The Existence Of Eventually Positive Solutions For Fourth Order Quasilinear Differential Equations

In this paper, we concern with the fourth order nonlinear diferential equation(p（t）|u"|^α-1u")"+q（t）|u|^β-1u=0.where α and β are positive numbers, p, q∈C[a,∞),a>0,and p（t）>0, q（t）>0for t≥a,under the conditions0<α <β and0<β <α we give necessary and sufcient criteriafor eventually positive solutions of the fourth order quasilinear diferential equations.In the paper,we introduce some bisic theory,including existence of eventually posi-tive solutions and the predecessors include some results for oscillatory of four quasilineardiferential equations, some preparatory lemmas which are used later in proofs of ourexistence theorems. and our main results and the detail proof.