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Oscillation criteria for second-order nonlinear damped differential equations. (English) Zbl 1059.34039

The paper deals with the oscillatory nature of solutions of the second-order nonlinear functional-differential equation

r(t)ψ(x(t))x ' (t) ' +p(t)k(t,x(t),x ' (t))x ' (t)+ft,xτ 01 (t),...,xτ 0m (t),x ' τ 11 (t),...,x ' τ 1m (t)=0·(1)

Several oscillation theorems for (1) and for its particular case when k(t,x,x ' )=1 are established. Since (1) has a very general form, it might be very hard to satisfy conditions of the theorems. Some assumptions on the functions in (1) like k(t,x,y)y α for some α0 and for all x,y real, p(t)0, or f(t,x 01 ,...,x 0m ,x 11 ,...,x 1m )sgnx 01 q(t) i=1 m α i x 0i for x 0i x 01 >0, i=1,2,...,m, restrict significantly possible applications of oscillation criteria. The proofs of all results are based on the fundamental Lemma 1 whose proof is not provided, although all the routine computations which can be found in numerous related papers on oscillation are meticulously carried out. The paper concludes with a quite general example of oscillatory equation (1) with k(t,x,x ' )1·

34K11Oscillation theory of functional-differential equations