Kyungpook Mathematical Journal 2018; 58(2): 291-305
A General Uniqueness Theorem concerning the Stability of AQCQ Type Functional Equations
Yang-Hi Lee and Soon-Mo Jung*
Department of Mathematics Education, Gongju National University of Education, 32553 Gongju, Republic of Korea, e-mail : yanghi2@hanmail.net, Mathematics Section, College of Science and Technology, Hongik University, 30016 Sejong, Republic of Korea, e-mail: smjung@hongik.ac.kr
*Corresponding Author.
Received: August 24, 2017; Accepted: April 12, 2018; Published online: June 23, 2018.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

In this paper, we prove a general uniqueness theorem which is useful for proving the uniqueness of the relevant additive mapping, quadratic mapping, cubic mapping, quartic mapping, or the additive-quadratic-cubic-quartic mapping when we investigate the (generalized) Hyers-Ulam stability.

Keywords: uniqueness, stability, Hyers-Ulam stability, generalized Hyers-Ulam stability, AQCQ type functional equation, AQCQ mapping
1. Introduction

From now on, we assume that V and W are vector spaces and a is a fixed real number larger than 1. If a mapping f : VW is given, then we set

$f1(x):=fo(ax)-a3fo(x),f2(x):=fe(ax)-a4fe(x),f3(x):=fo(ax)-afo(x),f4(x):=fe(ax)-a2fe(x),Af(x,y):=f(x+y)-f(x)-f(y),Q2f(x,y):=f(x+y)+f(x-y)-2f(x)-2f(y),Cf(x,y):=f(2x+y)-3f(x+y)+3f(y)-f(-x+y)-6f(x),Q4f(x,y):=f(2x+y)-4f(x+y)+6f(y)-4f(-x+y)+f(-2x+y)-24f(x)$

for x, yV, where fo and fe denote the odd and even part of f, respectively.

If a mapping f : VW satisfies the equation Af(x, y) = 0, Q2f(x, y) = 0, Cf(x, y) = 0, or Q4f(x, y) = 0 for all x, yV, then it is called an additive mapping, a quadratic mapping, a cubic mapping, or a quartic mapping, respectively. We notice that the real-valued mappings f(x) = ax, g(x) = ax2, h(x) = ax3, and k(x) = ax4 are solutions to Af(x, y) = 0, Q2f(x, y) = 0, Cg(x, y) = 0, and Q4h(x, y) = 0, respectively.

We call a mapping f : VW an additive-quadratic-cubic-quartic mapping if it is expressed as the sum of an additive mapping, a quadratic mapping, a cubic mapping, and a quartic mapping, and vice versa. An additive-quadratic-cubic-quartic type functional equation is just the functional equation, each of whose solutions is an additive-quadratic-cubic-quartic mapping. For example, the real-valued mapping f(x) = ax4 + bx3 + cx2 + dx defined on ℝ is a solution to the additive-quadratic-cubic-quartic type functional equation.

Whenever we investigate the stability problems for additive-quadratic-cubic-quartic type functional equations or others, we encounter some uniqueness problems. To our best of knowledge, however, no author has succeeded in proving even a uniqueness theorem for these cases ([1, 2, 7, 12, 14, 15, 17]) except our papers ([8, 9, 10, 11]). The ideas of the present paper are strongly based on the previous papers [8, 9, 10, 11]. But this paper includes more general results than the previous three papers.

In this paper, a general uniqueness theorem will be proved which is useful for proving the uniqueness of the relevant additive mapping, quadratic mapping, cubic mapping, quartic mapping, or the additive-quadratic-cubic-quartic mapping when we investigate the (generalized) Hyers-Ulam stability. In Section 3, we make use of our uniqueness theorem to improve the stability theorems presented in [5, 13], where the uniqueness of exact solutions have not been proved.

2. Main Result

We assume that V is a real vector space and Y is a real normed space. The following somewhat surprising theorem states that if for any given mapping f, there exists a mapping F (near f) with some properties (which are possessed by quadratic, cubic, quartic, or possessed by additive-quadratic-cubic-quartic mappings), then F is uniquely determined.

