EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTION FOR NONLINEAR FRACTIONAL Q-DIFFERENCE EQUATION WITH INTEGRAL BOUNDARY CONDITIONS∗

This paper studies a class of nonlinear fractional q-difference equations with integral boundary conditions. By exploiting the properties of Green’s function and two fixed point theorems for a sum operator, the existence and uniqueness of positive solutions for the boundary value problem are established. Iterative schemes for approximating the solutions are also obtained. Explicit examples are given to illustrate main results.


Introduction
In this paper, we consider an integral boundary value problem of fractional qdifference equations given by Nonlinear fractional q-difference equations appear in the mathematical modeling of many phenomena in engineering and science and have attracted much attention during the past decades (see, for example, [6-8, 12, 13, 19]). Boundary value problems involving fractional q-difference equations have been studied by applying different methods (for instance, see [2-5, 9-11, 14, 16, 18, 22, 23] and the references therein). Ahmad etc [3][4][5] investigated existence of solutions for nonlinear fractional q-difference equation with different boundary conditions by some classical fixed point theorems. Etemad etc [10] studied the existence of solutions for a new class of fractional q-integro-difference equation involving Riemann-Liouville q-derivatives and a q-integral of different orders, supplemented with boundary conditions containing q-integrals of different orders. Li and Yang [16] investigated the existence of positive solutions and two iterative schemes approximating the solutions for a class of nonlinear fractional q-difference equations with integral boundary conditions by applying monotone iterative method. Zhao and Yang [22] obtained sufficient conditions for the existence and uniqueness of solutions for a singular coupled integral boundary value problem of nonlinear higher-order fractional q-difference equations by using a mixed monotone method and Guo-Krasnoselskii fixed point theorem. Motivated by aforementioned works, we obtain the existence and uniqueness of positive solutions for the problem (1.1) by using the method of [11]. We also present sequences approximating a unique solution to the given problem.

Preliminarie
For the convenience, we collect here the necessary definitions from the theory of fractional q-calculus. Let q ∈ (0, 1) and define The q-analogue of the power function ( More generally, if α ∈ R, then The q-gamma function is defined by The following expression is called the q-derivative of the function f (x). D q has the following properties and formulas: , The q-integral of a function f defined on the interval [0, b] is given by Definition 2.2 (Rajković etc [17]). The fractional q-derivative of the Riemann-Liouville type of order α 0 is defined by where m is the smallest integer greater than or equal to α.
Proof. By integrating the two sides on In view of Definition 2.2, we deduce By applying Definition 2.3 and Lemma 2.1, we have It follows from (2.2)-(2.4) and Definition 2.1, we have are constants to be determined. Using the boundary conditions given by (1.1) in (2.5), we find that c 3 = 0, c 2 = 0 and Therefore, it follows that Finally, in order to solve the problem (1.1), it is sufficient to find the solution of the following integral equation: [22]). The Green's function G 1 (t, qs) defined in Lemma 2.2 satisfies the following properties: Lemma 2.4. The Green's function G(t, qs) defined in Lemma 2.2 satisfies the following inequality Proof. By Lemma 2.3, we have On the other hand, from the expression of G 1 (t, qs), it is obvious that Definition 2.4 (Guo [15]). Let E be a real Banach space. A nonempty convex closed set P is called a cone provided that: (1) au ∈ P , for all u ∈ P ; a 0; (2) u, −u ∈ P implies u = 0.
For all x, y ∈ E, the notation x ∼ y means that there exist λ > 0 and µ > 0 such that λx y µx. Clearly, ∼ is an equivalence relation. Given h > θ(i.e., h θ and h = θ), we denote by P h the set P h = {x ∈ E : x ∼ h} . It is easy to see that P h ⊂ P .
Definition 2.5 (Guo [15]). Let γ be a real number with 0 < γ < 1. An operator A : P → P is said to be γ-concave if it satisfies A(tx) > t γ Ax for all t ∈ (0, 1), x ∈ P . An operator A : E → E is said to be homogeneous if it satisfies A(tx) = tAx for all t > 0, x ∈ E. An operator A : P → P is said to be sub-homogeneous if it satisfies A(tx) tAx for all t > 0, x ∈ P .
Theorem 2.1 (Theorem 2.2, [21]). Let P be a normal cone in a real Banach space E, A : P → P be an increasing γ-concave operator, and B : P → P be an increasing sub-homogeneous operator. Assume that (i) there is h > θ such that Ah ∈ P h and Bh ∈ P h ; (ii) there exists a constant δ 0 > 0 such that Ax δ 0 Bx for all x ∈ P . Then the operator equation Ax + Bx = x has a unique solution x * in P h . Moreover, constructing successively the sequence y n = Ay n−1 + By n−1 , n = 1, 2, · · · , for any initial value y 0 ∈ P h , we have y n → x * as n → ∞.
Theorem 2.2 (Theorem 2.1, [20]). Let P be a normal cone a real Banach space E, A : P → P be an increasing operator, and B : P → P be a decreasing operator. Assume that: (i) for any x ∈ P and t ∈ (0, 1), there exist ϕ i (t) ∈ (t, 1)(i = 1, 2) such that Bx; (ii) there exists h 0 ∈ P h such that Ah 0 + Bh 0 ∈ P h . Then the operator equation Ax + Bx = x has a unique solution x * in P h . Moreover, for any initial values x 0 , y 0 ∈ P h , constructing successively the sequences x n = Ax n−1 + By n−1 , y n = Ay n−1 + Bx n−1 ; n = 1, 2, · · · , we have x n → x * , y n → x * as n → ∞.

Main results and proofs
Consider the Banach space E = C[0, 1] with the norm x = sup{|x(t)| : t ∈ [0, 1]}. Define the cone on E: P = {x ∈ E : x(t) 0, t ∈ [0, 1]}, then P is a normal cone in E and the normality constant is 1. E can be equipped with a partial order given by x y, x, y ∈ E ⇔ x(t) y(t), t ∈ [0, 1].
In order to solve the q-analogue of the fractional differential problem (1.1), it is sufficient to find positive solutions of the following integral equation: qs)[f (s, u(s)) + g(s, u(s))]d q s.  Second, from (H2), for γ ∈ (0, 1) and u ∈ P , we have

Example
Then by Theorem 3.2, problem (4.2) has a unique positive solution in P h , where h(t) = t

Conclusions
We have discussed the existence and uniqueness of positive solutions for a class of nonlinear fractional q-difference equations with integral boundary conditions by two fixed point theorems of a sum operator in partial ordering Banach space. We also present sequences approximating a unique solution to the given problem. In particular, we skillfully use a sum operator to solve the inconsistency of monotonicity of nonlinear terms. In other words, our results do not require super-linearity, sublinearity or boundness of nonlinear terms. We yield several new results to boundary value problems of fractional q-difference equations.