# Stability of then-Dimensional Mixed-Type Additive and Quadratic Functional Equation in Non-Archimedean Normed Spaces

Authors: Yang-Hi Lee; Soon-Mo Jung; Themistocles M. Rassias
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

We will prove the stability of the functional equation2f(∑i=1nxi)+∑1≤i,j≤n,i≠jf(xi-xj)=(n+1)∑i=1nf(xi)+(n-1)∑i=1nf(-xi) in non-Archimedean normed spaces.


---

## Body

## 1. Introduction

A classical question in the theory of functional equations is “when is it true that a function, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?’’ Such a problem, calleda stability problem of the functional equation, was formulated by Ulam in 1940 (see [1]). In the following year, Hyers [2] gave a partial solution of Ulam’s problem for the case of approximate additive functions. Subsequently, his result was generalized by Aoki [3] for additive functions and by Rassias [4] for linear functions. Indeed, they considered the stability problem for unbounded Cauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians (see [5–23]).A non-Archimedean field is a field𝕂 equipped with a function (valuation) |·|:𝕂→[0,∞) such that(F1)
|r|=0 if and only if r=0;(F2)
|rs|=|r||s|;(F3)
|r+s|≤max{|r|,|s|} for all r,s∈𝕂.Clearly, it holds that|1|=|-1|=1 and |n|≤1 for all n∈ℕ.LetX be a vector space over a scalar field 𝕂 with a non-Archimedean and nontrivial valuation |·|. A function ∥·∥:X→ℝ is a non-Archimedean norm (valuation) if it satisfies the following conditions:(N1)
∥x∥=0 if and only if x=0;(N2)
∥rx∥=|r|∥x∥ for all r∈𝕂 and x∈X;(N3)
∥x+y∥≤max{∥x∥,∥y∥} for all x,y∈X.Then(X,∥·∥) is called a non-Archimedean space. Due to the fact that
(1.1)‖xn-xm‖≤max⁡m≤i<n‖xi+1-xi‖(n>m),
a sequence {xn} is Cauchy if and only if {xn+1-xn} converges to zero in a non-Archimedean space. A complete non-Archimedean space is a non-Archimedean space in which every Cauchy sequence is convergent.Recently, Moslehian and Rassias [24] proved the Hyers-Ulam stability of the Cauchy functional equation
(1.2)f(x+y)=f(x)+f(y),
and the quadratic functional equation
(1.3)f(x+y)+f(x-y)=2f(x)+2f(y)
in non-Archimedean normed spaces.We now consider then-dimensional mixed-type quadratic and additive functional equation(1.4)2f(∑i=1nxi)+∑1≤i,j≤n,i≠jf(xi-xj)=(n+1)∑i=1nf(xi)+(n-1)∑i=1nf(-xi),
whose solution is called a quadratic-additive function.In 2009, Towanlong and Nakmahachalasint [25] obtained a stability result for the functional equation (1.4), in which they constructed a quadratic-additive function F by composing an additive function A and a quadratic function Q, where A and Q approximate the odd part and the even part of the given function f, respectively.In this paper, we investigate a general stability problem for then-dimensional mixed-type quadratic and additive functional equation (1.4) in non-Archimedean normed spaces.


