Skof, in 1981, was the first author to solve the Ulam problem for quadratic mappings

http://jipam.vu.edu.au/

Volume 5, Issue 3, Article 52, 2004

THE ULAM STABILITY PROBLEM IN APPROXIMATION OF APPROXIMATELY QUADRATIC MAPPINGS BY QUADRATIC MAPPINGS

JOHN MICHAEL RASSIAS PEDAGOGICALDEPARTMENT, E. E.,

NATIONAL ANDCAPODISTRIANUNIVERSITY OFATHENS, SECTION OFMATHEMATICS ANDINFORMATICS,

4, AGAMEMNONOSSTR., AGHIAPARASKEVI, ATHENS15342, GREECE.

[email protected]

URL:http://www.primedu.uoa.gr/ jrassias/

Received 23 September, 2003; accepted 29 November, 2003 Communicated by D. Bainov

ABSTRACT. S.M. Ulam, 1940, proposed the well-known Ulam stability problem and in 1941, the problem for linear mappings was solved by D.H. Hyers. D.G. Bourgin, 1951, also investi- gated the Ulam problem for additive mappings. P.M. Gruber, claimed, in 1978, that this kind of stability problem is of particular interest in probability theory and in the case of functional equations of different types. F. Skof, in 1981, was the first author to solve the Ulam problem for quadratic mappings. During the years 1982-1998, the author established the Hyers-Ulam stability for the Ulam problem for different mappings. In this paper we solve the Ulam stability problem by establishing an approximation of approximately quadratic mappings by quadratic mappings. Today there are applications in actuarial and financial mathematics, sociology and psychology, as well as in algebra and geometry.

Key words and phrases: Ulam problem, Ulam type problem, General Ulam problem, Quadratic mapping, Approximately quadratic mapping, Square of the quadratic weighted mean.

2000 Mathematics Subject Classification. 39B.

1. INTRODUCTION

S.M. Ulam [24] proposed the general Ulam stability problem: "When is it true that by slightly changing the hypotheses of a theorem one can still assert that the thesis of the theorem re- mains true or approximately true?" D.H. Hyers [13] solved this problem for linear mappings.

D.G. Bourgin [3] also investigated the Ulam problem for additive mappings. P.M. Gruber [12]

claimed that this kind of stability problem is of particular interest in probability theory and in the case of functional equations of different types. Th.M. Rassias [20] employed Hyers’ ideas to new additive mappings, and later I. Fenyö ([7], [8]) established the stability of the Ulam prob- lem for quadratic and other mappings. Z. Gajda and R. Ger [10] showed that one can obtain

ISSN (electronic): 1443-5756

c 2004 Victoria University. All rights reserved.

165-03

analogous stability results for subadditive multifunctions. Other interesting stability results have been achieved also by the following authors: J. Aczél [1], C. Borelli and G.L. Forti ([2], [9]), P.W. Cholewa [4], St. Czerwik [5], H. Drljevic [6] and L. Paganoni [14]. F. Skof ([21] – [23]) was the first author to solve the Ulam problem for quadratic mappings. We ([15] – [19]) solved the above Ulam problem for different mappings. P. Gˇavru¸tˇa [11] answered a question of ours [17] concerning the stability of the Cauchy equation. Today there are applications in actuarial and financial mathematics, sociology and psychology, as well as in algebra and geometry.

In this paper we introduce the following quadratic functional equation (∗) Q(a₁x₁+a₂x₂) +Q(a₂x₁−a₁x₂) = a²₁+a²₂

[Q(x₁) +Q(x₂)]

with quadratic mappingsQ:X →Y such thatXandY are real linear spaces.

Denote

K_r =K_r(kx₁k,kx₂k)

=

2^r−1(kx₁k^r+kx₂k^r)−(kx₁+x₂k^r+kx₁−x₂k^r)

=







2^r−1(kx1k^r+kx2k^r)−(kx1+x2k^r+kx1−x2k^r), ifr >2 kx₁+x2k^r+kx₁−x2k^r−2^r−1(kx₁k^r+kx₂k^r), if1< r <2,

for every(x₁, x₂)∈X², whereX is a normed linear space. Note thatK_r≥0for any fixed real r: 1< r6= 2.Note also that

K_r =K_r(kxk,kxk) = 0, K_r(|a₁| kxk,|a₂| kxk) = β₁kxk^r, K_r m⁻¹|a₁| kxk, m⁻¹|a₂| kxk

