# A Fixed Point Approach to the Stability of ann-Dimensional Mixed-Type Additive and Quadratic Functional Equation

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

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## Abstract

We investigate the stability problems for a functional equation2f(∑j=1nxj)+∑1≤i,j≤n,i≠jf(xi-xj)=(n+1)∑j=1nf(xj)+(n-1)∑j=1nf(-xj) by using the fixed point method.


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## Body

## 1. Introduction

In 1940, Ulam [1] gave a wide-ranging talk before a Mathematical Colloquium at the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the following question concerning the stability of homomorphisms.LetG1be a group and letG2be a metric group with a metricd(·,·). Givenɛ>0, does there exist aδ>0such that if a functionh:G1→G2satisfies the inequalityd(h(xy),h(x)h(y))<δfor allx,y∈G1, then there is a homomorphismH:G1→G2withd(h(x),H(x))<ɛfor allx∈G1?If the answer is affirmative, we say that the functional equation for homomorphisms is stable. Hyers [2] was the first mathematician to present the result concerning the stability of functional equations. He answered the question of Ulam for the case where G1 and G2 are assumed to be Banach spaces. This result of Hyers is stated as follows.Letf:E1→E2be a function between Banach spaces such that(1.1)‖f(x+y)-f(x)-f(y)‖≤δfor someδ>0and for allx,y∈E1. Then the limitA(x)=limn→∞2-nf(2nx)exists for eachx∈E1, andA:E1→E2is the unique additive function such that∥f(x)-A(x)∥≤δfor everyx∈E1. Moreover, iff(tx)is continuous intfor each fixedx∈E1, then the functionAis linear.We remark that the additive functionA is directly constructed from the given function f and this method is called the direct method. The direct method is a very powerful method for studying the stability problems of various functional equations. Taking this famous result into consideration, the additive Cauchy equation f(x+y)=f(x)+f(y) is said to have the Hyers-Ulam stability on (E1,E2) if, for every function f:E1→E2 satisfying the inequality (1.1) for some δ≥0 and for all x,y∈E1, there exists an additive function A:E1→E2 such that f-A is bounded on E1.In 1950, Aoki [3] generalized the theorem of Hyers for additive functions, and in the following year, Bourgin [4] extended the theorem without proof. Unfortunately, it seems that their results failed to receive attention from mathematicians at that time. No one has made use of these results for a long time.In 1978, Rassias [5] actually rediscovered the result of Aoki; he proved the following theorem of Hyers for linear functions.Letf:E1→E2be a function between Banach spaces. Iffsatisfies the functional inequality(1.2)‖f(x+y)-f(x)-f(y)‖≤θ(‖x‖p+‖y‖p)for someθ≥0,pwith0≤p<1and for allx,y∈E1, then there exists a unique additive functionA:E1→E2such that∥f(x)-A(x)∥≤(2θ/(2-2p))∥x∥pfor eachx∈E1. If, in addition,f(tx)is continuous intfor each fixedx∈E1, then the functionAis linear.This result of Rassias attracted a number of mathematicians who began to be stimulated to investigate the stability problems of functional equations. By regarding a large influence of Ulam, Hyers, and Rassias on the study of stability problems of functional equations, the stability phenomenon proved by Aoki and Rassias is called theHyers-Ulam-Rassias  stability. For the last thirty years many results concerning the Hyers-Ulam-Rassias  stability of various functional equations have been obtained (see [6–16]). For some discussion of possible definitions of stability for functional equations, see [17].The following functional equation(1.3)2f(∑j=1nxj)+∑1≤i,j≤n,i≠jf(xi-xj)=(n+1)∑j=1nf(xj)+(n-1)∑j=1nf(-xj)
is called an n-dimensional mixed-type additive and quadratic functional equation, and each solution of (1.3) is called a quadratic-additive function. Recently, Towanlong and Nakmahachalasint [18] solved the n-dimensional mixed-type additive and quadratic functional equation.LetE1andE2be vector spaces. A functionf:E1→E2satisfies (1.3) for allx1,x2,…,xn∈E1if and only if there exist an additive functiona:E1→E2and a quadratic functionq:E1→E2such thatf(x)=a(x)+q(x)for allx∈E1.Moreover, they also investigated the Hyers-Ulam-Rassias stability of (1.3) by using the direct method (see [18]). Indeed, they tried to approximate the even and odd parts of each solution of a perturbed inequality by the even and odd parts of an “exact” solution of (1.3), respectively.In Theorems3.1 and 3.3 of this paper, we will apply the fixed point method and prove the Hyers-Ulam-Rassias stability of the n-dimensional mixed-type additive and quadratic functional equation. The advantage of this paper, in comparison with [18], is to approximate each solution of a perturbed inequality by an “exact” solution of (1.3), and we obtain sharper estimations in consequence of this advantage.Throughout this paper, letV be a (real or complex) vector space, Y a Banach space, and n an integer larger than 1.


