# On the Hyers-Ulam Stability of a General Mixed Additive and Cubic Functional Equation inn-Banach Spaces

Authors: Tian Zhou Xu; John Michael Rassias
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The objective of the present paper is to determine the generalized Hyers-Ulam stability of the mixed additive-cubic functional equation inn-Banach spaces by the direct method. In addition, we show under some suitable conditions that an approximately mixed additive-cubic function can be approximated by a mixed additive and cubic mapping.


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## Body

## 1. Introduction and Preliminaries

A basic question in the theory of functional equations is as follows: when is it true that a function, which approximately satisfies a functional equation, must be close to an exact solution of the equation?If the problem accepts a unique solution, we say the equation is stable (see [1]). The study of stability problems for functional equations is related to a question of Ulam [2] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [3]. The result of Hyers was generalized by Aoki [4] for approximate additive mappings and by Rassias [5] for approximate linear mappings by allowing the Cauchy difference operator CDf(x,y)=f(x+y)-[f(x)+f(y)] to be controlled by ϵ(∥x∥p+∥y∥p). In 1994, a generalization of Rassias’ theorem was obtained by Găvruţa [6], who replaced ϵ(∥x∥p+∥y∥p) by a general control function φ(x,y). On the other hand, several further interesting discussions, modifications, extensions, and generalizations of the original problem of Ulam have been proposed (see, e.g. [7–12] and the references therein).Recently, Park [9] investigated the approximate additive mappings, approximate Jensen mappings, and approximate quadratic mappings in 2-Banach spaces and proved the generalized Hyers-Ulam stability of the Cauchy functional equation, the Jensen functional equation, and the quadratic functional equation in 2-Banach spaces. This is the first result for the stability problem of functional equations in 2-Banach spaces.In [11, 12], we introduced the following mixed additive-cubic functional equation for fixed integers k with k≠0, ±1:(1.1)f(kx+y)+f(kx-y)=kf(x+y)+kf(x-y)+2f(kx)-2kf(x),
with f(0)=0, and investigated the generalized Hyers-Ulam stability of (1.1) in quasi-Banach spaces and non-Archimedean fuzzy normed spaces, respectively.In this paper, we investigate, approximate mixed additive-cubic mappings inn-Banach spaces. That is, we prove the generalized Hyers-Ulam stability of a general mixed additive-cubic equation (1.1) in n-Banach spaces by the direct method.The concept of 2-normed spaces was initially developed by Gähler [13, 14] in the middle of 1960s, while that of n-normed spaces can be found in [15, 16]. Since then, many others have studied this concept and obtained various results; see for instance [15, 17–19].We recall some basic facts concerningn-normed spaces and some preliminary results.Definition 1.1.
Letn∈ℕ, and let X be a real linear space with dim X≥n and ∥·,…,·∥:Xn→ℝ a function satisfying the following properties:(N1)
∥x1,x2,…,xn∥=0 if and only if x1,x2,…,xn are linearly dependent,(N2)
∥x1,x2,…,xn∥ is invariant under permutation,(N3)
∥αx1,x2,…,xn∥=|α|∥x1,x2,…,xn∥,(N4)
∥x+y,x2,…,xn∥≤∥x,x2,…,xn∥+∥y,x2,…,xn∥ for all α∈ℝ and x,y,x1,x2,…,xn∈X. Then the function ∥·,…,·∥ is called an n-norm on X and the pair (X,∥·,…,·∥) is called an n-normed space.Example 1.2.
Forx1,x2,…,xn∈ℝn, the Euclidean n-norm ∥x1,x2,…,xn∥E is defined by
(1.2)‖x1,x2,…,xn‖E=|det⁡(xij)|=abs(|x11⋯x1n⋮⋱⋮xn1⋯xnn|),
where xi=(xi1,…,xin)∈ℝn for each i=1,2,…,n.Example 1.3.
The standardn-norm on X, a real inner product space of dimension dim X≥n, is as follows:
(1.3)‖x1,x2,…,xn‖S=|〈x1,x1〉⋯〈x1,xn〉⋮⋱⋮〈xn,x1〉⋯〈xn,xn〉|1/2,
where 〈·,·〉 denotes the inner product on X. If X=ℝn, then this n-norm is exactly the same as the Euclidean n-norm ∥x1,x2,…,xn∥E mentioned earlier. For n=1, this n-norm is the usual norm ∥x1∥=〈x1,x1〉1/2.Definition 1.4.
A sequence{xk} in an n-normed space X is said to converge to some x∈X in the n-norm if
(1.4)lim⁡k→∞‖xk-x,y2,…,yn‖=0,
for every y2,…,yn∈X.Definition 1.5.
A sequence{xk} in an n-normed space X is said to be a Cauchy sequence with respect to the n-norm if
(1.5)lim⁡k,l→∞‖xk-xl,y2,…,yn‖=0,
for every y2,…,yn∈X. If every Cauchy sequence in X converges to some x∈X, then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be an n-Banach space.Now we state the following results as lemma (see [9] for the details).Lemma 1.6.
LetX be an n-normed space. Then,(1)
Forxi∈X(i=1,…,n) and γ, a real number,
(1.6)‖x1,…,xi,…,xj,…,xn‖=‖x1,…,xi,…,xj+γxi,…,xn‖
for all 1≤i≠j≤n,(2)
|∥x,y2,…,yn∥-∥y,y2,…,yn∥|≤∥x-y,y2,…,yn∥ for all x,y,y2,…,yn∈X,(3)
if∥x,y2,…,yn∥=0 for all y2,…,yn∈X, then x=0,(4)
for a convergent sequence{xj} in X,
(1.7)lim⁡j→∞‖xj,y2,…,yn‖=‖lim⁡j→∞xj,y2,…,yn‖
for all y2,…,yn∈X.


