Existence of Positive Solutions to Boundary Value Problems for Fractional Differential Equation with P-Laplacian Operators

Abstract

For the study of some complex physical, biological and other phenomena, it is difficult for the traditional integer differential equations to describe these processes accurately. In this paper, by means of the Guo-Krasnoselskii fixed point theorem, we study the existence of positive solutions to boundary value problems for a class of fractional differential equations with p-Laplacian operators. { ( D 0 + β ( ϕ p ( D 0 + α u ( t ) ) ) ) = f ( t , u ( t ) ) , t ∈ [ 0 , 1 ] , ( ϕ p ( D 0 + α u ( 0 ) ) ) ′ = ( ϕ p ( D 0 + α u ( 1 ) ) ) , u ( 0 ) = u ( 1 ) = u ′ ( 0 ) = u ′ ( 1 ) = 0 , where β ∈ ( 1 , 2 ] , α ∈ ( 3 , 4 ] are real numbers, both D 0 + β and D 0 + α are in the range of standard Riemann-Liouville derivatives, ϕ p ( s ) = | s | p − 2 s , p ∈ ( 1 , + ∞ ) , ϕ p − 1 = ϕ q , 1 p + 1 q = 1 , and f ∈ C ( [ 0 , 1 ] × [ 0 , + ∞ ) → [ 0 , + ∞ ) ) .