## 2. Preliminaries

LetX be a nonempty set. A function d:X2→[0,∞] is called a generalized metric on X if and only if d satisfies the following:  (M1)
d(x,y)=0 if and only if x=y;(M2)
d(x,y)=d(y,x) for all x,y∈X;(M3)
d(x,z)≤d(x,y)+d(y,z) for all x,y,z∈X.We remark that the only difference between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity.We now introduce one of the fundamental results of the fixed point theory. For the proof, we refer to [19].Theorem 2.1.
Let(X,d) be a complete generalized metric space. Assume that Λ:X→X is a strict contraction with the Lipschitz constant L<1. If there exists a nonnegative integer n0 such that d(Λn0+1x,Λn0x)<∞ for some x∈X, then the following statements are true.  (i)
The sequence{Λnx} converges to a fixed point x* of Λ.(ii)
x* is the unique fixed point of Λ in X*={y∈X∣d(Λn0x,y)<∞}.(iii)
Ify∈X*, then(2.1)d(y,x*)≤11-Ld(Λy,y).In 1991, Baker applied the fixed point method to prove the Hyers-Ulam stability of a nonlinear functional equation (see [20]). Thereafter, Radu noticed that many theorems concerning the Hyers-Ulam stability of various functional equations follow from the fixed point alternative (Theorem 2.1). Indeed, he applied the fixed point method to prove the existence of a solution of the inequality (1.1) and investigated the Hyers-Ulam stability of the additive Cauchy equation (see [21] and also [22–26]). For a somewhat different fixed point approach to stability of functional equations, see [27, 28].


