We investigate the stability problems for the n-dimensional mixed-type additive and quadratic functional equation 2f(∑j=1nxj)+∑1≤i,j≤n,i≠jf(xi-xj)=(n+1)∑j=1nf(xj)+(n-1)∑j=1nf(-xj) in random normed spaces by applying the fixed point method.

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.

Let G1 be a group, and let G2 be a metric group with a metric d(·,·). Given ɛ>0, does there exist a δ>0 such that if a function h:G1→G2 satisfies the inequality d(h(xy),h(x)h(y))<δ for all x,y∈G1, then there is a homomorphism H:G1→G2 with d(h(x),H(x))<ɛ for all x∈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.

Let f:E1→E2 be a function between Banach spaces such that‖f(x+y)-f(x)-f(y)‖≤δ for some δ>0 and for all x,y∈E1. Then the limit A(x)=limn→∞2-nf(2nx) exists for each x∈E1, and A:E1→E2 is the unique additive function such that ∥f(x)-A(x)∥≤δ for every x∈E1. Moreover, if f(tx) is continuous in t for each fixed x∈E1, then function A is linear.

We remark that the additive function A 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] addressed the Hyers’s stability theorem and attempted to weaken the condition for the bound of the norm of Cauchy difference and generalized the theorem of Hyers for linear functions.

Let f:E1→E2 be a function between Banach spaces. If f satisfies the functional inequality‖f(x+y)-f(x)-f(y)‖≤θ(‖x‖p+‖y‖p)for some θ≥0, p with 0≤p<1 and for all x,y∈E1, then there exists a unique additive function A:E1→E2 such that ∥f(x)-A(x)∥≤(2θ/(2-2p))∥x∥p for each x∈E1. If, in addition, f(tx) is continuous in t for each fixed x∈E1, then the function A is 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 Rassias is called the Hyers-Ulam-Rassias stability. For the last thirty years, many results concerning the Hyers-Ulam-Rassias stability of various functional equations have been obtained (see [6–17]).

In this paper, applying the fixed point method, we prove the Hyers-Ulam-Rassias stability of the n-dimensional mixed-type additive and quadratic functional equation 2f(∑j=1nxj)+∑1≤i,j≤n,i≠jf(xi-xj)=(n+1)∑j=1nf(xj)+(n-1)∑j=1nf(-xj)
in random normed spaces. Every solution of (1.3) is called a quadratic-additive function.

Throughout this paper, let n be an integer larger than 1.

2. Preliminaries

We introduce some terminologies, notations, and conventions usually used in the theory of random normed spaces (see [18, 19]). The set of all probability distribution functions is denoted byΔ+∶={F:[0,∞]→[0,1]∣Fisleft-continuousandnondecreasingon[0,∞),F(0)=0,andF(∞)=1}.
Let us define D+:={F∈Δ+∣limt→∞F(t)=1}. The set Δ+ is partially ordered by the usual pointwise ordering of functions, that is, F≤G if and only if F(t)≤G(t) for all t≥0. The maximal element for Δ+ in this order is the distribution function ɛ0:[0,∞]→[0,1] given byɛ0(t)={0ift=0,1ift>0.

Definition 2.1 (See [<xref ref-type="bibr" rid="B28">18</xref>]).

A function τ:[0,1]×[0,1]→[0,1] is called a continuous triangular norm (briefly, continuous t-norm) if τ satisfies the following conditions:

τ is commutative and associative;

τ is continuous;

τ(a,1)=a for all a∈[0,1];

τ(a,b)≤τ(c,d) for all a,b,c,d∈[0,1] with a≤c and b≤d.

Typical examples of continuous t-norms are τP(a,b)=ab, τM(a,b)=min{a,b}, and τL(a,b)=max{a+b-1,0}.

Definition 2.2 (See [<xref ref-type="bibr" rid="B29">19</xref>]).

Let X be a vector space, τ a continuous t-norm, and let Λ:X→D+ be a function satisfying the following conditions:

Λx(t)=ɛ0(t) for all t>0 if and only if x=0;

Λαx(t)=Λx(t/|α|) for all x∈X, α≠0, and for all t≥0;

Λx+y(t+s)≥τ(Λx(t),Λy(s)) for all x,y∈X and all t,s≥0.

