where $b \ge 0$. We give conditions on $q$ which ensure that (1) has a pair of solutions $y_1(t),\;y_2(t)$ such that the $n$-th derivative ($n\ge 1$) of the function $p(t)= y_1^2(t) +y_2^2(t)$ has the sign $(- 1)^\{n+1\}$ for sufficiently large $t$ and that the higher differences of a sequence related to the zeros of solutions of (1) are ultimately regular in sign.},
author = {Došlá, Zuzana, Háčik, Miloš, Muldoon, Martin E.},
journal = {Archivum Mathematicum},
keywords = {n-times monotonic functions; completely monotonic functions; ultimately monotonic functions and sequences; regularly varying functions; Appell differential equation; generalized Airy equation; higher differences; ultimate monotonicity; regularly varying functions; higher monotonicity},
language = {eng},
number = {1-2},
pages = {83-96},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Further higher monotonicity properties of Sturm-Liouville functions},
url = {http://eudml.org/doc/247440},
volume = {029},
year = {1993},
}

TY - JOUR
AU - Došlá, Zuzana
AU - Háčik, Miloš
AU - Muldoon, Martin E.
TI - Further higher monotonicity properties of Sturm-Liouville functions
JO - Archivum Mathematicum
PY - 1993
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 029
IS - 1-2
SP - 83
EP - 96
AB - Suppose that the function $q(t)$ in the differential equation (1) $y^{\prime \prime }+q(t)y=0 $ is decreasing on $(b,\infty )$ where $b \ge 0$. We give conditions on $q$ which ensure that (1) has a pair of solutions $y_1(t),\;y_2(t)$ such that the $n$-th derivative ($n\ge 1$) of the function $p(t)= y_1^2(t) +y_2^2(t)$ has the sign $(- 1)^{n+1}$ for sufficiently large $t$ and that the higher differences of a sequence related to the zeros of solutions of (1) are ultimately regular in sign.
LA - eng
KW - n-times monotonic functions; completely monotonic functions; ultimately monotonic functions and sequences; regularly varying functions; Appell differential equation; generalized Airy equation; higher differences; ultimate monotonicity; regularly varying functions; higher monotonicity
UR - http://eudml.org/doc/247440
ER -

References

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  10. Higher monotonicity properties of certain Sturm-Liouville functions. III, Canad. J. Math. 22 (1970), 1238-1265. (1970) MR0274845
  11. Higher monotonicity properties of certain Sturm-Liouville functions, V, Proc. Roy. Soc. Edinburgh 77A (1977), 23-37. (1977) Zbl0361.34027MR0445033
  12. Regularly varying functions, Lecture Notes in Math., no. 508, Springer, 1976. (1976) Zbl0324.26002MR0453936
  13. Monotonicity properties of zeros of the differential equation y ' ' + q ( x ) y = 0 , Arch. Math. (Brno) 6 (1970), 37-74. (1970) MR0296420
  14. Multiply monotone functions and their Laplace transforms, Duke Math. J. 23 (1956), 189-207. (1956) Zbl0070.28501MR0077581

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