Existence of positive solutions for the fractional q-difference boundary value problem

In this paper, we investigate the existence of positive solutions for a class of fractional boundary value problems involving q-difference. By using the fixed point theorem of cone mappings, two existence results are obtained. Examples are given to illustrate the abstract results.

Theorem A Let τ = q n with n ∈ N. Suppose that f (t, u) is a nonnegative continuous function on [0, 1] × R + . If there exist two positive constants r 2 > r 1 > 0 such that the function f satisfies (P1) β [α-1] q + M max (t,u)∈[0,1]×[0,r 1 ] f (t, u) ≤ r 1 ; qs) is the Green's function which will be specified later, then the BVP (1.1) has a solution satisfying u(t) > 0 for t ∈ (0, 1]. Clearly, the conditions (P1) and (P2) are strong in application. In 2015, Li et al. [9] studied a class of fractional Schrödinger equations with q-difference of the form where ρ(t) is the trapping potential, n is the mass of a particle, is the Planck constant, ℵ is the energy of a particle. Let λ = n and h(t) = ℵρ(t). They transformed Eq. (1.2) to subject to the boundary conditions By applying a fixed point theorem in cones, they proved several theorems for the existence of positive solutions of the problem (1.3)-(1.4). Here, we just list two important results of [9].
Theorem B Suppose that (H1) and one of (H2) and (H3) hold, where where r > 0 is constant.
In the present work, we consider the fractional boundary value problem (Fr-BVP) with q-difference of the form , f may be singular at t = 0 and/or 1. Here, δ(t) is a scaling function of u in the nonlinearity f .
For the sake of simplicity, denote Throughout this paper, we always assume that the functions f and ω satisfy the following conditions.
(A1) ω ∈ C[0, 1] and there exists ξ > 0 such that ω(t) ≥ ξ for t ∈ I; δ m ) > 0 for any t ∈ I and there exist constants σ 1 ≥ σ 2 > 1 such that, for every τ ∈ (0, 1], for any t ∈ I and x ∈ R + ; (A4) f (t, δ m ) > 0 for any t ∈ I and there exist constants 0 < σ 3 ≤ σ 4 < 1 such that, for every τ ∈ (0, 1], for any t ∈ I and x ∈ R + . (1.10) (3) The condition (1.9) is equivalent to By using the fixed point theorem of cone mappings, we obtain the following theorems. The rest of this paper is organized as follows. In Sect. 2 we introduce some preliminaries and notations which are useful in our proof. In Sect. 3, we will prove Theorems 1.1 and 1.2. Examples are given in Sect. 4 to illustrate the abstract results.

Preliminaries
In this section, we introduce some definitions and notations on fractional q-difference equations. Some related lemmas are also given in this section. For q ∈ (0, 1) and a, b, α ∈ R, we denote abq n abq n+α .
The q-analogue of the power function (ab) n is defined by The q-gamma function is given by and it satisfies Γ q (α + 1) = [α] q Γ q (α).
Let be a function defined on [0, 1]. The q-derivative of is The q-derivative of of high order is given by The following definitions of fractional q-calculus are cited from [3].

Definition 2.1
The fractional q-integral of the Riemann-Liouville type of order α ≥ 0 for the function is defined by

Definition 2.2
The fractional q-derivative of the Riemann-Liouville type of order α ≥ 0 for the function is defined by where m := α is the smallest integer greater than or equal to α.
We refer the reader to the papers [3,10] and the monographs [1,2] for more details on the definitions of fractional q-calculus. In is the Green's function of the linear Fr-BVP (2.1).

Lemma 2.2 ([3]) The Green's function G(t, qs) has the following properties:
Let η ∈ (0, 1). Define a cone K in E by Then K is a nonempty closed convex cone of E. Proof For fixed u ∈ E with u(t) ≥ 0 for all t ∈ I * , choosing a constant a ∈ (0, 1) such that a u < 1. Then, for any t ∈ I * , by (1.8) and (1.10), we have So, for any t ∈ I * , by (2.2), we have This implies that the operator Q : K → E is well defined. By
Hence, Q : K → K . By the Ascoli-Arzela theorem, one can prove that Q : K → K is completely continuous.
At last, we state a fixed point theorem of cone mapping to end this section, which is useful in the proof of our main results. Lemma 2.6 ( [5]) Let E be a Banach space, P ⊂ E a cone in E. Assume that Ω 1 and Ω 2 are two bounded and open subset of E with θ ∈ Ω 1 , Ω 1 ⊂ Ω 2 . If Q : P ∩ (Ω 2 \ Ω 1 ) → P is a completely continuous operator such that either (i) Qu ≤ u , ∀u ∈ P ∩ ∂Ω 1 and Qu ≥ u , ∀u ∈ P ∩ ∂Ω 2 , or (ii) Qu ≥ u , ∀u ∈ P ∩ ∂Ω 1 and Qu ≤ u , ∀u ∈ P ∩ ∂Ω 2 , Then Q has at least one fixed point in P ∩ (Ω 2 \ Ω 1 ).

Proof of the main results
In this section, we will apply Lemma 2.6 to prove the existence of positive solutions of the Fr-BVP (1.7). For any 0 < r < R, let Proof of Theorem 1.1 On the one hand, defining an operator Q : K → E as in (2.3), we prove that there exists a constant r ∈ (0, 1] such that In fact, for u ∈ K with u ≤ 1, we have So, by Lemma 2.2, we have 1 σ 2 -1 , then r ∈ (0, 1). For any u ∈ K ∩ ∂Ω r , we have If β 1 ≤ 1, choosing r = 1, then, for any u ∈ K ∩ ∂Ω r , we have On the other hand, we prove that there exists a constant R > 1 such that In fact, for u ∈ K with u(t) ≥ 1 for t ∈ I * , we have Thus, for every u ∈ K ∩ ∂Ω R , by Lemma 2.5, we have Qu ∈ K and, for any η ∈ (0, 1), Hence, by Lemma 2.6, Q has at least one fixed point u * ∈ K ∩ (Ω R \ Ω r ) satisfying 0 < r ≤ u * ≤ R. Hence for η ∈ (0, 1), min t∈[η,1] u * (t) ≥ η α-1 u * > 0 and it is a positive solution of the Fr-BVP (1.7).