A triple (X,Λ,τ) is called a random normed space (briefly, RN-space).

If (X,∥·∥) is a normed space, we can define a function Λ:X→D+ by
Λx(t)=tt+‖x‖
for all x∈X and t>0. Then (X,Λ,τM) is a random normed space, which is called the induced random normed space.

Definition 2.3.

Let (X,Λ,τ) be an RN-space.

A sequence {xn} in X is said to be convergent to a point x∈X if, for every t>0 and ɛ>0, there exists a positive integer N such that Λxn-x(t)>1-ɛ whenever n≥N.

A sequence {xn} in X is called a Cauchy sequence if, for every t>0 and ɛ>0, there exists a positive integer N such that Λxn-xm(t)>1-ɛ whenever n≥m≥N.

An RN-space (X,Λ,τ) is called complete if and only if every Cauchy sequence in X converges to a point in X.

Definition 2.4.

Let X be a nonempty set. A function d:X2→[0,∞] is called a generalized metric on X if and only if d satisfies

d(x,y)=0 if and only if x=y;

d(x,y)=d(y,x) for all x,y∈X;

d(x,z)≤d(x,y)+d(y,z) for all x,y,z∈X.

We now introduce one of the fundamental results of the fixed point theory. For the proof, we refer to [20] or [21].

Theorem 2.5 (See [<xref ref-type="bibr" rid="B23">20</xref>, <xref ref-type="bibr" rid="B27">21</xref>]).

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:

the sequence {Λnx} converges to a fixed point x* of Λ;

x* is the unique fixed point of Λ in X*={y∈X∣d(Λn0x,y)<∞};

if y∈X*, then

d(y,x*)≤11-Ld(Λy,y).

In 2003, Radu [22] noticed that many theorems concerning the Hyers-Ulam stability of various functional equations follow from the fixed point alternative (Theorem 2.5). 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 also [23–26]). Furthermore, Miheţ and Radu [27] applied the fixed point method to prove the stability theorems of the additive Cauchy equation in random normed spaces.

In 2009, Towanlong and Nakmahachalasint [28] established the general solution and the stability of the n-dimensional mixed-type additive and quadratic functional equation (1.3) by using the direct method. According to [28], a function f:E1→E2 is a quadratic-additive function, where E1 and E2 are vector spaces, if and only if there exist an additive function a:E1→E2 and a quadratic function q:E1→E2 such that f(x)=a(x)+q(x) for all x∈E1.

3. Hyers-Ulam-Rassias Stability

Throughout this paper, let X be a real vector space and let (Y,Λ,τM) be a complete RN-space. For a given function f:X→Y, we use the following abbreviation: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∈X.

We will now prove the stability of the functional equation (1.3) in random normed spaces by using fixed point method.

Theorem 3.1.

Let X be a real vector space, (Z,Λ′,τM) an RN-space, (Y,Λ,τM) a complete RN-space, and let φ:(X∖{0})n→Z be a function. Assume that φ satisfies one of the following conditions:

Λ′αφ(x1,x2,…,xn)(t)≤Λ′φ(nx1,nx2,…,nxn)(t) for some 0<α<n;

Λ′φ(nx1,nx2,…,nxn)(t)≤Λ′αφ(x1,x2,…,xn)(t) for some α>n2

for all x1,x2,…,xn∈X∖{0} and t>0. If a function f:X→Y satisfies f(0)=0 and
ΛDf(x1,x2,…,xn)(t)≥Λ′φ(x1,x2,…,xn)(t)
for all x1,x2,…,xn∈X∖{0} and t>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
Λf(x)-F(x)(t)≥{M(x,2(n-α)t)ifφsatisfies(i),M(x,2(α-n2)t)ifφsatisfies(ii)
for all x∈X∖{0} and t>0, where M(x,t):=τM(Λ′φ(x̂)(t),Λ′φ(-x̂)(t)), and x̂=(x,x,…,x).Proof.

