Positive Solutions for Nonlinear Caputo Type Fractional q-Difference Equations with Integral Boundary Conditions

Since Al-Salam [1] and Agarwal [2] introduced the fractional q-difference calculus, the theory of fractional q-difference calculus itself and nonlinear fractional q-difference equation boundary value problems have been extensively investigated by many researchers. For some recent developments on fractional q-difference calculus and boundary value problems of fractional q-difference equations, see [3–16] and the references therein. For example, authors [17–20] considered some anti-periodic boundary value problems of nonlinear fractional q-difference equations. By applying the generalized Banach contraction principle, the monotone iterative method, and the Krasnoselskii’s fixed point theorem. In [21], the authors investigated Caputo q-fractional initial value problems independently of the paper [3] where some open problems raised there. In [22], Mittag-Leffler stabilitry of q-fractional systems was investigated. In [23,24], some important q-fractional inequalities were proved. Those inequalities are necessary for the development of q-fractional systems. Zhao et al. [25] showed some existence results of positive solutions to nonlocal q-integral boundary value problems of nonlinear fractional q-derivative equations. Under different conditions, Graef and Kong [26,27] investigated the existence of positive solutions for boundary value problems with fractional q-derivatives in terms of different ranges of λ, respectively. By applying some standard fixed point theorems, Agarwal et al. [28] and Ahmad et al. [29] showed some existence results for sequential q-fractional integrodifferential equations with q-antiperiodic boundary conditions and nonlocal four-point boundary conditions, respectively. In [30], by applying a mixed monotone method and the Guo-Krasnoselskii fixed point theorem, Zhao and Yang obtained the existence and uniqueness of positive solutions for the singular coupled integral boundary value problem of nonlinear higher-order fractional q-difference equations.


Introduction
Since Al-Salam [1] and Agarwal [2] introduced the fractional q-difference calculus, the theory of fractional q-difference calculus itself and nonlinear fractional q-difference equation boundary value problems have been extensively investigated by many researchers.For some recent developments on fractional q-difference calculus and boundary value problems of fractional q-difference equations, see [3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein.For example, authors [17][18][19][20] considered some anti-periodic boundary value problems of nonlinear fractional q-difference equations.By applying the generalized Banach contraction principle, the monotone iterative method, and the Krasnoselskii's fixed point theorem.In [21], the authors investigated Caputo q-fractional initial value problems independently of the paper [3] where some open problems raised there.In [22], Mittag-Leffler stabilitry of q-fractional systems was investigated.In [23,24], some important q-fractional inequalities were proved.Those inequalities are necessary for the development of q-fractional systems.Zhao et al. [25] showed some existence results of positive solutions to nonlocal q-integral boundary value problems of nonlinear fractional q-derivative equations.
Under different conditions, Graef and Kong [26,27] investigated the existence of positive solutions for boundary value problems with fractional q-derivatives in terms of different ranges of λ, respectively.By applying some standard fixed point theorems, Agarwal et al. [28] and Ahmad et al. [29] showed some existence results for sequential q-fractional integrodifferential equations with q-antiperiodic boundary conditions and nonlocal four-point boundary conditions, respectively.In [30], by applying a mixed monotone method and the Guo-Krasnoselskii fixed point theorem, Zhao and Yang obtained the existence and uniqueness of positive solutions for the singular coupled integral boundary value problem of nonlinear higher-order fractional q-difference equations.
In [31], Ferreira considered the nonlinear fractional q-difference boundary value problem as follows: where D α q is the q-derivative of Riemann-Liouville type of order α.By applying a fixed point theorem in cones, sufficient conditions for the existence of positive solutions were enunciated.
In [32], Ahmad et al. studied the following nonlocal boundary value problems of nonlinear fractional q-difference equations where c D α q denotes the Caputo fractional q-derivative of order α, and ).The existence of solutions for the problem were shown by applying some well-known tools of fixed point theory such as the Banach contraction principle, the Krasnoselskii fixed point theorem, and the Leray-Schauder nonlinear alternative.
In [33], Zhou and Liu investigated the following fractional q-difference system where c D α q and c D α q denote the Caputo fractional q-derivative of order α and β, respectively.The uniqueness and existence of solution were obtained based on the nonlinear alternative of Leray-Schauder type and Banach's fixed-point theorem.
In [34], the author considered the following coupled integral boundary value problem for systems of nonlinear semipositone fractional q-difference equations where λ, µ, ν are three parameters with 0 < µ < [β] q and 0 < ν < [α] q , α, β ∈ (n − 1, n] are two real numbers and n ≥ 3, D α q , D β q are the fractional q-derivative of the Riemann-Liouville type, and f , g are sign-changing continuous functions.By applying the nonlinear alternative of Leray-Schauder type and the Krasnoselskii's fixed point theorems, sufficient conditions for the existence of one or multiple positive solutions were obtained.
In [35], Li and Yang considered the following nonlinear fractional q-difference equation with integral boundary conditions where α ∈ (n − 1, n] are a real number and n ≥ 3 is an integer, D α q are the fractional q-derivative of the Riemann-Liouville type, µ > 0 and 0 < q < 1 are two constants, g, h are two given continuous functions, and f : . By applying monotone iterative method and some inequalities associated with the Green's function, the existence results of positive solutions and two iterative schemes approximating the solutions were established.
Motivated by the wide applications of coupled boundary value problems and the results mentioned above, we consider the following nonlinear Caputo type fractional q-difference equations with integral boundary conditions where c D α is the Caputo type fractional q-derivative of fractional order α, a, b, c, d are real constants with a > b > 0 and c The main aim of this paper is to investigate the existence of positive solutions for a class of nonlinear Caputo type fractional q-difference equations with integral boundary conditions by means of the Guo-Krasnoselskii fixed point theorem and Leggett-Williams fixed point theorem.Furthermore, some examples are given to illustrate our main results.

