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volume 5, issue 3, article 52, 2004.

Received 23 September, 2003;

accepted 29 November, 2003.

Communicated by:D. Bainov

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Journal of Inequalities in Pure and Applied Mathematics

THE ULAM STABILITY PROBLEM IN APPROXIMATION OF APPROXIMATELY QUADRATIC MAPPINGS BY

QUADRATIC MAPPINGS

JOHN MICHAEL RASSIAS

Pedagogical Department, E. E.,

National and Capodistrian University of Athens, Section of Mathematics and Informatics, 4, Agamemnonos Str., Aghia Paraskevi, Athens 15342, Greece.

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

c

2000Victoria University ISSN (electronic): 1443-5756 165-03

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The Ulam Stability Problem In Approximation Of Approximately Quadratic

Mappings By Quadratic Mappings John Michael Rassias

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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004

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 investigated the Ulam problem for additive mappings. P.M. Gru- ber, 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.

2000 Mathematics Subject Classification:39B

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

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 remains 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 problem for quadratic and other mappings. Z. Gajda and R. Ger [10] showed that one can obtain 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 mappings Q : X → Y such that X and Y are real linear spaces.

Denote

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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 ≥ 0 for any fixed realr : 1< r6= 2.Note also that

K_r =K_r(kxk,kxk) = 0, Kr(|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 =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 =Kr(1,0) =

2^r−1−2 =

( 2^r−1−2, ifr >2 2−2^r−1, if1< r <2,

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β₃ =K_r m⁻¹,0

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

IfX andY are normed linear spaces andY complete, then we establish an approximation of approximately quadratic mappingsf : X → Y by quadratic mappings Q : 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 ofx1,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, where1 6=m =a²₁ +a²₂ =|a|² >0anda₁ 6= a₂. Note thatm^r−2 <1if (r, m)∈I₁, andm^2−r <1if(r, m)∈I₂.

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

2(m−1)Q(0) = 0, or

(1.1) Q(0) = 0.

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Definition 1.1. Let XandY be real linear spaces. Let a = (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. Let XandY be real linear spaces. Let a = (a₁, a₂) ∈ R² − {(0,0)} : 0 < m = a²₁ +a²₂ 6= 1anda₁ 6= a₂. Then a mapping Q¯ : X → Y is called the square of the quadratic weighted mean of Q with 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+a²₂xi

, if (r, m=a²₁+a²₂)∈I₂ for allx∈X.

For everyx∈RsetQ(x) = x². Then the mappingQ¯ :R→Ris quadratic, such thatQ¯(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 employed in mathematical statistics: x²_w = ^a²¹^x_a²¹2^+a²²^x²²

1+a²₂

with weightsw1 =a²₁ andw2 = a²₂, datax1 =x2 = x, andQ(aix) = (aix)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,

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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),

ifI1holds. 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),

ifI₁holds. Therefore from (1.4) and (1.5) we have

(1.6) Q(x) =m⁻²Q(mx),

which is (1.3) forn = 1, ifI₁ holds. Similarly, from (1.1), (1.2) and (∗), with x_i = ^a_mⁱx (i= 1,2), we obtain

(1.7) Q(x) = ¯Q(x)

ifI₂holds. Besides from (1.1), (1.2) and (∗), withx₁ = _m^x,x₂ = 0,we get Qa1

mx

+Qa2

mx

=mQ(m⁻¹x),

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or

(1.8) Q(x) =¯ m²Q(m⁻¹x)

ifI₂holds. 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:

Q m⁻⁽ⁿ⁺¹⁾x

=m⁻²Q(m⁻ⁿx) (1.11)

=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).

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2. Quadratic Functional Stability

Theorem 2.1. Let X and Y be normed linear spaces. Assume thatY is com- plete. Assume in addition that mapping f : X → Y satisfies the functional inequality (∗∗). Define I₁ = {(r, m) ∈ R² : 1 < r < 2, m > 1, or r > 2, 0 < m < 1}, and I₂ = {(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 : 16=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 =Kr(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∈X andn∈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

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respect toa = (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 ∈N₀andc≥0(constant independent ofx∈X).

Existence.

Proof. It is useful for the following, to observe that, from (∗∗) withx₁ =x₂ = 0 and0< m6= 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 (2.4)

≤ σc

|m²−m^r| 1−m^n|r−2|

kxk^r

=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.

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Forn= 0,it is trivial.

Definef¯: X → Y, the square of the quadratic weighted mean of f with respect to a = (a₁, a₂) by replacingQ, Q¯ of (1.2) with f, 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+a²₂xi

, if(r, m=a²₁+a²₂ =|a|²)∈I₂ for allx∈X.

From (2.3), (2.5) and (∗∗), withxi =aix(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,

ifI₁holds. 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,

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ifI₁holds. 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,

ifI₂holds. 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, ifI₂holds. 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), withmⁿxin place ofx, and

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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, ifI₁holds.

Similarly assume (2.4) is true if (r, m) ∈ I2. From (2.11), with m⁻ⁿx in 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, ifI2holds.

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

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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 , ifi > j >0, andh₁ =m^jx, we have:

kf_i(x)−f_j(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, ifI₁holds: 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,

ifI2holds: 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.

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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⁻²ⁿcKr(mⁿkx1k, mⁿkx2k), holds for all vectors(x₁, x₂)∈X², and alln ∈Nwithf_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)

≤m²ⁿcK_r m⁻ⁿkx₁k, m⁻ⁿkx₂k ,

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holds for all vectors(x₁, x₂)∈X², and alln ∈Nwithf_n(x) =m²ⁿf(m⁻ⁿx) : I2 holds. 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 (∗), com- pleting the proof thatQis a quadratic mapping inX.

It is now clear from (2.4) with n → ∞, as well as formula (2.1) that (2.2) holds inX. This completes the existence proof of the above Theorem2.1.

Uniqueness

LetQ⁰ :X →Y be a quadratic mapping satisfying (2.2), as well asQ. Then Q⁰ =Q.

Proof. Remember both Q and Q⁰ satisfy (1.3) for (r, m) ∈ I₁, too. Then for everyx∈Xandn∈N,

kQ(x)−Q⁰(x)k (2.17)

=

m⁻²ⁿQ(mⁿx)−m⁻²ⁿQ⁰(mⁿx)

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

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≤m⁻²ⁿ 2σc

m²−m^r kmⁿxk^r

=m^n(r−2) 2σc

m²−m^r kxk^r→0, asn→ ∞, ifI1holds: m^r−2 <1.

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

(2.18)

=

m²ⁿQ(m⁻ⁿx)−m²ⁿQ⁰(m⁻ⁿx)

≤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−m²kxk^r →0, asn→ ∞, ifI₂holds: 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 Theorem2.1in caser= 2?

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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.

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