Journal of Inequalities in Pure and
Applied Mathematics
THE ULAM STABILITY PROBLEM IN APPROXIMATION OF
APPROXIMATELY QUADRATIC MAPPINGS BY
QUADRATIC MAPPINGS
volume 5, issue 3, article 52,
2004.
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/
Received 23 September, 2003;
accepted 29 November, 2003.
Communicated by: D. Bainov
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
165-03
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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. 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.
The Ulam Stability Problem In
Approximation Of
Approximately Quadratic
Mappings By Quadratic
Mappings
John Michael Rassias
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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.
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1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Quadratic Functional Stability . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
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. Gǎvruţǎ [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(a1 x1 + a2 x2 ) + Q(a2 x1 − a1 x2 ) = a21 + a22 [Q(x1 ) + Q(x2 )]
with quadratic mappings Q : X → Y such that X and Y are real linear
spaces.
Denote
The Ulam Stability Problem In
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Approximately Quadratic
Mappings By Quadratic
Mappings
John Michael Rassias
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
Kr = Kr (kx1 k , kx2 k)
= 2r−1 (kx1 kr + kx2 kr ) − (kx1 + x2 kr + kx1 − x2 kr )
( r−1
2 (kx1 kr + kx2 kr ) − (kx1 + x2 kr + kx1 − x2 kr ) , if r > 2
=
kx1 + x2 kr + kx1 − x2 kr − 2r−1 (kx1 kr + kx2 kr ) ,
if 1 < r < 2,
for every (x1 , x2 ) ∈ X 2 , where X is a normed linear space. Note that Kr ≥ 0
for any fixed real r : 1 < r 6= 2. Note also that
Kr = Kr (kxk , kxk) = 0,
Kr
Kr (|a1 | kxk , |a2 | kxk) = β1 kxkr ,
m−1 |a1 | kxk , m−1 |a2 | kxk = β1 m−r kxkr ,
Kr (kxk , 0) = β2 kxkr and
Kr m−1 kxk , 0 = β3 kxkr ,
The Ulam Stability Problem In
Approximation Of
Approximately Quadratic
Mappings By Quadratic
Mappings
John Michael Rassias
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where
β1 = Kr (|a1 | , |a2 |)
= 2r−1 (|a1 |r + |a2 |r ) − (|a1 + a2 |r + |a1 − a2 |r )
(
2r−1 (|a1 |r + |a2 |r ) − (|a1 + a2 |r + |a1 − a2 |r ) , if r > 2
=
|a1 + a2 |r + |a1 − a2 |r − 2r−1 (|a1 |r + |a2 |r ) ,
if 1 < r < 2,
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(
β2 = Kr (1, 0) = 2r−1 − 2 =
2r−1 − 2, if r > 2
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2 − 2r−1 , if 1 < r < 2,
J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
β3 = Kr m−1 , 0 = β2 m−r ,
Note that a1 6= a2 , and 1 6= m = a21 + a22 > 0.
If X and Y are normed linear spaces and Y complete, then we establish an
approximation of approximately quadratic mappings f : X → Y by quadratic
mappings Q : X → Y , such that the corresponding approximately quadratic
functional inequality
(∗∗) f (a1 x1 + a2 x2 ) + f (a2 x1 − a1 x2 ) − a21 + a22 [f (x1 ) + f (x2 )]
≤ cKr (kx1 k , kx2 k)
holds with a constant c ≥ 0 (independent of x1 ,x2 ∈ X), and any fixed pair
a = (a1 , a2 ) ∈ R2 − {(0, 0)} and any fixed real r > 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, where 1 6= m = a21 + a22 = |a|2 > 0 and a1 6= a2 . Note that mr−2 < 1 if
(r, m) ∈ I1 , and m2−r < 1 if (r, m) ∈ I2 .
It is useful for the following, to observe that, from (∗) with x1 = x2 = 0, and
0 < m 6= 1 we get
2(m − 1)Q(0) = 0,
The Ulam Stability Problem In
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Approximately Quadratic
Mappings By Quadratic
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John Michael Rassias
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(1.1)
Q(0) = 0.
