Positive Solutions of a Two-Point Boundary Value Problem for Singular Fractional Differential Equations in Banach Space

This paper investigates the existence of positive solutions to a two-point boundary value problem (BVP) for singular fractional differential equations in Banach space and presents a number of new results. First, by constructing a novel cone and using the fixed point index theory, a sufficient condition is established for the existence of at least two positive solutions to the approximate problem of the considered singular BVP. Second, using Ascoli-Arzela theorem, a sufficient condition is obtained for the existence of at least two positive solutions to the considered singular BVP from the convergent subsequence of the approximate problem. Finally, an illustrative example is given to support the obtained new results.


Introduction
Fractional differential equations have been widely investigated recently due to its wide applications [1][2][3] in biology, physics, medicine, control theory, and so forth.As a matter of fact, fractional derivatives provide a more excellent tool for the description of memory and hereditary properties of various materials and processes than integer derivatives.As an important issue for the theory of fractional differential equations, the existence of positive solutions to kinds of boundary value problems (BVPs) has attracted many scholars' attention, and lots of excellent results have been obtained [4][5][6][7][8][9][10][11] by means of fixed point theorems, upper and lower solutions technique, and so forth.
It is noted that as a special class of fractional differential equations, the singular fractional differential equations with kinds of boundary values have been studied in a series of recent works [7,12,13].In [7], Jiang et al. studied a singular nonlinear semipositone fractional differential system with coupled boundary conditions and presented some sufficient conditions for the existence of a positive solution by using the fixed point theory in cone and constructing some available integral operators together with approximating technique.Zhang et al. [13] considered a class of two-point BVP for singular fractional differential equations with a negatively perturbed term and established some results on the multiplicity of positive solutions by using the approximating technique.In [12], Agarwal et al. investigated the existence of positive solutions for a two-point singular fractional boundary value problem and proposed some existence criteria by using sequential techniques.It should be pointed out that the nonlinearities of [7,13] are singular at  = 0,1, while the nonlinearity of [12] is singular at  = 0. To our best knowledge, there are fewer results on two-point BVPs for singular fractional differential equations with the nonlinearity being singular at both  = 0,1 and  = 0. Motivated by this, we consider the following two-point BVP of singular fractional differential equations in Banach space: where 1 <  ≤ 2 is a real number,  = [0, 1],  :  × E → E is continuous,  denotes the null element in the Banach space E with the norm ‖ ⋅ ‖,   0 + is the standard Riemann-Liouville fractional derivative, and (, ) may be singular at  = 0,1 and  = .Firstly, we establish a sufficient condition for the existence of at least two positive solutions to the approximate problem of BVP (1) by constructing a novel cone and using the fixed point index theory.Secondly, using Ascoli-Arzela theorem, we obtained a sufficient condition for the existence of at least two positive solutions to BVP (1) from the convergent subsequence of the approximate problem.Finally, we give an illustrative example to support the obtained new results.
The main features of this paper are as follows.(i) A class of fractional-order two-point boundary value problems with the nonlinearity being singular at both  = 0, 1 and  =  is firstly studied in this paper, which generalizes the existing singular fractional differential equations [7,12,13] and has wider applications.(ii) A sequential-based method is proposed for singular fractional differential equations with the nonlinearity being singular at both  = 0, 1 and  = 0, which enriches the theory of fractional differential equations.
The rest of this paper is organized as follows.Section 2 contains the definition of Riemann-Liouville fractional derivative and some notation.The main results are presented in Section 3, which is followed by an illustrative example in Section 4.

Preliminaries
We first recall some well-known results about Riemann-Liouville derivative.For details, please refer to [14,15] and the references therein.
Definition 1.The Riemann-Liouville fractional integral of order  > 0 of a function  : (0, ∞) →  is given by provided the right side is pointwise defined on (0, ∞).
Definition 2. The Riemann-Liouville fractional derivative of order  > 0 of a continuous function  : (0, ∞) →  is given by where  is the smallest integer greater than or equal to , provided that the right side is pointwise defined on (0, ∞).
One can easily obtain the following properties from the definition of Riemann-Liouville derivative.
The following lemmas will be used in the proof of the main results.
For convenience, let us list the following assumptions.
According to [18], BVP ( 1) is equivalent to where Consider the operator  associated with the singular boundary value problem (1), which is defined by Set  * (, ) =  2− (, ).As in [19], we have Define Then, one can see that () is a fixed point of the operator  * if and only if  −2 () is a solution of BVP (1).
Choose  ∈   with ‖‖ = 1.We consider the following approximate problem of ( 15): It is easy to check that  is a cone in [, E].Let   = { ∈  : ‖‖ < }.By (H1) and (H2), we can conclude that which implies that the operator  * is well defined.In addition, from the definition of , we can prove that By ( 16), we have (  ( Based on (H1) and (H2), we have where where   (⋅, ⋅) denotes the Hausdorff metrics, which implies that where   = [, 1 − ].By (H4), we can obtain Consequently, the operator   is a strict set contraction from   into .
Firstly, we prove that On one hand, by the absolute continuity of integration [19], for any  > 0, there exists  1 > 0 such that On the other hand, from (H2), we set and  = min{ 1 ,  1 /}.For 0 <  < , we have which implies that (49) holds.Similarly, one can prove that (50) holds.
From (H3), we define By the absolute continuity of integration, for  ∈ (0, 1/2) and the previous  > 0, there exists 0 < Since Thus,   () = 0.It follows from Lemma 8 that there is a convergent subsequence of {  }.Without loss of generality, we assume that {  } itself converges to some  ∈ .Then, the dominated convergence theorem and ( 16) imply that  () = ∫ Similarly, {  } has a convergent subsequence, which converges to ().Then,  −2 () is also a positive solution to BVP (1).Finally, we show  ̸ = .We only need to prove that the operator  * has no fixed point in   .
In fact, if it is not true, then we assume that  is a fixed point of the operator  * in   .Then  () = ∫