The Existence of Positive Solution to Three-Point Singular Boundary Value Problem of Fractional Differential Equation
نویسندگان
چکیده
and Applied Analysis 3 Definition 2.2. The fractional derivative of order q > 0 of a continuous function x : 0, ∞ → R is given by Dx t 1 Γ ( n − q) ( d dt )n∫ t 0 x s t − s q−n 1 ds, 2.2 where n q 1, provided that the right side is pointwise defined on 0,∞ . Lemma 2.3 see 7 . 1 If x ∈ L 0, 1 , ρ > σ > 0, then DIx t Iρ−σx t . 2 If ρ > 0, λ > 0, then Dρtλ−1 Γ λ /Γ λ − ρ tλ−ρ−1. Lemma 2.4 see 12 . Assume that x ∈ C 0, 1 ∩ L 0, 1 with a fractional derivative of order q > 0 that belongs to C 0, 1 ∩ L 0, 1 . Then IDx t x t A1tq−1 A2tq−2 · · · ANtq−N, 2.3 Ai ∈ R, i 1, 2, . . . ,N, where N is the smallest integer greater than or equal to q. Lemma 2.5. If y ∈ C 0, 1 ∩ L 0, 1 and 1 < q ≤ 2, ξ ∈ 0, 1 , α ∈ R satisfy that αΓ q ξ q−1 /2 / Γ q 1 /2 , then the problems Du t y t 0, 0 < t < 1, u 0 0, u 1 αD q−1 /2u t ∣ ∣ ∣ t ξ 2.4 have the unique solution
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تاریخ انتشار 2009