Green-Liouville approximation and correct solvability in L p ( ℝ ) of the general Sturm-Liouville equation

Nina Chernyavskaya; Leonid Shuster

Czechoslovak Mathematical Journal (2024)

  • Issue: 1, page 247-272
  • ISSN: 0011-4642

Abstract

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We consider the equation - ( r ( x ) y ' ( x ) ) ' + q ( x ) y ( x ) = f ( x ) , x ∈ ℝ , where f ∈ L p ( ℝ ) , p ∈ ( 1 , ∞ ) and r > 0 , 1 r ∈ L 1 loc ( ℝ ) , q ∈ L 1 loc ( ℝ ) . For particular equations of this form, we suggest some methods for the study of the question on requirements to the functions r and q under which the above equation is correctly solvable in the space L p ( ℝ ) , p ∈ ( 1 , ∞ ) .

How to cite

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Chernyavskaya, Nina, and Shuster, Leonid. "Green-Liouville approximation and correct solvability in $L_p(\mathbb {R})$ of the general Sturm-Liouville equation." Czechoslovak Mathematical Journal (2024): 247-272. <http://eudml.org/doc/299223>.

@article{Chernyavskaya2024,
abstract = {We consider the equation \[ -(r(x) y^\{\prime \}(x))^\{\prime \}+q(x)y(x)=f(x),\quad x\in \mathbb \{R\}, \] where $f\in L_p(\mathbb \{R\})$, $p\in (1,\infty )$ and \[ r>0,\quad \frac\{1\}\{r\}\in L\_1^\{\rm loc\}(\mathbb \{R\}),\quad q\in L\_1^\{\rm loc\}(\mathbb \{R\}). \] For particular equations of this form, we suggest some methods for the study of the question on requirements to the functions $r$ and $q$ under which the above equation is correctly solvable in the space $L_p(\mathbb \{R\}),$$p\in (1,\infty ).$},
author = {Chernyavskaya, Nina, Shuster, Leonid},
journal = {Czechoslovak Mathematical Journal},
keywords = {Green-Liouville approximation; correct solvability; general Sturm-Liouville equation},
language = {eng},
number = {1},
pages = {247-272},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Green-Liouville approximation and correct solvability in $L_p(\mathbb \{R\})$ of the general Sturm-Liouville equation},
url = {http://eudml.org/doc/299223},
year = {2024},
}

TY - JOUR
AU - Chernyavskaya, Nina
AU - Shuster, Leonid
TI - Green-Liouville approximation and correct solvability in $L_p(\mathbb {R})$ of the general Sturm-Liouville equation
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 247
EP - 272
AB - We consider the equation \[ -(r(x) y^{\prime }(x))^{\prime }+q(x)y(x)=f(x),\quad x\in \mathbb {R}, \] where $f\in L_p(\mathbb {R})$, $p\in (1,\infty )$ and \[ r>0,\quad \frac{1}{r}\in L_1^{\rm loc}(\mathbb {R}),\quad q\in L_1^{\rm loc}(\mathbb {R}). \] For particular equations of this form, we suggest some methods for the study of the question on requirements to the functions $r$ and $q$ under which the above equation is correctly solvable in the space $L_p(\mathbb {R}),$$p\in (1,\infty ).$
LA - eng
KW - Green-Liouville approximation; correct solvability; general Sturm-Liouville equation
UR - http://eudml.org/doc/299223
ER -

References

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