AbstractIn this paper, we investigate the Hyers–Ulam–Rassias stability problems of a new quadratic functional equationfx−y−z+fx+fy+fz=fx−y+fy+z+fz−x.Furthermore, the stability results will be applied to the study of an interesting asymptotic property of the quadratic functions

Jung, Soon-Mo

Elsevier - Publisher Connector 

Ž .Journal of Mathematical Analysis and Applications 232, 384]393 1999
Article ID jabr.1999.6282, available online at http:rrwww.idealibrary.com on
On the Hyers]Ulam]Rassias Stability of a Quadratic
Functional Equation
Soon-Mo Jung
Mathematics Section, College of Science and Technology, Hong-Ik Uni¤ersity, 339-800
Chochiwon, South Korea
E-mail: smjung@wow.hongik.ac.kr
Submitted by William A. Kirk
Received September 8, 1998
In this paper, we investigate the Hyers]Ulam]Rassias stability problems of a
new quadratic functional equation
f x y y y z q f x q f y q f z s f x y y q f y q z q f z y x .Ž . Ž . Ž . Ž . Ž . Ž . Ž .
Furthermore, the stability results will be applied to the study of an interesting
asymptotic property of the quadratic functions. Q 1999 Academic Press
Key Words: Hyers]Ulam]Rassias stability; quadratic functional equation
1. INTRODUCTION
w xIn 1940, S. M. Ulam 14 gave a wide ranging talk before the mathemat-
ics club of the University of Wisconsin in which he discussed a number of
important unsolved problems. Among those was the question concerning
the stability of group homomorphisms:
Ž .Let G be a group and let G be a metric group with the metric d ? , ? .1 2
Gi¤en « ) 0, does there exist a d ) 0 such that if a function
Ž Ž . Ž . Ž ..h : G “ G satisfes the inequality d h xy , h x h y - d for all1 2
x, y gG , then there exists a homomorphism H : G “ G with1 1 2
Ž Ž . Ž ..d h x , H x - « for all x g G ?1
The case of approximately additive functions was solved by D. H. Hyers
w x6 under the assumption that G and G are Banach spaces. Taking this1 2
Ž . Ž . Ž .fact into account, the additive functional equation f x q y s f x q f y
Ž .is said to have Hyers]Ulam stability on G , G . The result of Hyers was1 2
384
0022-247Xr99 $30.00
Copyright Q 1999 by Academic Press
All rights of reproduction in any form reserved.
A QUADRATIC FUNCTIONAL EQUATION 385
Ž w x.further generalized by Th. M. Rassias see 12 . Since then, the stability
problems of several functional equations have been extensively investi-
gated. The terminology Hyers]Ulam]Rassias stability originates from
these historical backgrounds. These terminologies are also applied to the
case of other functional equations. For more detailed definitions of such
w xterminologies, we can refer to 5, 8, 9 .
Ž . 2It is easy to see that the quadratic function f x s x is a solution of
each of the following functional equations:
f x q y q f x y y s 2 f x q 2 f y , 1Ž . Ž . Ž . Ž . Ž .
f x q y q z q f x q f y q f z s f x q y q f y q z q f z q xŽ . Ž . Ž . Ž . Ž . Ž . Ž .
2Ž .
and
f x y y y z q f x q f y q f z s f x y y q f y q z q f z y x .Ž . Ž . Ž . Ž . Ž . Ž . Ž .
3Ž .
So, it is natural that each equation is called a quadratic functional
Ž .equation. In particular, every solution of the quadratic equation 1 is said
to be a quadratic function.
It is well known that a function f between real vector spaces is
quadratic if and only if there exists a unique symmetric biadditive function
Ž . Ž . Ž w x.B such that f x s B x, x for all x see 1, 7, 11 .
Ž .The functional equation 2 was solved by Pl. Kannappan. In fact, he
Ž .proved that a functional on a real vector space is a solution of Eq. 2 if
and only if there exist a symmetric biadditive function B and an additive
Ž . Ž . Ž . Ž w x.function A such that f x s B x, x q A x for any x see 11 , while the
w x Ž .author investigated in his paper 10 the Hyers]Ulam stability of Eq. 2 on
restricted domains and applied the result to the study of an interesting
asymptotic behavior of the quadratic functions.
Ž .The signs of variables x, y, z in Eq. 2 may be changed and transformed
Ž .into the functional equation 3 which has still the quadratic property.
Ž . Ž .Indeed, the functional equation 2 is different from Eq. 3 , since every
Ž .additive function not vanishing everywhere is a solution of 2 but it is not
Ž .a solution of 3 .
