Existence Theory for q-Antiperiodic Boundary Value Problems of Sequential q-Fractional Integrodifferential Equations

and Applied Analysis 3 Lemma 4. For a given h ∈ C([0, 1],R), the boundary value problem c D α q ( c D γ q +λ) x (t) = h (t) , 0 ≤ t ≤ 1, 0 < q < 1, x (0) = −x (1) , (t (1−γ) D q x (t)) 󵄨󵄨󵄨󵄨t=0 = −D q x (1) (15) is equivalent to the q-integral equation x (t) = ∫ t

The aim of the present study is to establish some existence and uniqueness results for the problem (1) by means of Krasnoselskii's fixed-point theorem, Leray-Schauder nonlinear alternative, and Banach's contraction principle. Though the tools employed in this work are standard, yet their exposition in the framework of the given problem is new.
Fractional calculus has developed into a popular mathematical modelling tool for many real world phenomena occurring in physical and technical sciences, see, for example, [1][2][3][4]. A fractional-order differential operator distinguishes itself from an integer-order differential operator in the sense that it is nonlocal in nature and can describe the memory and hereditary properties of some important and useful materials and processes. This feature has fascinated many researchers and several results ranging from theoretical analysis to asymptotic behavior and numerical methods for fractional differential equations have been established. For some recent work on the topic, see [5][6][7][8][9][10][11][12] and references therein.
The mathematical modeling of linear control systems, concerning the controllability of systems consisting of a set of well-defined interconnected objects, is based on the linear systems of divided difference functional equations. The controllability in mathematical control theory studies the concepts such as controllability of the state, controllability of the output, controllability at the origin, and complete controllability. The -difference equations play a key role in the control theory as these equations are always completely controllable and appear in the -optimal control problem [13]. The variational -calculus is known as a generalization of the continuous variational calculus due to the presence of an extra-parameter whose nature may be physical or economical. The study of the -uniform lattice rely on the -Euler equations. In other words, it suffices to solve the -Euler-Lagrange equation for finding the extremum of the functional involved instead of solving the Euler-Lagrange equation [14]. One can find more details in a series of papers [15][16][17][18][19][20][21].

Abstract and Applied Analysis
The subject of fractional -difference ( -fractional) equations is regarded as fractional analogue of -difference equations and has recently gained a considerable attention. For examples and details, we refer the reader to the works [22][23][24][25][26][27][28][29][30][31][32][33] and references therein while some earlier work on the subject can be found in [34][35][36]. The present work is motivated by recent interest in the study of fractional-order differential equations.

Preliminaries on Fractional -Calculus
Let us describe the notations and terminology forfractional calculus [35].
Definition 1 (see [35]). Let be a function defined on [0, 1]. The fractional -integral of the Riemann-Liouville type of order ≥ 0 is ( 0 )( ) = ( ) and Observe that the above -integral reduces to the following one for = 1.
Further details of -integrals and fractional -integrals can be found respectively in Section 1.3 and Section 4.2 of the text [35].

Remark 2.
The semigroup property holds for -fractional integration (Proposition 4.3 [35]): Further, it has been shown in Lemma 6 of [37] that Before giving the definition of fractional -derivative, we recall the concept of -derivative.
Let be a real valued function defined on a -geometric set (| | ̸ = 1). Then the -derivative of a function is defined as For 0 ∈ , the -derivative at zero is defined for | | < 1 by Definition 3 (see [35]). The Caputo fractional -derivative of order > 0 is defined by where ⌈ ⌉ is the smallest integer greater than or equal to .
Next we enlist some properties involving Riemann-Liouville -fractional integral and Caputo fractionalderivative (Theorem 5.2 [35]): Now we establish a lemma that plays a key role in the sequel.

Lemma 4. For a given ℎ ∈ ([0, 1], R), the boundary value problem
is equivalent to the -integral equation Proof. It is well known that the solution of -fractional equation in (15) can be written as Differentiating (17), we obtain Using the boundary conditions (15) in (17) and (18) and solving the resulting expressions for 0 and 1 , we get Substituting the values of 0 and 1 in (17) yields the solution (16). The converse follows in a straightforward manner. This completes the proof.
In view of Lemma 4, we define an operator U : C → C as Observe that the problem (1) has solutions only if the operator equation = U has fixed points.

Main Results
For the forthcoming analysis, the following conditions are assumed.
For computational convenience, we set Our first existence result is based on Krasnoselskii's fixed point theorem. Proof. Consider the set = { ∈ C : ‖ ‖ ≤ }, where is given by Define operators U 1 and U 2 on as For , ∈ , we find that Thus, U 1 + U 2 ∈ . Continuity of and imply that the operator U 1 is continuous. Also, U 1 is uniformly bounded on as Now, we prove the compactness of the operator U 1 . In view of ( 1 ), we define Consequently, for 1 , 2 ∈ [0, 1], we have which is independent of and tends to zero as 2 → 1 . Thus, U 1 is relatively compact on . Hence, by the Arzelá-Ascoli Theorem, U 1 is compact on . Now, we shall show that U 2 is a contraction. From ( 1 ) and for , ∈ , we have )] where we have used (22). In view of the assumption Λ < 1, the operator U 2 is a contraction. Thus, all the conditions of Lemma 5 are satisfied. Hence, by the conclusion of Lemma 5, the problem (1) has at least one solution on [0, 1].
Our next result is based on Leray-Schauder nonlinear alternative.

Lemma 7 (nonlinear alternative for single valued maps, see [39]). Let be a Banach space, a closed, convex subset of
, an open subset of , and 0 ∈ . Suppose that U : → is a continuous, compact (i.e., U( ) is a relatively compact subset of ) map. Then either (i) U has a fixed point in , or (ii) there is a ∈ (the boundary of in ) and ∈ (0, 1) with = U( ). ( 4 ) there exists a constant > 0 such that Then the boundary value problem (1) has at least one solution on [0, 1].
Proof. Consider the operator U : C → C defined by (20). The proof consists of several steps.
(i) It is easy to show that U is continuous.

Abstract and Applied Analysis 7
For a positive number , let = { ∈ C : ‖ ‖ ≤ } be a bounded set in ([0, 1] × R) and ∈ . Then, we have Abstract and Applied Analysis This shows that U ∈ .
Let 1 , 2 ∈ [0, 1] with 1 < 2 and ∈ , where is a bounded set of ([0, 1], R). Then, we obtain Obviously the right-hand side of the above inequality tends to zero independently of ∈ as 2 − 1 → 0. Therefore, it follows by the Arzelá-Ascoli theorem that U : C → C is completely continuous.
(iv) Let be a solution of the given problem such that = U for ∈ (0, 1). Then, for ∈ [0, 1], it follows by the procedure used to establish (ii) that Consequently, we have In view of ( 4 ), there exists such that ‖ ‖ ̸ = . Let us set Abstract and Applied Analysis 9 Note that the operator U : → ([0, 1], R) is continuous and completely continuous. From the choice of , there is no ∈ such that = U( ) for some ∈ (0, 1). In consequence, by the nonlinear alternative of Leray-Schauder type (Lemma 7), we deduce that Uhas a fixed point ∈ which is a solution of the problem (1). This completes the proof.
Finally we show the existence of a unique solution of the given problem by applying Banach's contraction mapping principle (Banach fixed-point theorem).
Example 11. Consider the following -fractionalantiperiodic boundary value problem: