Multiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian

Abstract

Abstract In this article, we investigate the Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian { D 0 + β ( ϕ p ( D 0 + α u ( t ) ) ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , ξ u ( 0 ) − η u ′ ( 0 ) = 0 , γ u ( 1 ) + δ u ′ ( 1 ) = 0 , D 0 + α u ( 0 ) = 0 , where 1 < α ≤ 2 , 0 < β ≤ 1 , D 0 + α , D 0 + β are the standard Caputo fractional derivatives, ϕ p ( s ) = | s | p − 2 s , p > 1 , ϕ p − 1 = ϕ q , 1 / p + 1 / q = 1 , ξ , η , γ , δ ≥ 0 , ρ : = ξ γ + ξ δ + η γ > 0 , and f : [ 0 , 1 ] × [ 0 , + ∞ ) → [ 0 , + ∞ ) is continuous. By means of the properties of the Green’s function, Leggett-Williams fixed-point theorems, and fixed-point index theory, several new sufficient conditions for the existence of at least two or at least three positive solutions are obtained. As an application, an example is given to demonstrate the main result. MSC:34A08, 34B18, 35J05.