=β₁m^−rkxk^r, K_r(kxk,0) = β₂kxk^r and K_r m⁻¹kxk,0

=β₃kxk^r, where

β1 =Kr(|a1|,|a2|)

=

2^r−1(|a1|^r+|a2|^r)−(|a1+a2|^r+|a1−a2|^r)

=







2^r−1(|a₁|^r+|a₂|^r)−(|a₁+a₂|^r+|a₁−a₂|^r), ifr >2

|a₁+a₂|^r+|a₁−a₂|^r−2^r−1(|a₁|^r+|a₂|^r), if1< r <2,

β₂ =K_r(1,0) =

2^r−1−2 =







2^r−1−2, ifr >2 2−2^r−1, if1< r <2, β₃ =K_r m⁻¹,0

=β₂m^−r, Note thata₁ 6=a₂, and16=m=a²₁+a²₂ >0.

IfXandY are normed linear spaces andY complete, then we establish an approximation of approximately quadratic mappingsf : X → Y by quadratic mappingsQ :X → Y, such that the corresponding approximately quadratic functional inequality

(∗∗)

f(a1x1+a2x2) +f(a2x1−a1x2)− a²₁+a²₂

[f(x1) +f(x2)]

≤cK_r(kx₁k,kx₂k)

holds with a constant c ≥ 0 (independent of x₁,x₂ ∈ X), and any fixed pair a = (a₁, a₂) ∈ R²− {(0,0)}and any fixed realr >1 :

I₁ ={(r, m)∈R² : 1< r <2, m >1 and r >2,0< m <1},or I₂ ={(r, m)∈R² : 1< r <2,0< m <1 and r >2, m >1},

hold, where16=m =a²₁+a²₂ =|a|² >0anda1 6=a2. Note thatm^r−2 <1if(r, m)∈I1, and m^2−r<1if(r, m)∈I2.

It is useful for the following, to observe that, from (∗) withx₁ =x₂ = 0, and0< m6= 1we get

2(m−1)Q(0) = 0, or

(1.1) Q(0) = 0.

Definition 1.1. LetXandY be real linear spaces. Leta= (a1, a2)∈R²− {(0,0)}: 0< m= a²₁+a²₂ 6= 1anda1 6=a2. Then a mappingQ :X →Y is called quadratic with respect toa, if (∗) holds for every vector(x₁, x₂)∈X².

Definition 1.2. LetXandY be real linear spaces. Leta= (a1, a2)∈R²− {(0,0)}: 0< m= a²₁ +a²₂ 6= 1and a1 6= a2. Then a mappingQ¯ : X → Y is called the square of the quadratic weighted mean ofQwith respect toa = (a₁, a₂), if

(1.2) Q(x) =¯







Q(a1x)+Q(a2x)

a²₁+a²₂ , if (r, m=a²₁+a²₂)∈I₁ (a²₁+a²₂)

h Q

a1

a²₁+a²₂x

+Q a2

a²₁+a²₂x i

, if (r, m=a²₁+a²₂)∈I2

for allx∈X.

For every x ∈ R set Q(x) = x². Then the mapping Q¯ : R → R is quadratic, such that Q¯(x) = x². Denoting by

q

x²_w the quadratic weighted mean, we note that the above- mentioned mappingQis an analogous case to the square of the quadratic weighted mean em- ployed in mathematical statistics: x²_w = ^a21x_a²¹2^+a22x22

1+a²₂ with weightsw₁ = a²₁ and w₂ = a²₂, data x₁ =x₂ =x, andQ(a_ix) = (a_ix)²,(i= 1,2).

Now, claim that forn ∈N₀ ={0,1,2, . . .}that

(1.3) Q(x) =

( m⁻²ⁿQ(mⁿx), if (r, m)∈I₁ m²ⁿQ(m⁻ⁿx), if (r, m)∈I₂, for allx∈Xandn ∈N₀.

