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Oscillatory behavior of solutions of certain second order nonlinear differential equations. (English) Zbl 0855.34039

The authors study oscillatory behavior of solutions of the nonlinear second order differential equation (*) [a(t)(y ' ) σ ] ' +q(t)f(y)=r(t), where a is an eventually positive function, the nonlinearity f satisfies uf(u)>0, f ' (u) for all u0, and the power σ is a positive ratio of the type (odd/odd) or (even/odd). A typical result is the following statement:

Theorem: Let σ be the quotient of two odd integers and suppose that the following assumptions are satisfied: |r(s)|ds<, - q(s)ds<, there exist 0<μν such that μf ' (u)ν and

ds a(s) 1/σ == ds a(s)·

If y is a nonoscillatory solution of (*) such that lim inf t |y(t)|>0 and there exists L>0 so that |y ' (t)|L 1/(σ-1) , then

a(s)y ' (s) σ+1 f ' y ( s ) f y ( s ) 2 ds<,lim t a(t)y ' (t) σ fy ( t )=0

and

a(t)y ' (t) σ fy ( t )+ t a(s)y ' (s) σ+1 f ' y ( s ) f y ( s ) 2 ds+ t q(s)-r(s) fy ( s )ds

for t sufficiently large.

Reviewer: O.Došlý (Brno)

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