Stability of the equation of the p-Wright affine functions

We prove some stability results for the equation of the p-Wright affine functions.


Introduction and preliminaries
Let 0 < p < 1 be a fixed real number. We say that a function f mapping a real nonempty interval I into the set of reals R is p-Wright convex provided (see, e.g., [4,9,16,22]) x,y∈ I.
If f satisfies the functional equation then we say that it is p-Wright affine (see [4]). Note that for p = 1/2 Eq. (1.1) becomes the Jensen's functional equation For p = 1/3 Eq. (1.1) takes the form which has been investigated by Najati and Park [18]; in particular, they proved some results on its stability and applied them in the investigation of the generalized (σ, τ )-Jordan derivations on Banach algebras. The cases of more arbitrary p were studied in [4,5,15] (see also [9,13]). We prove some results concerning the Hyers-Ulam stability and superstability of (1.1). For more information and numerous references on the stability J. Brzdȩk AEM of functional equations we refer to, e.g., [10,14,17,21]; for some examples of various recent outcomes showing new directions in this area of research see, e.g., [3,7,8,11,12,19,20]. The method of the proof of the main result corresponds to some observations in [6,7,20] and the main tool in it is a fixed point result that can be derived from [1, Theorem 1] (see also [2,Theorem 2]). To present it we need the following four hypotheses (R + denotes the set of nonnegative reals).
(H1) X is a normed space over a field F ∈ {R, C} (C denotes the set of complex numbers) and Y is a Banach space.
(H3) T : Y X → Y X is an operator satisfying the inequality Now we are in a position to present the above mentioned fixed point theorem. Theorem 1.1. Assume that hypotheses (H1)-(H4) are satisfied. Suppose that there are functions ε : X → R + and ϕ : X → Y such that and Then there exists a unique fixed point ψ of T with Vol. 85 (2013) p-Wright affine functions 499

Stability
The next theorem is the main result in this paper and concerns the stability of Eq. (1.1); it corresponds in particular to some results in [18].
and G is given by: where g 0 and T are defined by (2.6) and (2.7).
Proof. Note that (2.1) with y = 0 gives Then (2.5) implies the inequality which means that Hence, according to Theorem 1.1, there exists a unique solution G 0 : X → Y of the equation Now we show that, for every x, y ∈ X, n ∈ N 0 (nonnegative integers), Clearly, the case n = 0 is just (2.1). Next, fix m ∈ N 0 and assume that (2.10) holds for every x, y ∈ X with n = m. Then x,y∈ X. Thus, by induction we have shown that (2.10) holds for every x, y ∈ X and n ∈ N 0 . Letting n → ∞ in (2.10), we obtain that Write G(x) := G 0 (x) + g(0) for x ∈ X. Then it is easily seen that x,y∈ X and (2.2) holds. It remains to show the statement concerning the uniqueness of G. So suppose that M 0 ∈ (0, ∞) and G 1 : X → Y is a solution to (1.1) with Note that G(0) = g(0) = G 1 (0), and, by (2.2), We show that, for each j ∈ N 0 , (2.14) The case j = 0 is exactly (2.13). So fix l ∈ N 0 and assume that (2.14) holds for j = l. Then, in view of (2.11) and (2.12), Thus we have shown (2.14). Now, letting j → ∞ in (2.14) we get G 1 = G.

A complementary observation on superstability
The following very simple observation on the superstability of Eq. (1.1) complements Theorem 2.1.
for every x, y ∈ X. Then g is a solution to (1.1).
Thus, by induction we have shown that (3.3) holds for every x, y ∈ X and n ∈ N 0 . Letting n → ∞ in (3.3), we obtain that g is a solution to (1.1), because |p| k + |1 − p| k < 1.