Lyapunov type inequalities for second-order differential equations with mixed nonlinearities
Abstract In this paper, we present some new Lyapunov and Hartman type inequalities for second-order equations with mixed nonlinearities: x ′′ ⁢ ( t ) + p ⁢ ( t ) ⁢ | x ⁢ ( t ) | β - 1 ⁢ x ⁢ ( t ) + q ⁢ ( t ) ⁢ | x ⁢ ( t ) | γ - 1 ⁢ x ⁢ ( t ) = 0 , $x^{\prime\prime}(t)+p(t)|x(t)|^{\beta-1}x(t)+q(t)|x(t)|^{\gamma-1}x(t)=0,$ where p ⁢ ( t ) ${p(t)}$ , q ⁢ ( t ) ${q(t)}$ are real-valued functions and 0 < γ < 1 < β < 2 ${0<\gamma<1<\beta<2}$ . No sign restrictions are imposed on the potential functions p ⁢ ( t ) ${p(t)}$ and q ⁢ ( t ) ${q(t)}$ . The inequalities obtained generalize the existing results for the special cases of this equation in the literature.