Separated boundary value problems via quantum Hilfer and Caputo operators

Abstract

This paper describes a new class of boundary value fractional-order differential equations of the $ q $-Hilfer and $ q $-Caputo types, with separated boundary conditions. The presented problem is converted to an equivalent integral form, and fixed-point theorems are used to prove the existence and uniqueness of solutions. Moreover, several special cases demonstrate how the proposed problems advance beyond the existing literature. Examples are provided to support the analysis presented.

Keywords

• hilfer fractional derivative
• caputo derivative
• separated boundary condition
• existence
• fixed point theorems