Solvability of Some Fractional Boundary Value Problems with a Convection Term YongfangWei and

Fractional differential equations (FDEs) present new models for many applications in physics, biomathematics, environmental issues, control theory, image processing, chemistry, mechanics, and so on [1–17]. Recently, researchers focus on studying various aspects of fractional differential equations, such as stability analysis, existence, multiplicity, and uniqueness of solutions [1–40]. Among all these topics, the existence and multiplicity results of positive solutions represent a topic of high interest in fractional calculus. Some authors studied the existence and uniqueness of solutions for fractional differential equations with Caputo or Riemann-Liouville derivatives based on the Banach contraction principle and investigate the stability results for various fractional problems [4, 5, 16, 17]. Others studied the existence and multiplicity results of positive solutions or the iterative scheme. By the use of the Krasnoesel-skii’s fixed-point theorem, Zhang [34] obtained some existence results of positive solutions of following problem without the convection term: CDα0+y (x) = f (x, y (x)) , 0 < x < 1, y (0) + y󸀠 (0) = 0, y (1) + y󸀠 (1) = 0. (1)

Some authors studied the existence and uniqueness of solutions for fractional differential equations with Caputo or Riemann-Liouville derivatives based on the Banach contraction principle and investigate the stability results for various fractional problems [4,5,16,17].Others studied the existence and multiplicity results of positive solutions or the iterative scheme.By the use of the Krasnoesel-skii's fixed-point theorem, Zhang [34] obtained some existence results of positive solutions of following problem without the convection term: 0+  () =  (,  ()) , 0 <  < 1,  (0) +   (0) = 0,  (1) +   (1) = 0. ( Wang and Liu [29] deduced the Green function and some interesting properties for the Dirichlet BVPs where    0+ is the Riemann-Liouville (R-L) fractional derivative,  ∈ (1, 2), and  > 0. And they established an iterative scheme to approximate the unique positive solution under the singular conditions.
Meng and Stynes [23] considered the following linear two-point fractional differential equation BVPs with general Robin type boundary condition: where    0+ denotes the Caputo derivative,  ∈ (1,2], ,  0 ,  1 are constant, and  ∈ [0, 1].Meng used two parameter Mittag-Leffler functions to establish explicitly Green's function for the problems.They obtained the nonnegativity of Green's function.
This paper is devoted to the research of the solvability of the following nonlinear fractional BVPs: where 1 <  ≤ 2 and ,  0 ,  1 ∈ R are constants.   0+ is the Caputo's fractional derivative.Compared to the existing literature, the interesting point here is that the convection term is involved in the study of the solvability of fractional differential boundary value problems.By applying some fixed-point theorems, some existence and multiplicity results of positive solutions are given.Some examples are presented in last section to illustrate the main theorems.
In the sequel, the following conditions will be used:

Background Material and Definitions
In order to solve the BVPs (4), (5), the following definitions and lemmas are needed.
By the use of some interesting properties of the Mittag-Leffler function, Meng and Stynes proved the following result.
We pointed out here that this lemma comes from Theorem 5.1 and Remark 5.2 of Ref. [23] with some equivalent changes.
The following theorems are fundamental for proving the main results.Lemma 6 (see [41]).Let  be a Banach space,  ⊆  a cone, and Ω 1 , Ω 2 two bounded sets of  with 0 ∈ Ω

Existence and Multiplicity
In this section, by applying Lemma 6 and Lemma 7, some solvability results for BVPs (4), ( 5) will be obtained.
Thus, the set (Ω) is bounded.By the fact that the Green function (, ) is continuous on [0, 1] × [0, 1], one has the fact that it is uniformly continuous.Therefor, for given  > 0, there exists  > 0 such that for each  ∈ Ω, Hence, the set (Ω) is equicontinuous.Thus, using the Arzela-Ascoli theorem, we claim that  :  →  is a completely continuous operator.

Proof. Define two open sets
And the proof also holds when 0 <  2 <  1 .Therefore, by Lemma 6, the proof is complete.
Proof.We just need to prove that all the conditions of Lemma 7 hold.

Two Examples
Now, we present two examples to check the main results.
Taking into account that the functions (, ) and (, ) are all continuous and nonnegative, the operator  :  →  is continuous and nonnegative, i.e.,  :  → .Suppose Ω ⊂  is a bounded set, and for all  ∈ Ω there holds ‖‖ ≤  1 for some  1 > 0.