Positive solution for q-fractional four-point boundary value problems with p-Laplacian operator

This paper investigates a class of four-point boundary value problems of fractional q-difference equations with p-Laplacian operator Dqβ(φp(Dqαu(t)))=f(t,u(t)), t∈(0,1), u(0)=0, u(1)=au(ξ), Dqαu(0)=0, and Dqαu(1)=bDqαu(η), where Dqα and Dqβ are the fractional q-derivative of the Riemann-Liouville type, p-Laplacian operator is defined as φp(s)=|s|p−2s, p>1, and f(t,u) may be singular at t=0,1 or u=0. By applying the upper and lower solutions method associated with the Schauder fixed point theorem, some sufficient conditions for the existence of at least one positive solution are established. Furthermore, two examples are presented to illustrate the main results.