Existence of eventually positive solutions of fourth order quasilinear differential equations

We consider the fourth order quasilinear differential equation(p(t)|u″|α−1u″)″+q(t)|u|β−1u=0,(p(t)|u″|α−1u″)″+q(t)|u|β−1u=0, where α and β   are positive constants, p,q∈C[a,∞)p,q∈C[a,∞), a>0a>0, and p(t)>0p(t)>0, q(t)>0q(t)>0 for t⩾at⩾a, and establish necessary and sufficient integral conditions for the existence of eventually positive solutions of the fourth order quasilinear differential equations.