Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0926.34025
Cecchi, Mariella; Došlá, Zuzana; Marini, Mauro
On third order differential equations with property A and B. (English)
[J] J. Math. Anal. Appl. 231, No.2, 509-525 (1999). ISSN 0022-247X

This article concerns oscillatory and asymptotic properties at infinity for differential equations of the type $$y^{(3)}- q(x)y'\pm r(x)f(y)= 0,\quad x\in \bbfR_+,\tag{E$_\pm$}$$ with $q, r\in C(\bbfR_+,\bbfR_+)$, $f\in C(\bbfR,\bbfR)$, $r(x)>0$, and $tf(t)>0$ for all $t\ne 0$. In the linear case $f(t)\equiv t$, $(\text{E}_\pm)$ will be denoted $(\text{L}_\pm)$. Seven theorems give connections between oscillation and properties A or B of $(\text{L}_\pm)$ or $(\text{E}_\pm)$, as defined by {\it I. T. Kiguradze} and {\it Z. A. Chanturiya} [Mathematics and its Applications, Soviet Series 89, Dordrecht: Kluwer Academic Publ. (1993; Zbl 0782.34002)]. The main theorems in the linear case state that $(\text{L}_+)[(\text{L}_-)]$ has at least one nontrivial oscillatory solution if and only if it has property A [property B, respectively], extending results of {\it M. Greguš} [Third order linear differential equations. Mathematics and its Applications. D. Reidel Publ. Comp. (1987; Zbl 0602.34005)] and {\it M. Gera} [Acta Math. Univ. Comenianae 46/47, 189-203 (1985; Zbl 0612.34029)]. Corollaries provide sufficient conditions on $q$, $r$ for equivalence of (i) property A for $(\text{L}_+)$ and property B for its adjoint equations; and (ii) property B for $(\text{L}_-)$ and property A for its adjoint.\par In the second part of the paper these results are applied to generate sufficient conditions for $(\text{E}_+)$ to have property A and for $(\text{E}_-)$ to have property B. Related results of the authors are given in [Ann. Mat. Pura Appl., IV. Ser. 173, 373-389 (1997) (to appear) and Nonlinear Anal., Theory Methods Appl. 30, No. 3, 1583-1594 (1997; Zbl 0892.34032)].
[Charles A.Swanson (Vancouver)]

MSC 2000:: *34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34C11 Qualitative theory of solutions of ODE: Growth, etc.

Keywords: linear; semilinear; oscillation; asymptotic behavior; properties A or B

Citations: Zbl 0782.34002; Zbl 0602.34005; Zbl 0612.34029; Zbl 0892.34032

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]