Generalized Hyers-Ulam Stability of the Second-Order Linear Differential Equations

The stability problem of functional equations started with the question concerning stability of group homomorphisms proposed by Ulam 1 during a talk before a Mathematical Colloquium at the University of Wisconsin, Madison. In 1941, Hyers 2 gave a partial solution of Ulam’s problem for the case of approximate additive mappings in the context of Banach spaces. In 1978, Rassias 3 generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences ‖f x y − f x − f y ‖ ≤ ‖x‖p ‖y‖p , > 0, p ∈ 0, 1 . This phenomenon of stability that was introduced by Rassias 3 is called the Hyers-Ulam-Rassias stability or the generalized Hyers-Ulam stability . Let X be a normed space over a scalar field K, and let I be an open interval. Assume that for any function f : I −→ X satisfying the differential inequality