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On stability of the functional equation of pWright affine functions in (2, α)Banach spaces
Journal of the Egyptian Mathematical Society volume 27, Article number: 21 (2019)
Abstract
Stability of functional equations has recent applications in many fields. We show that the stability results obtained by J. Brzdęk and concerning the functional equation of the pWright affine function:
f(px_{1}+(1−p)x_{2})+f((1−p)x_{1}+px_{2})=f(x_{1})+f(x_{2}),
can be proved also in (2, α)Banach spaces, for some real number α∈(0,1). This is done using some fixedpoint theorem.
Introduction
The rabid development of the theory of functional equations has been strongly promoted by its applications in various fields, e.g., networks and communication (see, e.g., [4, 16, 17, 21, 31]). They have applications in computer graphics [29], in information theory [2, 28], in digital filtering [34], and in decision theory [1, 35]. Stability of functional equations is nowadays a popular subject with many interesting applications (see, e.g., [5–8, 12, 23, 30] for more details). Stability can be seen from different perspectives, see [30], and hundreds of researchers are dealing with such amazing topic. It has applications in optimization theory (see, e.g., [26]), it is somehow related to the notion of shadowing (see, e.g., [22]), and it has applications in the economy (see [13]).
The starting point of the stability of functional equations was due to S.M. Ulam who posed an open problem in 1940. The problem posed by Ulam can be stated as follows (see, e.g., [30]):
Let G_{1} be a group and (G_{2},d) a metric group. Given ε>0, does there exist δ>0 such that if g:G_{1}→G_{2} satisfies
for all x,y∈G_{1}, then a homomorphism f:G_{1}→G_{2} exists such that
for all x,y∈G_{1}?
It should be noted that Hyers’s introduced a partial answer to Ulam’s problem in Banach spaces(see, e.g., [23]).
Stability is useful because it can be considered as an efficient tool for evaluating the error people usually face when replacing functions that satisfy some equations only approximately, by the exact solutions to those equations. Roughly speaking, an equation is said to be stable in some class of functions if any function from that class, satisfying the equation approximately (in some sense), is near (in some way) to an exact solution of the equation. In the last few decades, several stability problems of various (functional, difference, differential, integral) equations have been investigated by many mathematicians (see, e.g., [9–11, 24, 27] for more details), but mainly in classical spaces.
Since the notion of an approximate solution and the idea of nearness of two functions can be understood in many ways, depending on the needs and tools available in a particular situation. One of such nonclassical measures of a distance can be introduced by the notion of a 2norm. As far as we know, the concept of the linear 2normed space was introduced first by Gähler et al. in [20], and it seems that the first work on the HyersUlam stability of functional equations in complete 2normed spaces (that is, 2Banach spaces), see, e.g, [19]. See also [14, 33] for some details in 2Banach spaces. In this article, we investigate the stability of the functional equation of the pWright affine functions investigated in [3] but in (2, α)Banach spaces.
The article is organized as follows: in the “Preliminaries” section, we recall some definitions and the functional equation of our interest; the “Fixedpoint theorem” section introduces the fixedpoint theorem used in the stability; in the “Stability” section, we investigate the stability of the functional equation of the pWright affine functions; the “An observation on superstability” section introduces a simple observation on superstability; and the “Conclusion” section concludes our work.
Preliminaries
Throughout the article, we use \(\mathbb {R}_{+}\) to denote the set of nonnegative reals, \(\mathbb {R}\) denotes the set of reals, \(\mathbb {N}\) denotes the set of positive integers, and \(\mathbb {K}\) to denote the field of real or complex numbers. 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 \(\mathbb {R}\) is pWright convex provided (see, e.g., [15])
If f satisfies the functional equation
then we say that it is pWright affine (see [15]). Note that for p=1/2, Eq. (1) becomes the Jensen’s functional equation
For p=1/3, Eq. (1) takes the form
which has been investigated by Najati and Park in [32]. The cases of more arbitrary p were studied in [15] (see also [25]). We prove some results concerning the HyersUlam stability of (1). The method of the proof of the main result corresponds to some observations in [12] and the main tool in it is a fixed point. Let us recall first (see, for instance, [18]) some definitions.
Definition 1
By a linear 2normed space, we mean a pair (X,∥.,.∥) such that X is an at least twodimensional real linear space and
is a function satisfying the following conditions:

(1)
∥x_{1},x_{2}∥=0 if and only if x_{1} and x_{2} are linearly dependent;

(2)
∥x_{1},x_{2}∥=∥x_{2},x_{1}∥ for x_{1},x_{2}∈X

(3)
∥x_{1},x_{2}+x_{3}∥≤∥x_{1},x_{2}∥+∥x_{1},x_{3}∥ for x_{i}∈X,i=1,2,3

(4)
∥βx_{1},x_{2}∥=β∥x_{1},x_{2}∥ for \(\beta \in \mathbb {R}\) and x_{1},x_{2}∈X.
A generalized version of a linear 2normed spaces is the (2,α)normed space defined in the following manner:
Definition 2
Let α be a fixed real number with 0<α≤1, and let X be a linear space over \(\mathbb {K}\) with dimX>1. A function
is called a (2,α)norm on X if and only if it satisfies the following conditions:

(C1)
∥x_{1},x_{2}∥_{α}=0 if and only if x_{1} and x_{2} are linearly dependent;

(C2)
∥x_{1},x_{2}∥_{α}=∥x_{2},x_{1}∥_{α} for x_{1},x_{2}∈X

(C3)
∥x_{1},x_{2}+x_{3}∥_{α}≤∥x_{1},x_{2}∥_{α}+∥x_{1},x_{3}∥_{α} for x_{i}∈X,i=1,2,3

(C4)
∥λx_{1},x_{2}∥_{α}=λ^{α}∥x_{1},x_{2}∥_{α} for \(\lambda \in \mathbb {R}\) and x_{1},x_{2}∈X
The pair (X,∥.,.∥_{α}) is called a (2,α)normed space.
Definition 3
A sequence \((x_{n})_{n\in \mathbb {N}}\) of elements of a linear (2,α)normed space X is called a Cauchy sequence if there are linearly independent y,z∈X such that
whereas \((x_{n})_{n\in \mathbb {N}}\) is said to be convergent if there exists an x∈X (called a limit of this sequence and denoted by \({\lim }_{n\rightarrow \infty } x_{n}\)) with
A linear (2,α)normed space in which every Cauchy sequence is convergent is called a (2,α)Banach space.
Let us also mention that in linear (2,α)normed spaces, every convergent sequence has exactly one limit and the standard properties of the limit of a sum and a scalar product are valid. Next, it is easily seen that we have the following property.
Lemma 1
If X is a linear (2,α)normed space, x,y,z∈X, y,z are linearly independent, and
then x=0.
Let us yet recall a lemma from [33].
Lemma 2
If X is a linear (2,α)normed space and \((x_{n})_{n\in \mathbb {N}}\) is a convergent sequence of elements of X, then
We introduce a simple example of a (2,α)normed space.
Example 1
For x=(x_{1},x_{2}), \(y=(y_{1},y_{2})\in X=\mathbb {R}^{2}\), the (2,α)norm on X is defined by
where α is a fixed real number with 0<α≤1.
The main tool used in this article is the following fixedpoint theorem. It is a version of theorem 1 in [10]. In order to write it, we need the following assumptions.
Fixedpoint theorem
Let us introduce the following three assumptions:

