摘要：In this paper, the author is concerned with the fractional equation D 0 + α C u ( t ) = f ( t , u ( t ) , C D 0 + α 1 u ( t ) , C D 0 + α 2 u ( t ) ) , t ∈ ( 0 , 1 ) , with the anti-periodic boundary value conditions u ( 0 ) = − u ( 1 ) , t β 1 − 1 C D 0 + β 1 u ( t ) t → 0 = − t β 1 − 1 C D 0 + β 1 u ( t ) t = 1 , t β 2 − 2 C D 0 + β 2 u ( t ) t → 0 = − t β 2 − 2 C D 0 + β 2 u ( t ) t = 1 , where D 0 + γ C denotes the Caputo fractional derivative of order γ, the constants α, α 1 , α 2 , β 1 , β 2 satisfy the conditions 2 < α ≤ 3 , 0 < α 1 ≤ 1 < α 2 ≤ 2 , 0 < β 1 < 1 < β 2 < 2 . Different from the recent studies, the function f involves the Caputo fractional derivative D 0 + α 1 C u ( t ) and D 0 + α 2 C u ( t ) . In addition, the author put forward new anti-periodic boundary value conditions, which are more suitable than those studied in the recent literature. By applying the Banach contraction mapping principle and the Leray-Schauder degree theory, some existence results of solutions are obtained. MSC:34A08, 34B15.
关键词：fractional differential equations ; anti-periodic boundary value problems ; existence results ; fixed point theorem
