On solutions of differential equations with ``common zero'' at infinity

Árpád Elbert; Jaromír Vosmanský

Archivum Mathematicum (1997)

  • Volume: 033, Issue: 1-2, page 109-120
  • ISSN: 0044-8753

Abstract

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The zeros c k ( ν ) of the solution z ( t , ν ) of the differential equation z ' ' + q ( t , ν ) z = 0 are investigated when lim t → ∞ q ( t , ν ) = 1 , ∫ ∞ | q ( t , ν ) - 1 | d t < ∞ and q ( t , ν ) has some monotonicity properties as t → ∞ . The notion c κ ( ν ) is introduced also for κ real, too. We are particularly interested in solutions z ( t , ν ) which are “close" to the functions sin t , cos t when t is large. We derive a formula for d c κ ( ν ) / d ν and apply the result to Bessel differential equation, where we introduce new pair of linearly independent solutions replacing the usual pair J ν ( t ) , Y ν ( t ) . We show the concavity of c κ ( ν ) for | ν | ≥ 1 2 and also for | ν | < 1 2 under the restriction c κ ( ν ) ≥ π ν 2 ( 1 - 2 ν ) .

How to cite

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Elbert, Árpád, and Vosmanský, Jaromír. "On solutions of differential equations with ``common zero'' at infinity." Archivum Mathematicum 033.1-2 (1997): 109-120. <http://eudml.org/doc/248025>.

@article{Elbert1997,
abstract = {The zeros $c_k(\nu )$ of the solution $z(t, \nu )$ of the differential equation $z^\{\prime \prime \}+ q(t, \nu )\, z=0$ are investigated when $\lim \limits _\{t\rightarrow \infty \} q(t, \nu )=1$, $\int ^\infty | q(t, \nu )-1|\, dt <\infty $ and $q(t, \nu )$ has some monotonicity properties as $t\rightarrow \infty $. The notion $c_\kappa (\nu )$ is introduced also for $\kappa $ real, too. We are particularly interested in solutions $z(t, \nu )$ which are “close" to the functions $\sin t$, $\cos t$ when $t$ is large. We derive a formula for $d c_\kappa (\nu )/d\nu $ and apply the result to Bessel differential equation, where we introduce new pair of linearly independent solutions replacing the usual pair $J_\nu (t)$, $Y_\nu (t)$. We show the concavity of $c_\kappa (\nu )$ for $|\nu |\ge \frac\{1\}\{2\}$ and also for $|\nu |<\frac\{1\}\{2\}$ under the restriction $c_\kappa (\nu )\ge \pi \nu ^2 (1-2\nu )$.},
author = {Elbert, Árpád, Vosmanský, Jaromír},
journal = {Archivum Mathematicum},
keywords = {common zeros; dependence on parameter; Bessel functions; higher monotonicity; common zeros; dependence on parameter; Bessel functions; higher monotonicity},
language = {eng},
number = {1-2},
pages = {109-120},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On solutions of differential equations with ``common zero'' at infinity},
url = {http://eudml.org/doc/248025},
volume = {033},
year = {1997},
}

TY - JOUR
AU - Elbert, Árpád
AU - Vosmanský, Jaromír
TI - On solutions of differential equations with ``common zero'' at infinity
JO - Archivum Mathematicum
PY - 1997
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 033
IS - 1-2
SP - 109
EP - 120
AB - The zeros $c_k(\nu )$ of the solution $z(t, \nu )$ of the differential equation $z^{\prime \prime }+ q(t, \nu )\, z=0$ are investigated when $\lim \limits _{t\rightarrow \infty } q(t, \nu )=1$, $\int ^\infty | q(t, \nu )-1|\, dt <\infty $ and $q(t, \nu )$ has some monotonicity properties as $t\rightarrow \infty $. The notion $c_\kappa (\nu )$ is introduced also for $\kappa $ real, too. We are particularly interested in solutions $z(t, \nu )$ which are “close" to the functions $\sin t$, $\cos t$ when $t$ is large. We derive a formula for $d c_\kappa (\nu )/d\nu $ and apply the result to Bessel differential equation, where we introduce new pair of linearly independent solutions replacing the usual pair $J_\nu (t)$, $Y_\nu (t)$. We show the concavity of $c_\kappa (\nu )$ for $|\nu |\ge \frac{1}{2}$ and also for $|\nu |<\frac{1}{2}$ under the restriction $c_\kappa (\nu )\ge \pi \nu ^2 (1-2\nu )$.
LA - eng
KW - common zeros; dependence on parameter; Bessel functions; higher monotonicity; common zeros; dependence on parameter; Bessel functions; higher monotonicity
UR - http://eudml.org/doc/248025
ER -

References

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  6. Principal pairs of solutions of linear second order oscillatory differential equations, Differential and Integral Equations 5 (1992), 945-960. MR1167505
  7. Monotonicity properties of zeros of the differential equation y ' ' + q ( x ) y = 0 , Arch. Math.(Brno) 6 (1970), 37-74. MR0296420
  8. Zeros of solutions of linear differential equations as continuous functions of the parameter κ , Partial Differential Equations, Pitman Research Notes in Mathematical Series, 273, Joseph Wiener and Jack K. Hale, Longman Scientific & Technical, 1992, 253-257. 
  9. A treatise on the Theory of Bessel Functions, 2 ed. Cambridge University Press, London, 1944. Zbl0849.33001MR0010746

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