On Liapunov type inequality for certain second order quasi-linear differential equations
A Lyapunov-type inequality is established for differential equations of the form \[ [|y'(t)|^\sigma \text{sgn }y']' +q(t)f(y(t))=0,\quad t\geq t_0\geq0, \tag{\(*\)} \] where \(\sigma>0\) is a constant, \(q\) is a real-valued continuous function on \([t_0,\infty), t_0\geq0\) and \(f\) is a real-valued continuous function on \((-\infty,\infty)\) with \(0<y f(y)\leq r|y|^{\sigma+1}, y\neq0\). The main result of the paper runs as follows: If \(y(t)\) is a solution of (\(*\)) with \(y(a) = y(b) = 0\) (\(a < b\)) and \(y(t)\neq0\) for \(t\in (a,b)\), then \(\int_a^b q^+(t)dt>\frac{2^{\sigma+1}}{r(b-a)^\sigma}\), where \(q^+(t)=\max\{q(t),0\}\). Some results related to the number of zeros of a solution of (\(*\)) in an interval \([t_0,T]\) and the distance between two consecutive zeros of a solution of (\(*\)) are obtained.
distance between two zeros of a solution
second order quasi-linear differential equations
scientific article; zbMATH DE number 5077535