### Theorem 2.1

Assume that a > 1 is a fixed real number and Φ : V {0} → [0,∞) is a function satisfying one of the following properties

$limn→∞ 1anΦ(anx)=0,$$limn→∞ anΦ (xan)=limn→∞ 1a2nΦ(anx)=0,$$limn→∞ a2nΦ (xan)=limn→∞ 1a3nΦ(anx)=0,$$limn→∞ a3nΦ (xan)=limn→∞ 1a4nΦ(anx)=0,$$limn→∞ a4nΦ (xan)=0$

for all xV {0}. Suppose f : VY is an arbitrary mapping. If a mapping F : VY satisfies the inequality

$‖f(x)-F(x)‖ ≤Φ(x)$

for all xV {0} and if F satisfies each of the following equalities

$F1(ax)=aF1(x),F2(ax)=a2F2(x),F3(ax)=a3F3(x),F4(ax)=a4F4(x)$

for all xV, then F is given as

$F(x)={limn→∞ [1a4-a2 (f4(anx)a4n-f2(anx)a2n)+1a3-a (f3(anx)a3n-f1(anx)an)]in the case of (2.1),limn→∞ [1a4-a2 (f4(anx)a4n-f2(anx)a2n)+1a3-a (f3(anx)a3n-anf1 (xan))]in the case of (2.2),limn→∞ [1a4-a2 (f4(anx)a4n-a2nf2 (xan))+1a3-a (f3(anx)a3n-anf1 (xan))]in the case of (2.3),limn→∞ [1a4-a2 (f4(anx)a4n-a2nf2 (xan))+1a3-a (a3nf3 (xan)-anf1 (xan))]in the case of (2.4),limn→∞ [1a4-a2 (a4nf4 (xan)-a2nf2 (xan))+1a3-a (a3nf3 (xan)-anf1 (xan))]in the case of (2.5)$

for all xV {0}. In other words, F is the unique mapping satisfying (2.6) and (2.7).

Proof

Suppose a mapping F satisfies (2.6) and (2.7) for a given f : VY .

(i) We consider F1(x) = Fo(ax) – a3Fo(x). When Φ : V {0} → [0,∞) has the property (2.1), we make use of (2.7) to get

$‖F1(x)-1anf1(anx)‖=1an‖F1(anx)-f1(anx)‖≤1an‖Fo(an+1x)-fo(an+1x)‖+a3an‖fo(anx)-Fo(anx)‖≤12an (Φ(an+1x)+Φ(-an+1x)+a3Φ(anx)+a3Φ(-anx))→0, as n→∞,$

for all xV {0}. Thus, $F1(x)=limn→∞ 1anfo(anx)$ for all xV {0} provided Φ has the property (2.1).

When Φ : V {0} → [0,∞) has the property (2.2), (2.3), (2.4), or (2.5), then we use (2.7) to show that

$‖F1(x)-anf1 (xan)‖=an‖F1 (xan)-f1 (xan)‖≤an‖Fo (axan)-fo (axan)‖+an+3‖fo (xan)-Fo (xan)‖≤an2 (Φ (axan)+Φ (-axan)+a3Φ (xan)+a3Φ (-xan))→0, as n→∞,$

for all xV {0}. Hence, $Fo(x)=limn→∞anf1 (xan)$ for all xV {0} provided Φ has the property (2.2), (2.3), (2.4), or (2.5).

(ii) We consider the mapping F2(x) = Fe(ax)–a4Fe(x). When Φ : V {0} → [0,∞) has the property (2.1) or (2.2), then we apply (2.7) to verify

$‖F2(x)-1a2nf2(anx)‖=1a2n‖F2(anx)-f2(anx)‖≤1a2n‖(Fe-fe)(an+1x)‖+1a2n‖a4(Fe-fe)(anx)‖≤12a2n (Φ(an+1x)+Φ(-an+1x)+a4Φ(anx)+a4Φ(-anx))→0, as n→∞,$

for all xV {0}. Then, $F2(x)=limn→∞ 1a2n f2 (anx)$ is true for all xV {0} provided Φ has the property (2.1) or (2.2).

