Classification of initial data for the Riccati equation

N. Chernyavskaya; L. Shuster

Bollettino dell'Unione Matematica Italiana (2002)

  • Volume: 5-B, Issue: 2, page 511-525
  • ISSN: 0392-4041

Abstract

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We consider a Cauchy problem y ′ ⁢ x + y 2 ⁢ x = q ⁢ x , y ⁢ x x = x 0 = y 0 where x 0 , y 0 ∈ R and q ⁢ x ∈ L 1 loc ⁢ R is a non-negative function satisfying the condition: ∫ - ∞ x q ⁢ t ⁢ d ⁢ t > 0 , ∫ x ∞ q ⁢ t ⁢ d ⁢ t > 0 ⁢  for  ⁢ x ∈ R . We obtain the conditions under which y ⁢ x can be continued to all of R . This depends on x 0 , y 0 and the properties of q ⁢ x .

How to cite

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Chernyavskaya, N., and Shuster, L.. "Classification of initial data for the Riccati equation." Bollettino dell'Unione Matematica Italiana 5-B.2 (2002): 511-525. <http://eudml.org/doc/195415>.

@article{Chernyavskaya2002,
abstract = {We consider a Cauchy problem $$y'(x)+y^\{2\}(x)= q(x),\qquad y(x)|\_\{x=x\_\{0\}\}=y\_\{0\}$$ where $x_\{0\}$ , $y_\{0\}\in \mathbb\{R\}$ and $q(x)\in L_\{1\}^\{\text\{loc\}\}(R)$ is a non-negative function satisfying the condition: $$\int\_\{-\infty\}^\{x\} q(t)\, dt> 0, \quad \int\_\{x\}^\{\infty\} q(t) \, dt> 0 \qquad \text\{ for \} x\in \mathbb\{R\}.$$ We obtain the conditions under which $y(x)$ can be continued to all of $\mathbb\{R\}$. This depends on $x_\{0\}$ , $y_\{0\}$ and the properties of $q(x)$.},
author = {Chernyavskaya, N., Shuster, L.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {511-525},
publisher = {Unione Matematica Italiana},
title = {Classification of initial data for the Riccati equation},
url = {http://eudml.org/doc/195415},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - Chernyavskaya, N.
AU - Shuster, L.
TI - Classification of initial data for the Riccati equation
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/6//
PB - Unione Matematica Italiana
VL - 5-B
IS - 2
SP - 511
EP - 525
AB - We consider a Cauchy problem $$y'(x)+y^{2}(x)= q(x),\qquad y(x)|_{x=x_{0}}=y_{0}$$ where $x_{0}$ , $y_{0}\in \mathbb{R}$ and $q(x)\in L_{1}^{\text{loc}}(R)$ is a non-negative function satisfying the condition: $$\int_{-\infty}^{x} q(t)\, dt> 0, \quad \int_{x}^{\infty} q(t) \, dt> 0 \qquad \text{ for } x\in \mathbb{R}.$$ We obtain the conditions under which $y(x)$ can be continued to all of $\mathbb{R}$. This depends on $x_{0}$ , $y_{0}$ and the properties of $q(x)$.
LA - eng
UR - http://eudml.org/doc/195415
ER -

References

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  2. CHERNYAVSKAYA, N.- SHUSTER, L., Estimates for the Green function of a general Sturm-Liouville operator and their applications, Proc. Amer. Math. Soc., 127, no. 5 (1999), 1413-1426. Zbl0918.34032MR1625725
  3. CHERNYAVSKAYA, N.- SHUSTER, L., Asymptotics on the diagonal of the Green function of a Sturm-Louiville operator and its applications, J. London Math. Soc., 61 (2) (2000), 506-530. Zbl0959.34019MR1760676
  4. CHERNYAVSKAYA, N.- SHUSTER, L., On the WKB-method, Different. Uravnenija25, 10 (1989), 1826-1829. Zbl0702.34053MR1025660
  5. CHERNYAVSKAYA, N.- SHUSTER, L., Estimates for Green's function of the Sturm-Liouville operator, J. Diff. Eq., 111 (1994), 410-421. Zbl0852.34023MR1284420
  6. CHERNYAVSKAYA, N.- SHUSTER, L., Weight summability of solutions of the Sturm-Liouville equation, J. Diff. Eq., 151, 456-473, 1999 preprint AMSPPJ0128-34-003 (1998). Zbl0921.34030MR1669697
  7. DAVIES, E. B.- HARRELL, E. M., Conformally flat Riemannian metrics, Schrödinger operators and semiclassical approximation, J. Diff. Eq., 66, 2 (1987), 165-188. Zbl0616.34020MR871993
  8. GOURSAT, E., A Course in Mathematical Analysis, Vol. II, Part 2, Differential Equations, New York, 1959. Zbl0144.04501
  9. HARTMAN, P., Ordinary Differential Equations, Wiley, New York, 1964. Zbl0125.32102MR171038
  10. MYNBAEV, K.- OTELBAEV, M., Weighted Fuctional Spaces and the Spectrum of Differential Operators, Nauka, Moscow, 1988. Zbl0651.46037MR950172
  11. STEKLOV, W. A., Sur une méthode nouvelle pour résoudre plusiers problèmes sur le développement d'une fonction arbitraire en séries infinies, Comptes Rendus, Paris, 144 (1907), 1329-1332. JFM38.0437.02

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