2. Analysis of a New Stability

In this chapter, we consider the non-homogeneous fractional delay oscillation equation with order \(\kappa \) and investigate the existence of a unique solution in matrix-valued fuzzy Banach spaces for this equation using the alternative fixed point theorem. We introduce the Wright controller to investigate the Hyers–Ulam–Wright stability for the NH-FD-O equation with order \(\kappa \). Using the fractional integral with respect to the \(\varPsi \) function and the \(\varPsi \)-Hilfer fractional derivative, we consider Volterra fractional equations. Considering the Gauss Hypergeometric function as a control function, we introduce the concept of the Hyers-Ulam-Rassias-Kummer stability of this fractional equation and study the existence, uniqueness, and approximation for two classes of fractional Volterra integro-differential and fractional Volterra integral. We apply the Cădariu-Radu method derived from the Diaz-Margolis alternative fixed point theorem. Also we investigate a conformable fractional differential equation (CFD-E) with a constant coefficient on a compact interval. Considering the Diaz-Margolis alternative fixed point theorem (D-MAFPT) and using the Cădariu-Radu method (CRM), we prove the existence of a unique solution and the Hyers-Ulam-Rassias-\(OS_1\)-multiple stability (H-U-R-\(OS_1\)-MS) for the CFD equation. We consider a nonlinear fractional integral differential equation (N-FIDE) with arbitrary order and we show that there is a unique solution for this type of equation according to Babenko’s strategy and considering the multivariate-Mittag-Leffler function. All results are proved according to Banach’s contractive principle (BCP) and Leray-Schauder’s fixed point theorem (LS-FTP).