Additive and Fréchet functional equations on restricted domains with some characterizations of inner product spaces

We investigate the Hyers–Ulam stability of an equation involving a single variable of the form ∥f(x)−αf(kn(x))−βf(kn+1(x))∥⩽u(x) where f is an unknown operator from a nonempty set X into a Banach space Y, and it preserves the addition operation, besides other certain conditions. The theory is employed and stability theorems are proven for various functional equations involving several variables. By comparing this method with the available techniques, it was noticed that this method does not require any restriction on the parity, on the domain, and on the range of the function. Our findings suggest that it is very much easy and more appropriate to apply the proposed method while investigating the stability of functional equations, in particular for several variables.

/pdf/the-stability-of-functional-equations-with-a-new-direct-g3zy5qb9.pdf

The Stability of Functional Equations with a New Direct Method

More than ten years after Justyna Sikorska [8] attempted to solve the Heyers-Ulam stability of a single variable equation by using direct method. In this paper, we will improve the results of Justyna Sikorska by using a more efficient approach. Relations between the generalized functional equation, the dependence of their different parameters and several properties are also further explored. To achieve the problem, we try to develop some new techniques to overcome the fundamental difficulties caused by the different properties of the function and the presence of several variables in the equation. Furthermore, we continue to construct and study a couple of functional equations by making a new direct method.

/pdf/a-note-on-the-stability-of-functional-equations-via-a-e6mlyy98.pdf

A Note on the Stability of Functional Equations via a Celebrated Direct Method

/pdf/additive-double-r-functional-inequalities-in-b-homogeneous-f-56edkk2nxq.pdf

Additive double ρ-functional inequalities in β-homogeneous F-spaces

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Additive double ρ-functional inequalities in β-homogeneous F-spaces