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
Definition 1.1. Let Xand Y be real linear spaces. Let a = (a1 , a2 ) ∈ R2 −
{(0, 0)} : 0 < m = a21 + a22 6= 1 and a1 6= a2 . Then a mapping Q : X → Y is
called quadratic with respect to a, if (∗) holds for every vector (x1 , x2 ) ∈ X 2 .
Definition 1.2. Let Xand Y be real linear spaces. Let a = (a1 , a2 ) ∈ R2 −
{(0, 0)} : 0 < m = a21 + a22 6= 1 and a1 6= a2 . Then a mapping Q̄ : X → Y
is called the square of the quadratic weighted mean of Q with respect to a =
(a1 , a2 ), if
(1.2) Q̄(x)
 Q(a1 x)+Q(a2 x)
,
if (r, m = a21 + a22 ) ∈ I1

a21 +a22

=
i
h 
 (a2 + a2 ) Q
a2
a1
x
+
Q
x
, if (r, m = a21 + a22 ) ∈ I2
2
2
2
2
1
2
a +a
a +a
1
2
1
The Ulam Stability Problem In
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Mappings By Quadratic
Mappings
John Michael Rassias
2
for all x ∈ X.
For every x ∈ R set Q(x) = x2 .q
Then the mapping Q̄ : R → R is quadratic,
such that Q̄ (x) = x2 . Denoting by x2w the quadratic weighted mean, we note
that the above-mentioned mapping Q is an analogous case to the square of the
a2 x2 +a2 x2
quadratic weighted mean employed in mathematical statistics: x2w = 1 a12 +a22 2
1
2
with weights w1 = a21 and w2 = a22 , data x1 = x2 = x, and Q (ai x) = (ai x)2 ,
(i = 1, 2).
Now, claim that for n ∈ N0 = {0, 1, 2, . . . } that
( −2n
m Q(mn x), if (r, m) ∈ I1
(1.3)
Q(x) =
m2n Q(m−n x), if (r, m) ∈ I2 ,
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
for all x ∈ Xand n ∈ N0 .
For n = 0, it is trivial. From (1.1), (1.2) and (∗), with xi = ai x (i = 1, 2),
we obtain
Q(mx) = m [Q(a1 x) + Q(a2 x)] ,
or
(1.4)
Q̄(x) = m−2 Q(mx),
if I1 holds. Besides from (1.1), (1.2) and (∗), with x1 = x, x2 = 0, we get
Q(a1 x) + Q(a2 x) = mQ(x),
or
(1.5)
The Ulam Stability Problem In
Approximation Of
Approximately Quadratic
Mappings By Quadratic
Mappings
John Michael Rassias
Q̄(x) = Q(x),
if I1 holds. Therefore from (1.4) and (1.5) we have
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(1.6)
Q(x) = m−2 Q(mx),
which is (1.3) for n = 1, if I1 holds. Similarly, from (1.1), (1.2) and (∗), with
xi = ami x (i = 1, 2), we obtain
(1.7)
Q(x) = Q̄(x)
x
if I2 holds. Besides from (1.1), (1.2) and (∗), with x1 = m
, x2 = 0, we get
a a 1
2
Q
x +Q
x = mQ(m−1 x),
m
m
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
or
(1.8)
Q̄(x) = m2 Q(m−1 x)
if I2 holds. Therefore from (1.7) and (1.8) we have
(1.9)
Q(x) = m2 Q(m−1 x),
which is (1.3) for n = 1, if I2 holds.
Assume (1.3) is true and from (1.6), with mn x in place of x, we get:
(1.10)
2
Q mn+1 x = m2 Q(mn x) = m2 (mn )2 Q(x) = mn+1 Q(x).
Similarly, with m−n x in place of x, we get:
(1.11)
Q m−(n+1) x = m−2 Q(m−n x)
−2
= m (m
−n 2
) Q(x) = m
The Ulam Stability Problem In
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Mappings
John Michael Rassias
−(n+1) 2
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Q(x).
