| ABSTRACT |
The notion of Hyers–Ulam stability emerged from a question posed by Ulam concerning the stability behavior of approximate functional relations in metric structures. Hyers provided the first rigorous solution within Banach spaces, establishing a foundational result that later evolved into a broad stability theory. Subsequent contributions, particularly by Rassias, extended the original framework by introducing variable perturbation bounds, thereby giving rise to what is currently termed Generalized Hyers–Ulam (GHU) stability.
In recent years, the concept has been extended beyond functional equations to a wide spectrum of mathematical models, including ordinary differential equations, delay systems, fractional models, and stochastic dynamics. The present work focuses on examining the GHU stability of general higher-order ordinary differential equations through multiple analytical methodologies. The techniques employed include direct norm estimation, contraction mapping principles, Grönwall-type integral inequalities, operator-theoretic methods, and Laplace transform analysis.
The study establishes sufficient conditions guaranteeing existence, uniqueness, and generalized stability of solutions for both linear and nonlinear higher-order equations. Explicit bounds describing the deviation between approximate and exact solutions are obtained. A comparative evaluation of the adopted methods is also provided. The findings strengthen the theoretical framework of stability analysis and enhance its applicability to engineering and scientific models involving perturbations. |
| References |
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