When Φ : V {0} → [0,∞) satisfies (2.3), (2.4) or (2.5), we get

$‖F2(x)-a2nf2 (xan)‖=a2n‖F2 (xan)-f2 (xan)‖≤a2n‖(Fe-fe) (xan-1)‖+a2n‖a4(Fe-fe) (xan)‖≤a2n2 (Φ (xan-1)+Φ (-xan-1)+a4Φ (xan)+a4Φ (-xan))→0, as n→∞,$

for all xV {0}. Thus, $F2(x)=limn→∞ a2n f2 (xan)$ for all xV {0} provided Φ has the property (2.3), (2.4) or (2.5). (iii) We now consider F3(x) = Fo(ax) – aFo(x). When Φ : V {0} → [0,∞) has the property (2.1), (2.2) or (2.3), then we make use of (2.7) to see

$‖F3(x)-1a3nf3(anx)‖=1a3n‖F3(anx)-f3(anx)‖≤1a3n‖(Fo-fo)(an+1x)‖+1a3n‖a(Fo-fo)(anx)‖≤12a3n (Φ (an+1x)+Φ(-an+1x)+aΦ(anx)+aΦ(-anx))→0, as n→∞,$

for all xV {0}. Hence, $F3(x)=limn→∞ 1a3n f3 (anx)$ holds for all xV {0} provided Φ has the property (2.1), (2.2), or (2.3).

When Φ : V {0} → [0,∞) has the property (2.4) or (2.5), we then obtain

$‖F3(x)-a3nf3 (xan)‖=a3n‖F3 (xan)-f3 (xan)‖≤a3n‖(Fo-fo) (xan-1)‖+a2n‖a(Fo-fo) (xan)‖≤a3n2 (Φ (xan-1)+Φ (-xan-1)+aΦ (xan)+aΦ (-xan))→0, as n→∞,$

for all xV {0}. Therefore, we have $F3(x)=limn→∞ a3n f3 (xan)$ for all xV {0} provided Φ satisfies (2.4) or (2.5).

(iv) Finally, we consider F4(x) = Fe(ax) – a2Fe(x). If Φ : V {0} → [0,∞) has the property (2.1), (2.2), (2.3) or (2.4), then it follows from (2.7) that

$‖F4(x)-1a4nf4(anx)‖=1a4n‖F4(anx)-f4(anx)‖≤1a4n‖(Fe-fe)(an+1x)‖+1a4n‖a2(fe-Fe)(anx)‖≤12a4n (Φ (an+1x)+Φ(-an+1x)+a2Φ(anx)+a2Φ(-anx))→0, as n→∞,$

for all xV {0} provided Φ has the property (2.1), (2.2), (2.3), or (2.4). That is, $F4(x)=limn→∞ 1a4n f4 (anx)$ holds for all xV {0}.

Now, we deal with the case when Φ : V {0} → [0,∞) satisfies (2.5). It holds that

$‖F4(x)-a4nf4 (xan)‖=a4n‖F4 (xan)-f4 (xan)‖≤a4n‖(Fe-fe) (xan-1)‖+a4n‖a2(fe-Fe) (xan)‖≤a4n2 (Φ (xan-1)+Φ (-xan-1)+a2Φ (xan)+a2Φ (-xan))→0, as n→∞,$

for all xV {0}. Thus, it holds that $F4(x)=limn→∞ a4n f4 (xan)$ for all xV {0} provided Φ satisfies (2.5).

Consequently, since $F(x)=F4(x)-F2(x)a4-a2+F3(x)-F1(x)a3-a$, F is expresses as one of equalities in (2.8) and F is uniquely determined in each case.

3. Applications

Theorem 2.1 seems to be impractical for applications in general cases. Thus, it is necessary to introduce some corollaries which are easily applicable to the uniqueness problems for the generalized Hyers-Ulam stability. For the exact definition of the generalized Hyers-Ulam stability, we refer the reader to [3, 6].

### Corollary 3.1

Assume that a > 1 is a fixed real number and φ : V {0} → [0,∞) satisfies either

$Φ(x):=∑i=0∞1aiφ(aix)<∞ for all x∈V{0}$

or

$Φ(x):=∑i=0∞a4iφ(xai)<∞ for all x∈V{0}.$

Suppose f : VY is an arbitrary mapping. If a mapping F : VY satisfies (2.6) for all xV {0} and (2.7) for all xV, then F is uniquely determined.

Proof

When φ satisfies (3.1), it is obvious that

$limn→∞ 1anΦ(anx)=limn→∞∑i=0∞1an+iφ(an+ix)=limn→∞∑i=n∞1aiφ(aix)=0,$

i.e., Φ has the property (2.1) for all xV {0}. For the case of (3.2), it is clear that

$limn→∞ a4nΦ (xan)=limn→∞∑i=0∞a4n+4iφ(xan+i)=limn→∞∑i=n∞a4iφ(xai)=0,$

i.e., Φ has the property (2.5) for all xV {0}. Hence, our assertion is true in view of Theorem 2.1.