Forn = 0, it is trivial. From (1.1), (1.2) and (∗), withx_i =a_ix (i= 1,2), we obtain Q(mx) = m[Q(a₁x) +Q(a₂x)],

or

(1.4) Q(x) =¯ m⁻²Q(mx),

ifI₁ holds. Besides from (1.1), (1.2) and (∗), withx₁ =x,x₂ = 0, we get Q(a₁x) +Q(a₂x) = mQ(x),

or

(1.5) Q(x) =¯ Q(x),

ifI₁ holds. Therefore from (1.4) and (1.5) we have

(1.6) Q(x) = m⁻²Q(mx),

which is (1.3) for n = 1, if I1 holds. Similarly, from (1.1), (1.2) and (∗), with xi = ^a_mⁱx (i= 1,2), we obtain

(1.7) Q(x) = ¯Q(x)

ifI₂ holds. Besides from (1.1), (1.2) and (∗), withx₁ = _m^x,x₂ = 0,we get Q

a1

mx

+Q a2

mx

=mQ(m⁻¹x), or

(1.8) Q(x) =¯ m²Q(m⁻¹x)

ifI₂ holds. Therefore from (1.7) and (1.8) we have

(1.9) Q(x) =m²Q(m⁻¹x),

which is (1.3) forn= 1, ifI2 holds.

Assume (1.3) is true and from (1.6), withmⁿxin place ofx, we get:

(1.10) Q mⁿ⁺¹x

=m²Q(mⁿx) = m²(mⁿ)²Q(x) = mⁿ⁺¹2

Q(x).

Similarly, withm⁻ⁿxin place ofx, we get:

(1.11) Q m⁻⁽ⁿ⁺¹⁾x

=m⁻²Q(m⁻ⁿx) =m⁻²(m⁻ⁿ)²Q(x) = m⁻⁽ⁿ⁺¹⁾2

Q(x).

These formulas (1.10) and (1.11) by induction, prove formula (1.3).

2. QUADRATICFUNCTIONAL STABILITY

Theorem 2.1. Let X and Y be normed linear spaces. Assume that Y is complete. Assume in addition that mapping f : X → Y satisfies the functional inequality (∗∗). Define I₁ = {(r, m)∈R² : 1< r <2, m >1, orr > 2, 0< m <1}, andI₂ ={(r, m)∈R² : 1< r <2, 0 < m < 1, orr > 2, m > 1}for any fixed paira = (a₁, a₂)of realsa_i 6= 0 (i = 1,2)and any fixed realr >1 : 1 6=m =a²₁+a²₂ =|a|² >0,a₁ 6=a₂. Besides define

0< β₁ =K_r(|a₁|,|a₂|)

=

2^r−1(|a₁|^r+|a₂|^r)−(|a₁+a₂|^r+|a₁ −a₂|^r)

=







2^r−1(|a₁|^r+|a₂|^r)−(|a₁+a₂|^r+|a₁−a₂|^r), ifr >2

|a₁+a₂|^r+|a₁−a₂|^r−2^r−1(|a₁|^r+|a₂|^r), if1< r <2, β₂ =K_r(1,0) =|2^r−1−2|, andσ =β₁+mβ₂ >0. Also define

f_n(x) =







m⁻²ⁿf(mⁿx), if(r, m)∈I₁ m²ⁿf(m⁻ⁿx), if(r, m)∈I₂ for allx∈Xandn∈N₀ ={0,1,2, . . .}.

Then the limit

(2.1) Q(x) = lim

n→∞f_n(x)

exists for all x ∈ X and Q : X → Y is the unique quadratic mapping with respect to a = (a₁, a₂), such that

kf(x)−Q(x)k ≤ σc

|m²−m^r|kxk^r (2.2)

=kxk^r







σc/(m²−m^r), if(r, m)∈I₁ σc/(m^r−m²), if(r, m)∈I₂ holds for allx∈X andn∈N0 andc≥0(constant independent ofx∈X).

Existence.

Proof. It is useful for the following, to observe that, from (∗∗) withx₁ =x₂ = 0and0< m 6=

1, we get

2|m−1| kf(0)k ≤0, or

(2.3) f(0) = 0.

Now claim that forn∈N₀

kf(x)−fn(x)k ≤ σc

|m²−m^r| 1−m^n|r−2|

kxk^r (2.4)

=kxk^r







σc

m²−m^r 1−m^n(r−2)

, if (r, m)∈I1 :m^r−2 <1

σc

m^r−m² 1−m^n(2−r)

, if (r, m)∈I₂ :m^2−r <1.

Forn = 0,it is trivial.