Preliminaries
For the convenience of the reader, we present some necessary definitions and lemmas of fractional q-calculus theory to facilitate analysis of the q-fractional boundary value problem (1).These details can be found in the recent literature; see [36,37] and references therein.
Let q ∈ (0, 1) and define The q-analogue of the power (a Note that, if b = 0, then a (α) = a α .Here, we point out that the following equality holds The q-gamma function is defined by The q-derivative of a function f is defined by and q-derivatives of higher order by The q-integral of a function f defined in the interval [0, b] is given by Similarly as done for derivatives, an operator I n q can be defined, namely, The fundamental theorem of calculus applies to these operators I q and D q , i.e., and if f is continuous at x = 0, then Basic properties of the two operators can be found in the book [36].We now point out five formulas that will be used later ( i D q denotes the derivative with respect to variable i) [13].

Definition 2 ([38]
).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.

Definition 3 ([38]
).The fractional q-derivative of the Caputo type of order α ≥ 0 is defined by where m is the smallest integer greater than or equal to α.
For the sake of simplicity, we always assume that the following condition (H1) holds: and Now, we will obtain the Green's function of the boundary value problem (1) and some of its properties.Lemma 5. Let y ∈ C[0, 1], and then the boundary value problem has a unique solution u in the form where Proof.By applying (6) to the boundary conditions au(0) − bu(1) = 1 0 g(s)u(s)d q s and c(D q u)(0) − d(D q u)(1) = 1 0 h(s)u(s)d q s, we obtain Substituting d 1 and d 2 in Equation (10) into the (4), we have Multiplying both sides of the first and second equations of (11) by g(t) and h(t), respectively, and integrating the resulting equations obtained with respect to t from 0 to 1, we obtain Solving for 1 0 g(s)u(s)d q s and 1 0 h(s)u(s)d q s from the above equations, we have Substituting 1 0 g(s)u(s)d q s and 1 0 h(s)u(s)d q s in Equation ( 12) into the (11), we have H(t, qs)x(s)d q s, which implies that (8) has a unique solution (9).This completes the proof of the lemma.

Lemma 6.
The function H(t, s) has the following property: and For given s ∈ (0, 1), g 1 , g 2 are increasing with respect to t for t ∈ [0, 1].Hence, we have min and max Therefore, we get Furthermore, we obtain This completes the proof of the lemma.
Lemma 7. T : P → P is completely continuous.
Proof.The operator T : P → P is continuous in view of continuity of H(t, s) and f (t, u).By means of the Arzela-Ascoli theorem, T : P → P is completely continuous.
In order to obtain the main results in this paper, we will use the following cone compression and expansion fixed point theorem.
In order to state our main results, we need to introduce the following notations.

Main Results
In this section, we establish the existence of positive solutions for boundary value problem (1) by using the Guo-Krasnoselskii fixed point theorem and the Leggett-Williams fixed point theorem.
Theorem 1. Assume (H1) holds.Furthermore, suppose one of the following conditions is satisfied.
(H2) There exist two constants µ and ν with 0 Then, the problem (1) has at least one positive solution.

Conclusions
In this paper, a class of nonlinear Caputo type fractional q-difference equations with integral boundary conditions are studied.By using some well-known fixed point theorems, the existence of one or multiple positive solutions are established for nonlinear Caputo type fractional q-difference equations with integral boundary conditions.Finally, two examples are presented to illustrate the effectiveness of the obtained results.