Nontrivial Solution of Fractional Differential System Involving Riemann-Stieltjes Integral Condition

and Applied Analysis 3 2. Preliminaries and Lemmas For the convenience of the reader, we present here some definitions from fractional calculus which are to be used in the later sections. Definition 2.1 see 18–20 . The Riemann-Liouville fractional integral of order α > 0 of a function x : 0, ∞ → R is given by Ix t 1 Γ α ∫ t 0 t − s α−1x s ds 2.1 provided that the right-hand side is pointwise defined on 0, ∞ . Definition 2.2 see 18–20 . The Riemann-Liouville fractional derivative of order α > 0 of a function x : 0, ∞ → R is given by Dα t x t 1 Γ n − α ( d dt )n ∫ t 0 t − s n−α−1x s ds, 2.2 where n α 1, α denotes the integer part of number α, provided that the right-hand side is pointwise defined on 0, ∞ . Lemma 2.3 see 18–20 . (1) If x ∈ L 0, 1 , ν > σ > 0, then IIx t I x t , Dσ t Ix t Iν−σx t , Dσ t Ix t x t . 2.3 (2) If ν > 0, σ > 0, then Dν t tσ−1 Γ σ Γ σ − ν t σ−ν−1. 2.4 Lemma 2.4 see 18–20 . Assume that x ∈ L1 0, 1 with a fractional derivative of order α > 0 that belongs to L1 0, 1 . Then IαDα t x t x t c1tα−1 c2tα−2 · · · cntα−n, 2.5 where ci ∈ R i 1, 2, . . . , n , n is the smallest integer greater than or equal to α. Let x t Iv t , v t ∈ C 0, 1 ; by standard discussion, we easily reduce the system 1.1 to the following modified problems: −Dα−β t v t λf ( t, Iv t , v t , y t ) , −D t y t g ( t, Iv t ) , t ∈ 0, 1 , v 0 0, v 1 ∫1 0 v s dA s , y 0 0, y 1 ∫1 0 y s dB s , 2.6 and the system 2.6 is equivalent to the system 1.1 . 4 Abstract and Applied Analysis Lemma 2.5 see 21 . Let h ∈ L1 0, 1 , if 1 < α − β, γ ≤ 2, then the unique solution of the linear problems −Dα−β t v t h t , t ∈ 0, 1 , v 0 0, v 1 0, −D t y t h t , t ∈ 0, 1 , y 0 0, y 1 0, 2.7