zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Interval criteria for oscillation of second-order linear ordinary differential equations. (English) Zbl 0924.34026
New oscillation criteria are established for the second-order differential equation $$ (p(t)y')'+q(t)y=0 \tag{*} $$ where $1/p,q\in L_{\text{loc}}([t_0,\infty), \Bbb R)$ and $p>0$ a.e. on $[t_0,\infty)$. Integral conditions on the functions $p,q$ are given which guarantee the existence of (disjoint) intervals $[a_i,b_i]$, $a_i<b_i\leq a_{i+1}$, $a_i\to\infty$ as $i\to \infty$, such that any nontrivial solution to (*) has at least one zero in $(a_i,b_i)$, which implies oscillation of (*). These integral conditions use “$H$-function” technique introduced by {\it Ch. G. Philos} [Arch. Math. 53, 483--492 (1989; Zbl 0661.34030)] and by {\it H. J. Li} [J. Math. Anal. Appl. 194, 217--234 (1995; Zbl 0836.34033)]. Some of them are extensions of Kamenev’s and Philos’ type criteria; see {\it I. V. Kamenev} [Mat. Zametki 23, 249--251 (1978; Zbl 0408.34031)]. Examples illustrating the oscillation criteria are given, too.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
Full Text: DOI
[1] Butler, G. J.; Erbe, L. H.; Mingarelli, A. B.: Riccati techniques and variational principles in oscillation theory for linear systems. Trans. amer. Math. soc. 303, 263-282 (1987) · Zbl 0648.34031
[2] Byers, R.; Harris, B. J.; Kwong, M. K.: Weighted means and oscillation conditions for second order matrix differential equations. J. differential equations 61, 164-177 (1986) · Zbl 0609.34042
[3] El-Sayed, M. A.: An oscillation criterion for a forced second-order linear differential equation. Proc. amer. Math. soc. 118, 813-817 (1993) · Zbl 0777.34023
[4] Erbe, L. H.; Kong, Q.; Ruan, S.: Kamenev type theorems for second order matrix differential systems. Proc. amer. Math. soc. 117, 957-962 (1993) · Zbl 0777.34024
[5] Fite, W. B.: Concerning the zeros of the solutions of certain differential equations. Trans. amer. Math. soc. 19, 341-352 (1918) · Zbl 46.0702.02
[6] Hartman, P.: On nonoscillatory linear differential equations of second order. Amer. J. Math. 74, 389-400 (1952) · Zbl 0048.06602
[7] Hartman, P.: Ordinary differential equations. (1982) · Zbl 0476.34002
[8] Kamenev, I. V.: Integral criterion of linear differential equations of second order. Mat. zametki 23, 249-251 (1978) · Zbl 0386.34032
[9] Kwong, M. K.: On Lyapunov’s inequality for disfocality. J. math. Anal. appl. 83, 486-494 (1981) · Zbl 0504.34020
[10] Kwong, M. K.; Zettl, A.: Integral inequalities and second linear oscillation. J. differential equations 45, 16-33 (1982) · Zbl 0498.34022
[11] Li, H. J.: Oscillation criteria for second order linear differential equations. J. math. Anal. appl. 194, 217-234 (1995) · Zbl 0836.34033
[12] Philos, Ch.G.: Oscillation theorems for linear differential equations of second order. Arch. math. (Basel) 53, 483-492 (1989) · Zbl 0661.34030
[13] Wintner, A.: A criterion of oscillatory stability. Quart. appl. Math. 7, 115-117 (1949) · Zbl 0032.34801