Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions

Abstract We consider the following nonlinear fractional differential equation with infinite-point boundary value conditions { D 0 + α u ( t ) + q ( t ) f ( t , u ( t ) ) = 0 , 0 t 1 , u ( 0 ) = u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( i ) ( 1 ) = ∑ j = 1 ∞ α j u ( ξ j ) , where α > 2 , n − 1 α ≤ n , i ∈ [ 1 , n − 2 ] is a fixed integer, α j ≥ 0 , 0 ξ 1 ξ 2 ⋯ ξ j − 1 ξ j ⋯ 1 ( j = 1 , 2 , … ) , Δ − ∑ j = 1 ∞ α j ξ j α − 1 > 0 , Δ = ( α − 1 ) ( α − 2 ) ⋯ ( α − i ) . The nonlinear term f permits singularities with respect to both the time and space variables. By introducing height functions of the nonlinear term on some bounded sets and considering integrations of these height functions, several local existence and multiplicity of positive solutions theorems are obtained.