(A1)
E is a nonempty set, (Y,∥.,.∥_{α}) is a (2,α)Banach space, Y_{0} is a subset of Y containing two linearly independent vectors, \(j \in \mathbb {N}\),
$$f_{i}:E \rightarrow E, \,\, g_{i}:Y_{0}\rightarrow Y_{0}, \,\,\text{and}\,\,\, L_{i}:E\times Y_{0} \rightarrow \mathbb{R}_{+} \,\, \text{for} \,\, i = 1,\cdots, j;$$ 
(A2)
T:Y^{E}→Y^{E} is an operator satisfying the inequality
$$\begin{array}{*{20}l} \ \mathrm{T}\xi(x) \mathrm{T} \mu(x),y \_{\alpha} &\;\leq \sum\limits_{i=1}^{j}L_{i}(x,y) \ \xi(f_{i}(x))\mu(f_{i}(x)),g_{i}(y)\_{\alpha}, \\ &\; \xi,\mu \in Y^{E}, \,x \in E, y\in Y_{0} \end{array} $$(2) 
(A3)
\(\Lambda :{\mathbb {R}}^{E\times Y_{0}}\rightarrow {\mathbb {R}}^{E\times Y_{0}} \) is an operator defined by
$$\begin{array}{*{20}l} \Lambda \delta(x,y)&\;:=\sum\limits_{i=1}^{j}L_{i}(x,y) \delta(f_{i}(x),g_{i}(y)), \delta \in {\mathbb{R}}^{E\times Y_{0}}, \\ &\;x \in E, \,\,\,y\in Y_{0} \end{array} $$(3)
Now, its the position to present the abovementioned fixedpoint theorem.
Theorem 1
Let hypotheses (A1)–(A3) hold and function
fulfill the following two conditions:
Then, there exists a unique fixed point ψ of T for which
Moreover,
We skip the proof as it is illustrated in [12].
Stability
In this section, we introduce the main result in this article that concerns the stability of Eq. (1); it corresponds in particular to some results in [12].
Theorem 2
Let (A1) be valid, \(p\in \mathbb {K}, A,k\in (0,\infty), p^{\alpha k}+1p^{\alpha k}<1\), and
satisfy
Then there exists a unique solution G:X→Y of Eq. (1) such that
and G is given by
where g_{0} and T are defined by (13) and (14). Moreover, G is the unique solution of Eq. (1) such that there exists a constant M∈(0,∞) with
Proof
Note that (8) with x_{2}=0 gives
Write
and
Then (12) implies the inequality
which means that
Further, note that (A3) holds with k=2, f_{1}(x)=px,f_{2}(x)=(1−p)x,L_{i}(x)=1 for i=1,2,x∈E. Define Λ as in (A3). Clearly, with \(\varepsilon (x):= A(\x_{1},y\_{\alpha }^{k})\) for x∈E, we have
Hence, according to Theorem 2, there exists a unique solution G_{0}:X→Y of the equation
such that
moreover,
Now we show that, for every \(x_{1}, x_{2} \in E, n \in \mathbb {N}_{0}\) (nonnegative integers),
It is easy to see that the case n=0 is just (8). Next, fix \(m \in \mathbb {N}_{0}\) and assume that (21) holds for every x_{1},x_{2}∈E with n=m. Then
which is clearly
Thus, by induction, we have shown that (21) holds for every x_{1},x_{2}∈E and \(n\in \mathbb {N}_{0}\). Letting n→∞ in (21), we obtain that
Write G(x_{1}):=G_{0}(x_{1})+g(0) for x_{1}∈E. Then it is easily seen that
and (9) holds. It remains to show the uniqueness of G. So suppose that M_{0}∈(0,∞) and G_{1}:X→Y is a solution to (1) with
Note that
and, by (9),
The case j=0 is exactly (29). So fix \(l\in \mathbb {N}_{0}\) and assume that (29) holds for j=l. Then, in view of (27) and (28),
Thus, we have shown (29). Now, letting j→∞ in (29), we get G_{1}=G. □
An observation on superstability
The following is a very simple observation on the superstability of Eq. (1) complements Theorem 2.
Theorem 3
Let (A1) be valid, \(p\in \mathbb {F}, A,k\in (0,\infty), p^{2\alpha k}+1p^{2\alpha k}<1\), and
satisfy
for every x_{1},x_{2}∈E,y∈Y_{0}. Then g is a solution to (1).
Proof
It is easy to see that (30) with x_{2}=0 gives
We show that, for every x_{1},x_{2}∈E, \(y\in Y_{0}, n\in \mathbb {N}_{0},\)
It is easy to see that the case n=0 is just (30). Next, fix \(m\in \mathbb {N}_{0}\) and assume that (32) holds for every x_{1},x_{2}∈E, with n=m. Then, by (31),
which is clearly
for every x_{1},x_{2}∈E,y∈Y_{0}. Therefore, by induction, we have shown that (32) holds for every x_{1},x_{2}∈E and \(n\in \mathbb {N}_{0}\). Letting n→∞ in (32), we obtain that g is a solution to (1). □
Conclusion
In this paper, we managed to generalize some recent results concerning the stability of the functional equation of the pWright affine functions in (2, α)Banach spaces, for some real number α∈(0,1]. The main tool in the investigation is a fixedpoint theory. This work may be further generalized to be in (n, α)Banach spaces, for some natural number n.
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Elhady, Es. On stability of the functional equation of pWright affine functions in (2, α)Banach spaces. J Egypt Math Soc 27, 21 (2019). https://doi.org/10.1186/s427870190024y
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DOI: https://doi.org/10.1186/s427870190024y