Cauchy problem for fractional $(p,q)$-difference equations

In this research article, we deal with the global convergence of successive approximations (s.a) as well as the existence of solutions to a fractional $(p,q)$-difference equation. Then, we discuss the existence result of the solutions of Caputo-type $(p,q)$-difference fractional vector-order equations in a Banach space. Also, we prove a theorem on the global convergence of successive approximations to the unique solution of our problem. Finally, the application of the main results is demonstrated by presenting numerical examples.