## 2. Approximate Mixed Additive-Cubic Mappings

In this section, we investigate the generalized Hyers-Ulam stability of the generalized mixed additive-cubic functional equation inn-Banach spaces. Let X be a linear space and Y an n-Banach space. For convenience, we use the following abbreviation for a given mapping f:X→Y:(2.1)Df(x,y):=f(kx+y)+f(kx-y)-kf(x+y)-kf(x-y)-2f(kx)+2kf(x)
for all x,y∈X.Theorem 2.1.
LetX be a linear space and Y an n-Banach space. Let f:X→Y be a mapping with f(0)=0 for which there is a function φ:Xn+1→[0,∞) such that
(2.2)∑j=0∞12jφ(2jx,2jy,u2,…,un)<∞,(2.3)‖Df(x,y),u2,…,un‖Y≤φ(x,y,u2,…,un)
for all x,y,u2,…,un∈X. Then, there is a unique additive mapping A:X→Y such that
(2.4)‖f(2x)-8f(x)-A(x),u2,…,un‖Y≤∑j=0∞12j+1φ̃(2jx,u2,…,un)
for all x,u2,…,un∈X, where
(2.5)φ̃(x,u2,…,un):=1|k3-k|{(0,(k+1)x2,u2,…,un)(|k|+1)[φ(x,(2k+1)x,u2,…,un)+φ(x,(2k-1)x,u2,…,un)]+φ(3x,x,u2,…,un)+(8k2+1)φ(x,x,u2,…,un)+φ(x,3kx,u2,…,un)+φ(x,kx,u2,…,un)+k2φ(2x,2x,u2,…,un)+φ(2x,2kx,u2,…,un)+2φ(x,(k+1)x,u2,…,un)+2φ(x,(k-1)x,u2,…,un)+2φ(2x,x,u2,…,un)+2φ(2x,kx,u2,…,un)+8φ(x2,kx2,u2,…,un)+8|k|φ(x2,(2k-1)x2,u2,…,un)+8|k|φ(x2,(2k+1)x2,u2,…,un)+8φ(x2,3kx2,u2,…,un)+|k|+1|k-1|φ(0,(k+1)x,u2,…,un)+8k2+1|k-1|φ(0,(k-1)x,u2,…,un)+2|k-1|φ(0,x,u2,…,un)+|k||k-1|φ(0,(3k-1)x,u2,…,un)+k2|k-1|φ(0,2(k-1)x,u2,…,un)+k2+|k|-1|k-1|φ(0,2kx,u2,…,un)+8|k||k-1|φ(0,(3k-1)x2,u2,…,un)+8|k||k-1|φ(0,(k+1)x2,u2,…,un)+8k2+2|k|-8|k-1|φ(0,kx,u2,…,un)}.Proof.
Lettingx=0 in (2.3), we get
(2.6)‖f(y)+f(-y),u2,…,un‖Y≤1|k-1|φ(0,y,u2,…,un)
for all y,u2,…,un∈X. Putting y=x in (2.3), we have
(2.7)‖f((k+1)x)+f((k-1)x)-kf(2x)-2f(kx)+2kf(x),u2,…,un‖Y≤φ(x,x,u2,…,un)
for all x,u2,…,un∈X. Thus
(2.8)‖f(2(k+1)x)+f(2(k-1)x)-kf(4x)-2f(2kx)+2kf(2x),u2,…,un‖Y≤φ(2x,2x,u2,…,un)
for all x,u2,…,un∈X. Letting y=kx in (2.3), we get
(2.9)‖f(2kx)-kf((k+1)x)-kf(-(k-1)x)-2f(kx)+2kf(x),u2,…,un‖Y≤φ(x,kx,u2,…,un)
for all x,u2,…,un∈X. Letting y=(k+1)x in (2.3), we have
(2.10)‖f((2k+1)x)+f(-x)-kf((k+2)x)-kf(-kx)-2f(kx)+2kf(x),u2,…,un‖Y≤φ(x,(k+1)x,u2,…,un)
for all x,u2,…,un∈X. Letting y=(k-1)x in (2.3), we have
(2.11)‖f((2k-1)x)-(k+2)f(kx)-kf(-(k-2)x)+(2k+1)f(x),u2,…,un‖Y≤φ(x,(k-1)x,u2,…,un)
for all x,u2,…,un∈X. Replacing x and y by 2x and x in (2.3), respectively, we get
(2.12)‖f((2k+1)x)+f((2k-1)x)-2f(2kx)-kf(3x)+2kf(2x)-kf(x),u2,…,un‖Y≤φ(2x,x,u2,…,un)
for all x,u2,…,un∈X. Replacing x and y by 3x and x in (2.3), respectively, we get
(2.13)‖f((3k+1)x)+f((3k-1)x)-2f(3kx)-kf(4x)-kf(2x)+2kf(3x),u2,…,un‖Y≤φ(3x,x,u2,…,un)
for all x,u2,…,un∈X. Replacing x and y by 2x and kx in (2.3), respectively, we have
(2.14)‖f(3kx)+f(kx)-kf((k+2)x)-kf(-(k-2)x)-2f(2kx)+2kf(2x),u2,…,un‖Y≤φ(2x,kx,u2,…,un)
for all x,u2,…,un∈X. Setting y=(2k+1)x in (2.3), we have
(2.15)‖f((3k+1)x)+f(-(k+1)x)-kf(2(k+1)x)-kf(-2kx)-2f(kx)+2kf(x),u2,…,un‖Y≤φ(x,(2k+1)x,u2,…,un)
for all x,u2,…,un∈X. Letting y=(2k-1)x in (2.3), we have
(2.16)‖f((3k-1)x)+f(-(k-1)x)-kf(-2(k-1)x)-kf(2kx)-2f(kx)+2kf(x),u2,…,un‖Y≤φ(x,(2k-1)x,u2,…,un)
