# A General Uniqueness Theorem concerning the Stability of Additive and Quadratic Functional Equations

Authors: Yang-Hi Lee; Soon-Mo Jung
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We prove a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional
equations. This uniqueness theorem can replace the repeated proofs for uniqueness of the relevant solutions of given equations while
we investigate the stability of functional equations.


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## Body

## 1. Introduction

LetG1 and G2 be abelian groups. For any mapping f:G1→G2, we define
(1)Afx,y∶=fx+y-fx-fy,Qfx,y∶=fx+y+fx-y-2fx-2fy
for all x,y∈G1. A mapping f:G1→G2 is called an additive mapping (or a quadratic mapping) if f satisfies the functional equation Af(x,y)=0 (or Qf(x,y)=0) for all x,y∈G1. We notice that the mappings g,h:R→R given by g(x)=ax and h(x)=ax2 are solutions of Ag(x,y)=0 and Qh(x,y)=0, respectively.A mappingf:G1→G2 is called a quadratic-additive mapping if and only if f is represented by the sum of an additive mapping and a quadratic mapping. A functional equation is called a quadratic-additive type functional equation if and only if each of its solutions is a quadratic-additive mapping. The mapping f:R→R given by f(x)=ax2+bx is a solution of the quadratic-additive type functional equation. In the study of the stability problems of quadratic-additive type functional equations, the uniqueness problem frequently occurs under various conditions. However, no author has succeeded in proving the relevant uniqueness (see [1–3]).In this paper, we prove a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations. In Section4, we apply our uniqueness theorem to complement stability theorems of the papers [4, 5] where the uniqueness has not been proved. Indeed, this uniqueness theorem can save us much trouble in proving the uniqueness of relevant solutions repeatedly appearing in the stability problems for various quadratic-additive type functional equations.


## 2. Main Result

LetX be a real vector space and Y a real normed space. For any mapping f:X→Y, fe and fo denote the even part and the odd part of f, respectively.In the following theorem, we prove that if, for any given mappingf, there exists a mapping F (near f) with some properties possessed by additive or quadratic or quadratic-additive mappings, then the mapping F is uniquely determined.Theorem 1.
Leta>1 be a real constant, let Φ:X∖{0}→[0,∞) be a function satisfying one of the following conditions,
(2)limn→∞1anΦanx=0,(3)limn→∞anΦxan=limn→∞1a2nΦanx=0,(4)limn→∞a2nΦxan=0,
for all x∈X∖{0}, and let f:X→Y be an arbitrarily given mapping. If there exists a mapping F:X→Y such that
(5)fx-Fx≤Φx
for all x∈X∖{0} and
(6)Feax=a2Fex,Foax=aFox
for all x∈X, then F is determined by
(7)Fx=limn→∞1a2nfeanx+1anfoanxifΦsatisfies2limn→∞1a2nfeanx+anfoxanifΦsatisfies3limn→∞a2nfexan+anfoxanifΦsatisfies4
for all x∈X∖{0}. In other words, F is the unique mapping satisfying (5) and (6).Proof.
Assume thatF is a mapping satisfying (5) and (6) for a given mapping f:X→Y.
First, we consider the case whenΦ:X∖{0}→[0,∞) satisfies condition (2) or (3). It then follows from (2), (3), (5), and (6) that
(8)Fex-1a2nfeanx=1a2nFeanx-feanx≤12a2nFanx-fanx+12a2nF-anx-f-anx≤12a2nΦanx+Φ-anx⟶0,xxxxxxxxxxxxxxxxxxxxixxxasn⟶∞,
for all x∈X∖{0}. That is, it holds that Fe(x)=limn→∞(1/a2n)fe(anx) for all x∈X∖{0}.
IfΦ:X∖{0}→[0,∞) satisfies condition (2), then we use (2), (5), and (6) to get
(9)Fox-1anfoanx=1anFoanx-foanx≤12anFanx-fanx+12anf-anx-F-anx≤12anΦanx+Φ-anx⟶0,xxxxxxxxxxxxxxxxxxxxxasn⟶∞,
for all x∈X∖{0}. Thus, it holds that Fox=limn→∞(1/an)fo(anx) for all x∈X∖{0} when Φ satisfies (2). Therefore, we get
(10)Fx=limn→∞1a2nfeanx+limn→∞1anfoanx
for all x∈X∖{0} when Φ satisfies (2).
On the other hand, ifΦ:X∖{0}→[0,∞) satisfies condition (3), then we have
(11)Fox-anfoxan=anFoxan-foxan≤an2Fxan-fxan+an2f-xan-F-xan≤an2Φxan+Φ-xan⟶0,xxxxxxxxxxxxxxxxxxxxasn⟶∞,
for all x∈X∖{0}. Therefore, we have Fo(x)=limn→∞anfo(x/an) for all x∈X∖{0} when Φ satisfies (3). Hence, we get
(12)Fx=limn→∞1a2nfeanx+limn→∞anfoxan
for all x∈X∖{0} provided Φ satisfies (3).
Now, we deal with the case whenΦ:X∖{0}→[0,∞) satisfies condition (4). Then, we have
(13)Fox-anfoxan=anFoxan-foxan≤an2Fxan-fxan+an2f-xan-F-xan≤an2Φxan+Φ-xan⟶0,xxxxxxxxxxxxxxxxxxxxasn⟶∞,
for all x∈X∖{0}. Thus, it follows that Fo(x)=limn→∞anfo(x/an) for all x∈X∖{0} provided Φ satisfies (4). Similarly, if Φ satisfies condition (4), then we have
(14)Fex-a2nfexan=a2nFexan-fexan≤a2n2Fxan-fxan+a2n2F-xan-f-xan≤a2n2Φxan+Φ-xan⟶0,xxxxxxxxxxxxxxxxxxxxxasn⟶∞,
for all x∈X∖{0}. It thus holds that Fe(x)=limn→∞a2nfe(x/an) for all x∈X∖{0} when Φ satisfies (4). Therefore, if Φ satisfies (4), then we get
(15)Fx=limn→∞a2nfexan+limn→∞anfoxan
for all x∈X∖{0}. Altogether, F is uniquely determined for each case.