### Corollary 3.2

Assume that a > 1 is a fixed real number and the functions φ, ψ : V {0} → [0,∞) satisfy each of the following conditions

$∑i=0∞aiψ (xai)<∞,∑i=0∞1a2iφ(aix)<∞,Φ˜(x):=∑i=0∞aiφ (xai)<∞,Ψ˜(x):=∑i=0∞1a2iψ(aix)<∞$

for all xV {0}. Suppose f : VY is an arbitrary mapping. If a mapping F : VY satisfies the inequality

$‖f(x)-F(x)‖ ≤Φ˜(x)+Ψ˜(x)$

for all xV {0} and if F satisfies each condition of (2.7) for all xV, then F is uniquely determined.

Proof

We set Φ(x) := Φ̃(x) + Ψ̃(x) and then use (3.3) to obtain

$1a4nΦ(a2nx)=∑i=0∞1a4n-iφ(a2n-ix)+∑i=0∞1a4n+2iψ(a2n+ix)$

for all xV {0}. We make change of the summation indices in the preceding equality with j = i – 2n and k = 2n + i to get

$1a4nΦ(a2nx)=1a2n∑j=-2n∞ajφ (xaj)+∑k=2n∞1a2kψ(akx)=1a2n∑i=12n1aiφ(aix)+1a2n∑i=0∞aiφ(xai)+∑i=2n∞1a2iψ(aix)=1an∑i=1n-1aian1a2iφ(aix)+∑i=n2naia2n1a2iφ(aix)+1a2nΦ˜(x)+∑i=2n∞1a2iψ(aix)≤1an∑i=1∞1a2iφ(aix)+∑i=n∞1a2iφ(aix)+1a2nΦ˜(x)+∑i=2n∞1a2iψ(aix)$

for any xV {0}. Hence, we obtain

$limn→∞ 1a4nΦ(a2nx)=0$

for all xV {0}.

On the other hand, we make use of the above equality to prove

$limn→∞ 1a4n+2Φ(a2n+1x)=1a2limn→∞ 1a4nΦ(a2nax)=0$

for all xV {0}. Using the two equalities above, we get

$limn→∞ 1a2nΦ(anx)=0$

for all xV {0}.

Similarly, it holds that

$a2nΦ (xa2n)=∑i=0∞a2n+iφ (xa2n+i)+∑i=0∞1a2i-2nψ(ai-2nx)$

for all xV {0}. If we make change of the summation indices in the last equality with j = i + 2n and k = i – 2n, then we get

$a2nΦ (xa2n)=∑j=-2n∞ajφ (xaj)+1a2n∑k=-2n∞1a2kψ(akx)=∑i=2n∞aiφ (xai)+1a2n∑i=12na2iψ (xai)+1a2n∑i=0∞1a2iψ(aix)=∑i=2n∞aiφ (xai)+1an∑i=1n-1aianaiψ (xai)+∑i=n2naia2naiψ (xai)+1a2nΨ˜(x)≤∑i=2n∞aiφ (xai)+1an∑i=1∞aiψ (xai)+∑i=n∞aiψ (xai)+1a2nΨ˜(x)$

for any xV {0}. Thus, we obtain

$limn→∞ a2nΦ (xa2n)=0,limn→∞ a2n+1Φ (xa2n+1)=alimn→∞ a2nΦ (1a2nxa)=0$

for any xV {0}. Hence, it holds that

$limn→∞ anΦ (xan)=0$

for xV {0}.

Theorem 2.1 implies that our conclusion for this corollary is true.

### Corollary 3.3

Assume that a > 1 is a fixed real number and the functions φ, ψ : V {0} → [0,∞) satisfy each of the following conditions

$∑i=0∞a2iψ (xai)<∞,∑i=0∞1a3iφ(aix)<∞,Φ˜(x):=∑i=0∞a2iφ (xai)<∞,Ψ˜(x):=∑i=0∞1a3iψ(aix)<∞$

for all xV {0}. Suppose f : VY is an arbitrary mapping. If a mapping F : VY satisfies the inequality

$‖f(x)-F(x)‖ ≤Φ˜(x)+Ψ˜(x)$

for all xV {0} and each of the conditions in (2.7) for all xV, then F is uniquely determined.

