A Study of Nonlinear Fractional q-Difference Equations with Nonlocal Integral Boundary Conditions

and Applied Analysis 3 Proof. Using Lemma 5, we canwrite the solution of fractional q-difference equation in (13) as x (t) = ∫ t


Introduction
Several kinds of boundary value problems of fractionalorder have recently been investigated by many researchers. Fractional derivatives appear naturally in the mathematical modelling of dynamical systems involving fractals and chaos. In fact, the concept of fractional calculus has played a key role in improving the work based on integer-order (classical) calculus in several diverse disciplines of science and engineering. This might have been due to the fact that fractionaldifferential operators help to understand the hereditary phenomena in many materials and processes in a better way than the corresponding integer-order differential operators. Examples include physics, chemistry, biology, biophysics, blood flow phenomena, control theory, signal and image processing, and economics [1][2][3][4]. For some recent results on the topic, see a series of papers [5][6][7][8][9][10][11][12] and the references therein.
The purpose of the present paper is to study the following nonlocal boundary value problem of nonlinear fractionaldifference equations: where ∈ ([0, 1] × R, R), is the fractional -derivative of the Caputo type, and is a real number.
The paper is organized as follows. Section 2 contains some necessary background material on the topic, while the main results are presented in Section 3. We make use of Banach's contraction principle, Krasnoselskii's fixed point theorem, and Leray-Schauder nonlinear alternative to establish the existence results for the problem at hand.

Preliminaries on Fractional -Calculus
Here we recall some definitions and fundamental results on fractional -calculus.
Definition 3 (see [29]). The fractional -derivative of the Riemann-Liouville type of order ≥ 0 is defined by ( 0 )( ) = ( ) and where [ ] is the smallest integer greater than or equal to .
Definition 4 (see [29]). The fractional -derivative of the Caputo type of order ≥ 0 is defined by where [ ] is the smallest integer greater than or equal to . Now we state some known results involving -derivatives and -integrals.

Lemma 7.
Let ℎ ∈ ([0, 1], R) be a given function. Then the unique solution of the boundary value problem, is given by Proof. Using Lemma 5, we can write the solution of fractional -difference equation in (13) as Using the boundary conditions of (13) in (16), we get 1 = 0 and where 1 is given by (15). Substituting the values of 0 , 1 in (16), we obtain (14).
By virtue of Lemma 7, we define an operator G : C → C as and note that the given problem (2) has solutions only if the operator equation G = has fixed points.

Main Results
In the sequel, we assume that For computational convenience, we introduce the notations: where Our first existence result is based on Leray-Schauder nonlinear alternative. Lemma 8 (nonlinear alternative for single valued maps [30]). Let be a Banach space, a closed, convex subset of , and an open subset of with 0 ∈ . Suppose that G : → is a continuous, compact (i.e., G( ) is a relatively compact subset of ) map. Then either (i) G has a fixed point in , or (ii) there is ∈ (the boundary of in ) and ∈ (0, 1) with = G( ).
It is obvious that the right hand side of the above inequality tends to zero independently of ∈ as 2 → 1 . Therefore the operator G is completely continuous by the Arzelá-Ascoli theorem.
Thus the operator G satisfies the hypothesis of Lemma 8 and hence by its conclusion, either condition (i) or condition (ii) holds. We claim that the conclusion (ii) is not possible. Let = { ∈ ([0, 1], R) : ‖ ‖ < } with given by (21). Then we will show that ‖G ‖ < . Indeed, by means of ( 4 ), we get Assume that there exist ∈ and ∈ (0, 1) such that = G . Then for such a choice of and , we get a contradiction: Thus it follows by Lemma 8 that G has a fixed point ∈ which is a solution of the problem (2). This completes the proof.
Our next result deals with existence and uniqueness of solutions for the problem (2) and is based on Banach's fixed point theorem.
Theorem 11. Suppose that the assumption ( 1 ) holds and that Λ < 1, where Λ is given by (19). Then the boundary value problem (2) has a unique solution.
Proof. Let us consider the set = { ∈ C : ‖ ‖ ≤ }, where is given by and introduce the operators G 1 and G 2 on as In order to show the hypothesis of Krasnoselskii's fixed point theorem, we proceed as follows.
(i) For , ∈ , we find that This implies that G 1 + G 2 ∈ .
(ii) From the continuity of , it follows that the operator G 1 is continuous. Also, G 1 is uniformly bounded on as .
Abstract and Applied Analysis 7 Next, for any ∈ , and 1 , 2 ∈ [0, 1] with 1 < 2 , we have which is independent of and tends to zero as 2 → 1 . Thus, G 1 is equicontinuous. So G 1 is relatively compact on . Hence, by the Arzelá-Ascoli theorem, G 1 is compact on .
(iii) From ( 1 ) and (33) it follows that G 2 is a contraction mapping.
Thus all the conditions of Lemma 13 are satisfied. Hence, by the conclusion of Lemma 13, the problem (2) has at least one solution on [0, 1].
As a special case, for ( ) ≡ 1, there always exists a positive such that (34) holds true. In consequence, we have the following corollary.
all the conditions of Corollary 12 are satisfied. Therefore, the conclusion of Corollary 12 applies to the problem (41).