Positive solutions for a nonlocal fractional differential equation

Abstract In this paper, we study the following singular boundary value problem of a nonlocal fractional differential equation { D 0 + α u ( t ) + q ( t ) f ( t , u ( t ) ) = 0 , 0 t 1 , n − 1 α ≤ n , u ( 0 ) = u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( 1 ) = ∫ 0 1 u ( s ) d A ( s ) , where α ≥ 2 , D 0 + α is the standard Riemann–Liouville derivative, ∫ 0 1 u ( s ) d A ( s ) is given by Riemann–Stieltjes integral with a signed measure, q may be singular at t = 0 and/or t = 1 , f ( t , x ) may also have singularity at x = 0 . The existence and multiplicity of positive solutions are obtained by means of the fixed point index theory in cones.

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