Define f¯ : X → Y, the square of the quadratic weighted mean of f with respect to a = (a₁, a₂)by replacingQ,Q¯ of (1.2) withf,f, respectively, as follows:¯

(2.5) f¯(x) =







f(a1x)+f(a2x)

a²₁+a²₂ , if(r, m=a²₁ +a²₂ =|a|²)∈I1

(a²₁+a²₂) h

f a1

a²₁+a²₂x

+f a2

a²₁+a²₂x i

, if(r, m=a²₁ +a²₂ =|a|²)∈I2

for allx∈X.

From (2.3), (2.5) and (∗∗), withx_i =a_ix(i= 1,2), we obtain kf(mx)−m[f(a1x)+f(a2x)]k ≤σckxk^r, or

(2.6)

m⁻²f(mx)−f¯(x)

≤ β₁c m² kxk^r,

ifI₁ holds. Besides from (2.3), (2.5) and (∗∗), withx₁ =x,x₂ = 0, we get kf(a₁x)+f(a₂x)−mf(x)k ≤cK_r(kxk,0) = β₂ckxk^r, or

(2.7)

f¯(x)−f(x) ≤ β₂c

m kxk^r, ifI₁ holds. Therefore from (2.6) and (2.7) we have

(2.8)

f(x)−m⁻²f(mx) ≤ σc

m² kxk^r = σc

m²−m^r 1−m^r−2 kxk^r, which is (2.4) forn= 1, ifI₁ holds.

Similarly, from (2.3), (2.5) and (∗∗), withx_i = ^a_mⁱx(i= 1,2), we obtain

(2.9)

f(x)−f¯(x) ≤ β₁c

m^r kxk^r,

ifI₂ holds. Besides from (2.3), (2.5) and (∗∗), withx₁ = _m^x,x₂ = 0, we get

fa₁

mx

+fa₂ mx

−mf(m⁻¹x)

≤cK_r m⁻¹kxk,0

=β₃ckxk^r, or

(2.10)

f¯(x)−m²f(m⁻¹x)

≤mβ₃ckxk^r = mβ₂c m^r kxk^r, ifI₂ holds. Therefore from (2.9) and (2.10) we have

(2.11)

f(x)−m²f(m⁻¹x) ≤ σc

m^r kxk^r= σc

m^r−m² 1−m^2−r kxk^r, which is (2.4) forn= 1, ifI₂ holds.

Assume (2.4) is true if (r, m) ∈ I₁. From (2.8), withmⁿx in place of x, and the triangle inequality, we have

kf(x)−f_n+1(x)k (2.12)

=

f(x)−m^−2(n+1)f mⁿ⁺¹x

≤

f(x)−m⁻²ⁿf(mⁿx) +

m⁻²ⁿf(mⁿx)−m^−2(n+1)f mⁿ⁺¹x

≤ σc m² −m^r

1−m^n(r−2)

+m⁻²ⁿ 1−m^r−2 m^nr

kxk^r

= σc

m²−m^r 1−m^(n+1)(r−2) kxk^r, ifI₁ holds.

Similarly assume (2.4) is true if(r, m) ∈I₂. From (2.11), withm⁻ⁿxin place ofx, and the triangle inequality, we have

kf(x)−f_n+1(x)k (2.13)

=

f(x)−m²⁽ⁿ⁺¹⁾f m⁻⁽ⁿ⁺¹⁾x

≤

f(x)−m²ⁿf m⁻ⁿx +

m²ⁿf(m⁻ⁿx)−m²⁽ⁿ⁺¹⁾f m⁻⁽ⁿ⁺¹⁾x

≤ σc m^r−m²

(1−m^n(2−r))+m²ⁿ(1−m^2−r)m^−nr kxk^r

= σc

m^r−m² 1−m^(n+1)(2−r) kxk^r, ifI2 holds.

Therefore inequalities (2.12) and (2.13) prove inequality (2.4) for anyn∈N₀.

Claim now that the sequence {f_n(x)} converges. To do this it suffices to prove that it is a Cauchy sequence. Inequality (2.4) is involved if (r, m) ∈ I₁. In fact , if i > j > 0, and h₁ =m^jx, we have:

kfi(x)−fj(x)k=

m⁻²ⁱf(mⁱx)−m^−2jf(m^jx) (2.14)

=m^−2j

m^−2(i−j)f m^i−jh₁

−f(h₁)

≤m^−2j σc

m²−m^r 1−m^{(i−j)(r−2)} kxk^r

< σc

m²−m^rm^−2jkxk^r −−−→

j→∞ 0, ifI₁ holds:m^r−2 <1.

