The Existence of Positive Solution to Three‐Point Singular Boundary Value Problem of Fractional Differential Equation

Abstract

We investigate the existence of positive solution to nonlinear fractional differential equation three‐point singular boundary value problem: D q u ( t ) + f ( t , u ( t )) = 0, 0 < t < 1, u (0) = 0, u (1) = α D ( q −1)/2 u ( t )| t = ξ , where 1 < q ≤ 2 is a real number, ξ ∈ (0, 1/2], α ∈ (0, + ∞ ) and α Γ ( q ) ξ ( q −1)/2 < Γ (( q + 1)/2), D q is the standard Riemann‐Liouville fractional derivative, and f ∈ C ((0, 1] × [0, + ∞ ), [0, + ∞ )), lim t →+0 f ( t , ⋅) = + ∞ (i.e., f is singular at t = 0). By using the fixed‐point index theory, the existence result of positive solutions is obtained.