# Stability of ann-Dimensional Mixed-Type Additive and Quadratic Functional Equation in Random Normed Spaces

Authors: Yang-Hi Lee; Soon-Mo Jung
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We investigate the stability problems for then-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.


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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 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] 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.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 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 [6–17]).In this paper, applying the fixed point method, we prove the Hyers-Ulam-Rassias stability of then-dimensional mixed-type additive and quadratic 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)
in random normed spaces. Every solution of (1.3) is called a quadratic-additive function.Throughout this paper, letn 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 by(2.1)Δ+∶={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 by(2.2)ɛ0(t)={0ift=0,1ift>0.Definition 2.1 (See [18]).
A functionτ:[0,1]×[0,1]→[0,1] is called a continuous triangular norm (briefly, continuoust-norm) if τ satisfies the following conditions:(a)
τ is commutative and associative;(b)
τ is continuous;(c)
τ(a,1)=a for all a∈[0,1];(d)
τ(a,b)≤τ(c,d) for all a,b,c,d∈[0,1] with a≤c and b≤d.
Typical examples of continuoust-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 [19]).
LetX be a vector space, τ a continuous t-norm, and let Λ:X→D+ be a function satisfying the following conditions:(R1)
Λx(t)=ɛ0(t) for all t>0 if and only if x=0;(R2)
Λαx(t)=Λx(t/|α|) for all x∈X, α≠0, and for all t≥0;(R3)
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
(2.3)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.(i)
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.(ii)
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.(iii)
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.
LetX be a nonempty set. A function d:X2→[0,∞] is called a generalized metric on X if and only if d satisfies(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 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 [20, 21]).
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(2.4)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, letX 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:(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∈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.
LetX 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:(i)
Λ′αφ(x1,x2,…,xn)(t)≤Λ′φ(nx1,nx2,…,nxn)(t) for some 0<α<n;(ii)
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
(3.2)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
(3.3)DF(x1,x2,…,xn)=0
for all x1,x2,…,xn∈X∖{0} and
(3.4)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 whereφ 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
(3.5)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 operatorJ:S→S defined by
(3.6)Jf(x):=f(nx)-f(-nx)2n+f(nx)+f(-nx)2n2.
Then we can apply induction on m to prove
(3.7)Jmf(x)=f(nmx)-f(-nmx)2nm+f(nmx)+f(-nmx)2n2m
for all x∈X and m∈ℕ.
Letf,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
(3.8)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
(3.9)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
(3.10)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
(3.11)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
(3.12)F(x)=lim⁡m→∞(f(nmx)-f(-nmx)2nm+f(nmx)+f(-nmx)2n2m)
for all x∈X.
By Theorem2.5, (3.11), and the definition of d, we have
(3.13)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 thatF is a quadratic-additive function. It follows from (R3) and the definition of τM that
(3.14)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(3.15)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
(3.16)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
(3.17)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 thatφ 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
(3.18)Jg(x):=n2(g(xn)-g(-xn))+n22(g(xn)+g(-xn))
for all g∈S and x∈X. Notice that
(3.19)Jmg(x)=nm2(g(xnm)-g(-xnm))+n2m2(g(xnm)+g(-xnm))
for all x∈X and m∈ℕ.
Letf,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
(3.20)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
(3.21)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
(3.22)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
(3.23)F(x)=lim⁡m→∞(nm2(f(xnm)-f(-xnm))+n2m2(f(xnm)+f(-xnm)))
for all x∈X. Since
(3.24)d(f,F)≤11-n2/αd(f,Jf)≤12(α-n2),
the second inequality in (3.4) holds true.
Next, we will show thatF 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
(3.25)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
(3.26)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
(3.27)ΛDF(x1,x2,…,xn)(t)=1
for any x1,x2,…,xn∈X∖{0} and t>0. By (R1), this implies that
(3.28)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 Theorem3.1, we can also prove the preceding theorem if the domains of relevant functions include 0.Theorem 3.2.
LetX 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.
LetX 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:(iii)
φ(nx1,…,nxn)≤αφ(x1,…,xn) for some 1<α<n;(iv)
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
(3.29)‖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
(3.30)DF(x1,x2,…,xn)=0
for all x1,x2,…,xn∈X∖{0} and
(3.31)‖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(3.32)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
(3.33)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(3.34)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 Theorem3.1, there exists a unique function F:X→Y such that(3.35)DF(x1,x2,…,xn)=0
for all x1,x2,…,xn∈X∖{0} and
(3.36)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.
LetX 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
(3.37)‖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
(3.38)‖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(3.39)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.
LetX 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
(3.40)‖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
(3.41)‖f(x)-F(x)‖≤{nθ‖x‖p2(n-np)ifx∈X∖{0},0ifx=0.Proof.
If we putZ:=ℝ, α:=np, and define
(3.42)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
(3.43)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
(3.44)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 Theorem3.2, there exists a unique quadratic-additive function F:X→Y satisfying
(3.45)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
(3.46)‖f(x)-F(x)‖t≤{nθ‖x‖p2(n-np)tifx∈X∖{0},0ifx=0
for all t>0, which ends the proof.



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*Source: 547865-2012-01-17.xml*