Nontrivial solutions for fractional q-difference boundary value problems

In this paper, we investigate the existence of nontrivial solutions to the nonlinear q-fractional boundary value problem


Introduction
The q-difference calculus or quantum calculus is an old subject that was first developed by Jackson [9,10]. It is rich in history and in applications as the reader can confirm in the paper [6].
The origin of the fractional q-difference calculus can be traced back to the works by Al-Salam [3] and Agarwal [1]. More recently, perhaps due to the explosion in research within the fractional calculus setting (see the books [13,14]), new developments in this theory of fractional q-difference calculus were made, specifically, q-analogues of the integral and differential fractional operators properties such as q-Laplace transform, q-Taylor's formula [4,15], just to mention some.
To the best of the author knowledge there are no results available in the literature considering the problem of existence of nontrivial solutions for fractional q-difference boundary value problems. As is well-known, the aim of finding nontrivial solutions is of main importance in various fields of science and engineering (see the book [2] and references therein). Therefore, we find it pertinent to investigate on such a demand within this q-fractional setting.
This paper is organized as follows: in Section 2 we introduce some notation and provide to the reader the definitions of the q-fractional integral and differential operators together with some basic properties. Moreover, some new general results within this theory are given. In Section 3 we consider a Dirichlet type boundary value problem. Sufficient conditions for the existence of nontrivial solutions are enunciated.

Preliminaries on fractional q-calculus
Let q ∈ (0, 1) and define The q-analogue of the power function (a − b) n with n ∈ N 0 is More generally, if α ∈ R, then Note that, if b = 0 then a (α) = a α . The q-gamma function is defined by and satisfies Γ q (x + 1) = [x] q Γ q (x). The q-derivative of a function f is here 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, it can be defined an operator I n q , 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 [11]. We point out here four formulas that will be used later, namely, the integration by parts formula and ( i D q denotes the derivative with respect to variable i) Let n ∈ N 0 . We show that Indeed, expanding both sides of the inequality (5) we obtain Since inequality (5) implies inequality (4) we are done with the proof.
The following definition was considered first in [1] where m is the smallest integer greater or equal than α.
Let us now list some properties that are already known in the literature. Its proof can be found in [1,15].
. The next result is important in the sequel. Since we didn't find it in the literature we provide a proof here.
Theorem 2.4. Let α > 0 and p be a positive integer. Then, the following equality holds: Proof. Let α be any positive number. We will do a proof using induction on p.
Suppose that p = 1. Using formula (3) we get: Therefore, Suppose now that (6) holds for p ∈ N. Then, The theorem is proved.
EJQTDE, 2010 No. 70, p. 5 We shall consider now the question of existence of nontrivial solutions to the following problem: subject to the boundary conditions where 1 < α ≤ 2 and f : [0, 1] × R → R is a nonnegative continuous function (this is the q-analogue of the fractional differential problem considered in [5]).
Then T has at least one fixed point in C ∩ (Ω 2 \Ω 1 ).
Let us put p = 2. In view of item 2 of Lemma 2.3 and Theorem 2.4 we see that for some constants c 1 , c 2 ∈ R. Using the boundary conditions given in (8) we take c 1 = 1 Γq(α) 1 0 (1 − qt) (α−1) f (t, y(t))d q t and c 2 = 0 to get If we define a function G by then, the following result follows.
Remark 3.3. If we let α = 2 in the function G, then we get a particular case of the Green function obtained in [16], namely, Some properties of the function G needed in the sequel are now stated and proved.
Lemma 3.4. Function G defined above satisfies the following conditions: Proof. We start by defining two functions g 1 ( It is clear that g 2 (x, qt) ≥ 0. Now, in view of Remark 2.1 we get, Moreover, for t ∈ (0, 1] we have that EJQTDE, 2010 No. 70, p. 7 which implies that g 1 (x, t) is decreasing with respect to x for all t ∈ (0, 1]. Therefore, Now note that G(0, qt) = 0 ≤ G(qt, qt) for all t ∈ [0, 1]. Therefore, by (10) and the definition of g 2 (it is obviously increasing in x) we conclude that G(x, qt) ≤ G(qt, qt) for all 0 ≤ x, t ≤ 1. This finishes the proof.
Let B = C[0, 1] be the Banach space endowed with norm u = sup t∈[0,1] |u(t)|. Define the cone C ⊂ B by Remark 3.5. It follows from the nonnegativeness and continuity of G and f that the operator T : C → B defined by satisfies T (C) ⊂ C and is completely continuous.
For our purposes, let us define two constants where τ 1 ∈ {0, q m } and τ 2 = q n with m, n ∈ N 0 , m > n. Our existence result is now given.
Proof. Since the operator T : C → C is completely continuous we only have to show that the operator equation y = T y has a solution satisfying r 1 ≤ y ≤ r 2 .