Some existence, uniqueness results on positive solutions for a fractional differential equation with infinite-point boundary conditions

Abstract. We investigate a class of Riemann–Liouville’s fractional differential equation with infinite-point boundary conditions. We give some new properties of the Green’s function associated with the fractional differential equation boundary value problem. Based upon these new properties and by using Schauder’s fixed point theorem, we establish some existence results on positive solutions for the boundary value problem. Further, by using a fixed point theorem of general concave operators, we also present an existence and uniqueness result on positive solutions for the boundary value problem.


Introduction
Recently, there are several researchers who studied fractional differential equation with infinite-point boundary conditions, see [2, 4-7, 19, 21] and some references therein.In these papers, the existence of positive solutions was considered by using different methods, which include the fixed point theorem of cone expansion-compression, Avery-Peterson's fixed point theorem, Leggett-Williams's fixed point theorem, Leray-Schauder's nonlinear alternative, Leray-Schauder degree theory, and so on.However, we can see that the results on the infinite-point boundary value problems for fractional differential equations are still very few, and the uniqueness results on positive solutions for this type of boundary value problem are seldom obtained.So, this type of boundary value problem is worthwhile to discuss.In this article, we investigate the following boundary value problem of fractional differential equation with a boundary condition involving infinite points:
Lemma 1.Some properties of the function G(t, s) are the following: In our paper, we will give some new properties of the Green's function G(t, s) and use these new properties to study the existence, uniqueness of positive solutions for problem (1).By using Schauder's fixed point theorem, we first establish some existence results on positive solutions for problem (1).Second, we use a fixed point theorem of general concave operators to obtain an existence and uniqueness result on positive solutions for problem (1).

New properties of the Green's function
In this section, we give some new properties of the Green's function G(t, s).First, we define a function g by Lemma 3. The following inequality holds: Proof.From α > 2 and Lemma 2 we have Note that i 1, g(t) t α−1 for 0 t 1.
Lemma 5.For 0 < s < t < 1, the following conclusion holds: Hence, the proof is finished.
From Lemmas 4 and 5 we can easily obtain the following result.
Theorem 1.For t, s ∈ [0, 1], the following inequality holds: Further, from Lemma 1 and Theorem 1 we can also obtain the following conclusion.
First, we present a condition: (H1) there are two constants where Proof.Define an operator A : E → E by From [19] we know that u is a solution of problem (1) if and only if u is a fixed point of A. For every u ∈ S, by Theorem 2 and (H1), and Hence, A(Ω) ⊆ Ω.Further, it follows from the continuity of G(t, s) and f (t, u) that A : Ω → Ω is completely continuous.Therefore, A has a fixed point u * in S by using Schauder's fixed point theorem.By Theorem 3, u * (t) τ 1 t α−1 0, t ∈ [0, 1], so, we claim that u * (t) is a positive solution.
From Theorem 4 we can easily obtain the following corollaries.
Example 2. In problem (3), we replace f (t, u) by Similar to Example 1, we can also easily prove that all the assumptions of Corollary 2 are satisfied.
4 Uniqueness of positive solutions for problem (1) Let (E, • ) be a real Banach space.θ is the zero element of E, and P ⊂ E is a cone.Then E is partially ordered by cone P , i.e., x y if and only if y − x ∈ P .P is called normal if there exists a constant K > 0 such that, for x, y ∈ E with θ x y, x K y .For x, y ∈ E, the notation x ∼ y means that there exist λ > 0 and µ > 0 such that λx y µx.Clearly, ∼ is an equivalence relation.For h > θ (i.e., h θ and h = θ), we denote by P h the set P h = {x ∈ E | x ∼ h}.Lemma 6. (See Lemma 2.1 and Theorem 2.1 in [18].)Let h > θ and P be a normal cone.Assume: (D1) A : P → P is increasing, and Ah ∈ P h ; (D2) for x ∈ P and t ∈ (0, 1), there exists ϕ(t) ∈ (t, 1) such that A(tx) ϕ(t)Ax. Then: (i) there are u 0 , v 0 ∈ P h and r ∈ (0, 1) such that rv 0 u 0 < v 0 , u 0 Au 0 Av 0 v 0 ; (ii) operator equation x = Ax has a unique solution in P h .Remark 1.An operator A is said to be generalized concave if it satisfies condition (D2).
Remark 2. In Theorem 5, we can see that λ, µ are not constants and depend on x, h.In Ω, τ 1 , τ 2 are two constants.So, Ω = P h .
Consequently, r 1 h Ah r 2 h and thus Ah ∈ P h .Finally, by Lemma 6, there are u 0 , v 0 ∈ P h and r ∈ (0, 1) such that rv 0 u 0 < v 0 , u 0 Au 0 Av 0 v 0 ; and we can claim that A has a unique fixed point in P h .That is, u 0 (t)

Theorem 4 .
Let condition (H1) be satisfied.Then problem (1) has at least one positive solution in Ω.