for all x,u2,…,un∈X. Letting y=3kx in (2.3), we have
(2.17)‖f(4kx)+f(-2kx)-kf((3k+1)x)-kf(-(3k-1)x)-2f(kx)+2kf(x),u2,…,un‖Y≤φ(x,3kx,u2,…,un)
for all x,u2,…,un∈X. By (2.6), (2.7), (2.13), (2.15), and (2.16), we get
(2.18)‖kf(2(k+1)x)+kf(-2(k-1)x)+6f(kx)-2f(3kx)-kf(4x)+2kf(3x)-6kf(x),u2,…,un‖Y≤φ(x,(2k+1)x,u2,…,un)+φ(x,(2k-1)x,u2,…,un)+φ(3x,x,u2,…,un)+φ(x,x,u2,…,un)+1|k-1|φ(0,(k+1)x,u2,…,un)+1|k-1|φ(0,(k-1)x,u2,…,un)+|k||k-1|φ(0,2kx,u2,…,un)
for all x,u2,…,un∈X. By (2.6), (2.10), and (2.11), we have
(2.19)‖f((2k+1)x)+f((2k-1)x)-kf((k+2)x)-kf(-(k-2)x)-4f(kx)+4kf(x),u2,…,un‖Y≤φ(x,(k+1)x,u2,…,un)+φ(x,(k-1)x,u2,…,un)+1|k-1|φ(0,x,u2,…,un)+|kk-1|φ(0,kx,u2,…,un)
for all x,u2,…,un∈X. It follows from (2.12) and (2.19) that
(2.20)‖kf((k+2)x)+kf(-(k-2)x)-2f(2kx)+4f(kx)-kf(3x)+2kf(2x)-5kf(x),u2,…,un‖Y≤φ(x,(k+1)x,u2,…,un)+φ(x,(k-1)x,u2,…,un)+φ(2x,x,u2,…,un)+1|k-1|φ(0,x,u2,…,un)+|kk-1|φ(0,kx,u2,…,un)
for all x,u2,…,un∈X. By (2.14) and (2.20), we have
(2.21)‖f(3kx)-4f(2kx)+5f(kx)-kf(3x)+4kf(2x)-5kf(x),u2,…,un‖Y≤φ(x,(k+1)x,u2,…,un)+φ(x,(k-1)x,u2,…,un)+φ(2x,x,u2,…,un)+φ(2x,kx,u2,…,un)+1|k-1|φ(0,x,u2,…,un)+|kk-1|φ(0,kx,u2,…,un)
for all x,u2,…,un∈X. By (2.6), (2.15), (2.16), and (2.17), we have
(2.22)‖kf(-(k+1)x)-kf(-(k-1)x)-k2f(2(k+1)x)+k2f(-2(k-1)x)+k2f(2kx)-(k2-1)f(-2kx)+f(4kx)-2f(kx)+2kf(x),u2,…,un‖Y≤|k|φ(x,(2k+1)x,u2,…,un)+|k|φ(x,(2k-1)x,u2,…,un)+φ(x,3kx,u2,…,un)+|kk-1|φ(0,(3k-1)x,u2,…,un)
for all x,u2,…,un∈X. It follows from (2.6), (2.8), (2.9), and (2.22) that
(2.23)‖f(4kx)-2f(2kx)-k3f(4x)+2k3f(2x),u2,…,un‖Y≤|k|φ(x,(2k+1)x,u2,…,un)+|k|φ(x,(2k-1)x,u2,…,un)+φ(x,3kx,u2,…,un)+φ(x,kx,u2,…,un)+k2φ(2x,2x,u2,…,un)+|kk-1|φ(0,(3k-1)x,u2,…,un)+|kk-1|φ(0,(k+1)x,u2,…,un)+k2|k-1|φ(0,2(k-1)x,u2,…,un)+k2-1|k-1|φ(0,2kx,u2,…,un)
for all x,u2,…,un∈X. Hence,
(2.24)‖f(2kx)-2f(kx)-k3f(2x)+2k3f(x),u2,…,un‖Y≤|k|φ(x2,(2k+1)x2,u2,…,un)+|k|φ(x2,(2k-1)x2,u2,…,un)+φ(x2,3kx2,u2,…,un)+φ(x2,kx2,u2,…,un)+k2φ(x,x,u2,…,un)+|kk-1|φ(0,(3k-1)x2,u2,…,un)+|kk-1|φ(0,(k+1)x2,u2,…,un)+k2|k-1|φ(0,(k-1)x,u2,…,un)+k2-1|k-1|φ(0,kx,u2,…,un)
for all x,u2,…,un∈X. By (2.9), we have
(2.25)‖f(4kx)-kf(2(k+1)x)-kf(-2(k-1)x)-2f(2kx)+2kf(2x),u2,…,un‖Y≤φ(2x,2kx,u2,…,un)
for all x,u2,…,un∈X. From (2.23) and (2.25), we have
(2.26)‖kf(2(k+1)x)+kf(-2(k-1)x)-k3f(4x)+(2k3-2k)f(2x)‖Y≤|k|φ(x,(2k+1)x,u2,…,un)+|k|φ(x,(2k-1)x,u2,…,un)+φ(x,3kx,u2,…,un)+φ(x,kx,u2,…,un)+k2φ(2x,2x,u2,…,un)+φ(2x,2kx,u2,…,un)+|kk-1|φ(0,(3k-1)x,u2,…,un)+|kk-1|φ(0,(k+1)x,u2,…,un)+k2|k-1|φ(0,2(k-1)x,u2,…,un)+k2-1|k-1|φ(0,2kx,u2,…,un)
for all x,u2,…,un∈X. Also, from (2.18) and (2.26), we get
(2.27)‖2f(3kx)-6f(kx)+(k-k3)f(4x)-2kf(3x)+(2k3-2k)f(2x)+6kf(x),u2,…,un‖Y≤(|k|+1)[φ(x,(2k+1)x,u2,…,un)+φ(x,(2k-1)x,u2,…,un)]+φ(3x,x,u2,…,un)+φ(x,x,u2,…,un)+φ(x,3kx,u2,…,un)+φ(x,kx,u2,…,un)+k2φ(2x,2x,u2,…,un)+φ(2x,2kx,u2,…,un)+|k|+1|k-1|φ(0,(k+1)x,u2,…,un)+1|k-1|φ(0,(k-1)x,u2,…,un)+k2+|k|-1|k-1|φ(0,2kx,u2,…,un)+|kk-1|φ(0,(3k-1)x,u2,…,un)+k2|k-1|φ(0,2(k-1)x,u2,…,un)
for all x,u2,…,un∈X.
