Solvability of sequential fractional order multi-point boundary value problems at resonance
Abstract In this paper, we study the following sequential fractional differential equation a ( t ) C D 0 + α u ( t ) ′ = f ( t , u ( t ) , u ′ ( t ) , C D 0 + α u ( t ) ) , t ∈ J , u ( 0 ) = 0 , C D 0 + α u ( 0 ) = 0 , u ( 1 ) = ∑ j = 1 m - 1 σ j u ( ξ j ) , where C D 0 + α is the Caputo fractional derivative, 1 α ⩽ 2 , σ j ∈ R , ξ j ∈ ( 0 , 1 ) , j = 1 , 2 , … , m - 1 , m ∈ N , m > 1 , J = [ 0 , 1 ] and ∑ j = 1 m - 1 σ j ξ j = 1 . By the coincidence degree continuation theorem, we get some existence results about the boundary value problem at resonance.