| Title:
|
Methods of analysis of the condition for correct solvability in $L_p (\mathbb R)$ of general Sturm-Liouville equations (English) |
| Author:
|
Chernyavskaya, Nina A. |
| Author:
|
Shuster, Leonid A. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
4 |
| Year:
|
2014 |
| Pages:
|
1067-1098 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider the equation $$\label {1} - (r(x)y'(x))'+q(x)y(x)=f(x),\quad x\in \mathbb R \eqno {(*)} $$ where $f\in L_p(\mathbb R)$, $p\in (1,\infty )$ and \begin {gather} r>0,\quad q\ge 0,\quad \frac {1}{r}\in L_1^{\rm loc}(\mathbb R),\quad q\in L_1^{\rm loc}(\mathbb R), \nonumber \\ \lim _{|d|\to \infty }\int _{x-d}^x \frac {{\rm d} t}{r(t)}\cdot \int _{x-d}^x q(t) {\rm d} t=\infty . \nonumber \end {gather} In an earlier paper, we obtained a criterion for correct solvability of ($*$) in $L_p(\mathbb R),$ $p\in (1,\infty ).$ In this criterion, we use values of some auxiliary implicit functions in the coefficients $r$ and $q$ of equation ($*$). Unfortunately, it is usually impossible to compute values of these functions. In the present paper we obtain sharp by order, two-sided estimates (an estimate of a function $f(x)$ for $x\in (a,b)$ through a function $g(x)$ is sharp by order if $c^{-1}|g(x)|\le |f(x)|\le c|g(x)|,$ $x\in (a,b),$ $c=\rm const$) of auxiliary functions, which guarantee efficient study of the problem of correct solvability of ($*$) in $L_p(\mathbb R),$ $p\in (1,\infty ).$ (English) |
| Keyword:
|
correct solvability |
| Keyword:
|
Sturm-Liouville equation |
| MSC:
|
34B24 |
| idZBL:
|
Zbl 06433715 |
| idMR:
|
MR3304799 |
| DOI:
|
10.1007/s10587-014-0154-1 |
| . |
| Date available:
|
2015-02-09T17:42:34Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144162 |
| . |
| Reference:
|
[1] Chernyavskaya, N. A., El-Natanov, N., Shuster, L. A.: Weighted estimates for solutions of a Sturm-Liouville equation in the space $L_1(\mathbb R)$.Proc. R. Soc. Edinb., Sect. A, Math. 141 (2011), 1175-1206. MR 2855893 |
| Reference:
|
[2] Chernyavskaya, N., Shuster, L.: A criterion for correct solvability in $L_p(\mathbb R)$ of a general Sturm-Liouville equation.J. Lond. Math. Soc., II. Ser. 80 (2009), 99-120. MR 2520380, 10.1112/jlms/jdp012 |
| Reference:
|
[3] Chernyavskaya, N., Shuster, L.: A criterion for correct solvability of the Sturm-Liouville equation in the space $L_p(R)$.Proc. Am. Math. Soc. 130 (2002), 1043-1054. Zbl 0994.34014, MR 1873778, 10.1090/S0002-9939-01-06145-7 |
| Reference:
|
[4] Chernyavskaya, N., Shuster, L.: Regularity of the inversion problem for a Sturm-Liouville equation in $L_p(\mathbb R)$.Methods Appl. Anal. 7 (2000), 65-84. MR 1796006 |
| Reference:
|
[5] Chernyavskaya, N., Shuster, L.: Estimates for the Green function of a general Sturm-Liouville operator and their applications.Proc. Am. Math. Soc. 127 (1999), 1413-1426. Zbl 0918.34032, MR 1625725, 10.1090/S0002-9939-99-05049-2 |
| Reference:
|
[6] Chernyavskaya, N., Shuster, L.: Solvability in $L_p$ of the Neumann problem for a singular non-homogeneous Sturm-Liouville equation.Mathematika 46 (1999), 453-470. MR 1832636, 10.1112/S0025579300007919 |
| Reference:
|
[7] Chernyavskaya, N., Shuster, L.: Solvability in $L_p$ of the Dirichlet problem for a singular nonhomogeneous Sturm-Liouville equation.Methods Appl. Anal. 5 (1998), 259-272. Zbl 0924.34012, MR 1659147 |
| Reference:
|
[8] Mynbaev, K. T., Otelbaev, M. O.: Weighted Function Spaces and the Spectrum of Differential Operators.Nauka, Moskva Russian (1988). MR 0950172 |
| Reference:
|
[9] Titchmarsh, E. C.: The Theory of Functions. (2. ed.).University Press, Oxford (1939). MR 0197687 |
| . |