## 3. Hyers-Ulam-Rassias Stability

LetV be a (real or complex) vector space and let Y be a Banach space. For a given function f:V→Y, we use the following abbreviation: (3.1)Df(x1,x2,…,xn)≔2f(∑j=1nxj)+∑1≤i,j≤n,i≠jf(xi-xj)-(n+1)∑j=1nf(xj)-(n-1)∑j=1nf(-xj)
for all x1,x2,…,xn∈V.In the following theorem, we prove the stability of the functional equation (1.3) by using the fixed point method.Theorem 3.1.
Letφ:(V∖{0})n→[0,∞) be a given function. Assume that a function f:V→Y satisfies f(0)=0 and the inequality
(3.2)‖Df(x1,x2,…,xn)‖≤φ(x1,x2,…,xn)
for all x1,x2,…,xn∈V∖{0}. If there exists a constant 0<L<1 such that φ has the property
(3.3)φ(nx1,nx2,…,nxn)≤nLφ(x1,x2,…,xn)
for all x1,x2,…,xn∈V∖{0}, then there exists a unique function F:V→Y such that DF(x1,x2,…,xn)=0 for all x1,x2,…,xn∈V∖{0} and
(3.4)‖f(x)-F(x)‖≤n+14n2(1-L)ψ(x)
for all x∈V∖{0}, where the function ψ:V∖{0}→Y is defined by
(3.5)ψ(x):=φ(x,x,…,x)+φ(-x,-x,…,-x)
for all x∈V∖{0}. In particular, F is represented by
(3.6)F(x)=lim⁡m→∞(f(nmx)-f(-nmx)2nm+f(nmx)+f(-nmx)2n2m)
for all x∈V.Proof.
LetS be the set of all functions g:V→Y with g(0)=0. We introduce a generalized metric on S by
(3.7)d(g,h):=inf⁡{K∈[0,∞]∣‖g(x)-h(x)‖≤Kψ(x)∀x∈V∖{0}},
where ψ is given in (3.5). It is not difficult to show that (S,d) is a complete generalized metric space (see [29] or [30, 31]).
Now we consider the operatorJ:S→S defined by
(3.8)Jg(x):=g(nx)-g(-nx)2n+g(nx)+g(-nx)2n2
for all x∈V. We can apply induction on m to prove
(3.9)Jmg(x)=g(nmx)-g(-nmx)2nm+g(nmx)+g(-nmx)2n2m
for all x∈V and m∈ℕ.
Letg,h∈S and let K∈[0,∞] be an arbitrary constant with d(g,h)≤K. By the definition of d, (3.3), and (3.5), we have
(3.10)‖Jg(x)-Jh(x)‖≤(n+1)‖g(nx)-h(nx)‖2n2+(n-1)‖g(-nx)-h(-nx)‖2n2≤Kψ(nx)n≤KLψ(x)
for all x∈V∖{0}, which implies that
(3.11)d(Jg,Jh)≤Ld(g,h)
for any g,h∈S. That is, J is a strict contraction with the Lipschitz constant L.
Moreover, by (3.2), we see that
(3.12)‖f(x)-Jf(x)‖=‖n-14n2Df(-x,-x,…,-x)-n+14n2Df(x,x,…,x)‖≤n+14n2ψ(x)
for all x∈V∖{0}, that is, d(f,Jf)≤(n+1)/4n2<∞ (see the definition of d). Therefore, according to Theorem 2.1, the sequence {Jmf} converges to the unique “fixed point” F:V→Y of J in the set T={g∈S|d(f,g)<∞} and F is represented by (3.6) for all x∈V. Notice that F(0)=0. By Theorem 2.1, we have
(3.13)d(f,F)≤11-Ld(f,Jf)≤n+14n2(1-L),
which implies the validity of (3.4).
By a somewhat tedious manipulation, it follows from (3.2), (3.3), and (3.6) that
(3.14)‖DF(x1,x2,…,xn)‖=lim⁡m→∞‖Df(nmx1,nmx2,…,nmxn)-Df(-nmx1,-nmx2,…,-nmxn)2nm+Df(nmx1,nmx2,…,nmxn)+Df(-nmx1,-nmx2,…,-nmxn)2n2m‖≤lim⁡m→∞nm+12n2m[φ(nmx1,…,nmxn)+φ(-nmx1,…,-nmxn)]=0
for all x1,x2,…,xn∈V∖{0}.We can prove the following corollary by a way similar to that presented in the proof of the preceding theorem.Corollary 3.2.
Letφ:Vn→[0,∞) be a given function such that φ(x1,x2,…,xn)=φ(-x1,-x2,…,-xn) for all x1,x2,…,xn∈V. Assume that a function f:V→Y satisfies f(0)=0 and the inequality (3.2) for all x1,x2,…,xn∈V. If there exists a constant 0<L<1 such that φ has the property (3.3) for all x1,x2,…,xn∈V, then there exists a unique quadratic-additive function F:V→Y such that