Ž .In Section 2 of this paper, we will show that the functional equation 3
Ž . Ž .is equivalent to Eq. 1 . For each case of p ) 0 p / 2 , p s 0 and p - 0
in the following functional inequality:
f x y y y z q f x q f y q f z y f x y yŽ . Ž . Ž . Ž . Ž .
p p p5 5 5 5 5 5yf y q z y f z y x F « x q y q z ,Ž . Ž . Ž .
SOON-MO JUNG386
the Hyers]Ulam]Rassias stability of the inequality will be discussed in
Section 3. In particular, we will construct a function which serves as a
counterexample for the case p s 2. In Section 4, by using a result
concerning the case p s 0 in the above inequality, the Hyers]Ulam
Ž .stability of Eq. 3 on restricted domains will be proved and the result will
be applied to the study of an asymptotic behavior of the quadratic
functions. Indeed, the following asymptotic behavior of quadratic functions
will be proved in Corollary 7:
Let X and Y be a normed space and a Banach space, respecti¤ely. A
5 Ž .function f : X “ Y is a quadratic function if and only if f x y y y z
Ž . Ž . Ž . Ž . Ž . Ž .5 5 5q f x q f y q f z y f x y y y f y q z y f z y x “ 0 as x
5 5 5 5q y q z “ ‘.
By N and R we denote the sets of positive integers and of real numbers,
respectively.
Ž .2. SOLUTIONS OF EQ. 3
Ž .It is an interesting thing to see that the functional equation 3 is
Ž . Ž .equivalent to the quadratic equation 1 . That is, every solution of Eq. 3
is a quadratic function.
THEOREM 1. If ¤ector spaces X and Y are common domain and range of
Ž . Ž . Ž .the f 's in both the functional equations 1 and 3 , Eq. 3 is equi¤alent to
Ž .Eq. 1 .
Ž . Ž .Proof. If we put x s y s z s 0 in 3 , we get f 0 s 0. By putting
Ž . Ž .y s z s 0 in Eq. 3 we see that every solution of Eq. 3 is even.
Ž . Ž .Replacing z by yy in 3 and using the evenness of f and f 0 s 0, we
Ž . Ž .can transform Eq. 3 into Eq. 1 .
Ž .Conversely, assume a function f : X “ Y is a solution of Eq. 1 .
Ž .Clearly, by 1 , we have
y q z y q z
f x y y y z q f x s 2 f x y q 2 fŽ . Ž . ž / ž /2 2
and
y q z y y z
f y q f z s 2 f q 2 f .Ž . Ž . ž / ž /2 2
A QUADRATIC FUNCTIONAL EQUATION 387
Ž .According to 1 and the last two equalities, and by using the fact
Ž . Ž .f 2 x s 4 f x , we obtain
f x y y y z q f x q f y q f zŽ . Ž . Ž . Ž .
y q z y y z
s 2 f x y q 2 f q f y q zŽ .ž / ž /2 2
s f x y y q f y q z q f z y x ,Ž . Ž . Ž .
Ž .since every solution of the equation 1 is even. This implies the equiva-
Ž . Ž .lence of the functional equations 1 and 3 .
Ž .3. HYERS]ULAM]RASSIAS STABILITY OF EQ. 3
In this section, let X and Y be a normed space and a Banach space,
respectively. For convenience, we use the following abbreviations:
D f x , y s f x q y q f x y y y 2 f x y 2 f yŽ . Ž . Ž . Ž . Ž .1
and
D f x , y , z s f x y y y z q f x q f y q f zŽ . Ž . Ž . Ž . Ž .3
y f x y y y f y q z y f z y x .Ž . Ž . Ž .
Assume that real constants d , « ) 0 and p / 2 are given. We will investi-
Ž .gate the Hyers]Ulam]Rassias stability of the functional equation 3 .
THEOREM 2. Assume that a function f : X “ Y satisfies the inequality
p p p5 5 5 5 5 5D f x , y , z F « x q y q z 4Ž . Ž .Ž .3
for some p ) 0, p / 2, and for all x, y, z g X. Then there exists a unique
quadratic function Q: X “ Y such that
8« p5 5f x y Q x F x 5Ž . Ž . Ž .p< <2 y 4
Ž .for all x g X. Moreo¤er, if f is measurable or f tx is continuous in t g R for
each fixed x g X, then the Q satisfies
Q tx s t 2 Q x 6Ž . Ž . Ž .
for all x g X and t g R.
SOON-MO JUNG388
Ž . Ž .Proof. Putting x s y s z s 0 in the inequality 4 yields f 0 s 0. If we
Ž .let y s z s 0 in 4 and then substitute y for x, we get
p5 5f y y f yy F « y 7Ž . Ž . Ž .
Ž .for any y g X. Further, we set y s 0 in 4 and then we replace z by yy
to obtain
p p5 5 5 5f x q y y f yx y y F « x q y 8Ž . Ž . Ž .Ž .