Proof

We set Φ(x) := Φ̃(x) + Ψ̃(x) and then use (3.5) to show

$1a6nΦ(a2nx)=∑i=0∞1a6n-2iφ(a2n-ix)+∑i=0∞1a6n+3iψ(a2n-ix)$

for all xV {0}. We change the summation indices in the preceding equality with j = i – 2n and k = 2n + i to get

$1a6nΦ(a2nx)=1a2n∑j=-2n∞a2jφ (xaj)+∑k=2n∞1a3kψ(akx)=1a2n∑i=12n1a2iφ(aix)+1a2n∑i=0∞a2iφ(xai)+∑i=2n∞1a3iψ(aix)=1an∑i=1n-11a2i+nφ(aix)+∑i=n2n1a2n+2iφ(aix)+1a2nΦ˜(x)+∑i=2n∞1a3iψ(aix)≤1an∑i=1∞1a3iφ(aix)+∑i=n∞1a3iφ(aix)+1a2nΦ˜(x)+∑i=2n∞1a3iψ(aix)$

for any xV {0}. Hence, we get

$limn→∞ 1a6nΦ(a2nx)=0$

for all xV {0}.

On the other hand, it follows from the above equality that

$limn→∞ 1a6n+3Φ(a2n+1x)=1a3limn→∞ 1a6nΦ(a2nax)=0$

for each xV {0}. By two equalities above, it holds that

$limn→∞ 1a3nΦ(anx)=0$

for all xV {0}.

Similarly, we have

$a4nΦ (xa2n)=∑i=0∞a4n+2iφ (xa2n+i)+∑i=0∞1a3i-4nψ(ai-2nx)$

for all xV {0}. If we change the summation indices in the last equality with j = i + 2n and k = i – 2n, then we get

$a4nΦ (xa2n)=∑j=2n∞a2jφ (xaj)+1a2n∑k=-2n∞1a3kψ(akx)=∑i=2n∞a2iφ (xai)+1a2n∑i=12na3iψ (xai)+1a2n∑i=0∞1a3iψ(aix)=∑i=2n∞a2iφ (xai)+1an∑i=1n-1aiana2iψ (xai)+∑i=n2naia2na2iψ (xai)+1a2nΨ˜(x)≤∑i=2n∞a2iφ (xai)+1an∑i=1∞a2iψ (xai)+∑i=n∞a2iψ (xai)+1a2nΨ˜(x)$

for any xV {0}. Thus, we obtain

$limn→∞ a4nΦ (xa2n)=0,limn→∞ a4n+2Φ (xa2n+1)=a2limn→∞ a4nΦ (1a2nxa)=0$

for every xV {0}. Hence, we have

$limn→∞ a2nΦ (xan)=0$

for each xV {0}. Theorem 2.1 implies that our assertion is true.

### Corollary 3.4

Assume that a > 1 is a fixed real number and the functions φ, ψ : V {0} → [0,∞) satisfy each of the following conditions

$∑i=0∞a3iψ (xai)<∞,∑i=0∞1a4iφ(aix)<∞,Φ˜(x):=∑i=0∞a3iφ (xai)<∞,Ψ˜(x):=∑i=0∞1a4iψ(aix)<∞$

for all xV {0}. Suppose f : VY is an arbitrary mapping. If a mapping F : VY satisfies the inequality

$‖f(x)-F(x)‖ ≤Φ˜(x)+Ψ˜(x)$

for all xV {0} as well as the conditions in (2.7) for all xV, then F is uniquely determined.

Proof

Let us set Φ(x) := Φ̃(x) + Ψ̃(x) and use (3.7) to get

$1a4nΦ(anx)=∑i=0∞1a4n-3iφ(an-ix)+∑i=0∞1a4n+4iψ(an+ix)$

for all xV {0}. We make change of the summation indices in the preceding equality with j = in and k = n + i to get

$1a4nΦ(anx)=1a4n∑j=-n∞a3jφ (xaj)+∑k=n∞1a4kψ(akx)=1a4n∑i=1n1a3iφ(aix)+1a4n∑i=0∞a3iφ (xai)+∑i=n∞1a4iψ(aix)=1a3n∑i=1naian1a4iφ(aix)+1a4nΦ˜(x)+∑i=n∞1a4iψ(aix)≤1a3n∑i=1∞1a4iφ(aix)+1a4nΦ˜(x)+∑i=n∞1a4iψ(aix)$

for any xV {0}. Hence, we get

$limn→∞ 1a4nΦ(anx)=0$

for all xV {0}.