On the other hand, it follows from (2.21) and (2.27) that(2.28)‖8f(2kx)-16f(kx)+(k-k3)f(4x)+(2k3-10k)f(2x)+16kf(x),u2,…,un‖Y≤(|k|+1)[φ(x,(2k+1)x,u2,…,un)+φ(x,(2k-1)x,u2,…,un)]+φ(3x,x,u2,…,un)+φ(x,x,u2,…,un)+φ(x,3kx,u2,…,un)+φ(x,kx,u2,…,un)+k2φ(2x,2x,u2,…,un)+φ(2x,2kx,u2,…,un)+2φ(x,(k+1)x,u2,…,un)+2φ(x,(k-1)x,u2,…,un)+2φ(2x,x,u2,…,un)+2φ(2x,kx,u2,…,un)+2|k-1|φ(0,x,u2,…,un)+2|k||k-1|φ(0,kx,u2,…,un)+|k|+1|k-1|φ(0,(k+1)x,u2,…,un)+1|k-1|φ(0,(k-1)x,u2,…,un)+k2+|k|-1|k-1|φ(0,2kx,u2,…,un)+|kk-1|φ(0,(3k-1)x,u2,…,un)+k2|k-1|φ(0,2(k-1)x,u2,…,un)
for all x,u2,…,un∈X. Therefore by (2.24) and (2.28), we get
(2.29)‖f(4x)-10f(2x)+16f(x),u2,…,un‖Y≤1|k3-k|×{(0,(k+1)x2,u2,…,un)(|k|+1)[φ(x,(2k+1)x,u2,…,un)+φ(x,(2k-1)x,u2,…,un)]+φ(3x,x,u2,…,un)+(8k2+1)φ(x,x,u2,…,un)+φ(x,3kx,u2,…,un)+φ(x,kx,u2,…,un)+k2φ(2x,2x,u2,…,un)+φ(2x,2kx,u2,…,un)+2φ(x,(k+1)x,u2,…,un)+2φ(x,(k-1)x,u2,…,un)+2φ(2x,x,u2,…,un)+2φ(2x,kx,u2,…,un)+8φ(x2,kx2,u2,…,un)+8|k|φ(x2,(2k-1)x2,u2,…,un)+8|k|φ(x2,(2k+1)x2,u2,…,un)+8φ(x2,3kx2,u2,…,un)+|k|+1|k-1|φ(0,(k+1)x,u2,…,un)+8k2+1|k-1|φ(0,(k-1)x,u2,…,un)+2|k-1|φ(0,x,u2,…,un)+|kk-1|φ(0,(3k-1)x,u2,…,un)+k2|k-1|φ(0,2(k-1)x,u2,…,un)+k2+|k|-1|k-1|φ(0,2kx,u2,…,un)+8|k||k-1|φ(0,(3k-1)x2,u2,…,un)+8|k||k-1|φ(0,(k+1)x2,u2,…,un)+8k2+2|k|-8|k-1|φ(0,kx,u2,…,un)}:=φ̃(x,u2,…,un)
for all x,u2,…,un∈X.
Now, letg:X→Y be the mapping defined by g(x):=f(2x)-8f(x) for all x,u2,…,un∈X. Then, (2.29) means that(2.30)‖f(4x)-10f(2x)+16f(x),u2,…,un‖Y≤φ̃(x,u2,…,un)
for all x,u2,…,un∈X. Also, we get
(2.31)‖g(2x)-2g(x),u2,…,un‖Y≤φ̃(x,u2,…,un)
for all x∈X. Replacing x by 2jx in (2.31) and dividing both sides of (2.31) by 2j+1, we get
(2.32)‖12jg(2jx)-12j+1g(2j+1x),u2,…,un‖Y≤12j+1φ̃(2jx,u2,…,un)
for all x,u2,…,un∈X and all integers j≥0. For all integers l,m with 0≤l<m, we have
(2.33)‖12lg(2lx)-12mg(2mx),u2,…,un‖Y≤∑j=lm-1‖12jg(2jx)-12j+1g(2j+1x),u2,…,un‖Y≤∑j=lm-112j+1φ̃(2jx,u2,…,un)
for all x,u2,…,un∈X. So, we get
(2.34)lim⁡l,m→∞‖12lg(2lx)-12mg(2mx),u2,…,un‖Y=0
for all x,u2,…,un∈X. This shows that the sequence {(1/2j)g(2jx)} is a Cauchy sequence in Y. Since Y is an n-Banach space, the sequence {(1/2j)g(2jx)} converges. So, we can define a mapping A:X→Y by
(2.35)A(x):=lim⁡j→∞12jg(2jx)
for all x∈X. Putting l=0, then passing the limit m→∞ in (2.33), and using Lemma 1.6(4), we get
(2.36)‖g(x)-A(x),u2,…,un‖Y≤∑j=0∞12j+1φ̃(2jx,u2,…,un)
for all x,u2,…,un∈X.