(3.15)‖f(x)-F(x)‖≤φ(x,x,…,x)2n(1-L)
for all x∈V.Proof.
Notice that(3.16)‖f(x)-Jf(x)‖=‖n-14n2Df(-x,-x,…,-x)-n+14n2Df(x,x,…,x)‖≤φ(x,x,…,x)2n
for all x∈V (see the proof of Theorem 3.1). By a similar method used in the proof of Theorem 3.1, we can show that there exists a unique quadratic-additive function F:V→Y satisfying (3.15).In the following theorem, we prove the Hyers-Ulam-Rassias stability of (1.3) under the condition (3.17) instead of (3.3).Theorem 3.3.
Givenφ:(V∖{0})n→[0,∞), assume that a function f:V→Y satisfies the condition f(0)=0 and the inequality (3.2) for all x1,x2,…,xn∈V∖{0}. If there exists a constant 0<L<1 such that
(3.17)Lφ(nx1,nx2,…,nxn)≥n2φ(x1,x2,…,xn)
for all x1,x2,…,xn∈V∖{0}, then there exists a unique function F:V→Y such that DF(x1,x2,…,xn)=0 for all x1,x2,…,xn∈V∖{0} and
(3.18)‖f(x)-F(x)‖≤L2n2(1-L)ψ(x)
for all x∈V∖{0}, where ψ is defined by (3.5). In particular, F is represented by
(3.19)F(x)=lim⁡m→∞(nm2(f(xnm)-f(-xnm))+n2m2(f(xnm)+f(-xnm)))
for all x∈V.Proof.
LetS and d be defined as in the proof of Theorem 3.1. Then (S,d) is a complete generalized metric space. We now consider the operator J:S→S defined by
(3.20)Jg(x):=n2(g(xn)-g(-xn))+n22(g(xn)+g(-xn))
for all g∈S and x∈V.
Notice that(3.21)Jmg(x)=nm2(g(xnm)-g(-xnm))+n2m2(g(xnm)+g(-xnm))
for all x∈V and m∈ℕ.
Letg,h∈S and let K∈[0,∞] be an arbitrary constant with d(g,h)≤K. By the definition of d, together with (3.5) and (3.17), we have
(3.22)‖Jg(x)-Jh(x)‖≤n2+n2‖g(xn)-h(xn)‖+n2-n2‖g(-xn)-h(-xn)‖≤n2Kψ(xn)≤LKψ(x)
for all x∈V∖{0}. Thus, we get
(3.23)d(Jg,Jh)≤Ld(g,h)
for any g,h∈S. That is, J is a strict contraction with the Lipschitz constant L.
Moreover, it follows from (3.2), (3.5), and (3.17) that
(3.24)‖f(x)-Jf(x)‖=‖12Df(xn,xn,…,xn)‖≤12φ(xn,xn,…,xn)≤L2n2φ(x,x,…,x)≤L2n2ψ(x)
for all x∈V∖{0}, which implies that d(f,Jf)≤L/2n2<∞. Therefore, according to Theorem 2.1, the sequence {Jmf} converges to the unique “fixed point” F of J in the set T={g∈S∣d(f,g)<∞} and F is represented by (3.19). In view of Theorem 2.1, we have
(3.25)d(f,F)≤11-Ld(f,Jf)≤L2n2(1-L).
Hence, the inequality (3.18) is true.
In a similar way presented in the proof of Theorem3.1, it follows from (3.2), (3.17), and (3.19) that
(3.26)‖DF(x1,x2,…,xn)‖=lim⁡m→∞‖n2m2(Df(x1nm,x2nm,…,xnnm)+Df(-x1nm,-x2nm,…,-xnnm))nm2(Df(x1nm,x2nm,…,xnnm)-Df(-x1nm,-x2nm,…,-xnnm))+n2m2(Df(x1nm,x2nm,…,xnnm)+Df(-x1nm,-x2nm,…,-xnnm))‖≤lim⁡m→∞n2m+nm2(φ(x1nm,x2nm,…,xnnm)+φ(-x1nm,-x2nm,…,-xnnm))=0
for all x1,x2,…,xn∈V∖{0}.We can prove the following corollary by a similar way as we did in the proof of Corollary3.2. We omit the proof.Corollary 3.4.
Letφ:Vn→[0,∞) be a given function such that φ(x1,x2,…,xn)=φ(-x1,-x2,…,-xn) for all x1,x2,…,xn∈V. Assume that a function f:V→Y satisfies f(0)=0 and the inequality (3.2) for all x1,x2,…,xn∈V. If there exists a constant 0<L<1 such that φ has the property (3.17) for all x1,x2,…,xn∈V, then there exists a unique quadratic-additive function F:V→Y such that
(3.27)‖f(x)-F(x)‖≤L2n2(1-L)φ(x,x,…,x)
for all x∈V.