Similarly, ifh₂ =m^−jxinI₂, we have:

kf_i(x)−f_j(x)k=

m²ⁱf(m⁻ⁱx)−m^2jf(m^−jx) (2.15)

=m^2j

m^2(i−j)f m^−(i−j)h₂

−f(h₂)

≤m^2j σc

m^r−m² 1−m^{(i−j)(2−r)} kxk^r

< σc

m^r−m²m^2jkxk^r −−−→

j→∞ 0, ifI₂ holds:m^2−r <1.

Then inequalities (2.14) and (2.15) define a mappingQ:X →Y, given by (2.1).

Claim that from (∗∗) and (2.1) we can get (∗), or equivalently that the afore-mentioned well- defined mappingQ:X →Y is quadratic.

In fact, it is clear from the functional inequality (∗∗) and the limit (2.1) for(r, m)∈ I1 that the following functional inequality

m⁻²ⁿkf(a1mⁿx1 +a2mⁿx2) +f(a2mⁿx1−a1mⁿx2)−(a²₁+a²₂) [f(mⁿx1) +f(mⁿx2)]

≤m⁻²ⁿcK_r(mⁿkx₁k, mⁿkx₂k), holds for all vectors (x₁, x₂) ∈ X², and all n ∈ N with f_n(x) = m⁻²ⁿf(mⁿx) : I₁ holds.

Therefore lim

n→∞f_n(a₁x₁+a₂x₂) + lim

n→∞f_n(a₂x₁−a₁x₂)−(a²₁+a²₂)h

n→∞lim f_n(x₁) + lim

n→∞f_n(x₂)i

≤

n→∞lim m^n(r−2)

cK_r(kx₁k,kx₂k) = 0, becausem^r−2 <1or

(2.16)

Q(a₁x₁+a₂x₂)+Q(a₂x₁−a₁x₂)− a²₁+a²₂

[Q(x₁)+Q(x₂)]

= 0, or mappingQsatisfies the quadratic equation (∗).

Similarly, from (∗∗) and (2.1) for(r, m)∈I₂we get that m²ⁿ

f(a1m⁻ⁿx1 +a2m⁻ⁿx2)+f(a2m⁻ⁿx1−a1m⁻ⁿx2)

−(a²₁+a²₂)

f(m⁻ⁿx1)+f(m⁻ⁿx2)

≤m²ⁿcKr m⁻ⁿkx1k, m⁻ⁿkx2k , holds for all vectors(x₁, x₂)∈X², and alln∈Nwithf_n(x) =m²ⁿf(m⁻ⁿx) :I₂holds. Thus

lim

n→∞f_n(a₁x₁+a₂x₂)+ lim

n→∞f_n(a₂x₁−a₁x₂)−(a²₁+a²₂) h

n→∞lim f_n(x₁)+ lim

n→∞f_n(x₂)i

≤

n→∞lim m^n(2−r)

cK_r(kx₁k,kx₂k) = 0, becausem^2−r <1, or (2.16) holds or mappingQsatisfies (∗).

Therefore (2.16) holds ifI_j (j = 1,2)hold or mappingQsatisfies (∗), completing the proof thatQis a quadratic mapping inX.

It is now clear from (2.4) withn → ∞, as well as formula (2.1) that (2.2) holds inX. This

completes the existence proof of the above Theorem 2.1.

Uniqueness

LetQ⁰ :X →Y be a quadratic mapping satisfying (2.2), as well asQ. ThenQ⁰ =Q.

Proof. Remember bothQandQ⁰ satisfy (1.3) for(r, m) ∈ I₁, too. Then for everyx ∈ Xand n∈N,

kQ(x)−Q⁰(x)k=

m⁻²ⁿQ(mⁿx)−m⁻²ⁿQ⁰(mⁿx) (2.17)

≤m⁻²ⁿ{kQ(mⁿx)−f(mⁿx)k+kQ⁰(mⁿx)−f(mⁿx)k}

≤m⁻²ⁿ 2σc

m²−m^r kmⁿxk^r

=m^n(r−2) 2σc

m²−m^r kxk^r →0, asn→ ∞, ifI₁ holds:m^r−2 <1.

Similarly for(r, m)∈I₂, we establish kQ(x)−Q⁰(x)k=

m²ⁿQ(m⁻ⁿx)−m²ⁿQ⁰(m⁻ⁿx) (2.18)

≤m²ⁿ

Q(m⁻ⁿx)−f(m⁻ⁿx) +

Q⁰(m⁻ⁿx)−f(m⁻ⁿx)

≤m²ⁿ 2σc m^r−m²

m⁻ⁿx

r

=m^n(2−r) 2σc

m^r−m² kxk^r →0, asn → ∞, ifI₂ holds:m^2−r <1.