Now we show thatA is additive. By Lemma 1.6, (2.2), (2.32), and (2.35), we have(2.37)‖A(2x)-2A(x),u2,…,un‖Y=lim⁡j→∞‖12jg(2j+1x)-12j-1g(2jx),u2,…,un‖Y=2lim⁡j→∞‖12j+1g(2j+1x)-12jg(2jx),u2,…,un‖Y≤lim⁡j→∞12jφ̃(2jx,u2,…,un)=0
for all x,u2,…,un∈X. By Lemma 1.6(3), A(2x)=2A(x) for all x∈X. Also, by Lemma 1.6(4), (2.2), (2.3), and (2.35), we get
(2.38)‖DA(x,y),u2,…,un‖Y=lim⁡j→∞12j‖Dg(2jx,2jy),u2,…,un‖Y=lim⁡j→∞12j‖Df(2j+1x,2j+1y)-8Df(2jx,2jy),u2,…,un‖Y≤lim⁡j→∞12j[‖Df(2j+1x,2j+1y),u2,…,un‖Y+8‖Df(2jx,2jy),u2,…,un‖Y]≤lim⁡j→∞12j[φ(2j+1x,2j+1y,u2,…,un)+8φ(2jx,2jy,u2,…,un)]=0
for all x,y,u2,…,un∈X. By Lemma 1.6(3), DA(x,y)=0 for all x,y∈X. Hence, the mapping A satisfies (1.1). By [11, Lemma 2.3], the mapping x→A(2x)-8A(x) is additive. Therefore, A(2x)=2A(x) implies that the mapping A is additive.
To prove the uniqueness ofA, let B:X→Y be another additive mapping satisfying (2.4). Fix x∈X. Clearly, A(2lx)=2lA(x) and B(2lx)=2lB(x) for all l∈ℕ. It follows from (2.4) that(2.39)‖A(x)-B(x),u2,…,un‖Y=‖A(2lx)2l-B(2lx)2l,u2,…,un‖Y≤12l[‖f(2l+1x)-8f(2lx)-A(2lx),u2,…,un‖Y+‖B(2lx)-f(2l+1x)+8f(2lx),u2,…,un‖Y]≤12l∑j=0∞12jφ̃(2j+lx,u2,…,un)≤∑j=0∞12j+lφ̃(2j+lx,u2,…,un)=∑j=l∞12jφ̃(2jx,u2,…,un)
for all x,u2,…,un∈X, and l∈ℕ. By (2.2), we see that the right-hand side of the above inequality tends to 0 as l→∞. Therefore, ∥A(x)-B(x),u2,…,un∥Y=0 for all u2,…,un∈X. By Lemma 1.6, we can conclude that A(x)=B(x) for all x∈X. So, A=B. This proves the uniqueness of A.Theorem 2.2.
LetX be a linear space and Y an n-Banach space. Let f:X→Y be a mapping with f(0)=0 for which there is a function φ:Xn+1→[0,∞) such that
(2.40)∑j=1∞2jφ(x2j,y2j,u2,…,un)<∞,‖Df(x,y),u2,…,un‖Y≤φ(x,y,u2,…,un)
for all x,y,u2,…,un∈X. Then, there is a unique additive mapping A:X→Y such that
(2.41)‖f(2x)-8f(x)-A(x),u2,…,un‖Y≤∑j=1∞2j-1φ̃(x2j,u2,…,un)
for all x,u2,…,un∈X, where φ̃(x,u2,…,un) is defined as in Theorem 2.1.Proof.
The proof is similar to the proof of Theorem2.1.Corollary 2.3.
LetX be a normed space and Y an n-Banach space. Let θ∈[0,∞),p,r2,…,rn∈(0,∞) such that p≠1, and let f:X→Y be a mapping with f(0)=0 such that
(2.42)‖Df(x,y),u2,…,un‖Y≤θ(‖x‖Xp+‖y‖Xp)‖u2‖Xr2⋯‖un‖Xrn
for all x,y,u2,…,un∈X. Then, there exists a unique additive mapping A:X→Y such that
(2.43)‖f(2x)-8f(x)-A(x),u2,…,un‖Y≤θϵ‖x‖Xp‖u2‖Xr2⋯‖un‖Xrn|(2-2p)(k3-k)|
for all x,u2,…,un∈X, where
(2.44)ϵ=(1+|k|+23-p|k|)[(2k+1)p+(2k-1)p]+2|k|+13+3p+3|k|p+16k2+3p|k|p+2p+1k2+2p(5+|k|p)+2|k+1|p+2|k-1|p+23-p(2+|k|+|k|p+3p|k|p)+(|k|+1)|k+1|p|k-1|+23-p|k||k-1||k+1|p+(1+8k2+2pk2)|k-1|p-1+2p|k|p(k2+|k|-1)|k-1|+2|k-1|+|k|(23-p+1)|k-1||3k-1|p+8k2+2|k|-8|k-1||k|p.Proof.
Defineφ(x,y)=θ(∥x∥Xp+∥y∥Xp)∥u2∥Xr2⋯∥un∥Xrn for all x,y,u2,…,un∈X, and apply Theorems 2.1 and 2.2.The following example shows that the assumptionp≠1 cannot be omitted in Corollary 2.3.Example 2.4.
LetX=ℂ be a linear space over ℝ. Define ∥·,·∥:X×X→ℝ by ∥x1,x2∥=|a1b2-a2b1|, where xj=aj+bji∈ℂ, aj,bj∈ℝ, j=1,2 (i=-1 is the imaginary unit). Then, (X,∥·,·∥) is a 2-normed linear space.
Letϕ:ℂ→ℂ defined by(2.45)ϕ(x)={x,for|x|<1,1,for|x|≥1.
Consider the functionf:ℂ→ℂ defined by(2.46)f(x)=∑m=0∞α-mϕ(αmx)
for all x∈ℂ, where α>|k|. Then, f satisfies the functional inequality
(2.47)‖Df(x,y),u‖≤4α2(|k|+1)α-1(|x|+|y|)|u|
for all x,y,u∈ℂ, but there do not exist an additive mapping A:ℂ→ℂ and a constant d>0 such that ∥f(x)-A(x),u∥≤d|x||u| for all x,u∈ℂ.
It is clear that|f(x)|≤α/(α-1) for all x∈ℂ. If |x|+|y|=0 or |x|+|y|≥1/α for all x,y∈ℂ, then the inequality (2.47) holds. Now suppose that 0<|x|+|y|<1/α. Then, there exists an integer n≥1 such that(2.48)1αn+1≤|x|+|y|<1αn.
