Oscillation criteria for second-order nonlinear damped differential equations
The paper deals with the oscillatory nature of solutions of the second-order nonlinear functional-differential equation \[ \begin{multlined} \left( r(t)\psi(x(t))x^{\prime}(t)\right) ^{\prime}+p(t)k(t,x(t),x^{\prime }(t))x^{\prime}(t)\\ +f\left( t,x\left[ \tau_{01}(t)\right] ,\ldots,x\left[ \tau_{0m} (t)\right] ,x^{\prime}\left[ \tau_{11}(t)\right] ,\ldots,x^{\prime}\left[ \tau_{1m}(t)\right] \right) =0.\end{multlined}\tag{1} \] Several oscillation theorems for (1) and for its particular case when \(k(t,x,x^{\prime})=1\) 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 \(k(t,x,y)\leq\left| y\right| ^{\alpha}\) for some \(\alpha\geq0\) and for all \(x,y\) real, \(p(t)\geq0,\) or \(f(t,x_{01},\ldots,x_{0m},x_{11},\ldots ,x_{1m}) \operatorname{sgn} x_{01}\geq q(t)\sum_{i=1}^{m}\alpha_{i}\left| x_{0i}\right| \) for \(x_{0i}x_{01}>0,\) \(i=1,2,\ldots,m,\) 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 \(k(t,x,x^{\prime})\equiv 1.\)
nonlinear functional-differential equation
scientific article; zbMATH DE number 2082602