Similarly, we obtain

$a3nΦ (xan)=∑i=0∞a3n+3iφ (xan+i)+∑i=0∞1a4i-3nψ(ai-nx)$

for all xV {0}. If we change the summation indices in the last equality with j = i + n and k = in, then we get

$a3nΦ (xan)=∑j=n∞a3jφ (xaj)+1a3n∑k=-n∞1a4kψ(akx)=∑i=n∞a3iφ (xai)+1a3n∑i=1na4iψ (xai)+1a3n∑i=0∞1a4iψ(aix)=∑i=n∞a3iφ (xai)+1a2n∑i=1naiana3iψ (xai)+1a3nΨ˜(x)≤∑i=n∞a3iφ (xai)+1a2n∑i=1∞a3iψ (xai)+1a3nΨ˜(x)$

for any xV {0}. Thus, we get

$limn→∞ a3nΦ (xan)=0$

for each xV {0}. Theorem 2.1 implies that our conclusion is true.

The following corollary states that if, for any given mapping f, there exists an additive, a quadratic, a cubic, a quartic, or an additive-quadratic-quartic mapping F near f, then F is uniquely determined.

### Corollary 3.5

Assume that a > 1 is a fixed rational number and a function φ : V {0} → [0,∞) satisfies the condition (3.1) or (3.2). Suppose f : VY is an arbitrary mapping. If an additive, a quadratic, a cubic, a quartic, or an additive-quadratic-cubic-quartic mapping F : VY satisfies the inequality (2.6), then F is uniquely determined.

The following corollaries are immediate consequences of Corollaries 3.2, 3.3, and 3.4, respectively, because each of additive, quadratic, cubic, quartic, and additive-quadratic-cubic-quartic mapping satisfies the conditions in (2.7) for any given rational number a > 1.

### Corollary 3.6

Assume that a > 1 is a fixed rational number and φ, ψ : V {0} → [0,∞) satisfy each of the conditions in (3.3). Suppose f : VY is an arbitrary mapping. If an additive, a quadratic, a cubic, a quartic, or an additive-quadratic-cubic-quartic mapping F : VY satisfies (3.4), then F is uniquely determined.

### Corollary 3.7

Assume that a > 1 is a fixed rational number and φ, ψ : V {0} → [0,∞) satisfy each of the conditions in (3.5). Suppose f : VY is an arbitrary mapping. If an additive, a quadratic, a cubic, a quartic, or an additive-quadratic-cubic-quartic mapping F : VY satisfies (3.6), then F is uniquely determined.

### Corollary 3.8

Assume that a > 1 is a fixed rational number and φ, ψ : V {0} → [0,∞) satisfy each of the conditions in (3.7). Suppose f : VY is an arbitrary mapping. If an additive, a quadratic, a cubic, a quartic, or an additive-quadratic-cubic-quartic mapping F : VY satisfies (3.8), then F is uniquely determined.

If we set Φ(x) := θ||x||p for some constants pR{1, 2, 3, 4} and θ > 0, then

$Φ has the property {(2.1)for p<1,(2.2)for 14.$

Hence, by Theorem 2.1, we have the following corollary concerning the Hyers-Ulam-Rassias stability. (For the exact definition of the Hyers-Ulam-Rassias stability, we refer to [4, 16, 18].)

### Corollary 3.9

Let p ∉ {1, 2, 3, 4} and θ > 0 be real constants, let X and Y be real normed spaces, and let f : XY be an arbitrary mapping. If a mapping F : XY satisfies the inequality

$‖f(x)-F(x)‖ ≤θ‖x‖p$

for all xX{0} as well as (2.7) for all xX, then F is uniquely determined.

Since each of additive, quadratic, cubic, quartic, or additive-quadratic-cubic-quartic mappings satisfies the conditions in (2.7), using Corollary 3.9, we can easily prove the following corollary.

### Corollary 3.10

Let p ∉ {1, 2, 3, 4} and θ > 0 be real constants, let X and Y be real normed spaces, and let f : XY be an arbitrary mapping. If an additive, a quadratic, a cubic, a quartic, or an additive-quadratic-cubic-quartic mapping F : XY satisfies the inequality

$‖f(x)-F(x)‖ ≤θ‖x‖p$

for all xX{0}, then F is uniquely determined.

Acknowledgements

Soon-Mo Jung was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2016R1D1A1B03931061).

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