## 3. Applications

In general, it is not easy to apply Theorem1 in practical applications. Hence, we introduce two corollaries which are easily applicable to investigating the uniqueness problems in the generalized Hyers-Ulam stability of functional equations. For the exact definition of the generalized Hyers-Ulam stability, we refer the reader to [6].Corollary 2.
Leta>1 be a real constant, let ϕ:X∖{0}→[0,∞) be a function satisfying one of the following conditions,
(16)Φx∶=∑i=0∞‍1aiϕaix<∞,(17)Φx∶=∑i=0∞‍a2iϕxai<∞,
for all x∈X∖{0}, and let f:X→Y be an arbitrarily given mapping. If there exists a mapping F:X→Y satisfying (5) for all x∈X∖{0} and (6) for all x∈X, then F is a unique mapping satisfying (5) and (6).Proof.
Ifϕ satisfies (16), then we have
(18)limn→∞1anΦanx=limn→∞∑i=0∞‍1an+iϕan+ix=limn→∞∑i=n∞‍1aiϕaix=0;
that is, Φ satisfies condition (2) for all x∈X∖{0}.
For the case of (17), it holds that
(19)limn→∞a2nΦxan=limn→∞∑i=0∞‍a2n+2iϕxan+i=limn→∞∑i=n∞‍a2iϕxai=0;
that is, Φ satisfies condition (4) for all x∈X∖{0}. Hence, our assertion is true in view of Theorem 1.Corollary 3.
Leta>1 be a real constant, let ϕ,ψ:X∖{0}→[0,∞) be functions satisfying each of the following conditions,
(20)∑i=0∞‍aiψxai<∞,∑i=0∞‍1a2iϕaix<∞,Φ~x∶=∑i=0∞‍aiϕxai<∞,Ψ~x∶=∑i=0∞‍1a2iψaix<∞,
for all x∈X∖{0}, and let f:X→Y be an arbitrarily given mapping. If there exists a mapping F:X→Y satisfying the inequality
(21)fx-Fx≤Φ~x+Ψ~x
for all x∈X∖{0} and the conditions in (6) for all x∈X, then F is a unique mapping satisfying (6) and (21).Proof.
If we setΦ(x)∶=Φ~(x)+Ψ~(x), then it follows from (20) that
(22)1a4nΦa2nx=∑i=0∞‍1a4n-iϕa2n-ix+∑i=0∞‍1a4n+2iψa2n+ix
for all x∈X∖{0}. We make change of the summation indices in the preceding equality with j=i-2n and k=2n+i to get
(23)1a4nΦa2nx=1a2n∑j=-2n∞ajϕxaj+∑k=2n∞‍1a2kψakx=1a2n∑i=12n‍1aiϕaix+1a2n∑i=0∞‍aiϕxai+∑i=2n∞‍1a2iψaix=1an∑i=1n-1‍aian1a2iϕaix+∑i=n2n‍aia2n1a2iϕaix+1a2nΦ~x+∑i=2n∞‍1a2iψaix≤1an∑i=1∞‍1a2iϕaix+∑i=n∞‍1a2iϕaix+1a2nΦ~x+∑i=2n∞‍1a2iψaix
for any x∈X∖{0}. Hence, we get
(24)limn→∞1a4nΦa2nx=0
for all x∈X∖{0}. On the other hand, we use the above equality to get
(25)limn→∞1a4n+2Φa2n+1x=1a2limn→∞1a4nΦa2nax=0
for all x∈X∖{0}. From the above two equalities, we conclude that
(26)limn→∞1a2nΦanx=0
for all x∈X∖{0}.
Similarly, we have(27)a2nΦxa2n=∑i=0∞‍a2n+iϕxa2n+i+∑i=0∞‍1a2i-2nψai-2nx
for all x∈X∖{0}. If we make change of the summation indices in the last equality with j=i+2n and k=i-2n, then we get
