Methods of analysis of the condition for correct solvability in $L_p\left(ℝ\right)$ of general Sturm-Liouville equations

Chernyavskaya, Nina A., and Shuster, Leonid A.. "Methods of analysis of the condition for correct solvability in $L_p (\mathbb {R})$ of general Sturm-Liouville equations." Czechoslovak Mathematical Journal 64.4 (2014): 1067-1098. <http://eudml.org/doc/269850>.

@article{Chernyavskaya2014,
abstract = {We consider the equation \[ - (r(x)y^\{\prime \}(x))^\{\prime \}+q(x)y(x)=f(x),\quad x\in \mathbb \{R\} \qquad \mathrm \{\{(*)\}\}\] 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|\rightarrow \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 ).$},
author = {Chernyavskaya, Nina A., Shuster, Leonid A.},
journal = {Czechoslovak Mathematical Journal},
keywords = {correct solvability; Sturm-Liouville equation; correct solvability; Sturm-Liouville equation},
language = {eng},
number = {4},
pages = {1067-1098},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Methods of analysis of the condition for correct solvability in $L_p (\mathbb \{R\})$ of general Sturm-Liouville equations},
url = {http://eudml.org/doc/269850},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Chernyavskaya, Nina A.
AU - Shuster, Leonid A.
TI - Methods of analysis of the condition for correct solvability in $L_p (\mathbb {R})$ of general Sturm-Liouville equations
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 4
SP - 1067
EP - 1098
AB - We consider the equation \[ - (r(x)y^{\prime }(x))^{\prime }+q(x)y(x)=f(x),\quad x\in \mathbb {R} \qquad \mathrm {{(*)}}\] 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|\rightarrow \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 ).$
LA - eng
KW - correct solvability; Sturm-Liouville equation; correct solvability; Sturm-Liouville equation
UR - http://eudml.org/doc/269850
ER -