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)
Oscillation of second-order nonlinear differential equations with nonlinear damping. (English) Zbl 1049.34040

The paper contains several oscillation criteria for the nonlinear differential equation

(r(t)k 1 (x,x ' )) ' +p(t)k 2 (x,x ' )x ' +q(t)f(x)=0,

where tt 0 0, under one of the leading assumptions:

(1) k 1 2 (u,v)α 1 vk 1 (u,v), uvk 2 (u,v)α 2 k 1 2 (u,v), p(t)0, α 1 >0, α 2 0;

(2) k 1 2 (u,v)α 1 vk 1 (u,v), uvk 2 (u,v)=α 2 k 1 2 (u,v), α 1 α 2 p(t)+r(t)>0, α 1 , α 2 >0;

(3) k 1 2 (u,v)α 1 vk 1 (u,v), k 1 (u,v)=vk 2 (u,v), α 1 >0.

The Philos class of test functions H(t,s) introduced here (p. 198) is governed by the formula -H(t,s)/s=h(t,s)H(t,s). The proofs of the results rely on an averaging technique of Kamenev-type for certain Riccati substitutions.

In case (1), one of the results reads as: Suppose that f(x)/xK>0 for all x0 and q(t)0. Assume also that

lim sup t+ H(t,t 0 ) -1 t 0 t [H(t,s)ρ(s)q(s)-α 1 r 2 (s)ρ(s) 4K(α 1 α 2 p(s)+r(s))Q 2 (t,s)]ds=+,

where Q(t,s)=h(t,s)-[ρ ' (s)/ρ(s)]H(t,s), for a certain positive, continuously differentiable function ρ. Then, the equation is oscillatory.

In case (2), the same conclusion is valid provided that the above condition is fulfilled.

In case (3), we have: Suppose that xf(x)0 and f ' (x)K>0 for all x0. Assume also that

lim sup t+ H(t,t 0 ) -1 t 0 t [H(t,s)ρ(s)q(s)-(α 1 /4K)r(s)ρ(s)Q 2 (t,s)]ds=+,

where Q(t,s)=h(t,s)+[p(s)/r(s)-ρ ' (s)/ρ(s)]H(t,s), for a certain positive, continuously differentiable function ρ. Then, the equation is oscillatory.

Another result deals with a nondifferentiable function f in the case of p with varying sign by imposing a differentiability condition on p (Theorem 3.5). The paper is elegantly written and an elaborated discussion of the relevant literature accompanies the computations.


MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory