Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions ∗

Abstract. In this paper, we investigate the existence of at least three positive solutions to a singular boundary value problem of Caputo’s fractional differential equations with a boundary condition involving values at infinite number of points. Firstly, we establish Green’s function and its properties. Then the existence of multiple positive solutions is obtained by Avery–Peterson’s fixed point theorem. Finally, an example is given to demonstrate the application of our main results.


Introduction
In this paper, we consider the following infinite-point fractional differential equations boundary value problem: where 2 < α, n − 1 < α n, i ∈ [1, n − 2] is a fixed integer, α j 0, 0 < ξ 1 < ξ 2 < • • • < ξ j−1 < ξ j < • • • < 1 (j = 1, 2, . . .), f is allowed to have singularities with respect to both time and space variables.Various theorems were then established for the existence and multiplicity of positive solutions.In [17], the author discussed the existence and multiplicity of positive solutions of the following problem: D α 0 + u(t) = a(t)f t, u(t) , t ∈ (0, 1), where 2 < α 3, m 1 is integer, ] is non-negative and not identically zero on any compact subset of (0, 1), f : [0, 1] × [0, +∞) → [0, +∞) is continuous and D α 0 + is the Riemann-Liouville differential fractional derivative of order α.Some results on the existence and multiplicity of positive solutions were obtained by the fixed point theorem.In [9], the authors investigated the existence of multiple positive solutions of the following fractional differential equation boundary value problem: . ., x i ) may be singular at t = 0 and c D α 0 + is the standard Caputo derivative.The authors obtained the existence result of at least three positive solutions for a two-point boundary value problem, in which the nonlinear terms contain derivatives up to order i by using Avery-Peterson's fixed point theorem.
Motivated by the results above, in this paper, we investigate the existence of positive solutions for a class of singular fractional differential equations subject to infinite-point boundary conditions.Compared with previous work in the field, our work presented in this paper has several new features.Firstly, values at infinite points are involved in the boundary conditions of the boundary value problem (1).Secondly, our study is on singular nonlinear differential boundary value problems, that is, f (t, u, v) is allowed to be singular at t = 0. Thirdly, the nonlinear term involves the first order derivative.Fourthly, the main tool used in this paper is Avery-Peterson's fixed point theorem.

Preliminaries and lemmas
For the convenience of the reader, we first present some basic definitions and lemmas, which are to be used in the proof of our results and can also be found in the recent literature such as [18,19].Firstly, we let E = C 1 [0, 1] be the Banach space with the maximum norm u = max u 0 , u 0 , where u 0 = max t∈[0,1] |u(t)|, u 0 = max t∈[0,1] |u (t)|.We also list below a condition to be used later in the paper.
L. Guo et al.
Lemma 1. (See [18,19].)Assume that u ∈ C n [0, 1], then where n is the least integer greater than or equal to α, then H is a relatively compact set if and only if : (a) H is equicontinuous and H (t) is a relatively compact set for any t ∈ J on E; (b) There exists t 0 ∈ J such that H(t 0 ) is a relatively compact set on E.
can be expressed by where in which and, obviously, Proof.By means of Lemma 1, we can reduce (2) to an equivalent integral equation On the other hand, u (1) = ∞ j=1 η j u(ξ j ), and so combining with we get Hence, Therefore, It is easy to check that G(t, s) and ∂G(t, s)/∂t are uniformly continuous on where Proof.By direct calculation, we get P (s) 0, s ∈ [0, 1], and so P (s) is nondecreasing with respect to s.For s ∈ [0, 1], we get and, obviously, Hence, for t, s ∈ [0, 1], we have Furthermore, for 0 s t 1, we get http://www.mii.lt/NAand, obviously, for 0 t s 1, we get G(t, s) 0.
On the other hand, for 0 s t 1, we have and for 0 t s 1, ∂G(t, s)/∂t 0 obviously holds. , Therefore, the proof of Lemma 4 is completed.
Nonlinear Anal.Model.Control, 21(5):635-650 Now we define a cone P on C 1 [0, 1] and an operator A : P → C 1 [0, 1] as follows: , a and b are the same as in Lemma 4, and Problems (1) has a positive solution if and only if u is a fixed point of A in P .
Lemma 6.A : P → P is completely continuous.