Existence Of Non-extreme Solutions Of Third-order Quasilinear Ordinary Differential Equations

This paper is concerned with non-extreme positive solutions of the third order quasilinear differential equation(p (t)|u' (t)|^α-1 u' (t))" + q (t)|u (t)|^β-1 u (t) = 0where α > 0 , β > 0 and p(t) and q(t) are continuous functions on an infinite interval [a, ∞) satisfying p(t) > 0 and q(t) > 0, (t > a). Where an eventually positive solutions u(t) of equation (1.1) is non-extreme if there exist positiveconstants C₁ > 0 and c₂ > 0 such that C₁ < u(t) < C₂ ds. Underthe assumptions α> 1 >β and α < 1 < β, necessary and sufficient integralconditions are established for the existence of non-extreme positive solutions .