Hyperstability of general linear functional equation

Our purpose is to investigate criteria for hyperstability of linear type functional equations. We prove that a function satisfying the equation approximately in some sense, must be a solution of it. We give some conditions on coefficients of the functional equation and a control function which guarantee hyperstability. Moreover, we show how our outcomes may be used to check whether the particular functional equation is hyperstable. Some relevant examples of applications are presented.

In the paper we prove, applying the fixed point approach, criteria for the θ-hyperstability of (1.1) under some natural assumptions on θ. In this way we obtain sufficient conditions for the θ-hyperstability of a wide class of functional equations and control functions θ. Moreover, we show how our outcomes may be used to check whether the particular functional equation is θ-hyperstable.
Our investigations have been motivated by a problem of optimality of some estimations arising in stability studies.
From now on, we assume that X, Y are normed spaces over a field F ∈ {R, C} and the coefficients in Eq. (1.1) are such that Notation Y D stands for the set of functions f : D → Y . The equation of the p-Wright affine function will be called shortly the p-Wright equation.

The main result
We start with the result concerning the hyperstability of Eq. (1.1). We show, under some suitable assumptions, that a function satisfying Eq. (1.1) approximately (in some sense) must be actually a solution to it.
Proof. Note that without loss of generality we can assume that Y is a Banach space, because otherwise we can replace it by its completion. The proof will be divided into two steps. First assume that A = 0. Assume that ∅ = I ⊂ N ≤m and the sequence {(c k 1 , . . . , c k n )} k∈N of the elements of F n 0 are such that the conditions (2.3), (2.4) and (2.5) hold. From (2.4) we get that there exists k 0 ∈ N such that For each k ∈ N k0 we define Taking x ∈ X 0 and substituting x j = c k j x, j ∈ N ≤n in (2.2) we have Thus (2.8) takes the form It is easy to prove by induction that for every x ∈ X 0 and l ∈ N 0 Therefore, using the fact that γ k < 1, we have Note that the operators T k and Λ k satisfy the assumptions of Theorem 1 in [5].
Applying this version of the fixed point theorem we obtain that there exists a unique fixed point holds and G k (x) = lim n→∞ (T n k g)(x) for x ∈ X 0 . Now, we show that G k is a solution of Eq. (2.6) (with A = 0). First we prove that for every l ∈ N 0 and every Clearly, the case l = 0 is just (2.2). Next, fix l ∈ N 0 and assume that (2.10) holds for every x 1 , . . . , x n ∈ X 0 . Then for every Vol. 90 (2016)

Hyperstability of general linear functional equation 531
Consequently, applying the inductive assumption and (2.1) we have . , x n ). Thus, by induction we have shown that (2.10) holds for l ∈ N 0 and x 1 , . . . , x n ∈ X 0 . Letting l → ∞ in (2.10), we obtain that G k satisfies Eq. (2.6) (with A = 0). Consequently, we get the sequence {G k } k∈N k 0 of functions satisfying (1.1) and (2.9) for k ∈ N k0 . Therefore g is a solution of (2.6), since it is a pointwise limit and consequently, according to our previous considerations, the function h satisfies (2.6) with A = 0, and hence g is a solution of (2.6), which finishes the proof.

Criteria for θ-hyperstability and applications
For the purpose of checking the θ-hyperstability we use the above Theorem 2.1. Namely, we give sufficient conditions for the θ-hyperstability of a wide class of functional equations and control functions θ. In the following two theorems (Theorems 3.2 and 3.8), criteria for determining whether a functional equation of the form (1.1) is θ-hyperstable are stated.
To present the first one we need the following natural assumptions on the control function θ.
Proof. Assume that i 0 and j 0 satisfy the conditions (3.1), (3.2) and put β k i := n j=1 a ij c k j . Take any l ∈ N such that Observe that c k j ∈ F 0 for j ∈ N ≤n , k ∈ N. It is easy to check that Since the conditions (3.2) and (3.3) are satisfied and we have β k i ∈ F 0 for i ∈ N ≤m and lim k→∞ |β k i | = +∞ for i = i 0 . According to our considerations and the conditions (a)-(c), the assumptions of Theorem 2.1 are fulfilled with I = {i 0 }, which completes the proof.
x,y∈ X 0 , then g is a solution of the p-Wright equation on X.
Proof. Take the matrix For equations with a greater number n of variables and a greater m, this method requires more calculations, but is still not complicated. As an example, consider the Fréchet equation. For the convenience of the reader, we first prove that a function satisfying the Fréchet equation on X 0 , fulfills it on the whole space X, since it is not as obvious as in the cases of linear and p-Wright equations.
Proof. Note that it is enough to show that g(0) = 0. Putting in (3.4) y = z = x, and then y = 2x, z = −x we get respectively, and hence Replacing y by x and z by −x in (3.4) yields Adding the above equalities we conclude that g(0) = 0, which completes the proof.
then g is a solution of the Fréchet equation on X.
Proof. Take a matrix and is not a solution of (3.5).
From Theorem 2.1, we derive another criterion for the θ-hyperstability of Eq. (1.1) with a particular form of the control function, namely for θ = θ 3 with C > 0, k j ∈ R such that n j=1 k j < 0. Note that in this case we do not need to assume that all k j are negative real numbers (in contrast to Theorem 3.2, where the condition (b) must be satisfied).

Final remarks
Note that our results correspond to the new ones concerning hyperstability. For example it has been proven that the Cauchy and linear equations are θ 1 -hyperstable with k 1 = k 2 < 0 (in [8] and [17], respectively). In our considerations k 1 , k 2 < 0 may be different.  has a solution, then g is a solution of (4.2) on X 0 . The hyperstability results have various interesting consequences. For instance, note that we get at once the following a bit surprising corollary. Our considerations can be used in further research on θ-hyperstability. It is interesting to investigate the hyperstability of other functional equations of the form (1.1), as well as seek other conditions guaranteeing the θ-hyperstability of specified equations and control functions θ.
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