## 2. Solutions of (1.4)

In this section, we prove the generalized Hyers-Ulam stability of then-dimensional mixed-type quadratic and additive functional equation (1.4). Assume that H is an additive group and X is a complete non-Archimedean space.For a given functionf:H→X, we use the abbreviations
(2.1)fe(x)∶=f(x)+f(-x)2,fo(x)∶=f(x)-f(-x)2,Af(x,y)∶=f(x+y)-f(x)-f(y),Qf(x,y)∶=f(x+y)+f(x-y)-2f(x)-2f(y),Dnf(x1,x2,…,xn)∶=2f(∑i=1nxi)+∑1≤i,j≤n,i≠jf(xi-xj)-(n+1)∑i=1nf(xi)-(n-1)∑i=1nf(-xi)
for all x,y,x1,x2,…,xn∈H and for an arbitrarily fixed n∈ℕ.Theorem 2.1.
Assume thatn≥2 is an integer. Let H and X be an additive group and a complete non-Archimedean space, respectively. A function f:H→X is a solution of (1.4) if and only if fe is quadratic, fo is additive, and fe(0)=0.Proof.
If a functionf:H→X is a solution of (1.4), then we have fe(0)=0,
(2.2)Qfe(x,y)=fe(x+y)+fe(x-y)-2fe(x)-2fe(y)=12Dnfe(x,y,0,…,0)+12(n-2)(n+3)fe(0)=0,Afo(x,y)=fo(x+y)-fo(x)-fo(y)=12Dnfo(x,y,0,…,0)=0
for all x,y∈H, that is, fe is quadratic and fo is additive.
Conversely, assume thatfe is quadratic, fo is additive, and fe(0)=0. We apply an induction on j to prove Dnfe(x1,x2,…,xn)=0 for all x1,x2,…,xn∈H. For j=2, we have
(2.3)Dnfe(x1,x2,0,…,0)=2fe(x1+x2)+2fe(x1-x2)-4fe(x1)-4fe(x2)-(n-2)(n+3)fe(0)=0.
If n>2 and Dnfe(x1,x2,…,xj,0,…,0)=0 for some integer j  (2≤j<n) and for all x1,x2,…,xj∈H, then a routine calculation yields
(2.4)Dnfe(x1,x2,…,xj+1,0,…,0)=Qfe(x1+⋯+xj,xj+1-xj)+12Dnfe(x1,…,xj-1,2xj,0,…,0)+12Dnfe(x1,…,xj-1,2xj+1,0,…,0)-∑k=1j-1(Qfe(xk,xj)+Qfe(xk,xj+1))-j2Qfe(xj+1,xj+1)-j2Qfe(xj,xj)=0
for all x1,x2,…,xj+1∈H. Hence, we conclude that
(2.5)Dnfe(x1,x2,…,xn)=0
for all x1,x2,…,xn∈H.
Sincefo is additive, a long calculation yields
(2.6)Dnfo(x1,x2,…,xn)=∑1≤i,j≤n,i≠jAfo(xi,-xj)+2∑i=1n-1Afo(∑j=1ixj,xi+1)=0.
Hence, it follows from (2.5) and (2.6) that
(2.7)Dnf(x1,x2,…,xn)=Dnfe(x1,x2,…,xn)+Dnfo(x1,x2,…,xn)=0
for all x1,x2,…,xn∈H; that is, f is a solution of (1.4).