Thus from (2.17), and (2.18) we findQ(x) =Q⁰(x)for allx∈X.

This completes the proof of the uniqueness and stability of equation (∗).

Open Problem. What is the situation in the above Theorem 2.1 in caser = 2?

REFERENCES

[1] J. ACZÉL, Lectures on Functional Equations and Their Applications, Academic Press, New York and London, 1966.

[2] C. BORELLIANDG.L. FORTI, On a general Hyers-Ulam stability result, Internat. J. Math. Math.

Sci., 18 (1995), 229–236.

[3] D.G. BOURGIN, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57 (1951),223-237 .

[4] P.W. CHOLEWA, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76–86.

[5] ST. CZERWIK, On the stability of the quadratic mapping in normed spaces , Abh. Math. Sem. Univ.

Hamburg, 62 (1992), 59–64 .

[6] H. DRLJEVIC, On the stability of the functional quadratic on A-orthogonal vectors, Publ. Inst.

Math. (Beograd) (N.S.), 36(50) (1984), 111–118.

[7] I. FENYÖ, Osservazioni su alcuni teoremi di D.H. Hyers, Istit. Lombardo Accad. Sci. Lett. Rend., A 114 (1980), 235–242 (1982).

[8] I. FENYÖ, On an inequality of P.W. Cholewa, in General Inequalities, 5. [Internat. Schriftenreihe Numer. Math., Vol. 80]. Birkhauser, Basel-Boston, MA, 1987, pp. 277–280.

[9] G.L. FORTI, Hyers-Ulam stability of functional equations in several variables, Aequationes Math- ematicae, 50 (1995), 143–190 .

[10] Z. GAJDAANDR. GER, Subadditive multifunctions and Hyers-Ulam stability, in General inequal- ities, 5. [Internat. Schriftenreihe Numer. Math., Vol. 80]. Birkhauser, Basel-Boston, MA, 1987.

[11] P. GAVRUTA, An answer to a question of John M. Rassias concerning the stability of Cauchy equation, in Advances in Equations and Inequalities, Hadronic Math. Series, U.S.A , 1999, pp.

67–71.

[12] M. GRUBER, Stability of isometries, Trans. Amer. Math. Soc., U.S.A., 245 (1978), 263–277.

[13] D.H. HYERS, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., 27 (1941), 222–224; The stability of homomorphisms and related topics, "Global Analysis-Analysis on Man- ifolds", Teubner - Texte zur Mathematik, 57 (1983), 140–153.

[14] L. PAGANONI, Soluzione di una equazione funzionale su dominio ristretto, Boll. Un. Mat. Ital., (5) 17-B (1980), 979–993.

[15] J.M. RASSIAS, On approximation of approximately linear mappings by linear mappings, J. Funct.

Anal., 46 (1982), 126–130.

[16] J.M. RASSIAS, On approximation of approximately linear mappings by linear mappings, Bull. Sc.

Math., 108 (1984), 445–446.

[17] J.M. RASSIAS, Solution of a problem of Ulam, J. Approx. Th., 57 (1989), 268–273.

[18] J.M. RASSIAS, On the stability of the general Euler-Lagrange functional equation, Demonstr.

Math., 29 (1996), 755–766.

[19] J.M. RASSIAS, Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings, J.

Math. Anal. & Applics., 220(1998), 613–639 .

[20] TH.M. RASSIAS, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc.,72 (1978), 297–300.

[21] F. SKOF, Proprieta locali e approssimazione di operatori. In Geometry of Banach spaces and related topics (Milan, 1983). Rend. Sem. Mat. Fis. Milano, 53 (1983), 113–129 (1986).

[22] F. SKOF, Approssimazione di funzioniδ-quadratiche su dominio ristretto, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 118 (1984), 58–70.

[23] F. SKOF, On approximately quadratic functions on a restricted domain, in Report of the third In- ternational Symposium of Functional Equations and Inequalities, 1986. Publ. Math. Debrecen, 38 (1991), 14.

[24] S.M. ULAM, A Collection of Mathematical Problems, Interscience Publishers, Inc., New York, 1968, p. 63.