Hence, αm|kx±y|<1,αm|x±y|<1,αm|x|<1 for all m=0,1,…,n-1. From the definition of f and (2.48), we obtain that
(2.49)‖Df(x,y),u‖=‖∑m=n∞α-mϕ(αm(kx+y))+∑m=n∞α-mϕ(αm(kx-y))-k∑m=n∞α-mϕ(αm(x+y))-k∑m=n∞α-mϕ(αm(x-y))-2∑m=n∞α-mϕ(αmkx)+2k∑m=n∞α-mϕ(αmx),u‖≤4α2(|k|+1)α-1(|x|+|y|)|u|.
Therefore, f satisfies (2.47). Now, we claim that the functional equation (1.1) is not stable for p=1 in Corollary 2.3. Suppose on the contrary that there exist an additive mapping A:ℂ→ℂ and a constant d>0 such that ∥f(x)-A(x),u∥≤d|x||u| for all x,u∈ℂ. Then, there exists a constant c∈ℂ such that A(x)=cx for all rational numbers x. So, we obtain that
(2.50)‖f(x),u‖≤(d+|c|)|x||u|
for all rational numbers x and all u∈ℂ. Let s∈ℕ with s+1>d+|c|. If x is a rational number in (0,α-s) and u=bi (b∈ℝ), then αmx∈(0,1) for all m=0,1,…,s, and we get
(2.51)‖f(x),u‖=‖∑m=0∞ϕ(αmx)αm,u‖≥∑m=0sϕ(αmx)αm|b|=(s+1)x|b|>(d+|c|)x|b|=(d+|c|)|x||u|,
which contradicts (2.50).Theorem 2.5.
LetX be a linear space and Y an n-Banach space. Let f:X→Y be a mapping with f(0)=0 for which there is a function φ:Xn+1→[0,∞) such that
(2.52)∑j=0∞18jφ(2jx,2jy,u2,…,un)<∞,(2.53)‖Df(x,y),u2,…,un‖Y≤φ(x,y,u2,…,un)
for all x,y,u2,…,un∈X. Then, there is a unique cubic mapping C:X→Y such that
(2.54)‖f(2x)-2f(x)-C(x),u2,…,un‖Y≤∑j=0∞18j+1φ̃(2jx,u2,…,un)
for all x,u2,…,un∈X, where φ̃(x,u2,…,un) is defined as in Theorem 2.1.Proof.
As in the proof of Theorem2.1, we have
(2.55)‖f(4x)-10f(2x)+16f(x),u2,…,un‖Y≤φ̃(x,u2,…,un)
for all x∈X, where φ̃(x,u2,…,un) is defined as in Theorem 2.1.
Now, leth:X→Y be the mapping defined by h(x):=f(2x)-2f(x). By (2.55), we have(2.56)‖h(2x)-8h(x),u2,…,un‖Y≤φ̃(x,u2,…,un)
for all x∈X. Replacing x by 2jx in (2.56) and dividing both sides of (2.56) by 8j+1, we get
(2.57)‖18jh(2jx)-18j+1h(2j+1x),u2,…,un‖Y≤18j+1φ̃(2jx,u2,…,un)
for all x,u2,…,un∈X and all integers j≥0. For all integers l,m with 0≤l<m, we have
(2.58)‖18lh(2lx)-18mh(2mx),u2,…,un‖Y≤∑j=lm-1‖18jh(2jx)-18j+1h(2j+1x),u2,…,un‖Y≤∑j=lm-118j+1φ̃(2jx,u2,…,un)
for all x,u2,…,un∈X. So, we get
(2.59)lim⁡l,m→∞‖18lh(2lx)-18mh(2mx),u2,…,un‖Y=0
for all x,u2,…,un∈X. This shows that the sequence {(1/8j)h(2jx)} is a Cauchy sequence in Y. Since Y is an n-Banach space, the sequence {(1/8j)h(2jx)} converges. So, we can define a mapping C:X→Y by
(2.60)C(x):=lim⁡j→∞18jh(2jx)
for all x∈X. Putting l=0, then passing the limit m→∞ in (2.58), and using Lemma 1.6(4), we get
(2.61)‖h(x)-C(x),u2,…,un‖Y≤∑j=0∞18j+1φ̃(2jx,u2,…,un)
for all x,u2,…,un∈X.
Now we show thatC is cubic. By Lemma 1.6, (2.52), (2.58), and (2.60), we have(2.62)‖C(2x)-8C(x),u2,…,un‖Y=lim⁡j→∞‖18jh(2j+1x)-18j-1h(2jx),u2,…,un‖Y=8lim⁡j→∞‖18j+1h(2j+1x)-18jh(2jx),u2,…,un‖Y≤lim⁡j→∞18jφ̃(2jx,u2,…,un)=0
for all x,u2,…,un∈X. By Lemma 1.6(3), C(2x)=8C(x) for all x∈X. Also, by Lemma 1.6(4), (2.52), (2.53), and (2.60), we get
(2.63)‖DC(x,y),u2,…,un‖Y=lim⁡j→∞18j‖Dh(2jx,2jy),u2,…,un‖Y=lim⁡j→∞18j‖Df(2j+1x,2j+1y)-2Df(2jx,2jy),u2,…,un‖Y≤lim⁡j→∞18j[‖Df(2j+1x,2j+1y),u2,…,un‖Y+2‖Df(2jx,2jy),u2,…,un‖Y]≤lim⁡j→∞18j[φ(2j+1x,2j+1y,u2,…,un)+2φ(2jx,2jy,u2,…,un)]=0
for all x,y,u2,…,un∈X. By Lemma 1.6(3), DC(x,y)=0 for all x,y∈X. Hence the mapping C satisfies (1.1). By [11, Lemma 2.3], the mapping x→C(2x)-2C(x) is cubic. Therefore, C(2x)=8C(x) implies that the mapping C is cubic.
