Janusz Brzdȩk

Remarks on stability of linear recurrence of higher order

Abstract We prove some stability results for linear recurrences with constant coefficients in normed spaces. As a consequence we obtain a complete solution of the problem of the Hyers–Ulam stability for such recurrences.

On functional which are orthogonally additive modulo Z

Let E be a real inner product space with dimension at least 2, D ⊂ E, f: E → R with f(x+y)−f(x)−f(y) ∈ Z for all orthogonal x,y ∈ E, and f(D) ⊂ (−γ,γ)+Z witn some real γ > 0. We prove that, under some additional assumptions, there are a unique linear functional A: E → R and a unique constant d ∈ R with f(x)−d∥x∥2−A(x) ∈ Z for x ∈ E. We also show some applications of this result to the determination of solutions F: E → C of the conditional equation: F(x+y) = F(x)F(y) for all orthogonal x,y ∈ E.

On almost additive functions

Let ( S , +) be a semigroup and ( H , +) be a group (neither necessarily commutative). Suppose that J ⊂ 2 s is a proper ideal in S such that and Ω( J ) = { M ⊂ S 2 : there exists U ( M )∈ J with M [x] ∈ J for x ∈ S / U ( M )}, where M [x] = { y ∈ S : ( y, x ) ∈ M }. We show that if f : S → H is a function satisfying then there exists exactly one additive function F : S → H with F ( x ) = f ( x ) J -almost everywhere in S . We also prove some results concerning regularity of the function F .

Remarks on Stability of the Linear Functional Equation in Single Variable

We present some observations concerning stability of the following linear functional equation (in single variable)

Remarks on Stability of the Equation of Homomorphism for Square Symmetric Groupoids

\varphi\bigl(f^m(x) \bigr)=\sum_{i=1}^m a_i(x)\varphi\bigl(f^{m-i}(x) \bigr)+F(x),

A Note on the Functions that Are Approximately p-Wright Affine

in the class of functions φ mapping a nonempty set S into a Banach space X over a field \(\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}\), where m is a fixed positive integer and the functions f:S→S, F:S→X and \(a_{i}:S\to\mathbb{K}\), i=1,…,m, are given. Those observations complement the results in our earlier paper (Brzdȩk et al. in J. Math. Anal. Appl. 373:680–689, 2011).

A fixed point approach to stability of functional equations

We present a survey of selected recent results of several authors concerning stability of the following polynomial functional equation (in single variable)