(28)a2nΦxa2n=∑j=2n∞‍ajϕxaj+1a2n∑k=-2n∞‍1a2kψakx=∑i=2n∞‍aiϕxai+1a2n∑i=12n‍a2iψxai+1a2n∑i=0∞‍1a2iψaix=∑i=2n∞‍aiϕxai+1an∑i=1n-1‍aianaiψxai+∑i=n2n‍aia2naiψxai+1a2nΨ~x≤∑i=2n∞‍aiϕxai+1an∑i=1∞‍aiψxai+∑i=n∞‍aiψxai+1a2nΨ~x
for any x∈X∖{0}. Thus, we obtain
(29)limn→∞a2nΦxa2n=0,limn→∞a2n+1Φxa2n+1=alimn→∞a2nΦ1a2nxa=0
for each x∈X∖{0}. Thus, it holds that
(30)limn→∞anΦxan=0
for each x∈X∖{0}. Theorem 1 implies that our conclusion for this corollary is true.In the following corollary, we prove that if for any given mappingf there exists an additive or a quadratic or a quadratic-additive mapping F near f, then the mapping F is uniquely determined.The proofs of the following two corollaries immediately follow from Corollaries2 and 3, respectively, because each of additive, quadratic, and quadratic-additive mappings satisfies both conditions in (6) for any given rational number a>1.Corollary 4.
Leta>1 be a rational number, let ϕ:X∖{0}→[0,∞) be a function satisfying condition (16) or (17) for all x∈X∖{0}, and let f:X→Y be an arbitrarily given mapping. If there exists an additive or a quadratic or a quadratic-additive mapping F:X→Y satisfying inequality (5), then F is uniquely determined.Corollary 5.
Leta>1 be a rational number, let ϕ,ψ:X∖{0}→[0,∞) be functions satisfying each of the conditions in (20) for all x∈X∖{0}, and let f:X→Y be an arbitrarily given mapping. If there exists an additive or a quadratic or a quadratic-additive mapping F:X→Y satisfying (21), then F is uniquely determined.Ifp<1 then Φ(x)∶=θxp satisfies (2), if 1<p<2 then Φ satisfies (3), or if p>2 then Φ satisfies (4). Hence, by Theorem 1, we have the following corollaries concerning the Hyers-Ulam-Rassias stability. For the detailed concept of the Hyers-Ulam-Rassias stability, we refer to [7–9].Corollary 6.
Letp∉{1,2} and θ>0 be real constants, let X, Y be real normed spaces, and let f:X→Y be an arbitrarily given mapping. If there exists a mapping F:X→Y satisfying the inequality
(31)fx-Fx≤θxp
for all x∈X∖{0} and (6) for all x∈X, then F is a unique mapping satisfying (6) and (31).Since each of additive, quadratic, and quadratic-additive mappings satisfies the conditions in (6), using Corollary 6, we can easily prove the following corollary.Corollary 7.
Letp∉{1,2} and θ>0 be real constants, let X, Y be real normed spaces, and let f:X→Y be an arbitrarily given mapping. If there exists an additive or a quadratic or a quadratic-additive mapping F:X→Y satisfying the inequality
(32)fx-Fx≤θxp
for all x∈X∖{0}, then F is uniquely determined.