Ž . Ž . Ž .for all x, y g X. Now, let z s yy in 4 . It then follows from 4 , 7 and
Ž .8 that the inequality
p p5 5 5 5D f x , y F 4« x q yŽ . Ž .1
w x Ž w x.holds for all x, y g X. According to S. Czerwik 4 or see 2 , it follows
from the last inequality that there exists a unique quadratic function Q:
Ž . Ž .X “ Y satisfying 5 and 6 .
Ž .In the case where the exponent p in the inequality 4 takes the value 0,
Ž .we can prove the Hyers]Ulam stability of Eq. 3 as we shall see in the
following theorem.
THEOREM 3. Assume that a function f : X “ Y satisfies the inequality
D f x , y , z F « 9Ž . Ž .3
for all x, y, z g X. Then there exists a unique quadratic function Q: X “ Y
such that the inequality
f x y Q x F 3«Ž . Ž .
Ž .holds for all x g X. Moreo¤er, if f is measurable or f tx is continuous in
Ž .t g R for each fixed x g X, then the Q satisfies 6 for all x g X and t g R.
Ž .Proof. By letting x s y s z s 0 in 9 we get
f 0 F « . 10Ž . Ž .
Ž .If we put y s z s 0 in 9 and then substitute y for x, we have
f y y f yy F 2« 11Ž . Ž . Ž .
Ž .for any y g X. If we set z s yy in 9 then we obtain
2 f x q f y q f yy y f x y y y f 0 y f yx y y F « 12Ž . Ž . Ž . Ž . Ž . Ž . Ž .
A QUADRATIC FUNCTIONAL EQUATION 389
Ž . Ž . Ž .for all x, y g X. By 12 , 11 and 10 , we get
D f x , y F 6«Ž .1
w x w xfor all x, y g X. According to a result of F. Skof 13 or P. W. Cholewa 3 ,
the assertion of our theorem is obvious.
Ž .Now, assume that the exponent p in 4 is negative. For this case, we try
Ž .to prove the stability of the equation 3 under an additional condition that
f is approximately even.
THEOREM 4. Suppose a function f : X “ Y satisfies the inequalities
p p p5 5 5 5 5 5D f x , y , z F « x q y q z ,Ž . Ž .3 13Ž .½ f x y f yx F dŽ . Ž .
 4for some p - 0 and for all x, y, z g X _ 0 . Then there exists a unique
quadratic function Q: X “ Y such that
2 4« p5 5f x y Q x F d q f 0 q xŽ . Ž . Ž .Ž . p3 4 y 2
 4 Ž .for all x g X _ 0 . If , moreo¤er, f is measurable or f tx is continuous in
Ž .t g R for each fixed x g X, then Q satisfies 6 for all x g X and t g R.
Ž .Proof. By letting z s yy in the first inequality in 13 and by using the
second inequality, we get
p p5 5 5 5D f x , y F 2d q f 0 q 2« x q yŽ . Ž . Ž .1
 4 w xfor all x, y g X _ 0 . According to 4 , this fact means the validity of the
assertion in our theorem.
Remark 1. It should be noted that the functional inequality
2 2 25 5 5 5 5 5D f x , y , z F « x q y q zŽ . Ž .3
is not stable in the sense of Hyers, Ulam and Rassias. We can slightly
w xmodify the example of Czerwik in 4 to show that there exists a function f :
R “ R which satisfies the above functional inequality for all x, y, z g R,
< Ž .but for which there is no quadratic function Q: R “ R such that f x y
Ž . < 2  4Q x rx is bounded on R_ 0 .
Indeed, we define
< <1r71 « for x G 1,Ž .
w x sŽ . 2½ < <1r71 « x for x - 1Ž .
SOON-MO JUNG390
and put
‘
yn nf x s 9 w 3 xŽ . Ž .Ý
ns0
w xfor all x g R. By the similar way as in 4 we can easily verify that f serves
as a counterexample for the Hyers]Ulam]Rassias stability of the func-
Ž .tional inequality 4 for the case p s 2.
Ž .4. HYERS]ULAM STABILITY OF EQ. 3 ON
RESTRICTED DOMAINS
w xIn the paper 10 , the author proved the Hyers]Ulam stability of the
Ž .functional equation 2 on restricted domains under some additional
conditions.
w xBy using the idea from 10 , we will prove the Hyers]Ulam stability of
Ž .the equation 3 on restricted domains.
Let X and Y be a normed space and a Banach space, respectively.
THEOREM 5. Let d ) 0 and « ) 0 be gi¤en. If a function f : X “ Y
Ž . 5 5 5 5 5 5satisfies the inequality 9 for all x, y, z g X with x q y q z G d, then
there exists a unique quadratic function Q: X “ Y such that
f x y Q x F 39« 14Ž . Ž . Ž .