These formulas (1.10) and (1.11) by induction, prove formula (1.3).
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
2.
Quadratic Functional 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, or
r > 2, 0 < m < 1}, and I2 = {(r, m) ∈ R2 : 1 < r < 2, 0 < m < 1,
or r > 2, m > 1} for any fixed pair a = (a1 , a2 ) of reals ai 6= 0 (i = 1, 2) and
any fixed real r > 1 : 1 6= m = a21 + a22 = |a|2 > 0, a1 6= a2 . Besides define
0 < β1 = Kr (|a1 | , |a2 |)
= 2r−1 (|a1 |r + |a2 |r ) − (|a1 + a2 |r + |a1 − a2 |r )
( r−1
2 (|a1 |r + |a2 |r ) − (|a1 + a2 |r + |a1 − a2 |r ) , if r > 2
=
|a1 + a2 |r + |a1 − a2 |r − 2r−1 (|a1 |r + |a2 |r ) ,
if 1 < r < 2,
β2 = Kr (1, 0) = |2r−1 − 2|, and σ = β1 + mβ2 > 0. Also define
( −2n
m f (mn x), if (r, m) ∈ I1
fn (x) =
m2n f (m−n x), if (r, m) ∈ I2
for all x ∈ X and n ∈ N0 = {0, 1, 2, . . . }.
Then the limit
The Ulam Stability Problem In
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Approximately Quadratic
Mappings By Quadratic
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John Michael Rassias
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(2.1)
Q(x) = lim fn (x)
n→∞
exists for all x ∈ X and Q : X → Y is the unique quadratic mapping with
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
respect to a = (a1 , a2 ), such that
(2.2)
σc
kxkr
|m2 − mr |
(
σc/(m2 − mr ), if (r, m) ∈ I1
= kxkr
σc/(mr − m2 ), if (r, m) ∈ I2
kf (x)−Q(x)k ≤
holds for all x ∈ X and n ∈ N0 and c ≥ 0 (constant independent of x ∈ X).
Existence.
Proof. It is useful for the following, to observe that, from (∗∗) with x1 = x2 = 0
and 0 < m 6= 1, we get
2 |m − 1| kf (0)k ≤ 0,
or
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f (0) = 0.
(2.3)
Now claim that for n ∈ N0
(2.4)
The Ulam Stability Problem In
Approximation Of
Approximately Quadratic
Mappings By Quadratic
Mappings
kf (x)−fn (x)k
σc
n|r−2|
≤
1
−
m
kxkr
2
r
|m − m |

n(r−2)
 m2σc
1
−
m
, if (r, m) ∈ I1 : mr−2 < 1
r
−m
r
= kxk
n(2−r)
 σc
1
−
m
, if (r, m) ∈ I2 : m2−r < 1.
r
2
m −m
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
For n = 0, it is trivial.
Define f¯ : X → Y , the square of the quadratic weighted mean of f with
respect to a = (a1 , a2 ) by replacing Q, Q̄ of (1.2) with f , f¯, respectively, as
follows:
(2.5) f¯(x)

f (a1 x)+f (a2 x)

,
if (r, m = a21 + a22 = |a|2 ) ∈ I1
 a21 +a22
=
i
h 
(a21 + a22 ) f 2a1 2 x + f 2a2 2 x , if (r, m = a21 + a22 = |a|2 ) ∈ I2
a +a
a +a
1
2
1
2
for all x ∈ X.
From (2.3), (2.5) and (∗∗), with xi = ai x (i = 1, 2), we obtain
The Ulam Stability Problem In
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Mappings By Quadratic
Mappings
John Michael Rassias
kf (mx)−m [f (a1 x)+f (a2 x)]k ≤ σc kxkr ,
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or
(2.6)
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−2
m f (mx)−f¯(x) ≤ β1 c kxkr ,
m2
if I1 holds. Besides from (2.3), (2.5) and (∗∗), with x1 = x, x2 = 0, we get
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kf (a1 x)+f (a2 x)−mf (x)k ≤ cKr (kxk , 0) = β2 c kxkr ,
or
(2.7)
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f¯(x)−f (x) ≤ β2 c kxkr ,
m
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
if I1 holds. Therefore from (2.6) and (2.7) we have
(2.8)
σc
f (x) − m−2 f (mx) ≤ σc kxkr =
1 − mr−2 kxkr ,
2
2
r
m
m −m
which is (2.4) for n = 1, if I1 holds.