## 4. Discussions

We now define(33)Dfx1,x2,…,xn∶=f∑i=1n‍xi+n-2∑i=1n‍fxi-∑1≤i<j≤n‍fxi+xj.
Nakmahachalasint [5, Theorem  3.1] investigated the generalized Hyers-Ulam stability of the functional equation Df(x1,x2,…,xn)=0 as we see in the following theorem.Theorem 8.
Letn>2 be an integer, let X be a real vector space, let Y be a Banach space, and let φ:Xn→[0,∞) be an even function. Define ϕ(x)∶=φ(x,x,-x,0,…,0) for all x∈X. If
(34)∑i=0∞‍12iϕ2ixconverges,limm→∞12mφ2mx1,2mx2,…,2mxn=0
or
(35)∑i=0∞‍4iϕx2iconverges,limm→∞4mφx12m,x22m,…,xn2m=0
for all x1,x2,…,xn∈X, and a mapping f:X→Y satisfies f(0)=0 and
(36)Dfx1,x2,…,xn≤φx1,x2,…,xn
for all x1,x2,…,xn∈X, then there exists a unique mapping F:X→Y satisfying DF(x1,x2,…,xn)=0 and if (34) holds,
(37)fex-Fex≤14∑i=0∞‍14iϕ2ix,fox-Fox≤12∑i=0∞‍12iϕ2ix
or if (35) holds,
(38)fex-Fex≤14∑i=1∞‍4iϕx2i,fox-Fox≤12∑i=1∞‍2iϕx2i
for all x∈X.We remark that neither (37) nor (38) assures the uniqueness of F near f because fe(x)-Fe(x) and fo(x)-Fo(x) may be very large even though f(x)-F(x) is very small.Using Corollary2, we improve Theorem 8 by giving proof for the uniqueness of F.Theorem 9.
Letn>2 be an integer, let X be a real vector space, let Y be a Banach space, and let φ:Xn→[0,∞) be an even function. Define ϕ(x)=φ(x,x,-x,0,…,0) for all x∈X. If one of (34) and (35) holds for all x1,x2,…,xn∈X and a mapping f:X→Y satisfies f(0)=0 and (36) for all x1,x2,…,xn∈X, then there exists a unique mapping F:X→Y satisfying DF(x1,x2,…,xn)=0 and if (34) holds,
(39)fx-Fx≤14∑i=0∞‍14iϕ2ix+12∑i=0∞‍12iϕ2ix
or if (35) holds,
(40)fx-Fx≤14∑i=1∞‍4iϕx2i+12∑i=1∞‍2iϕx2i
for all x∈X.Proof.
In Theorem8 or [5, Theorem  3.1], Nakmahachalasint showed that there exists a quadratic-additive mapping F:X→Y such that if (34) holds, then
(41)fx-Fx=fex+fox-Fex-Fox≤14∑i=0∞‍14iϕ2ix+12∑i=0∞‍12iϕ2ix
or if condition (35) holds, then
(42)fx-Fx=fex+fox-Fex-Fox≤14∑i=1∞‍4iϕx2i+12∑i=1∞‍2iϕx2i
for all x∈X. Hence, there exists a quadratic-additive mapping F:X→Y such that
(43)fx-Fx≤∑i=0∞‍12iϕ2ixif34holds,∑i=1∞‍4iϕx2iif35holds
for all x∈X. Moreover, Nakmahachalasint [5, Theorem  2.1] proved that the even part of each solution of Df(x1,x2,…,xn)=0 is a quadratic mapping, while its odd part is an additive mapping. Hence, every solution of Df(x1,x2,…,xn)=0 satisfies the conditions in (6).
