As a matter of fact, fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of a complex medium, polymer rheology, etc. involves derivatives of fractional order. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. In consequence, fractional differential equations have been of great interest; for example, see [2, 3, 4, 7, 16] and the references therein.

The q-difference calculus or quantum calculus is an old subject that was initially developed by Jackson [13, 14], basic definitions and properties of q-difference calculus can be found in the book mentioned in [15].

The fractional q-difference calculus had its origin in the works by Al-Salam [6] and Agarwal [1]. More recently, maybe due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional q-difference calculus were made, for example, q-analogues of the integral and differential fractional operators properties such as the q-Laplace transform, q-Taylor’s formula, Mittage-Leffler function [5, 17, 18], just to mention some.

Recently, the question of the existence of solutions for fractional q-difference boundary value problems have aroused considerable attention. There have been some papers dealing with the existence and multiplicity of solutions or positive solutions for boundary value problems involving nonlinear fractional q-difference equations, such as the Krasnosel’skii fixed-point theorem, the Leggett-Williams fixed-point theorem, and the Schauder fixed-point theorem, For examples, see [8, 9] and the references therein.

El-Shahed and Hassan [10] studied the existence of positive solutions of the following q-difference boundary value problem

Ferreira [11] and [12] considered the existence of positive solutions to nonlinear q-difference boundary value problems

and

respectively.

In this paper, we investigate the existence and uniqueness results for the following nonlinear fractional q-difference equations with three-point boundary conditions

where 0 < βξ^α−1 < 1, 0 < ξ < 1, Dqα is the fractional q-derivative of the Riemann-Liouville type of order α, and f : [0, 1] × [0,∞) → [0,∞) is continuous function.

For the convenience of the reader, we present some necessary definitions and lemmas of fractional q-calculus theory to facilitate analysis of problem (1.1). These details can be found in the recent literature; see [15] and references therein.

Let q ∈ (0, 1) and define

The q-analogue of the power (a − b)⁽ⁿ⁾ with n ∈ ℕ₀ 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]_qΓ_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

If a ∈ [0, b] and f is defined in the interval [0, b], its integral from a to b is defined by

Similarly as done for derivatives, an operator Iqn 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 [15]. We now point out three formulas that will be used later (_iD_q denotes the derivative with respect to variable i)

Denote that if α > 0 and a ≤ b ≤ t, then (t − a)^(α) ≥ (t − b)^(α) [11].

Definition 2.1

([19]) Let α ≥ 0 and f be function defined on [0, 1]. The fractional q-integral of the Riemann-Liouville type is Iq0f(x)=f(x) and

Definition 2.2

([19]) The fractional q-derivative of the Riemann-Liouville type of order α ≥ 0 is defined by Dq0f(x)=f(x) and

where m is the smallest integer greater than or equal to α.

Lemma 2.1

([19]) Let α β ≥ 0 and f be a function defined on [0, 1]. Then the next formulas hold:

• (a) (IqβIqαf)(x)=Iqα+βf(x),

• (b) (DqαIqαf)(x)=f(x).

Lemma 2.2

([11]) Let α > 0 and p be a positive integer. Then the following equality holds:

Lemma 2.3

Let y ∈ C[0, 1] and 1 < α ≤ 2, the unique solution of

is given by

where

Lemma 2.4

The function G(t, s) defined by (2.2) satisfies G(t, qs) ≥ 0 for all 0 ≤ s, t ≤ 1.

Lemma 2.5

Let ℐ : ℙ → ℂ be the operator defined by

Then ℐ : ℙ → ℙ is completely continuous.

In this section, our objective is to give and prove our main results.

Theorem 3.1

Suppose f(t, u) satisfies

Then the problem (1.1) has at least one positive solution.

Theorem 3.2

Suppose that f : [0, 1]×ℝ → [0,+∞) is a jointly continuous function satisfying the condition

Then the problem (1.1) has a unique positive solution if

In this section, we will present some examples to illustrate the main results.

Example 4.1

Consider the following q-fractional three-point boundary value problem

where α = 1.5 and β = ξ = q = 0.5. By simple computation, we can easily have

Thus, all the assumptions of Theorem 3.1 holds. Consequently, the conclusion of Theorem 3.1 implies that the problem (4.1) has at least one positive solution.

Example 4.2

Consider the following q-fractional three-point boundary value problem

where α = 1.5 and β = ξ = q = 0.5. Let

Clearly, L = 1/6 as |f(t, u) − f(t, v)| ≤ 1/6|u − v|. Further,

Thus, all the assumptions of Theorem 3.2 are satisfied. Therefore, the conclusion of Theorem 3.2 implies that the problem (4.2) has a unique positive solution.