In this paper, we study the existence of positive solutions for the nonlinear fractional boundary value problem with a p-Laplacian operator D 0 + β ( ϕ p ( D 0 + α u ( t ) ) ) = f ( t , u ( t ) ) , 0 < t < 1 , u ( 0 ) = u ′ ( 0 ) = u ′ ( 1 ) = 0 , D 0 + α u ( 0 ) = D 0 + α u ( 1 ) = 0 , where 2 < α ⩽ 3 , 1 < β ⩽ 2 , D 0 + α , D 0 + β are the standard Riemann-Liouville fractional derivatives, ϕ p ( s ) = | s | p − 2 s , p > 1 , ϕ p − 1 = ϕ q , 1 / p + 1 / q = 1 , and f ( t , u ) ∈ C ( [ 0 , 1 ] × [ 0 , + ∞ ) , [ 0 , + ∞ ) ) . By the properties of Green’s function, the Guo-Krasnosel’skii fixed-point theorem, the Leggett-Williams fixed-point theorem, and the upper and lower solutions method, some new results on the existence of positive solutions are obtained. As applications, examples are presented to illustrate the main results. MSC:34A08, 34B18, 35J05.

We consider the oscillation for a class of fractional differential equation[r(t)g(D-αy)(t)]'-p(t)f∫t∞‍(s-t)-αy(s)ds=0,fort>0,where0<α<1is a real number andD-αyis the Liouville right-sided fractional derivative of orderαofy. By generalized Riccati transformation technique, oscillation criteria for a class of nonlinear fractional differential equation are obtained.