Therefore, by (43), Corollary 2 implies that F is uniquely determined.Now we define(44)Dafx,y∶=2fax+y+afx-y+afy-x-a2+2afx-a2f-x-a+2fy-af-y
for a positive real constant a. The mapping f:R→R defined by f(x)=bx2+cx is a solution of the functional equation Daf(x,y)=0, where b,c are real constants. For the case a=1, Jin and Lee [4] investigated the stability of the equation D1f(x,y)=0 in Fuzzy spaces.Assume thatf:X→Y is a solution of the functional equation Daf(x,y)=0. Then we have f(0)=-Daf(0,0)/(2a2+2a)=0 and fe(0)=0. Moreover, we know that
(45)Dafey,a+1x-Dafex+y,x-aDafe-x,y-x+a+1Dafex,x=2aa+1fex+y+fex-y-2fex-2fey+fe0=2aa+1Qfex,y,Dafoxa,y-Dafoxa,0=2fox+y-fox-foy+2a+2fo0=2Afox,y
for all x,y∈X. Therefore, we get
(46)Qfex,y=Dafey,a+1x-Dafex+y,x2aa+1-Dafe-x,y-x2a+1+Dafex,x2a=0,Afox,y=Dafox/a,y-Dafox/a,02=0
for all x,y∈X; that is, fe is a quadratic mapping and fo is an additive mapping. Therefore, if a mapping f:X→Y satisfies the functional equation Daf(x,y)=0 for all x,y∈X, then f=fe+fo is a quadratic-additive mapping such that
(47)fea+1x=12Dafex,x+a+12fex=a+12fex,foa+1x=12Dafox,x+a+1fox=a+1fox
for all x∈X. But we do not know that all quadratic-additive mappings satisfy the equalities in (47) if a is not a rational number.Theorem 10.
Letθ>0 be an arbitrary real constant. If a mapping f:X→Y satisfies
(48)Dafx,y≤θxp+yp
for all x,y∈X with a positive real constant p∉{1,2}, then there exists a unique quadratic-additive mapping F:X→Y such that
(49)fx-Fx≤θxpa+1-a+1p+θxpa+12-a+1p
for all x∈X.Proof.
Notice thatf(0)=-Daf(0,0)/(2a2+2a)=0. Since Dafe(x,y)=(Daf(x,y)+Daf(-x,-y))/2, if 0<p<2, then it follows from (48) that
(50)fea+1nxa+12n-fea+1n+mxa+12(n+m)≤∑i=nn+m-1fea+1ixa+12i-fea+1i+1xa+12i+2≤∑i=nn+m-1‍12a+12i+2Dafea+1ix,a+1ix≤∑i=nn+m-1‍a+1ipθa+12i+2xp
for all x∈X. So, it is easy to show that the sequence {fe((a+1)nx)/(a+1)2n} is a Cauchy sequence for all x∈X. Since Y is complete, the sequence {fe((a+1)nx)/(a+1)2n} converges for all x∈X. Hence, we can define a mapping Fe:X→Y by
(51)Fex∶=limn→∞fea+1nxa+12n
for all x∈X.
Moreover, if we putn=0 and let m→∞ in (50), we obtain the inequality
(52)fex-Fex≤θxpa+12-a+1p
for all x∈X. From the definition of Fe, we get
(53)DaFex,y=limn→∞Dafea+1nx,a+1nya+12n≤limn→∞a+1p-2nθxp+yp=0
for all x,y∈X.
