In this paper, we consider the nonhomogeneous fractional delay oscillation equation with order κ and investigate the existence of a unique solution in matrix-valued fuzzy Banach spaces for this equation using the alternative fixed point theorem. In a fuzzy environment, we introduce a class of the matrix-valued fuzzy Wright controller to investigate the Hyers–Ulam–Wright stability for the NH-FD-O equation with order κ. Finally, an illustrative example to demonstrate the application of the main theorem is also considered.

The general fractional conformable derivative (GCD) and its attributes have been described by researchers in the recent times. Compared with other fractional derivative definitions, this derivative presents a generalization of the conformable derivative and follows the same derivation formulae. For electrical circuits, such as RLC, RC, and LC, we obtain a new class of fractional-order differential equations using this novel derivative, The use of GCD to depict electrical circuits has been shown to be more adaptable and lucrative than the usual conformable derivative.

The propose of the paper is devoted to study the existence of ?-periodic solutions for second order delay differential equation in abstract Banach space. Firstly, we build a new maximum principle for the ?-periodic solutions of the corresponding linear equation. Secondly, with the help of this maximum principle, we study the existence of the minimal and maximal periodic solutions for our concerns problem by means of perturbation method and monotone iterative technique of the lower and upper solutions. In addition, an example is presented to show the application of our main results.