We will first treat the case where φ satisfies the condition (i). Let S be the set of all functions g:X→Y with g(0)=0, and let us define a generalized metric on S by
d(g,h):=inf{u∈[0,∞]∣Λg(x)-h(x)(ut)≥M(x,t)∀x∈X∖{0},t>0}.
It is not difficult to show that (S,d) is a complete generalized metric space (see [29] or [30, 31]).

Consider the operator J:S→S defined by
Jf(x):=f(nx)-f(-nx)2n+f(nx)+f(-nx)2n2.
Then we can apply induction on m to prove
Jmf(x)=f(nmx)-f(-nmx)2nm+f(nmx)+f(-nmx)2n2m
for all x∈X and m∈ℕ.

Let f,g∈S and let u∈[0,∞] be an arbitrary constant with d(g,f)≤u. For some 0<α<n satisfying the condition (i), it follows from the definition of d, (R2), (R3), and (i) that
ΛJg(x)-Jf(x)(αutn)=Λ((n+1)(g(nx)-f(nx))/2n2)-((n-1)(g(-nx)-f(-nx))/2n2)(αutn)≥τM(Λ(n+1)(g(nx)-f(nx))/2n2((n+1)αut(2n2)),Λ(n-1)(g(-nx)-f(-nx))/2n2((n-1)αut(2n2)))≥τM(Λg(nx)-f(nx)(αut),Λg(-nx)-f(-nx)(αut))≥τM(Λ′φ(nx̂)(αt),Λ′φ(-nx̂)(αt))≥M(x,t)
for all x∈X∖{0} and t>0, which implies that
d(Jf,Jg)≤αnd(f,g).
That is, J is a strict contraction with the Lipschitz constant 0<α/n<1.

Moreover, by (R2), (R3), and (3.2), we see that
Λf(x)-Jf(x)(t2n)=Λ(-(n+1)Df(x̂)+(n-1)Df(-x̂))/4n2(t2n)≥τM(Λ(n+1)Df(x̂)/4n2((n+1)t4n2),Λ(n-1)Df(-x̂)/4n2((n-1)t4n2))≥τM(ΛDf(x̂)(t),ΛDf(-x̂)(t))≥M(x,t)
for all x∈X∖{0} and t>0. Hence, it follows from the definition of d that
d(f,Jf)≤12n<∞.
Now, in view of Theorem 2.5, the sequence {Jmf} converges to the unique “fixed point” F:X→Y of J in the set T={g∈S∣d(f,g)<∞} and F is represented by
F(x)=limm→∞(f(nmx)-f(-nmx)2nm+f(nmx)+f(-nmx)2n2m)
for all x∈X.

By Theorem 2.5, (3.11), and the definition of d, we have
d(f,F)≤11-α/nd(f,Jf)≤12(n-α),
that is, the first inequality in (3.4) holds true.

We will now show that F is a quadratic-additive function. It follows from (R3) and the definition of τM that
ΛDF(x1,x2,…,xn)(t)≥min{Λ2(F-Jmf)(∑j=1nxj)(t5),min{Λ(F-Jmf)(xi-xj)(t(5n(n-1)))∣1≤i,j≤n,i≠j},min{Λ(n+1)(Jmf-F)(xj)(t(5n))∣j=1,…,n},min{Λ(n-1)(Jmf-F)(-xj)(t(5n))∣j=1,…,n},ΛDJmf(x1,x2,…,xn)(t5)}
for all x1,x2,…,xn∈X∖{0}, t>0, and m∈ℕ. Due to the definition of F, the first four terms on the right-hand side of the above inequality tend to 1 as m→∞.