To prove the uniqueness ofC, let S:X→Y be another cubic mapping satisfying (2.54). Fix x∈X. Clearly, C(2lx)=8lA(x) and S(2lx)=8lS(x) for all l∈ℕ. It follows from (2.54) that(2.64)‖C(x)-S(x),u2,…,un‖Y=‖C(2lx)8l-S(2lx)8l,u2,…,un‖Y≤18l[‖f(2l+1x)-2f(2lx)-C(2lx),u2,…,un‖Y+‖S(2lx)-f(2l+1x)+2f(2lx),u2,…,un‖Y]≤18l∑j=0∞18jφ̃(2j+lx,u2,…,un)≤∑j=0∞18j+lφ̃(2j+lx,u2,…,un)=∑j=l∞18jφ̃(2jx,u2,…,un)
for all x,u2,…,un∈X, and l∈ℕ. By (2.52), we see that the right-hand side of the above inequality tends to 0 as l→∞. Therefore, ∥C(x)-S(x),u2,…,un∥Y=0 for all u2,…,un∈X. By Lemma 1.6, we can conclude that C(x)=S(x) for all x∈X. So C=S. This proves the uniqueness of C.Theorem 2.6.
LetX be a linear space and Y an n-Banach space. Let f:X→Y be a mapping with f(0)=0 for which there is a function φ:Xn+1→[0,∞) such that
(2.65)∑j=1∞8jφ(x2j,y2j,u2,…,un)<∞,‖Df(x,y),u2,…,un‖Y≤φ(x,y,u2,…,un)
for all x,y,u2,…,un∈X. Then, there is a unique cubic mapping C:X→Y such that
(2.66)‖f(2x)-2f(x)-C(x),u2,…,un‖Y≤∑j=1∞8j-1φ̃(x2j,u2,…,un)
for all x,u2,…,un∈X, where φ̃(x,u2,…,un) is defined as in Theorem 2.1.Proof.
The proof is similar to the proof of Theorem2.5.Corollary 2.7.
LetX be a normed space and Y an n-Banach space. Let θ∈[0,∞),p,r2,…,rn∈(0,∞) such that p≠3, and let f:X→Y be a mapping with f(0)=0 such that
(2.67)‖Df(x,y),u2,…,un‖Y≤θ(‖x‖Xp+‖y‖Xp)‖u2‖Xr2⋯‖un‖Xrn
for all x,y,u2,…,un∈X. Then, there exists a unique cubic mapping C:X→Y such that
(2.68)‖f(2x)-2f(x)-C(x),u2,…,un‖Y≤θϵ‖x‖Xp‖u2‖Xr2⋯‖un‖Xrn|(8-2p)(k3-k)|
for all x,u2,…,un∈X, where ϵ is defined as in Corollary 2.3.Proof.
Defineφ(x,y)=θ(∥x∥Xp+∥y∥Xp)∥u2∥Xr2⋯∥un∥Xrn for all x,y,u2,…,un∈X, and apply Theorems 2.5 and 2.6.The following example shows that the the generalized Hyers-Ulam stability problem for the case ofp=3 was excluded in Corollary 2.7.Example 2.8.
LetX=ℂ be a linear space over ℝ, and let ∥·,·∥:X×X→ℝ be defined as in Example 2.4. Then, (X,∥·,·∥) is a 2-normed linear space.
Letϕ:ℂ→ℂ be defined by(2.69)ϕ(x)={x3,for|x|<1,1,for|x|≥1.
Consider the functionf:ℂ→ℂ defined by(2.70)f(x)=∑m=0∞α-3mϕ(αmx)
for all x∈ℂ, where α>|k|. Then, f satisfies the functional inequality
(2.71)‖Df(x,y),u‖≤4α6(|k|+1)α3-1(|x|3+|y|3)|u|
for all x,y,u∈ℂ, but there do not exist a cubic mapping C:ℂ→ℂ and a constant d>0 such that ∥f(x)-C(x),u∥≤d|x|3|u| for all x,u∈ℂ.
It is clear that|f(x)|≤α3/(α3-1) for all x∈ℂ. If |x|3+|y|3=0 or |x|3+|y|3≥1/α3 for all x,y∈ℂ, then inequality (2.71) holds. Now suppose that 0<|x|3+|y|3<1/α3. Then, there exists an integer n≥1 such that(2.72)1α3(n+1)≤|x|3+|y|3<1α3n.
Hence, αm|kx±y|<1,αm|x±y|<1,αm|x|<1 for all m=0,1,…,n-1. From the definition of f and (2.72), we obtain that
(2.73)‖Df(x,y),u‖=‖∑m=n∞α-3mϕ(αm(kx+y))+∑m=n∞α-3mϕ(αm(kx-y))-k∑m=n∞α-3mϕ(αm(x+y))-k∑m=n∞α-3mϕ(αm(x-y))-2∑m=n∞α-3mϕ(αmkx)+2k∑m=n∞α-3mϕ(αmx),u‖≤4α6(|k|+1)α3-1(|x|3+|y|3)|u|.
Therefore, f satisfies (2.71). Now, we claim that the functional equation (1.1) is not stable for p=3 in Corollary 2.7. Suppose on the contrary that there exist a cubic mapping C:ℂ→ℂ and a constant d>0 such that ∥f(x)-C(x),u∥≤d|x|3|u| for all x,u∈ℂ. Then, there exists a constant β∈ℂ such that C(x)=βx3 for all rational numbers x. So, we obtain that
(2.74)‖f(x),u‖≤(d+|β|)|x|3|u|
for all rational numbers x and all u∈ℂ. Let s∈ℕ with s+1>d+|β|. If x is a rational number in (0,α-s) and u=bi (b∈ℝ), then αmx∈(0,1) for all m=0,1,…,s, and we get
(2.75)‖f(x),u‖=‖∑m=0∞ϕ(αmx)α3m,u‖≥∑m=0sϕ(αmx)α3m|b|=(s+1)x3|b|>(d+|β|)x3|b|=(d+|β|)|x|3|u|,
which contradicts (2.74).Theorem 2.9.
