Positive solutions for a class of fractional infinite-point boundary value problems

In this paper, we consider the following fractional differential equations (FDE for short) with infinite-point boundary value conditions: where \(\alpha \geq 2\), \(n-1 < \alpha \leq n\), \(0< \beta < \alpha -1\), \(\eta_{i}\in \mathbb{R}\), \(0 < \xi_{1}< \cdots < \xi_{i}<\xi_{i+1}< \cdots < 1\) (\(i=1,2,\ldots \)), \(\sum^{\infty }_{i=1}\eta_{i}\xi_{i}^{ \alpha -\beta -1}\neq 1\), \(D^{\alpha }_{0+}\) is the standard Riemann–Liouville derivative, \(f:[0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) is continuous, \(a(t)\in C((0,1),[0,+\infty ))\) may be singular at \(t=0,1\).

FDE serve as an excellent instrument for the description of memory and hereditary properties of various materials and processes. During the last few decades, much attention has been paid to the study of boundary value problems (BVP for short) of fractional differential equation, such as the nonlocal BVP [1, 3, 7, 13, 18], singular BVP [6, 8, 11, 19, 20, 25], semipositone BVP [14–16, 23], resonant BVP [2, 12], and impulsive BVP [10, 27]. Since only positive solutions are meaningful in most practical problems, some work has been done to study the existence of positive solutions for fractional boundary value problems by using the techniques of nonlinear analysis. In [16], the authors discussed the following fractional m-point BVP: where \(0< \beta < \alpha -1\), \(0 < \xi_{1}< \cdots < \xi_{m-2} < 1\) with \(\sum^{m-2}_{i=1}\eta_{i}\xi_{i}^{\alpha -\beta -1}< 1\). Some new positive properties of the corresponding Green function were established, and a number of theorems on the existence of positive solutions were obtained. However, there are few works on the uniqueness of solution for fractional boundary value problems [4, 5, 9, 17, 21, 22, 24, 26]. To obtain the uniqueness of solution, the main tool used in most of the works in literature is the Banach contraction map principle provided the nonlinearity satisfies the Lipschitz condition. Liang and Zhang [9] studied the uniqueness of positive solution for the following fractional three-point BVP: where \(0 < \eta < 1\) with \(0< \beta \eta^{\alpha -3}< 1\), \(f:[0,1] \times [0,+\infty )\rightarrow [0,+\infty )\) is continuous and nondecreasing with respect to the second variable, and there exists such that, for \(u,v \in [0,+\infty )\) with \(u\geq v\) and \(t\in [0,1]\), It should be noted that (1.2) implies \(f(t,u)-f(t,v)\leq \lambda (u-v)\) since \(\ln (1+x)\leq x, \forall x \geq 0\), that is, f satisfies the Lipschitz condition.

Set where

For the convenience in presentation, we here list the assumption to be used throughout the paper.

Let \(E=C[0,1]\) be endowed with the maximum norm \(\Vert u \Vert = \max_{0\leq t\leq 1}\vert u(t) \vert \), \(B_{r}=\{u\in E : \Vert u \Vert < r\}\). Define cones P, Q by

Let It is clear that T is a completely continuous linear operator and \(T(P)\subset Q\). By virtue of the Krein–Rutmann theorem and Lemma 2.3, we have the following lemma.

Let \(\lambda (t)=\sqrt{t}\), we have It is clear that where \(\Delta :=\Gamma (\frac{5}{2})[1-p(0)]\).

Let Then By direct calculations, we have and

It is clear that On the other hand, we have Noticing that we have which implies that It follows from Lemma 2.5 and Remark 3.2 that Therefore the assumptions of Theorem 3.1 are satisfied. Thus Theorem 3.1 ensures that BVP (4.1) has a unique positive solution.