By a somewhat tedious manipulation, we have
DJmf(x1,x2,…,xn)=12n2mDf(nmx1,…,nmxn)+12n2mDf(-nmx1,…,-nmxn)+12nmDf(nmx1,…,nmxn)-12nmDf(-nmx1,…,-nmxn).
Hence, it follows from (R2), (R3), definition of τM, (3.2), and (i) that
ΛDJmf(x1,…,xn)(t5)≥min{ΛDf(nmx1,…,nmxn)/2n2m(t20),ΛDf(-nmx1,…,-nmxn)/2n2m(t20),ΛDf(nmx1,…,nmxn)/2nm(t20),ΛDf(-nmx1,…,-nmxn)/2nm(t20)}≥min{ΛDf(nmx1,…,nmxn)(n2mt10),ΛDf(-nmx1,…,-nmxn)(n2mt10),ΛDf(nmx1,…,nmxn)(nmt10),ΛDf(-nmx1,…,-nmxn)(nmt10)ΛDf(nmx1,…,nmxn)(n2mt10)}≥min{Λ′φ(x1,…,xn)(n2mt(10αm)),Λ′φ(-x1,…,-xn)(n2mt(10αm)),Λ′φ(x1,…,xn)(nmt(10αm)),Λ′φ(-x1,…,-xn)(nmt(10αm))Λ′φ(x1,…,xn)(n2mt10αm)},
which tends to 1 as m→∞ for all x1,x2,…,xn∈X∖{0} and t>0. Therefore, (3.14) implies that
ΛDF(x1,x2,…,xn)(t)=1
for any x1,…,xn∈X∖{0} and t>0. By (R1), this implies that DF(x1,…,xn)=0 for all x1,…,xn∈X∖{0}, which ends the proof of the first part.

Now, assume that φ satisfies the condition (ii). Let (S,d) be the same as given in the first part. We now consider the operator J:S→S defined by
Jg(x):=n2(g(xn)-g(-xn))+n22(g(xn)+g(-xn))
for all g∈S and x∈X. Notice that
Jmg(x)=nm2(g(xnm)-g(-xnm))+n2m2(g(xnm)+g(-xnm))
for all x∈X and m∈ℕ.

Let f,g∈S and let u∈[0,∞] be an arbitrary constant with d(g,f)≤u. From (R2), (R3), the definition of d, and (ii), we have
ΛJg(x)-Jf(x)(n2utα)=Λ((n2+n)/2)(g(x/n)-f(x/n))+((n2-n)/2)(g(-x/n)-f(-x/n))(n2utα)≥τM(Λ((n2+n)/2)(g(x/n)-f(x/n))((n2+n)ut(2α)),Λ((n2-n)/2)(g(-x/n)-f(-x/n))((n2-n)ut(2α)))=τM(Λg(x/n)-f(x/n)(utα),Λg(-x/n)-f(-x/n)(utα))≥τM(M(xn,tα),M(-xn,tα))=τM(Λ′φ(x/n̂)(tα),Λ′φ(-x/n̂)(tα))=τM(Λ′αφ(x/n̂)(t),Λ′αφ(-x/n̂)(t))≥τM(Λ′φ(x̂)(t),Λ′φ(-x̂)(t))=M(x,t)
for all x∈X∖{0}, t>0, and for some α>n2 satisfying (ii), which implies that
d(Jf,Jg)≤n2αd(f,g).
That is, J is a strict contraction with the Lipschitz constant 0<n2/α<1.

Moreover, by (R2), (3.2), and (ii), we see that
Λf(x)-Jf(x)(t(2α))=Λ(1/2)Df(x/n̂)(t(2α))≥Λ′φ(x/n̂)(tα)=Λ′αφ(x/n̂)(t)≥Λ′φ(x̂)(t)≥M(x,t)
for all x∈X∖{0} and t>0. This implies that d(f,Jf)≤1/(2α)<∞ by the definition of d. Therefore, according to Theorem 2.5, the sequence {Jmf} converges to the unique “fixed point” F:X→Y of J in the set T={g∈S∣d(f,g)<∞} and F is represented by
F(x)=limm→∞(nm2(f(xnm)-f(-xnm))+n2m2(f(xnm)+f(-xnm)))
for all x∈X. Since
d(f,F)≤11-n2/αd(f,Jf)≤12(α-n2),
the second inequality in (3.4) holds true.