## 4. Applications

For a given functionf:V→Y, we will use the following abbreviation: (4.1)Af(x,y):=f(x+y)-f(x)-f(y)
for all x,y∈V.Corollary 4.1.
Letfk:V→Y be functions for which fk(0)=0 and there exist functions ϕk:V2→[0,∞) such that
(4.2)‖Afk(x,y)‖≤ϕk(x,y)
for all x,y∈V and k∈{1,2}. If there exists a constant 0<L<1 such that
(4.3)ϕ1(nx,ny)≤nLϕ1(x,y),(4.4)n2ϕ2(x,y)≤Lϕ2(nx,ny),
and ϕk(x,y)=ϕk(-x,-y) for all x,y∈V and k∈{1,2}, then there exist additive functions Fk:V→Y such that
(4.5)‖f1(x)-F1(x)‖≤Φ1(x)2n(1-L),(4.6)‖f2(x)-F2(x)‖≤LΦ2(x)2n2(1-L),
for all x∈V∖{0}, where Φk:V→Y are defined by
(4.7)Φk(x):=n(n-1)ϕk(x,-x)+2∑j=1n-1ϕk(jx,x)
for all x∈V and k∈{1,2}. In particular, the functions F1 and F2 are represented by
(4.8)F1(x)=lim⁡m→∞f1(nmx)nm,(4.9)F2(x)=lim⁡m→∞nmf2(xnm)
for all x∈V.Proof.
By a long manipulation, we obtain(4.10)Dfk(x1,x2,…,xn)=∑1≤i,j≤n,i≠jAfk(xi,-xj)+2∑j=1n-1Afk(∑i=1jxi,xj+1)
for all x1,x2,…,xn∈V and k∈{1,2}. If we put
(4.11)φk(x1,x2,…,xn):=∑1≤i,j≤n,i≠jϕk(xi,-xj)+2∑j=1n-1ϕk(∑i=1jxi,xj+1)
for all x1,x2,…,xn∈V and k∈{1,2}, then it follows from (4.2) and (4.10) that
(4.12)‖Dfk(x1,x2,…,xn)‖≤φk(x1,x2,…,xn)
for all x1,x2,…,xn∈V and k∈{1,2}. By (4.3) and (4.4), we know that φ1 and φ2 satisfy (3.3) and (3.17), respectively. Furthermore, φk(x1,x2,…,xn)=φk(-x1,-x2,…,-xn) for all x1,x2,…,xn∈V and k∈{1,2}.
Therefore, according to Corollary3.2, there exists a unique quadratic-additive function F1:V→Y satisfying (4.5) and F1 is represented by (3.6). By (4.2) and (4.3), we see
(4.13)lim⁡m→∞‖f1(nmx)+f1(-nmx)2nm‖=lim⁡m→∞‖f1(nmx)+f1(-nmx)-f1(0)2nm‖=lim⁡m→∞12nm‖Af1(nmx,-nmx)‖≤lim⁡m→∞12nmϕ1(nmx,-nmx)≤lim⁡m→∞Lm2ϕ1(x,-x)=0,lim⁡m→∞‖f1(nmx)+f1(-nmx)2n2m‖≤lim⁡m→∞Lm2nmϕ1(x,-x)=0
for all x∈V. From these and (3.6), we get (4.8).
Moreover, we have(4.14)‖Af1(nmx,nmy)nm‖≤ϕ1(nmx,nmy)nm≤Lmϕ1(x,y)
for all x,y∈V. Taking the limit as m→∞ in the aforementioned inequality, we get
(4.15)AF1(x,y)=0
for all x,y∈V.
Now, according to Corollary3.4, there exists a unique function F2:V→Y satisfying (4.6) and F2 is represented by (3.19). Due to (4.2) and (4.4), we get
