摘要：In the paper, we consider the existence criteria for positive solutions of the nonlinear p-Laplacian fractional differential equation whose nonlinearity contains the first-order derivative explicitly { ( φ p ( C D α u ( t ) ) ) ′ = φ p ( λ ) f ( t , u ( t ) , u ′ ( t ) ) , t ∈ ( 0 , 1 ) , k 0 u ( 0 ) − k 1 u ( 1 ) = 0 , m 0 u ( 0 ) − m 1 u ( 1 ) = 0 , x ( r ) ( 0 ) = 0 , r = 2 , 3 , … , [ α ] , where φ p is the p-Laplacian operator, i.e., φ p ( s ) = s p − 2 s , p > 1 , and φ q = φ p − 1 , 1 p + 1 q = 1 . D α C is the standard Caputo derivative and f ( t , u , u ′ ) : [ 0 , 1 ] × [ 0 , ∞ ) × ( − ∞ , + ∞ ) → [ 0 , ∞ ) satisfies the Carathéodory type condition. The nonlinear alternative of Leray-Schauder type and the fixed-point theorems in Banach space are used to investigate the existence of at least single, twin, triple, n or 2 n − 1 positive solutions for p-Laplacian fractional order differential equations. As an application, two examples are given to illustrate our theoretical results. MSC:34A08, 34B18, 34K37.
关键词：existence ; positive solution ; p -Laplacian fractional differential equation ; Caputo derivative ; fixed-point theorem ; Carathéodory type condition
