Positive Solutions for a Fractional Boundary Value Problem with a Perturbation Term

provided that the right-hand side is pointwise defined on (0,∞). Spurred by the extensively applicability of fractional derivatives in a variety of mathematical models in science and engineering [1–3], the subject of fractional differential equations with boundary value problems, which emerged as a new branch of differential equations, have attracted a great deal of attention for decades. As a small sampling of recent development, we refer the reader to [4–14]. When one seeks the existence of solution of boundary value problems for fractional differential equations, the usual method is converted to a Fredholm integral equation and find the fixed points by using various techniques of nonlinear analysis such as Banach contraction map principle [13, 15], linear operator theory [16, 17], Leggett-Williams fixed point theorem [12, 18], Schauder fixed point theorem and Leray-Schauder nonlinear alternative theory [19], andKrasnosel’skii fixed point theorem [20]. It should be noted that the Green’s functions play a vital role in the construction of an appropriate Fredholm integral equation. However, as a result of the unusual feature of the fractional calculus, the investigation on the Green’s functions for fractional boundary value problems is still in the initial stage. Recently, based on the spectral theory, the authors in [21] give an associated Green’s function for BVP (1) (2) as series of functions. This idea was also used in [22–24]. In the next section, we will study some new sharper upper and lower estimates for the Green’s function of BVP (1) (2) than the ones given in [21]. In Section 3, we employ the new estimate to obtain the existence of a positive solution of BVP (1) (2). The idea of this paper may trace to [21–27].

Here,    is the standard Riemann-Liouville derivative of order  > 0 of a continuous function  : (0, ∞) → R is given by provided that the right-hand side is pointwise defined on (0, ∞).
Spurred by the extensively applicability of fractional derivatives in a variety of mathematical models in science and engineering [1][2][3], the subject of fractional differential equations with boundary value problems, which emerged as a new branch of differential equations, have attracted a great deal of attention for decades.As a small sampling of recent development, we refer the reader to [4][5][6][7][8][9][10][11][12][13][14].When one seeks the existence of solution of boundary value problems for fractional differential equations, the usual method is converted to a Fredholm integral equation and find the fixed points by using various techniques of nonlinear analysis such as Banach contraction map principle [13,15], linear operator theory [16,17], Leggett-Williams fixed point theorem [12,18], Schauder fixed point theorem and Leray-Schauder nonlinear alternative theory [19], and Krasnosel'skii fixed point theorem [20].It should be noted that the Green's functions play a vital role in the construction of an appropriate Fredholm integral equation.However, as a result of the unusual feature of the fractional calculus, the investigation on the Green's functions for fractional boundary value problems is still in the initial stage.Recently, based on the spectral theory, the authors in [21] give an associated Green's function for BVP (1) (2) as series of functions.This idea was also used in [22][23][24].
In the next section, we will study some new sharper upper and lower estimates for the Green's function of BVP (1) (2) than the ones given in [21].In Section 3, we employ the new estimate to obtain the existence of a positive solution of BVP (1) (2).The idea of this paper may trace to [21][22][23][24][25][26][27].
It is well known that the function  0 (, ) is Green's function for BVP (1) (2) with () ≡ 0. In the following lemma we present some properties of Green's function  0 (, ), see [28] for details.Listed properties will be used later for estimating the upper bound and the lower bound on Green's function (, ) of BVP (1) (2).
The uniform convergence of (6) follows from the fact that ‖‖ < 1, where the operator  is defined by the following form: Indeed, the uniform convergence of ( 6) can be obtained by () < 1 (see [21,29]), where () is the spectral radius of .
Similar to the proof of Lemmas 2 and 3, we can obtain the following results.
By the properties of definite integral and  ∈ (2, 3), we assert that that is Thus, we obtain that This means that Lemmas 2-5 is more general and complements many known results.
Combining Lemmas 1 and 3, we obtain the following result.
Theorem 8. Assume that there exist and where Then BVP (1) (2) has at least one positive solution in Ω.
For any given  ∈ Ω, by ( 24) and ( 25), we conclude that and Therefore, (Ω) ⊂ Ω.By Schauder's fixed point theorem,  has a fixed point  in Ω which implies that BVP (1) (2) has at least one positive solution in Ω.
The following corollaries are direct results of Theorem 8.

Corollary 9.
Assume that there exist Let (, ) = √.It is easy to see that (30) and (31) hold when  1 is small enough and  2 is large enough.Then, by Corollary 9, BVP (34) has at least one solution.