Nonlinear differential equations are widely used in everyday scientific and engineering dynamics. Problems involving differential equations of fractional order with initial and phase changes are often employed. Using a novel norm that is comfortable for fractional and non-singular differential equations containing Atangana-Baleanu-Caputo fractional derivatives, we examined a new class of initial values issues in this study. The Perov fixed point theorems that are utilized in generalized Banach spaces form the foundation for the new findings. Examples of the numerical analysis are provided in order to safeguard and effectively present the key findings.