Next, we will show that F is a quadratic-additive function. As we did in the first part, we obtain the inequality (3.14). In view of the definition of F, the first four terms on the right-hand side of the inequality (3.14) tend to 1 as m→∞. Furthermore, a long manipulation yields
DJmf(x1,x2,…,xn)=n2m2Df(x1nm,…,xnnm)+n2m2Df(-x1nm,…,-xnnm)+nm2Df(x1nm,…,xnnm)-nm2Df(-x1nm,…,-xnnm).
Thus, it follows from (R2), (R3), definition of τM, (3.2), and (ii) that
ΛDJmf(x1,…,xn)(t5)≥min{Λ(n2m/2)Df(x1/nm,…,xn/nm)(t20),Λ(n2m/2)Df(-x1/nm,…,-xn/nm)(t20),Λ(nm/2)Df(x1/nm,…,xn/nm)(t20),Λ-(nm/2)Df(-x1/nm,…,-xn/nm)(t20)}≥min{Λ′φ(x1/nm,…,xn/nm)(t(10n2m)),Λ′φ(-x1/nm,…,-xn/nm)(t(10n2m)),Λ′φ(x1/nm,…,xn/nm)(t(10nm)),Λ′φ(-x1/nm,…,-xn/nm)(t(10nm))}≥min{Λ′α-mφ(x1,…,xn)(t(10n2m)),Λ′α-mφ(-x1,…,-xn)(t(10n2m)),Λ′α-mφ(x1,…,xn)(t(10nm)),Λ′α-mφ(-x1,…,-xn)(t(10nm))}=min{Λ′φ(x1,…,xn)(αmt(10n2m)),Λ′φ(-x1,…,-xn)(αmt(10n2m)),Λ′φ(x1,…,xn)(αmt(10nm)),Λ′φ(-x1,…,-xn)(αmt(10nm))},
which tends to 1 as m→∞ for all x1,x2,…,xn∈X∖{0} and t>0. Therefore, it follows from (3.14) that
ΛDF(x1,x2,…,xn)(t)=1
for any x1,x2,…,xn∈X∖{0} and t>0. By (R1), this implies that
DF(x1,x2,…,xn)=0
for all x1,x2,…,xn∈X∖{0}, which ends the proof.

By a similar way presented in the proof of Theorem 3.1, we can also prove the preceding theorem if the domains of relevant functions include 0.

Theorem 3.2.

Let X be a real vector space, (Z,Λ′,τM) an RN-space, (Y,Λ,τM) a complete RN-space, and let φ:Xn→Z be a function. Assume that φ satisfies one of the conditions (i) and (ii) in Theorem 3.1 for all x1,x2,…,xn∈X and t>0. If a function f:X→Y satisfies f(0)=0 and (3.2) for all x1,x2,…,xn∈X and t>0, then there exists a unique quadratic-additive function F:X→Y satisfying (3.4) for all x∈X and t>0.

Now, we obtain general Hyers-Ulam stability results of (1.3) in normed spaces. If X is a normed space, then (X,Λ,τM) is an induced random normed space. We get the following result.

Corollary 3.3.

Let X be a real vector space, Y a complete normed space, and let φ:(X∖{0})n→[0,∞) be a function. Assume that φ satisfies one of the following conditions:

φ(nx1,…,nxn)≤αφ(x1,…,xn) for some 1<α<n;

φ(nx1,…,nxn)≥αφ(x1,…,xn) for some α>n2

for all x1,x2,…,xn∈X∖{0}. If a function f:X→Y satisfies f(0)=0 and
‖Df(x1,x2,…,xn)‖≤φ(x1,x2,…,xn)
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
‖f(x)-F(x)‖≤{max{φ(x̂),φ(-x̂)}2(n-α)ifφsatisfies(iii),max{φ(x̂),φ(-x̂)}2(α-n2)ifφsatisfies(iv)
for all x∈X∖{0}.Proof.

Let us put
Z:=R,Λx(t):=tt+‖x‖,Λ′z(t):=tt+|z|
for all x,x1,x2,…,xn∈X∖{0}, z∈ℝ∖{0}, and t≥0. If φ satisfies the condition (iii) for all x1,x2,…,xn∈X∖{0} and for some 1<α<n, then
Λ′αφ(x1,…,xn)(t)=tt+αφ(x1,…,xn)≤tt+φ(nx1,…,nxn)=Λ′φ(nx1,…,nxn)(t)
for all x1,x2,…,xn∈X∖{0} and t>0, that is, φ satisfies the condition (i). In a similar way, we can show that if φ satisfies (iv), then it satisfies the condition (ii).

