Fixed point approach to the stability of generalized polynomials

Abstract

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.

General criteria for hyperstability of Cauchy-type equations

“…The hyperstability of arithmetically homogeneous functions is used in [10] to demonstrate the hyperstability of the monomial equation (therefore also of equation (1.1)) on Abelian semigroups. In [11], using a fixed point theorem, is proved that a class of control functions on commutative monoids ensure the hyperstability of monomial equation. Other types of control functions defined on abelian semigroups involving hyperstability for equation (1.1) are given in [13] and [1].…”

General criteria for hyperstability of Cauchy-type equations

“…The hyperstability of arithmetically homogeneous functions is used in [10] to demonstrate the hyperstability of the monomial equation (therefore also of equation (1.1)) on Abelian semigroups. In [11], using a fixed point theorem, is proved that a class of control functions on commutative monoids ensure the hyperstability of monomial equation. Other types of control functions defined on abelian semigroups involving hyperstability for equation (1.1) are given in [13] and [1].…”