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

Nina Chernyavskaya; Leonid Shuster

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 1, page 247-272
  • ISSN: 0011-4642

Abstract

top
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

top

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 74.1 (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},
volume = {74},
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
VL - 74
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

top
  1. Chernyavskaya, N. A., Shuster, L. A., On the WKB-method, Differ. Uravn. 25 (1989), 1826-1829 Russian. (1989) Zbl0702.34053MR1025660
  2. Chernyavskaya, N., Shuster, L., 10.1090/S0002-9939-97-04186-5, Proc. Am. Math. Soc. 125 (1997), 3213-3228. (1997) Zbl0884.34063MR1443146DOI10.1090/S0002-9939-97-04186-5
  3. Chernyavskaya, N., Shuster, L., 10.1090/S0002-9939-99-05049-2, Proc. Am. Math. Soc. 127 (1999), 1413-1426. (1999) Zbl0918.34032MR1625725DOI10.1090/S0002-9939-99-05049-2
  4. Chernyavskaya, N., Shuster, L., 10.4310/MAA.2000.v7.n1.a4, Methods Appl. Anal. 7 (2000), 65-84. (2000) Zbl0985.34019MR1796006DOI10.4310/MAA.2000.v7.n1.a4
  5. Chernyavskaya, N., Shuster, L., 10.1090/S0002-9939-01-06145-7, Proc. Am. Math. Soc. 130 (2002), 1043-1054. (2002) Zbl0994.34014MR1873778DOI10.1090/S0002-9939-01-06145-7
  6. Chernyavskaya, N. A., Shuster, L. A., 10.4171/ZAA/1285, Z. Anal. Anwend. 25 (2006), 205-235. (2006) Zbl1122.34021MR2229446DOI10.4171/ZAA/1285
  7. Chernyavskaya, N., Shuster, L., 10.1112/jlms/jdp012, J. Lond. Math. Soc., II. Ser. 80 (2009), 99-120. (2009) Zbl1188.34036MR2520380DOI10.1112/jlms/jdp012
  8. Chernyavskaya, N. A., Shuster, L. A., 10.1007/s10587-014-0154-1, Czech. Math. J. 64 (2014), 1067-1098. (2014) Zbl1349.34093MR3304799DOI10.1007/s10587-014-0154-1
  9. Chernyavskaya, N., Shuster, L., 10.1007/s40574-017-0144-y, Boll. Unione Mat. Ital. 11 (2018), 417-443. (2018) Zbl1404.34019MR3869579DOI10.1007/s40574-017-0144-y
  10. Chernyavskaya, N., Shuster, L., 10.48550/arXiv.2210.02911, Available at https://arxiv.org/abs/2210.02911 (2022), 29 pages. (2022) MR3376914DOI10.48550/arXiv.2210.02911
  11. Davies, E. B., Harrell, E. M., 10.1016/0022-0396(87)90030-1, J. Differ. Equations 66 (1987), 165-188. (1987) Zbl0616.34020MR0871993DOI10.1016/0022-0396(87)90030-1
  12. Hartman, P., Ordinary Differential Equations, John Wiley & Sons, New York (1964). (1964) Zbl0125.32102MR0171038
  13. Mynbaev, K. T., Otelbaev, M. O., Weighted Functional Spaces and the Spectrum of Differential Operators, Nauka, Moscow (1988), Russian. (1988) Zbl0651.46037MR0950172
  14. Olver, F. W. J., 10.1016/c2013-0-11254-8, Academic Press, New York (1974). (1974) Zbl0303.41035MR0435697DOI10.1016/c2013-0-11254-8
  15. Titchmarsh, E. C., The Theory of Functions, Oxford University Press, Oxford (1932). (1932) Zbl0005.21004MR3728294
  16. Whittaker, E. T., Watson, G. N., 10.1017/CBO9780511608759, Cambridge University Press, Cambridge (1962). (1962) Zbl0105.26901MR0178117DOI10.1017/CBO9780511608759

NotesEmbed ?

top

You must be logged in to post comments.