Ifp>2, then it follows from (48) that
(54)a+12nfexa+1n-a+12n+mfexa+1n+m≤∑i=nn+m-1‍a+12ifexa+1i-a+12i+1fexa+1i+1≤∑i=nn+m-1‍a+12i2Dafexa+1i+1,xa+1i+1≤∑i=nn+m-1‍a+12iθ2a+1(i+1)pxp
for all x∈X. So, it is easy to show that the sequence {(a+1)2nfe(x/(a+1)n)} is a Cauchy sequence for all x∈X. Since Y is complete, the sequence {(a+1)2nfe(x/(a+1)n)} converges for all x∈X. Hence, we can define a mapping Fe:X→Y by
(55)Fex∶=limn→∞a+12nfexa+1n
for all x∈X.
Moreover, if we putn=0 and let m→∞ in (54), we obtain inequality (52) for all x∈X. From the definition of Fe, we get
(56)DaFex,y=limn→∞a+12nDafexa+1n,ya+1n≤limn→∞a+12-pnθxp+yp=0
for all x,y∈X.
SinceDafo(x,y)=(Daf(x,y)-Daf(-x,-y))/2, if p<1, then it follows from (48) that
(57)foa+1nxa+1n-foa+1n+mxa+1n+m≤∑i=nn+m-1‍foa+1ixa+1i-foa+1i+1xa+1i+1≤12∑i=nn+m-1‍1a+1i+1Dafoa+1ix,a+1ix≤∑i=nn+m-1‍a+1ipθa+1i+1xp
for all x∈X. So, it is easy to show that the sequence {fo((a+1)nx)/(a+1)n} is a Cauchy sequence for all x∈X. Since Y is complete, the sequence {fo((a+1)nx)/(a+1)n} converges for all x∈X. Hence, we can define a mapping Fo:X→Y by
(58)Fox∶=limn→∞foa+1nxa+1n
for all x∈X.
Moreover, if we putn=0 and let m→∞ in (57), we obtain the inequality
(59)fox-Fox≤θxpa+1-a+1p
for all x∈X. From the definition of Fo, we get
(60)DaFox,y=limn→∞Dafoa+1nx,a+1nya+1n≤limn→∞a+1p-1nθxp+yp=0
for all x,y∈X.
Ifp>1, then it follows from (48) that
(61)a+1nfoxa+1n-a+1n+mfoxa+1n+m≤∑i=nn+m-1‍a+1ifoxa+1i-a+1i+1foxa+1i+1=∑i=nn+m-1‍a+1i+12Dafoxa+1i+1,xa+1i+1≤∑i=nn+m-1‍a+1i+1θa+1(i+1)pxp
for all x∈X. So, it is easy to show that the sequence {(a+1)nfo(x/(a+1)n)} is a Cauchy sequence for all x∈X. Since Y is complete, the sequence {(a+1)nfo(x/(a+1)n)} converges for all x∈X. Hence, we can define a mapping Fo:X→Y by
(62)Fox∶=limn→∞a+1nfoxa+1n
for all x∈X.
Moreover, if we putn=0 and let m→∞ in (61), we obtain inequality (59) for all x∈X. From the definition of F, we get
(63)DaFox,y=limn→∞a+1nDafoxa+1n,ya+1n≤limn→∞a+11-pnθxp+yp=0
for all x,y∈X.
For any case, we have thatDaF(x,y)=DaFe(x,y)+DaFo(x,y)=0 for all x,y∈X and
(64)fx-Fx≤fex-Fex+fox-Fox≤θxpa+1-a+1p+θxpa+12-a+1p;
that is, F is a quadratic-additive mapping satisfying (49). By Corollary 6 and (47), F is a unique quadratic-additive mapping satisfying (49).



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