Classification of initial data for the Riccati equation

Chernyavskaya, N. ; Shuster, L.

Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002), p. 511-525 / Harvested from Biblioteca Digitale Italiana di Matematica

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 $q(x)$.

Résumé

Consideriamo un problema di Cauchy $$y'(x)+y^{2}(x)= q(x),\qquad y(x)|_{x=x_{0}}=y_{0}$$ dove $x_{0}$ , $y_{0}\in \mathbb{R}$ e $q(x)\in L_{1}^{\text{loc}}(\mathbb{R})$ è una funzione non negativa che soddisfa la condizione: $$\int_{-\infty}^{x} q(t) \, dt > 0, \quad \int_{x}^{\infty} q(t) \, dt> 0 \qquad \text{ for } x\in \mathbb{R}.$$ Otteniamo le condizioni nelle quali $y(x)$ può essere continuata in tutto $\mathbb{R}$. Questo dipende da $x_{0}$, $y_{0}$ e dalle proprietà di $q(x)$.

Détail

Publié le : 2002-06-01

MR 1911203 | Zbl 1072.32001

Bibliographie

[1] Bellman, R.-Kalaba, R., Quasilinearization and Nonlinear Boundary-Value Problems, New York, 1965. | MR 178571 | Zbl 0139.10702

[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. | MR 1625725 | Zbl 0918.34032

[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. | MR 1760676 | Zbl 0959.34019

[4] Chernyavskaya, N.-Shuster, L., On the WKB-method, Different. Uravnenija25, 10 (1989), 1826-1829. | MR 1025660 | Zbl 0702.34053

[5] Chernyavskaya, N.-Shuster, L., Estimates for Green's function of the Sturm-Liouville operator, J. Diff. Eq., 111 (1994), 410-421. | MR 1284420 | Zbl 0852.34023

[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). | MR 1669697 | Zbl 0921.34030

[7] Davies, E. B.-Harrell, E. M., Conformally flat Riemannian metrics, Schrödinger operators and semiclassical approximation, J. Diff. Eq., 66, 2 (1987), 165-188. | MR 871993 | Zbl 0616.34020

[8] Goursat, E., A Course in Mathematical Analysis, Vol. II, Part 2, Differential Equations, New York, 1959. | Zbl 0144.04501

[9] Hartman, P., Ordinary Differential Equations, Wiley, New York, 1964. | MR 171038 | Zbl 0125.32102

[10] Mynbaev, K.-Otelbaev, M., Weighted Fuctional Spaces and the Spectrum of Differential Operators, Nauka, Moscow, 1988. | MR 950172 | Zbl 0651.46037

[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. | JFM 38.0437.02