Existence of solutions for multi-point boundary value problem of fractional q-difference equation

Fractional differential calculus have recently been addressed by many researchers of various fields of science and engineering such as physics, chemistry, biology, economics, control theory, and biophysics, etc. [1-4]. In particular, the existence of solutions to fractional boundary value problems is under strong research recently, see [5-7] and references therein. The fractional q-difference calculus had its origin in the works by Al-Salam [8] and Agarwal [9]. More recently, perhaps due to the explosion in research within the fractional differential calculus setting, 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 the q-Laplace transform, q-Taylor’s formula [10,11], just to mention some. The question of the existence of solutions for fractional q-difference boundary value problems is in its infancy, being few results available in the literature. Ferreira [12] considered the existence of positive solutions to nonlinear q-difference boundary value problem: (D q u)(t) = −f(t, u(t)), 0 < t < 1, 1 < α ≤ 2 u(0) = u(1) = 0.