Interval oscillation criteria for second-order nonlinear differential equations (Q2784879)

The authors consider the oscillatory behavior of solutions to the second-order nonlinear differential equation NEWLINE\[NEWLINE(r(t)y'(t))'+q(t)f(y(t))g(y'(t))=0,NEWLINE\]NEWLINE where \(t\geq t_0,\) the function \(r:[t_0,\infty)\to(0,\infty)\) is continuous, the function \(q:[t_0,\infty)\to(0,\infty)\) is continuous and \( q(t) \not= 0 \) on the set \( [T,\infty)\) for some \( T \geq t_0\); the function \(f:\mathbb{R}\to\mathbb{R} \) is continuous and satisfies \(yf(y) > 0 \) for \(y\not= 0; \) the function \( g : \mathbb{R} \to \mathbb{R}\) is continuous and obeys \( g(y) \geq k > 0 \) for \( y \not= 0. \)NEWLINENEWLINENEWLINEFirst, the case of a monotone function \( f \) is considered and sufficient conditions for the existence of an oscillatory solution to the original problem are derived. Then, the authors consider the case of a nonmonotone function \( f(y) \) such that the condition \( \frac{f(y)}{y} \geq \mu_0 > 0 \) for all \( y \not= 0\) holds, where \( \mu_0 \) is constant. The proposed criteria are based only on information on a sequence of subintervals of \([t_0,\infty).\) The authors demonstrate the application of the criteria of existence of oscillatory solutions for concrete equations.