Existence on positive solutions for boundary value problems of nonlinear fractional differential equations with p-Laplacian

Abstract

Abstract 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.