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Interval oscillation of a general class of second-order nonlinear differential equations with nonlinear damping. (English) Zbl 1064.34021

The authors are concerned with the oscillatory behavior of the second-order nonlinear differential equation with a nonlinear damping term

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

with p,q:[t 0 ,), r:[t 0 ,)(0,), f:, k 1 ,k 2 : 2 · It is also assumed that

k 1 2 (u,v)α 1 k 1 (u,v),

for some α 1 >0 and for all (u,v) 2 · Two cases are considered:

(a) f(x) is differentiable, xf(x)0 and f ' (x)μ 1 for some μ 1 >0 and all x0, and

vf(u)k 2 (u,v)α 2 k 1 2 (u,v)(b1)

for some α 2 >0 and for all (u,v) 2 ;

(b) f(x) is not necessarily differentiable, f(x)/xμ 2 for some μ 2 >0 and all x0, and

vuk 2 (u,v)α 3 k 1 2 (u,v)(b2)

for some α 1 >0 and for all (u,v) 2 · Using standard integral averaging technique, several interval oscillation criteria are obtained which require information on the behavior of the coefficients in equation (b) on a sequence of intervals (a n ,b n ) such that a n as n. Unfortunately, rather specific assumptions (b1) and (b2) significantly restrict possible the applicability of the theorems.

The statement of the fundamental Lemma 1.1 should be corrected as follows: “If there exists an interval (a,b)[t 0 ,) such that (1.2) holds, then, for all c(a,b), (1.3) is satisfied for every H𝒫” instead of the incorrect formulation “If there exist an interval (a,b)[t 0 ,) and a c(a,b) such that (1.2) holds, then (1.3) is satisfied for every H𝒫·” The statement of Theorem 3.1 should be corrected by adding the phrase “and there exists a c(a,b) such that (3.1) holds”.

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