(4.16)lim⁡m→∞n2m2‖f2(xnm)+f2(-xnm)‖=lim⁡m→∞n2m2‖Af2(xnm,-xnm)‖≤lim⁡m→∞n2m2ϕ2(xnm,-xnm)≤lim⁡m→∞Lm2ϕ2(x,-x)=0
as well as
(4.17)lim⁡m→∞nm2‖f2(xnm)+f2(-xnm)‖≤lim⁡m→∞Lm2nmϕ2(x,-x)=0
for all x∈V. From these and (3.19), we get (4.9).
Moreover, we have(4.18)‖nmAf2(xnm,ynm)‖≤nmϕ2(xnm,ynm)≤Lmnmϕ2(x,y)
for all x,y∈V. Taking the limit as m→∞ in the aforementioned inequality, we get
(4.19)AF2(x,y)=0
for all x,y∈V.Corollary 4.2.
LetX be a normed space and p∈(-∞,1)∪(2,∞). If a function f:X→Y satisfies f(0)=0 and the inequality
(4.20)‖Df(x1,x2,…,xn)‖≤θ(‖x1‖p+‖x2‖p+⋯+‖xn‖p)
for all x1,x2,…,xn∈X∖{0} and for some θ≥0, then there exists a unique function F:X→Y such that DF(x1,x2,…,xn)=0 for all x1,x2,…,xn∈X∖{0} and
(4.21)‖f(x)-F(x)‖≤{(n+1)θ‖x‖p2(n-np)ifp<1,nθ‖x‖pnp-n2ifp>2
for all x∈X∖{0}.Proof.
If we put(4.22)φ(x1,x2,…,xn):=θ(‖x1‖p+‖x2‖p+⋯+‖xn‖p)
for all x1,x2,…,xn∈X∖{0} and
(4.23)L:={np-1ifp<1,n2-pifp>2,
then our assertions follow from Theorems 3.1 and 3.3.Corollary 4.3.
LetX be a normed space. If a function f:X→Y satisfies f(0)=0 and the inequality
(4.24)‖Df(x1,x2,…,xn)‖≤θ∑1≤i≤n,xi≠0‖xi‖p
for all x1,x2,…,xn∈X and for some θ≥0 and p∈(-∞,1)∪(2,∞), then there exists a unique quadratic-additive function F:X→Y such that
(4.25)‖f(x)-F(x)‖≤{nθ‖x‖p2(n-np)ifp<1,nθ‖x‖p2(np-n2)ifp>2
for all x∈X∖{0}. In particular, if p<0, then f is a quadratic-additive function.Proof.
This corollary follows from Corollaries3.2 and 3.4 by putting
(4.26)φ(x1,x2,…,xn):=θ∑1≤i≤n,xi≠0‖xi‖p
for all x1,x2,…,xn∈X with L defined in (4.23).Corollary 4.4.
LetX be a normed space and let p1,p2,…,pn be real constants with p1+p2+⋯+pn∈(-∞,1)∪(2,∞). If a function f:X→Y satisfies f(0)=0 and the inequality
(4.27)‖Df(x1,x2,…,xn)‖≤θ‖x1‖p1‖x2‖p2⋯‖xn‖pn
for all x1,x2,…,xn∈X∖{0}, then there exists a unique function F:X→Y such that DF(x1,x2,…,xn)=0 for all x1,x2,…,xn∈X∖{0} and
(4.28)‖f(x)-F(x)‖≤{(n+1)θ‖x‖p2n(n-np)ifp<1,θ‖x‖pnp-n2ifp>2
for all x∈X∖{0}, where p:=p1+p2+⋯+pn.Proof.
If we put(4.29)φ(x1,x2,…,xn):=θ‖x1‖p1‖x2‖p2⋯‖xn‖pn
for all x1,x2,…,xn∈X∖{0} and L is defined in (4.23), then our assertion follows from Theorems 3.1 and 3.3.



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