Moreover, we get
ΛDf(x1,…,xn)(t)=tt+‖Df(x1,…,xn)‖≥tt+φ(x1,…,xn)=Λ′φ(x1,…,xn)(t)
for all x1,x2,…,xn∈X∖{0} and t>0, that is, f satisfies the inequality (3.2) for all x1,x2,…,xn∈X∖{0}.

According to Theorem 3.1, there exists a unique function F:X→Y such thatDF(x1,x2,…,xn)=0
for all x1,x2,…,xn∈X∖{0} and
Λf(x)-F(x)(t)≥{τM(Λ′φ(x̂)(2(n-α)t),Λ′φ(-x̂)(2(n-α)t))ifφsatisfies(iii),τM(Λ′φ(x̂)(2(α-n2)t),Λ′φ(-x̂)(2(α-n2)t))ifφsatisfies(iv)
for all x1,x2,…,xn∈X∖{0} and t>0, which ends the proof.

We now prove the Hyers-Ulam-Rassias stability of (1.3) in the framework of normed spaces.

Corollary 3.4.

Let X be a real normed space, p∈[0,1)∪(2,∞), and let Y be a complete normed space. If a function f:X→Y satisfies f(0)=0 and
‖Df(x1,x2,…,xn)‖≤θ(‖x1‖p+‖x2‖p+⋯+‖xn‖p)
for all x1,x2,…,xn∈X and for some θ≥0, then there exists a unique quadratic-additive function F:X→Y such that
‖f(x)-F(x)‖≤{nθ‖x‖p2(n-np)if0≤p<1,nθ‖x‖p2(np-n2)ifp>2
for all x∈X.

Proof.

If we put
φ(x1,x2,…,xn):=θ(‖x1‖p+‖x2‖p+⋯+‖xn‖p),
then the induced random normed space (X,Λx,τM) satisfies the conditions stated in Theorem 3.2 with α=np.

Corollary 3.5.

Let X be a real normed space, p∈(-∞,0), and let Y be a complete normed space. If a function f:X→Y satisfies f(0)=0 and
‖Df(x1,x2,…,xn)‖≤θ∑1≤i≤n,xi≠0‖xi‖p
for all x1,x2,…,xn∈X and for some θ≥0, then there exists a unique quadratic-additive function F:X→Y satisfying
‖f(x)-F(x)‖≤{nθ‖x‖p2(n-np)ifx∈X∖{0},0ifx=0.

Proof.

If we put Z:=ℝ, α:=np, and define
Λx(t):=tt+‖x‖,Λ′z(t):=tt+|z|,φ(x1,x2,…,xn):=θ∑1≤i≤n,xi≠0‖xi‖p
for all x,x1,x2,…,xn∈X and z∈Z, then we have
Λ′αφ(x1,x2,…,xn)(t)=tt+αφ(x1,…,xn)=tt+φ(nx1,…,nxn)=Λ′φ(nx1,nx2,…,nxn)(t),
that is, φ satisfies condition (i) given in Theorem 3.1 for all x1,x2,…,xn∈X and t>0. We moreover get
ΛDf(x1,x2,…,xn)(t)=tt+‖Df(x1,…,xn)‖≥tt+θ∑1≤i≤n,xi≠0‖xi‖p=tt+φ(x1,…,xn)=Λ′φ(x1,x2,…,xn)(t),
that is, f satisfies the inequality (3.2) for all x1,x2,…,xn∈X and t>0.

According to Theorem 3.2, there exists a unique quadratic-additive function F:X→Y satisfying
tt+‖f(x)-F(x)‖=Λf(x)-F(x)(t)≥M(x,2(n-np)t)={2(n-np)t2(n-np)t+nθ‖x‖pifx∈X∖{0},1ifx=0
for all t>0, or equivalently
‖f(x)-F(x)‖t≤{nθ‖x‖p2(n-np)tifx∈X∖{0},0ifx=0
for all t>0, which ends the proof.

Acknowledgment

The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2011-0004919).

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