LetX be a linear space and Y an n-Banach space. Let f:X→Y be a mapping with f(0)=0 for which there is a function φ:Xn+1→[0,∞) such that
(2.76)∑j=0∞12jφ(2jx,2jy,u2,…,un)<∞,(2.77)‖Df(x,y),u2,…,un‖Y≤φ(x,y,u2,…,un)
for all x,y,u2,…,un∈X. Then, there exist a unique additive mapping A:X→Y and a unique cubic mapping C:X→Y such that
(2.78)‖f(x)-A(x)-C(x),u2,…,un‖Y≤16∑j=0∞(12j+1+18j+1)φ̃(2jx,u2,…,un)
for all x,u2,…,un∈X, where φ̃(x,u2,…,un) is defined as in Theorem 2.1.Proof.
By Theorems2.1 and 2.5, there exist an additive mapping A′:X→Y and a cubic mapping C′:X→Y such that
(2.79)‖f(2x)-8f(x)-A′(x),u2,…,un‖Y≤∑j=0∞12j+1φ̃(2jx,u2,…,un),‖f(2x)-2f(x)-C′(x),u2,…,un‖Y≤∑j=0∞18j+1φ̃(2jx,u2,…,un)
for all x,u2,…,un∈X. Hence,
(2.80)‖f(x)+16A′(x)-16C′(x),u2,…,un‖Y≤16∑j=0∞(12j+1+18j+1)φ̃(2jx,u2,…,un)
for all x∈X. So, we obtain (2.78) by letting A(x)=-(1/6)A′(x) and C(x)=(1/6)C′(x) for all x∈X.
To prove the uniqueness ofA and C, let A′′,C′′:X→Y be another additive and cubic mapping satisfying (2.78). Fix x∈X. Let A1=A-A′′ and C1=C-C′′. So,(2.81)‖A1(x)+C1(x),u2,…,un‖Y≤‖f(x)-A(x)-C(x),u2,…,un‖Y+‖f(x)-A′′(x)-C′′(x),u2,…,un‖Y≤13∑j=0∞(12j+1+18j+1)φ̃(2jx,u2,…,un)
for all x,u2,…,un∈X. Then (2.76) implies that
(2.82)lim⁡n→∞18n‖A1(2nx)+C1(2nx),u2,…,un‖Y=0
for all x,u2,…,un∈X. Thus, C1=0. So, it follows from (2.81) that
(2.83)‖A1(x),u2,…,un‖Y≤13∑j=0∞(12j+1+18j+1)φ̃(2jx,u2,…,un)
for all u2,…,un∈X. Therefore, A1=0.Similarly to Theorem2.9, one can prove the following result.Theorem 2.10.
LetX be a linear space and Y an n-Banach space. Let f:X→Y be a mapping with f(0)=0 for which there is a function φ:Xn+1→[0,∞) such that
(2.84)∑j=0∞8jφ(x2j,y2j,u2,…,un)<∞,‖Df(x,y),u2,…,un‖Y≤φ(x,y,u2,…,un)
for all x,y,u2,…,un∈X. Then, there exist a unique additive mapping A:X→Y and a unique cubic mapping C:X→Y such that
(2.85)‖f(x)-A(x)-C(x),u2,…,un‖Y≤16∑j=1∞(2j-1+8j-1)φ̃(x2j,u2,…,un)
for all x,u2,…,un∈X, where φ̃(x,u2,…,un) is defined as in Theorem 2.1.Proof.
The proof is similar to the proof of Theorem2.9 and the result follows from Theorems 2.2 and 2.6.Theorem 2.11.
LetX be a linear space and Y an n-Banach space. Let f:X→Y be a mapping with f(0)=0 for which there is a function φ:Xn+1→[0,∞) such that
(2.86)∑j=1∞2jφ(x2j,y2j,u2,…,un)<∞,∑j=0∞18jφ(2jx,2jy,u2,…,un)<∞,‖Df(x,y),u2,…,un‖Y≤φ(x,y,u2,…,un)
for all x,y,u2,…,un∈X. Then, there exist a unique additive mapping A:X→Y and a unique cubic mapping C:X→Y such that
(2.87)‖f(x)-A(x)-C(x),u2,…,un‖Y≤16[∑j=1∞2j-1φ̃(x2j,u2,…,un)+∑j=0∞18j+1φ̃(2jx,u2,…,un)]
for all x,u2,…,un∈X, where φ̃(x,u2,…,un) is defined as in Theorem 2.1.Proof.
The proof is similar to the proof of Theorem2.9 and the result follows from Theorems 2.2 and 2.5.Corollary 2.12.
LetX be a normed space and Y an n-Banach space. Let θ∈[0,∞),r2,…,rn∈(0,∞), p∈(0,1)∪(1,3)∪(3,∞), and let f:X→Y be a mapping with f(0)=0 such that
(2.88)‖Df(x,y),u2,…,un‖Y≤θ(‖x‖Xp+‖y‖Xp)‖u2‖Xr2⋯‖un‖Xrn
for all x,y,u2,…,un∈X. Then, there exist a unique additive mapping A:X→Y and a unique cubic mapping C:X→Y such that
(2.89)‖f(x)-A(x)-C(x),u2,…,un‖Y≤16|k3-k|(1|2-2p|+1|8-2p|)θϵ‖x‖Xp‖u2‖Xr2⋯‖un‖Xrn
for all x,u2,…,un∈X, where ϵ is defined as in Corollary 2.3.Proof.
Defineφ(x,y)=θ(∥x∥Xp+∥y∥Xp)∥u2∥Xr2⋯∥un∥Xrn for all x,y,u2,…,un∈X, and apply Theorems 2.9–2.11.Remark 2.13.
The generalized Hyers-Ulam stability problem for the cases ofp=1 and p=3 was excluded in Corollary 2.12 (see Examples 2.4 and 2.8).



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*Source: 926390-2012-04-30.xml*