Upper and lower solutions for an integral boundary problem with two different orders ( p , q ) $\left ( p,q\right ) $ -fractional difference

Abstract

Abstract In this paper, a ( p , q ) $\left ( p,q\right ) $ -fractional nonlinear difference equation of different orders is considered and discussed. With the help of ( p , q ) $\left ( p,q\right ) $ -calculus for integrals and derivatives properties, we convert the main integral boundary value problem (IBVP) to an equivalent solution in the form of an integral equation, we use the upper–lower solution technique to prove the existence of positive solutions. We present an example of the IBVP to apply and demonstrate the results of our method.

Keywords

• IBVP
• Positive solutions
• Fixed point
• ( p , q ) $\left ( p,q\right ) $ -difference fractional derivatives