Multiple Positive Solutions of a Singular Semipositone Integral Boundary Value Problem for Fractional q-Derivatives Equation

and Applied Analysis 3 The q-integral of a function f defined in the interval [0, b] is given by

The fractional -difference calculus had its origin in the works by Al-Salam [36] and Agarwal [37]. More recently, perhaps due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional -difference calculus were made, specifically,analogues of the integral and differential fractional operators properties such as the Mittag-Leffler function, the -Laplace transform, and -Taylor's formula [3,13,22,38], just to mention some.
However, the theory of boundary value problems for nonlinear -difference equations is still in the initial stage and many aspects of this theory need to be explored.
Recently, Liang and Zhang [20] discussed the following nonlinear -fractional three-point boundary value problem: ( ) ( ) + ( , ( )) = 0, 0 < < 1, 2 < ≤ 3, By using a fixed point theorem in partially ordered sets, the authors obtained sufficient conditions for the existence and uniqueness of positive and nondecreasing solutions to the above boundary value problem.
In [21], Graef and Kong investigated the following boundary value problem with fractional -derivatives: where ≥ 0 is a parameter, and the uniqueness, existence, and nonexistence of positive solutions are considered in terms of different ranges of . Furthermore, Ahmad et al. [11] studied the following nonlinear fractional -difference equation with nonlocal boundary conditions ( ) ( ) = ( , ( )) , 0 ≤ ≤ 1, 1 < ≤ 2, where is the fractional -derivative of the Caputo type, and , , , ∈ R. The existence of solutions for the problem is shown by applying some well-known tools of fixed point theory such as Banach's contraction principle, Krasnoselskii's fixed point theorem, and the Leray-Schauder nonlinear alternative.
It is known that the fractional -derivative of Riemann-Liouville type played an important role in the development of the theory of fractional -derivatives and -integrals and for its applications in pure mathematics, and is a useful tool in the description of nonconservative models. The details can be found in [22].
Since -calculus has a tremendous potential for applications [3,22], we find it pertinent to investigate problems in this field. Motivated by the papers [20,21,30], we will deal with the following integral boundary value problem of nonlinear fractional -derivatives equation: ( ) ( ) + ( , ( )) = 0, ∈ (0, 1) , 2 < ≤ 3, subject to the boundary conditions where ∈ (0, 1), is parameter with 0 < < [ ] , is the -derivative of Riemann-Liouville type of order , : [0, 1] × R + 0 → R is continuous and semipositone and may be singular at = 0, in which R + 0 = (0, +∞), R = (−∞, +∞). In the present work, we investigate the existence of positive solutions for fractional -derivatives integral boundary value problem (7) and (8) involving the Riemann-Liouville's fractional derivative, which is different from [11]. We gave the corresponding Green's function of the boundary value problem (7) and (8), gave some properties of Green's function, and constructed a cone by properties of Green's function. Moreover the existence of at least two and three positive solutions to the boundary value problem (7) and (8) is enunciated.

Preliminaries on -Calculus and Lemmas
For the convenience of the reader, below we recall some known facts on fractional -calculus. The presentation here can be found in, for example, [1,3,12,19,21,22].
Let ∈ (0, 1) and define The -analogue of the power function ( − ) with ∈ N 0 := {0, 1, 2, . . .} is More generally, if ∈ R, then Clearly, if = 0 then ( ) = . The -gamma function is defined by and satisfies Γ ( The -derivative of a function is defined by and the -derivatives of higher order by Abstract and Applied Analysis 3 The -integral of a function defined in the interval [0, ] is given by If ∈ [0, ] and is defined in the interval [0, ], then its integral from to is defined by Similar to that for derivatives, an operator is given by The fundamental theorem of calculus applies to these operators and , that is, and if is continuous at = 0, then The following formulas will be used later, namely, the integration by parts formula: where denotes the derivative with respect to the variable .
The fractional -derivative of order ≥ 0 is defined by  Then, the following formulas hold: Lemma 5 (see [17]). Let > 0 and be a positive integer. Then, the following equality holds: Then the unique solution of the equation subject to BC (8) is given by where Proof. Let us put = 3. In view of Definition 1 and Lemma 4, we see that Then, it follows from Lemma 5 that the solution ( ) of (25) and BC (8) is given by for some constants 1 , 2 , 3 ∈ R. Since (0) = 0, we have 3 = 0. Differentiating both sides of (29) and with the help of (20) and (22), we obtain Then by the boundary conditions ( )(0) = 0, we get 2 = 0. Thus, (29) reduces to Using the boundary condition (1) = ∫ 1 0 ( ) , we get Hence, we have Integrate the above equation (33) from 0 to 1, and using (11), (19) and (20), we obtain then Combining this with (29) and (31) yields This completes the proof of the lemma.
Proof. The continuity of is easily checked. On the other hand, when 0 ≤ ≤ ≤ 1, in view of Lemma 3, we have Abstract and Applied Analysis 5 Further, since we have we get When 0 ≤ ≤ ≤ 1, since < [ ] , we have This completes the proof of the lemma.

The Main Results
In order to abbreviate our discussion, we give the following assumptions.