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Oscillation of certain second-order nonlinear differential equations. (English) Zbl 0893.34023

The author investigates oscillation properties of solutions of the nonlinear differential equation

a(t)(y ' (t)) σ ' +q(t)f(y(t))=0,(*)

where σ>0 is a quotient of odd integers, a(t)>0 and the nonlinearity f satisfies the usual sign condition yf(y)>0 and f ' (y)>0 for y0. A typical result is the following statement.

Theorem. Suppose that ds a(s) 1/σ = and

(i) 0< ε (dy/f(y) 1/σ ), -ε - (dy/f(y) 1/σ )< for any ε>0;

(ii) q(s)ds exists and lim t t (1/a(s) 1/σ )( s q(u)du) 1/σ ds=.

Then every solution of (*) is oscillatory.

Proofs of the results presented are essentially based on the generalized Riccati technique consisting in the fact that the quotient a(t)[y ' (y)] σ f(y(t)) satisfies certain Riccati-type differential equation.

The results of the paper extend, among others, oscillation criteria of P. J. Y. Wong and R. P. Agarwal [J. Math. Anal. Appl. 198, No. 2, 337-354 (1996; Zbl 0855.34039)] and in the linear case σ=1, f(y)y oscillation criteria of H. J. Li [J. Math. Anal. Appl. 194, No. 1, 217-234 (1995; Zbl 0836.34033)].

Reviewer: O.Došlý (Brno)

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