On the Generalized (<i>p</i>,<i>q</i>)-<i>ϕ</i>-Calculus with Respect to Another Function

Abstract

In the present paper, we generalized some of the operators defined in (p,q)-calculus with respect to another function. More precisely, the generalized (p,q)-ϕ-derivatives and (p,q)-ϕ-integrals were introduced with respect to the strictly increasing function ϕ with the help of different orders of the q-shifting, p-shifting, and (q/p)-shifting operators. Then, after proving some related properties, and as an application, we considered a generalized (p,q)-ϕ-difference problem and studied the existence property for its unique solutions with the help of the Banach contraction mapping principle.

Keywords

• (<i>p</i>,<i>q</i>)-<i>ϕ</i>-derivative
• (<i>p</i>,<i>q</i>)-<i>ϕ</i>-integral
• (<i>q</i>/<i>p</i>)-shifting operator
• <i>q</i>-operator
• existence of solution
• boundary value problem