Nontrivial Solutions for a System of Fractional q -Difference Equations Involving q -Integral Boundary Conditions

: In this paper, we study the existence of nontrivial solutions for a system of fractional q -difference equations involving q -integral boundary conditions, and we use the topological degree to establish our main results by considering the ﬁrst eigenvalue of some associated linear integral operators.

Motivated by the mentioned works above, in this paper we use topological degree theory to study nontrivial solutions for the following system of fractional q-difference equations with q-integral boundary conditions: where α ∈ (2, 3), v ∈ (1, 2), D α q is the α-order Riemann-Liouville's fractional q-derivative. Now, we list our assumptions for h, f i (i = 1, 2): Hypothesis 2 (H2). f i ∈ C ([0, 1] × R, R).
Finally, we state our main result in this paper:

Preliminaries
Let q ∈ (0, 1) and define The q-analogue of the power function (a − b) n with n ∈ N 0 is Definition 1 (see [3], Definition 2.2). Let α ≥ 0 and f be a function defined on [0, 1]. The fractional q-integral of the Riemann-Liouville type is I 0 q f (x) = f (x) and where m is the smallest integer greater or equal than α.
Lemma 1 (see [3], Lemma 2.3). Let α, β ≥ 0 and f be a function defined on [0, 1]. Then, the next formulas hold: Lemma 2 (see [3], Theorem 2.4). Let α > 0 and p be a positive integer. Then, the following equality holds: has a unique solution Proof. Using Definition 2 and Lemmas 2 and 3 we have Note from Hence, Consequently, we have This, together with (H1), implies that Thus, we have This completes the proof.  7)). The functions G i (i = 0, 1) has the following properties Lemma 5. The function G satisfies and G(t, qs) ≤ t α−1 ϕ 2 (qs), for t, s ∈ [0, 1], where This is the direct result from Lemma 2.5, so we omit the proof.
Then (E, · ) is a real Banach space and P is a cone on E. Moreover, E 2 = E × E is a Banach space with the norm (u, v) = u + v , and P 2 = P × P is a cone on E 2 . From Lemma 2.4 we can define operators T i (i = 1, 2) : E → E, and T : E 2 → E 2 as follows: where G is determined in Lemma 3. Please note that T i (i = 1, 2) and T are completely continuous operators, and (x, y) solves (8) if and only if (x, y) is a fixed point of the operator T. In addition, from Lemma 3 we can obtain that (9) is equivalent to For our purposes, we need to define the operator L by It is not difficult to prove that L : E → E is a linear completely continuous and T(P) ⊂ P. From Lemmas 2 and 3 in [7] we obtain that the spectral radius, denoted by r(L), is not equal to 0, and L has a positive eigenfunction ϕ * corresponding to its first eigenvalue λ 1 = (r(L)) −1 , i.e., ϕ * = λ 1 Lϕ * . Lemma 6. Let P 0 = {x ∈ P : x(t) ≥ t α−1 x , ∀t ∈ [0, 1]}. Then L(P) ⊂ P 0 .
Proof. If x ∈ P, and from (11) we have This completes the proof.
We recall the following topological degree theorems, which will play important roles in proving our main results.
Now (15) and (28)  Therefore the operator T has at least one fixed point in B R 1 \B r 1 . Equivalently, (8) has at least one nontrivial solution. This completes the proof.

Conclusions
In this paper, we use topological degree to study nontrivial solutions for the system of fractional q-difference Equation (8) with q-integral boundary conditions. There are only a few papers in the literature which consider systems of fractional q-difference equations with q-integral boundary conditions where the nonlinear terms may be unbounded from below. Our main theorem is obtained under some conditions concerning the first eigenvalues corresponding to the relevant linear operators. As a result, our main result generalizes and improves the corresponding ones in the works cited in this paper.