By means of a new stability result, established for symmetric and multi-additive mappings, and using the concepts of stability couple and of stability chain, we prove, by a recursive procedure, the generalized stability of two of Fréchet's polynomial equations. We also give a new functional characterization of generalized polynomials and a new approach to solving the generalized stability of the monomial equation. MSC: 39B82; 39B52; 20M15; 65Q30
Using a new fixed point theorem for linear operators which act on function spaces, we give an iterative method for proving the generalized stability in three essential cases and the hyperstability for polynomial equation ∆ n+1 y f (x) = 0 on commutative monoids.The proposed iterative fixed point method leads to final concrete unitary estimates, and also improves and complements the known stability results for generalized polynomials.
We provide three large classes of control functions that ensure the hyperstability of the Cauchy equation on restricted domains included in various types of commutative semigroups. Among other consequences, we obtain significant improvements on similar results known from the literature for several Aoki-Rassias-type control functions.
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