A Fractional q$q$-difference Equation with Integral Boundary Conditions and Comparison Theorem

Abstract

Abstract In this article, we mainly prove the existence of extremal solutions for a fractional <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> $q$ -difference equation involving Riemann–Lioville type fractional derivative with integral boundary conditions. A comparison theorem under weak conditions is also build, and then we apply the comparison theorem, monotone iterative technique and lower–upper solution method to prove the existence of extremal solutions. Moreover, we can construct two iterative schemes approximating the extremal solutions of the fractional <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> $q$ -difference equation with integral boundary conditions. In the last section, a simple example is presented to illustrate the main result.