Upper and Lower Solution Method for Fractional Boundary Value Problems on the Half-Line

where α ∈ R, α ̸ = 1, and η ∈ (0,∞) are given. Based on Leray-Schauder continuation theorem, some suitable conditions for the existence of solutions to (1) are established. On the other hand, fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary noninteger order. Fractional calculus is a wonderful technique to understand memory and hereditary properties of materials and processes. Some recent contributions to fractional differential equations are present in the monographs [12–19]. Very recently, Chen and Tang in [20] considered the following fractional differential boundary value problem on the half-line:

On the other hand, fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary noninteger order.Fractional calculus is a wonderful technique to understand memory and hereditary properties of materials and processes.Some recent contributions to fractional differential equations are present in the monographs [12][13][14][15][16][17][18][19].Very recently, Chen and Tang in [20] considered the following fractional differential boundary value problem on the half-line: 0 +  () =  (,  ()) ,  ∈ [0, ∞) , where 3 <  < 4 and   0 + is the standard Riemann-Liouville fractional derivative.By the recent Leggett-Williams normtype theorem, the existence of positive solutions is obtained.In 2011, [21] set up the global existence results of solutions of initial value problems on the half-axis as follows: where   0 + is the standard Riemann-Liouville fractional derivative.By constructing a special Banach space and employing fixed point theorems, some sufficient conditions for the existence of solutions are obtained.In [22], the authors studied the following boundary value problem of fractional order on the half-line:   0 +  () +  ()  (,  () ,  −1 0 +  ()) = 0,  ∈ [0, ∞) , where 1 <  ≤ 2,  ∈ ([0, ∞) × R 2 , R),  −1 0 + and   0 + are the standard Riemann-Liouville fractional derivatives.By Schauder's fixed point theorem on an unbounded domain, they obtain the existence result for (4).Some papers have recently been done for fractional boundary value problem on the half-line or unbounded domain, see [22][23][24][25][26][27][28][29][30][31].Inspired by the above-mentioned works, in this paper, we study the existence of solutions to the following fractional differential equations with boundary value problems: where () : [0, ∞) → [0, ∞), and   0 + is the standard Riemann-Liouville fractional derivative.By upper and lower solution method techniques, the sufficient conditions for solutions to (5) are obtained.Our main findings given in this paper have some new features.Firstly, the like Nagumo condition defined by us plays an important role in the nonlinear term involving the standard Riemann-Liouville derivatives.Secondly, to the best of our knowledge, no work has been done concerning fractional boundary value problems (5) and our method is different from that of [22,26,31].Thirdly, the nonlinear term  may take negative values, and (, , , ) depends on  allowed to be quadratic, referring to our example.The rest of this paper is organized as follows: in Section 2, we present some preliminaries and some lemmas that will be used in Section 3. The main result and proof will be given in Section 3. In addition, an example is given to demonstrate the application of our main result.

Preliminaries
We first present some basic definitions and preliminary results about fractional calculus; we refer the reader to [17,18] for more details.
Definition 1 (see [18]).The integral where  > 0, is called the Riemann-Liouville fractional integral of order  and Γ() is the Euler gamma function defined by Definition 2 (see [17]).A function () given in the interval [0, ∞), the expression where  = [] + 1, [] denotes the integer part of number , is called the Riemann-Liouville fractional derivative of order .
We consider the following two cases.
Definition 8. Let ,  ∈  ⋂  loc (0, ∞) be lower and upper solutions to (5) and suppose that (15) holds.A continuous function  : [0, ∞) × R 3 → R is said to satisfy the like Nagumo condition with respect to the pair of functions , , if there exist a nonnegative function  ∈ [0,∞) and a positive one We list some assumptions related to (), (), and  as follows.
Proof.Consider the following.
Step 1.  :  →  is well defined.For  ∈ , it follows from (34) that which implies lim We also have Combining (36) with (38), one has If we apply Lebesgue dominated convergence theorem with (37) and (39), then lim By virtue of ( 34 It follows from (39) that Therefore, we have Thus, we conclude that  ∈ .

Main Result
We are in the position to state the main existence result.
Proof.By Lemma 13, we know that  :  →  is completely continuous.By the Schauder fixed point theorem, we can easily obtain that  has at least one fixed point  ∈ .Thus,  is a solution of (32).Next, we will show that  satisfies the inequalities  () ≤  () ≤  () , which implies that  is a solution of (5).First of all, we will show that Note that lim  → ∞ ( −1 0 + () −  −1 0 + ()) < 0; then there are two cases.