Similarly, from (2.3), (2.5) and (∗∗), with xi =
(2.9)
ai
x
m
(i = 1, 2), we obtain
f (x)−f¯(x) ≤ β1 c kxkr ,
mr
x
if I2 holds. Besides from (2.3), (2.5) and (∗∗), with x1 = m
, x2 = 0, we get
a a 2
1
x +f
x − mf (m−1 x) ≤ cKr m−1 kxk , 0 = β3 c kxkr ,
f
m
m
The Ulam Stability Problem In
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Approximately Quadratic
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Mappings
John Michael Rassias
or
(2.10)
f¯(x)−m2 f (m−1 x) ≤ mβ3 c kxkr = mβ2 c kxkr ,
mr
if I2 holds. Therefore from (2.9) and (2.10) we have
(2.11)
σc
2−r
f (x)−m2 f (m−1 x) ≤ σc kxkr =
1
−
m
kxkr ,
mr
mr − m2
which is (2.4) for n = 1, if I2 holds.
Assume (2.4) is true if (r, m) ∈ I1 . From (2.8), with mn x in place of x, and
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
the triangle inequality, we have
kf (x) − fn+1 (x)k
= f (x) − m−2(n+1) f mn+1 x ≤ f (x) − m−2n f (mn x)
+ m−2n f (mn x) − m−2(n+1) f mn+1 x σc
≤ 2
1 − mn(r−2) + m−2n 1 − mr−2 mnr kxkr
r
m −m
σc
(n+1)(r−2)
= 2
1
−
m
kxkr ,
r
m −m
if I1 holds.
Similarly assume (2.4) is true if (r, m) ∈ I2 . From (2.11), with m−n x in
place of x, and the triangle inequality, we have
(2.12)
(2.13)
kf (x) − fn+1 (x)k
= f (x) − m2(n+1) f m−(n+1) x ≤ f (x) − m2n f m−n x + m2n f (m−n x) − m2(n+1) f m−(n+1) x σc
n(2−r)
2n
2−r
−nr
≤ r
(1
−
m
)+m
(1
−
m
)m
kxkr
2
m −m
σc
(n+1)(2−r)
1
−
m
kxkr ,
= r
m − m2
if I2 holds.
Therefore inequalities (2.12) and (2.13) prove inequality (2.4) for any n ∈
N0 .
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John Michael Rassias
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
Claim now that the sequence {fn (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 = mj x, we have:
(2.14)
kfi (x) − fj (x)k = m−2i f (mi x) − m−2j f (mj x)
= m−2j m−2(i−j) f mi−j h1 − f (h1 )
σc
≤ m−2j 2
1 − m(i−j)(r−2) kxkr
r
m −m
σc
< 2
m−2j kxkr −−−→ 0,
r
j→∞
m −m
if I1 holds: mr−2 < 1.
Similarly, if h2 = m−j x in I2 , we have:
(2.15)
kfi (x) − fj (x)k = m2i f (m−i x) − m2j f (m−j x)
= m2j m2(i−j) f m−(i−j) h2 − f (h2 )
σc
(i−j)(2−r)
≤ m2j r
1
−
m
kxkr
m − m2
σc
< r
m2j kxkr −−−→ 0,
j→∞
m − m2
if I2 holds: m2−r < 1.
Then inequalities (2.14) and (2.15) define a mapping Q : X → Y , given by
(2.1).
Claim that from (∗∗) and (2.1) we can get (∗), or equivalently that the aforementioned well-defined mapping Q : X → Y is quadratic.
