| Title:
|
On solutions of differential equations with ``common zero'' at infinity (English) |
| Author:
|
Elbert, Árpád |
| Author:
|
Vosmanský, Jaromír |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
33 |
| Issue:
|
1 |
| Year:
|
1997 |
| Pages:
|
109-120 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $c_\kappa (\nu )\ge \pi \nu ^2 (1-2\nu )$. (English) |
| Keyword:
|
common zeros |
| Keyword:
|
dependence on parameter |
| Keyword:
|
Bessel functions |
| Keyword:
|
higher monotonicity |
| MSC:
|
33C10 |
| MSC:
|
34A25 |
| MSC:
|
34C10 |
| MSC:
|
34M99 |
| idZBL:
|
Zbl 0914.34006 |
| idMR:
|
MR1464305 |
| . |
| Date available:
|
2008-06-06T21:32:45Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107601 |
| . |
| Reference:
|
[1] O. Borůvka: Linear Differential Transformations at the second order.The English University Press, London, 1971. MR 0463539 |
| Reference:
|
[2] Z. Došlá: Higher monotonicity properties of special functions: Application on Bessel case $|\nu |<1/2$.Comment. Math. Univ. Carolinae 31 (1990), 232-241. MR 1077894 |
| Reference:
|
[3] Á. Elbert and A. Laforgia: On the square of the zeros of Bessel functions.SIAM J. Math. Anal. 15 (1984), 206-212. MR 0728696 |
| Reference:
|
[4] Á. Elbert and A. Laforgia: Monotonicity properties of the zeros of Bessel functions.SIAM J. Math. Anal. 17 (1986), 1483-1488. MR 0860929 |
| Reference:
|
[5] Á. Elbert and M. E. Muldoon: On the derivative with respect to a parameter of a zero of a Sturm-Liouville function.SIAM J. Math. Anal. 25 (1994), 354-364. MR 1266563 |
| Reference:
|
[6] Á. Elbert, F. Neuman and J. Vosmanský: Principal pairs of solutions of linear second order oscillatory differential equations.Differential and Integral Equations 5 (1992), 945-960. MR 1167505 |
| Reference:
|
[7] J. Vosmanský: Monotonicity properties of zeros of the differential equation $y^{\prime \prime }+ q(x)\,y=0$.Arch. Math.(Brno) 6 (1970), 37-74. MR 0296420 |
| Reference:
|
[8] J. Vosmanský: Zeros of solutions of linear differential equations as continuous functions of the parameter $\kappa $.Partial Differential Equations, Pitman Research Notes in Mathematical Series, 273, Joseph Wiener and Jack K. Hale, Longman Scientific & Technical, 1992, 253-257. |
| Reference:
|
[9] G. N. Watson: A treatise on the Theory of Bessel Functions.2$^{\text{nd}}$ ed. Cambridge University Press, London, 1944. Zbl 0849.33001, MR 0010746 |
| . |
