zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Half-linear differential equations: linearization technique and its application. (English) Zbl 1128.34017
Oscillatory properties of the half-linear second-order differential equation $$(k(t)\varphi(x'))'+c(t)\varphi(x)=0,\quad\varphi(x)=|x|^{p-2}x,\ p>1\tag 1$$ are investigated, and these properties are compared with the oscillatory behavior of a certain associated linear second-order differential equation. At the first, essentials of the half-linear oscillation theory are recalled and also some auxiliary technical results are presented. The main results of the paper are statements where oscillatory properties of (N) are compared with those of a certain associated linear equation. As a corollary, a conjecture posed by the first author in the paper [{\it O. Došlý}, Perturbations of the half-linear Euler-Weber differential equations, J. Math. Anal. Appl. 323, 426--440 (2006; Zbl 1107.34030)] is proved and some new research are formulated.

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
WorldCat.org
Full Text: DOI
References:
[1] Cecchi, M.; Došlá, Z.; Marini, M.: Principal solutions and minimal sets of quasilinear differential equations. Dynam. systems appl. 13, 221-232 (2004) · Zbl 1123.34026
[2] Cecchi, M.; Došlá, Z.; Marini, M.: Half-linear equations and characteristic properties of the principal solution. J. differential equations 208, 494-507 (2005) · Zbl 1069.34048
[3] Cecchi, M.; Došlá, Z.; Marini, M.: Integral conditions for nonoscillation of second order nonlinear differential equations. Nonlinear anal. 64, 1278-1289 (2006) · Zbl 1114.34031
[4] Coppel, W. A.: Disconjugacy. Lecture notes in math. 220 (1971) · Zbl 0224.34003
[5] Došlá, Z.; Vrkoč, I.: On an extension of the Fubini theorem and its applications in odes. Nonlinear anal. A 57, 531-548 (2004) · Zbl 1053.34033
[6] Došlý, O.: Perturbations of the half-linear Euler -- Weber type differential equation. J. math. Anal. appl. 323, 426-440 (2006) · Zbl 1107.34030
[7] Došlý, O.: A remark on the linearization technique in half-linear oscillation theory. Opuscula math. 26, 305-315 (2006) · Zbl 1135.34314
[8] O. Došlý, Linearization techniques and oscillation of half-linear differential equations, submitted for publication
[9] Došlý, O.; Elbert, Á.: Integral characterization of the principal solution to half-linear second-order differential equations. Studia sci. Math. hungar. 36, 455-469 (2000) · Zbl 1012.34029
[10] Došlý, O.; Lomtatidze, A.: Oscillation and nonoscillation criteria for half-linear second order differential equations. Hiroshima math. J. 36, 203-219 (2006) · Zbl 1123.34028
[11] Došlý, O.; Peňa, S.: A linearization method in oscillation theory of half-linear differential equations. J. inequal. Appl. 2005, 535-545 (2005)
[12] Došlý, O.; Řehák, P.: Half-linear differential equations. North-holland math. Stud. 202 (2005)
[13] Elbert, Á.; Kusano, T.: Principal solutions of nonoscillatory half-linear differential equations. Adv. math. Sci. appl. 18, 745-759 (1998) · Zbl 0914.34031
[14] Elbert, Á.; Schneider, A.: Perturbations of the half-linear Euler differential equation. Results math. 37, 56-83 (2000) · Zbl 0958.34029
[15] Hartman, P.: Ordinary differential equation. (1964) · Zbl 0125.32102
[16] Jaroš, J.; Kusano, T.; Tanigawa, T.: Nonoscillation theory for second order half-linear differential equations in the framework of regular variation. Results math. 43, 129-149 (2003) · Zbl 1047.34034
[17] Jaroš, J.; Kusano, T.; Tanigawa, T.: Nonoscillatory half-linear differential equations and generalized karamata functions. Nonlinear anal. 64, 762-787 (2006) · Zbl 1103.34017
[18] Mirzov, J. D.: Principal and nonprincipal solutions of a nonoscillatory system. Tbiliss. GoS univ. Inst. prikl. Mat. trudy 31, 100-117 (1988) · Zbl 0735.34029
[19] Z. Pátíková, Hartman -- Wintner type criteria for half-linear second order differential equations, Math. Bohem., in press
[20] Řezníčková, J.: An oscillation criterion for half-linear second order differential equations. Miskolc math. Notes 5, 203-212 (2004)