Positive Solutions for Nonlinear Fractional Differential Equations with Boundary Conditions Involving Riemann-Stieltjes Integrals

and Applied Analysis 3 Inspired by the work of the above papers, the aim of this paper is to establish the existence and multiplicity of positive solutions of the BVP 1.1 . We discuss the boundary value problemwith the Riemann-Stieltjes integral boundary conditions, that is, the BVP 1.1 , which includes fractional order two-point, three-point, multipoint, and nonlocal boundary value problems as special cases. Moreover, the β · in 1.1 is a linear function on C 0, 1 denoting the Riemann-Stieltjes integral; the A in the Riemann-Stieltjes integral is of bounded variation, namely, dA can be a signed measure. By using the Krasnosel’skii fixed point theorem, the Leray-Schauder nonlinear alternative and the Leggett-Williams fixed point theorem, some existence and multiplicity results of positive solutions are obtained. The rest of this paper is organized as follows. In Section 2, we present some lemmas that are used to prove our main results. In Section 3, the existence and multiplicity of positive solutions of the BVP 1.1 are established by using some fixed point theorems. In Section 4, we give four examples to demonstrate the application of our theoretical results. 2. Basic Definitions and Preliminaries We begin this section with some preliminaries of fractional calculus. Let α > 0 and n α 1, where α is the largest integer smaller than or equal to α. For a function f : 0, ∞ → R, we define the fractional integral of order α of f as I 0 f t 1 Γ α ∫ t 0 t − s α−1f s ds, 2.1 provided the integral exists. The fractional derivative of order α > 0 of a continuous function f is defined by D 0 f t 1 Γ n − α ( d dt )n ∫ t 0 t − s n−α−1f s ds, 2.2 provided the right-hand side is pointwise defined on 0, ∞ . We recall the following properties 26, 27 which are useful for the sequel. For α > 0, β > 0, we have I 0 I β 0 f t I α β 0 f t , D α 0 I α 0 f t f t . 2.3 As an example, we can choose a function f such that f, D 0 f ∈ C 0, ∞ ∩ Lloc 0, ∞ . For α > 0, the general solution of the fractional differential equation D 0 u t 0 with u ∈ C 0, 1 ∩ L 0, 1 is given by u t c1tα−1 c2tα−2 · · · cntα−n, 2.4 where ci ∈ R i 1, 2, . . . , n . Hence for u ∈ C 0, 1 ∩ L 0, 1 , we have I 0 D α 0 u t u t c1t α−1 c2tα−2 · · · cntα−n. 2.5 4 Abstract and Applied Analysis Set G0 t, s 1 Γ α − n 2 ⎧ ⎨ ⎩ t 1 − s α−n 1 − t − s α−n , 0 ≤ s ≤ t ≤ 1, t 1 − s α−n , 0 ≤ t ≤ s ≤ 1. 2.6 Lemma 2.1 see 11 . Let y ∈ Cr 0, 1 Cr 0, 1 {y ∈ C 0, 1 , ty ∈ C 0, 1 , 0 ≤ r < 1} . Then the boundary value problem, Dα−n 2 0 v t y t , 0 < t < 1, n − 1 < α ≤ n, n ≥ 2, v 0 0, v 1 0, 2.7 has a unique solution v t ∫1 0 G0 t, s y s ds. 2.8 Lemma 2.2 see 11 . The function G0 t, s defined by 2.6 satisfies the following properties: i G0 t, s ≥ 0, G0 t, s ≤ G0 s, s for all t, s ∈ 0, 1 ; ii there exist a positive function ρ ∈ C 0, 1 and 0 < ξ < η < 1 such that min t∈ ξ,η G0 t, s ≥ ρ s G0 s, s , s ∈ 0, 1 , 2.9