## 3. Generalized Hyers-Ulam Stability of (1.4)

In the following theorem, we will investigate the stability problem of the functional equation (1.4).Theorem 3.1.
Assume thatn≥2 is an integer. Let H and X be an additive group and a complete non-Archimedean space, respectively. Assume that φ:Hn→[0,∞) is a function such that
(3.1)lim⁡m→∞φ(nmx1,nmx2,…,nmxn)|n|2m=0
for all x1,x2,…,xn∈H. Moreover, assume that the limit
(3.2)φ̃(x):=lim⁡m→∞max⁡0≤i<m{φ(nix,…,nix)|4||n|2i+2,φ(-nix,…,-nix)|4||n|2i+2}
exists for each x∈H. If a function f:H→X satisfies the inequality
(3.3)‖Dnf(x1,x2,…,xn)‖≤φ(x1,x2,…,xn)
for any x1,x2,…,xn∈H, then there exists a unique quadratic-additive function T:H→X such that
(3.4)‖f(x)-T(x)‖≤φ̃(x)
for each x∈H. In particular, T is given by
(3.5)T(x)=lim⁡m→∞(f(nmx)+f(-nmx)2n2m+f(nmx)-f(-nmx)2nm)
for all x∈H.Proof.
If we replacexi in (3.1) with 0 for each i∈{1,2,…,n}, then we have
(3.6)lim⁡m→∞φ(0,0,…,0)|n|2m=0.
Since |n|≤1, it holds that φ(0,0,…,0)=0 and
(3.7)‖(n-1)(n+2)f(0)‖=‖Dnf(0,0,…,0)‖≤φ(0,0,…,0)=0.
Hence, we conclude that f(0)=0.
LetJmf:H→Y be a function defined by
(3.8)Jmf(x)=f(nmx)+f(-nmx)2n2m+f(nmx)-f(-nmx)2nm
for all x∈H and m∈{0,1,2,…}. A tedious calculation, together with (F2), (N3), and (3.3), yields
(3.9)‖Jif(x)-Ji+1f(x)‖=‖-Dnf(nix,…,nix)4n2i+2-Dnf(-nix,…,-nix)4n2i+2-Dnf(nix,…,nix)4ni+1+Dnf(-nix,…,-nix)4ni+1‖≤max⁡{∥Dnf(nix,…,nix)∥|4||n|2i+2,∥Dnf(-nix,…,-nix)∥|4||n|2i+2,‖Dnf(nix,…,nix)‖|4||n|i+1,‖Dnf(-nix,…,-nix)‖|4||n|i+1}≤max⁡{φ(nix,…,nix)|4||n|2i+2,φ(-nix,…,-nix)|4||n|2i+2}
for all x∈H and i∈{0,1,2,…}. It follows from (3.1) and (3.9) that the sequence {Jmf(x)} is Cauchy. Since X is complete, we conclude that {Jmf(x)} is convergent.
Let us define(3.10)T(x):=lim⁡m→∞Jmf(x)
for any x∈H. It follows from (N3) and (3.9) that
(3.11)‖f(x)-Jmf(x)‖=‖∑i=0m-1(Jif(x)-Ji+1f(x))‖≤max⁡0≤i<m‖Jif(x)-Ji+1f(x)‖≤max⁡0≤i<m{φ(nix,…,nix)|4||n|2i+2,φ(-nix,…,-nix)|4||n|2i+2}
for all m∈{0,1,2,…} and x∈H. In view of (3.2), if we let m→∞ in (3.11), then we obtain the inequality (3.4).
Replacingxi in (3.3) with nmxi for i∈{1,2,…,n} and considering (F2) and (N3), we get
(3.12)‖DnJmf(x1,x2,…,xn)‖=‖Dnf(nmx1,…,nmxn)-Dnf(-nmx1,…,-nmxn)2nm+Dnf(nmx1,…,nmxn)+Dnf(-nmx1,…,-nmxn)2n2m‖≤max⁡{φ(-2mx1,…,-2mxn)|2||n|2mφ(nmx1,…,nmxn)|2||n|m,φ(-nmx1,…,-nmxn)|2||n|m,φ(nmx1,…,nmxn)|2||n|2m,φ(-2mx1,…,-2mxn)|2||n|2m}
for all m∈{0,1,2,…} and x1,x2,…,xn∈H. If we let m→∞ in the last inequality, then it follows from the condition (3.1) that DnT(x1,x2,…,xn)=0 for all x1,x2,…,xn∈H; that is, T is a quadratic-additive function.
Assume thatT′:H→X is another quadratic-additive function satisfying (3.4). By the definition of Dn, a routine calculation yields
(3.13)-DnT′(nix,…,nix)4n2i+2-DnT′(-nix,…,-nix)4n2i+2-DnT′(nix,…,nix)4ni+1+DnT′(-nix,…,-nix)4ni+1=-12n2(i+1)(T′(ni+1x)+T′(-ni+1x))+12n2i(T′(nix)+T′(-nix))-12ni+1(T′(ni+1x)-T′(-ni+1x))+12ni(T′(nix)-T′(-nix))
for each i∈{0,1,2,…} and x∈H. Hence, it follows from (3.8) that
(3.14)∑i=0k-1(-DnT′(nix,…,nix)4n2i+2-DnT′(-nix,…,-nix)4n2i+2-DnT′(nix,…,nix)4ni+1+DnT′(-nix,…,-nix)4ni+1)=T′(x)-JkT′(x)
for any k∈ℕ and x∈H. Since T′ is a solution of (1.4), it follows from the last equality that
(3.15)T′(x)=JkT′(x)
for any k∈ℕ and x∈H. Obviously, this equality also holds for T.
Consequently, by considering that|n|≤1, it follows from (N3), (3.1), (3.4), and (3.8) that
(3.16)‖T(x)-T′(x)‖=lim⁡k→∞‖JkT(x)-JkT′(x)‖≤lim⁡k→∞max⁡{‖JkT(x)-Jkf(x)‖,‖Jkf(x)-JkT′(x)‖}≤lim⁡k→∞|2|-1|n|-2kmax⁡{‖T(nkx)-f(nkx)‖,‖T(-nkx)-f(-nkx)‖,‖f(nkx)-T′(nkx)‖,‖f(-nkx)-T′(-nkx)‖}≤lim⁡k→∞lim⁡m→∞max⁡k≤i<m+k{φ(nix,…,nix)|8||n|2i+2,φ(-nix,…,-nix)|8||n|2i+2}=0
for all x∈H. Therefore, T=T′, which proves the uniqueness of T.Corollary 3.2.
LetX and Y be non-Archimedean normed spaces over 𝕂 with |n|<1. If Y is complete and f:X→Y satisfies the inequality
(3.17)‖Df(x1,x2,…,xn)‖≤θ∑i=1n‖xi‖r
for all x1,x2,…,xn∈X and for some r>2, then there exists a unique quadratic-additive function T:X→Y such that
(3.18)‖f(x)-T(x)‖≤nθ|4||n|2‖x‖r
for all x∈X.Proof.
Letφ(x1,x2,…,xn)=θ∑i=1n∥xi∥r. Since |n|<1 and r-2>0, we get
(3.19)lim⁡m→∞|n|-2mφ(nmx1,nmx2,…,nmxn)=lim⁡m→∞|n|m(r-2)φ(x1,x2,…,xn)=0
for all x1,x2,…,xn∈X. Therefore, the conditions of Theorem 3.1 are satisfied. Indeed, it is easy to see that φ̃(x)=nθ(|4|-1|n|-2)∥x∥r. By Theorem 3.1, there exists a unique quadratic-additive function T:X→Y such that (3.18) holds.



---

*Source: 401762-2012-04-02.xml*