The Ulam Stability Problem In
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Approximately Quadratic
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John Michael Rassias
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
In fact, it is clear from the functional inequality (∗∗) and the limit (2.1) for
(r, m) ∈ I1 that the following functional inequality
m−2n kf (a1 mn x1 + a2 mn x2 ) + f (a2 mn x1 − a1 mn x2 )
−(a21 + a22 ) [f (mn x1 ) + f (mn x2 )]
≤ m−2n cKr (mn kx1 k , mn kx2 k) ,
holds for all vectors (x1 , x2 ) ∈ X 2 , and all n ∈ N with fn (x) = m−2n f (mn x) :
I1 holds. Therefore
lim fn (a1 x1 + a2 x2 ) + lim fn (a2 x1 − a1 x2 )
n→∞
n→∞
h
i
2
2
− (a1 + a2 ) lim fn (x1 ) + lim fn (x2 ) n→∞
n→∞
≤ lim mn(r−2) cKr (kx1 k , kx2 k) = 0,
n→∞
because mr−2 < 1 or
(2.16) Q(a1 x1 +a2 x2 )+Q(a2 x1 −a1 x2 )− a21 +a22 [Q(x1 )+Q(x2 )] = 0,
or mapping Q satisfies the quadratic equation (∗).
Similarly, from (∗∗) and (2.1) for (r, m) ∈ I2 we get that
m2n f (a1 m−n x1 + a2 m−n x2 )+f (a2 m−n x1 − a1 m−n x2 )
−(a21 + a22 ) f (m−n x1 )+f (m−n x2 ) ≤ m2n cKr m−n kx1 k , m−n kx2 k ,
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
holds for all vectors (x1 , x2 ) ∈ X 2 , and all n ∈ N with fn (x) = m2n f (m−n x) :
I2 holds. Thus
lim fn (a1 x1 + a2 x2 )+ lim fn (a2 x1 − a1 x2 )
n→∞
n→∞
h
i
− (a21 + a22 ) lim fn (x1 )+ lim fn (x2 ) n→∞
n→∞
n(2−r)
≤ lim m
cKr (kx1 k , kx2 k) = 0,
n→∞
because m2−r < 1, or (2.16) holds or mapping Q satisfies (∗).
Therefore (2.16) holds if Ij (j = 1, 2) hold or mapping Q satisfies (∗), completing the proof that Q is a quadratic mapping in X.
It is now clear from (2.4) with n → ∞, as well as formula (2.1) that (2.2)
holds in X. This completes the existence proof of the above Theorem 2.1.
Uniqueness
Let Q0 : X → Y be a quadratic mapping satisfying (2.2), as well as Q. Then
0
Q = Q.
0
Proof. Remember both Q and Q satisfy (1.3) for (r, m) ∈ I1 , too. Then for
every x ∈ X and n ∈ N,
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(2.17)
0
kQ(x) − Q (x)k
= m−2n Q(mn x) − m−2n Q0 (mn x)
≤ m−2n {kQ(mn x) − f (mn x)k + kQ0 (mn x) − f (mn x)k}
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J. Ineq. Pure and Appl. Math. 5(3) Art. 52, 2004
2σc
kmn xkr
2
r
m −m
2σc
= mn(r−2) 2
kxkr → 0, as n → ∞,
r
m −m
≤ m−2n
if I1 holds: mr−2 < 1.
Similarly for (r, m) ∈ I2 , we establish
(2.18) kQ(x) − Q0 (x)k
= m2n Q(m−n x) − m2n Q0 (m−n x)
≤ m2n Q(m−n x) − f (m−n x) + Q0 (m−n x) − f (m−n x)
2σc m−n xr
≤ m2n r
m − m2
2σc
= mn(2−r) r
kxkr → 0, as n → ∞,
2
m −m
if I2 holds: m2−r < 1.
Thus from (2.17), and (2.18) we find Q(x) = Q0 (x) for all x ∈ 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 case r = 2?
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References
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[18] J.M. RASSIAS, On the stability of the general Euler-Lagrange functional
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