The stability of the functional equation f(x composite function y) = H(f(x), f(y)) (x, y in S) is investigated, where H is a homogeneous function and composite function is a square-symmetric operation on the set S. The results presented include and generalize the classical theorem of Hyers obtained in 1941 on the stability of the Cauchy functional equation.

This paper presents several characterizations of a local α-times integrated C-semigroup {T(t); 0 ≤ t < τ} by means of functional equation, subgenerator, and well-posedness of an associated abstract Cauchy problem.We also discuss properties concerning the nondegeneracy of T(·), the injectivity of C, the closability of subgenerators, the commutativity of T(·), and extension of solutions of the as...

In this paper the necessary and sufficient conditions for a mapping to be the dependence of the complete solution of some C first-order ordinary differential equation on the initial Cauchy condition are deduced. The result is obtained by studying the Cauchy atlas on a manifold of complete solutions. The proof is constructive the corresponding differential equation is obtained. The autonomous ca...

In 1940, Ulam proposed the general Ulam stability problem see 1 . Let G1 be a group and let G2 be a metric group with the metric d ·, · . Given ε > 0, does there exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality d h xy , h x h y < δ for all x, y ∈ G1 then there is a homomorphism H : G1 → G2 with d h x ,H x < ε for all x ∈ G1? In 1941, this problem was solved by Hyers 2 i...

* Correspondence: baak@hanyang. ac.kr Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea Full list of author information is available at the end of the article Abstract Using the fixed point method, we prove the Hyers-Ulam stability of a Cauchy-Jensen type additive set-valued functional equation, a Jensen type additive-quadratic se...