Ž .for all x g X. If , moreo¤er, f is measurable or f tx is continuous in t g R
Ž .for each fixed x g X, then the Q satisfies 6 for all x g X and t g R.
5 5 5 5 5 5 5 5Proof. Suppose that x q y q z - d. Choose a t g X with t G
2 d. Then, it holds
5 5 5 5 5 5 5 5 5 5 5 5x q y q t q z y t G d; x q y q t G d;
5 5 5 5 5 5 5 5 5 5 5 5z q x q t G d; x q t q t y x q t y z G d; 15Ž .
5 5 5 5 5 5 5 5 5 5 5 5z y t q x q x G d; x q z q t q t G d;
1r2 x y t q 1r2 x q t G d.Ž . Ž .
A QUADRATIC FUNCTIONAL EQUATION 391
Ž . Ž .From 9 , 15 and the following relation
f x y y y z q f x q f y q f z y f x y yŽ . Ž . Ž . Ž . Ž .
y f y q z y f z y x q f 0Ž . Ž . Ž .
s f x y y y z q f x q f y q t q f z y t y f x y y y tŽ . Ž . Ž . Ž . Ž .
y f y q z y f z y x y tŽ . Ž .
q f x y y y t q f x q f y q f t y f x y y y f y q tŽ . Ž . Ž . Ž . Ž . Ž .
y f t y xŽ .
q f z y x y t q f z q f x q f t y f z y x y f x q tŽ . Ž . Ž . Ž . Ž . Ž .
y f t y zŽ .
q f 2 x q z y t q f x q t q f t y x q f t y z y f 2 xŽ . Ž . Ž . Ž . Ž .
y f 2 t y x y z y f yx y zŽ . Ž .
y f 2 x q z y t y f z y t y f yx y f yx q f x q z y tŽ . Ž . Ž . Ž . Ž .
q f y2 x q f yx y z q tŽ . Ž .
q f 2 t y x y z q f yx y z q f yt q f ytŽ . Ž . Ž . Ž .
y f yx y z q t y f y2 t y f x q z y tŽ . Ž . Ž .
y2 f x y 2 f 1r2 x y t y 2 f 0 y 2 f y 1r2 x y tŽ . Ž . Ž . Ž .Ž . Ž .
q 2 f 1r2 x y t q 2 f y 1r2 x y t q 2 f yxŽ . Ž . Ž .Ž . Ž .
y f t y f t y f yt y f t q f 2 t q f 0 q f 0Ž . Ž . Ž . Ž . Ž . Ž . Ž .
q f t q f 1r2 t q f 0 q f y 1r2 t y f 1r2 tŽ . Ž . Ž . Ž . Ž .Ž . Ž . Ž .
y f y 1r2 t y f ytŽ . Ž .Ž .
y f 2 t y f t y f 0 y f yt q f t q f yt q f y2 tŽ . Ž . Ž . Ž . Ž . Ž . Ž .
q f 2 x q f x y t q f 0 q f yx y t y f x y tŽ . Ž . Ž . Ž . Ž .
y f yx y t y f y2 xŽ . Ž .
we obtain
D f x , y , z F 12« q f 0 F 13« , 16Ž . Ž . Ž .3
5 Ž .5 Ž .since we get f 0 F « by putting x s y s z s 0 in 9 .
Ž .It is clear that the inequality 16 holds for all x, y, z g X. Therefore,
the assertions of our theorem are immediate consequences of Theorem 3.
SOON-MO JUNG392
Ž . 3 5 5 5 5 5 5 4Define S s x, y, z g X : x - d, y - d, z - d for some given
Ž . 3 5 5 5 5 5 5 4 3d ) 0. The fact x, y, z g X : x q y q z G 3d ; X _S implies
that the following corollary is a consequence of Theorem 5.
Ž .COROLLARY 6. If a function f : X “ Y satisfies the inequality 9 for all
Ž . 3x, y, z g X _S, then there exists a unique quadratic function Q: X “ Y
Ž .which satisfies the inequality 14 for all x g X. Moreo¤er, if f is measurable
Ž . Ž .or f tx is continuous in t g R for each fixed x g X, then the Q satisfies 6
for all x g X and t g R.
By using Theorem 5, we can prove the following corollary concerning an
interesting asymptotic property of the quadratic functions.
COROLLARY 7. A function f : X “ Y is a quadratic function if and only if
5 Ž .5 5 5 5 5 5 5D f x, y, z “ 0 as x q y q z “ ‘.3
Proof. According to our asymptotic condition, there exists a sequence
Ž .« monotonically decreasing to zero such thatn
D f x , y , z F «Ž .3 n
5 5 5 5 5 5for all x, y, z g X with x q y q z G n. Hence, it follows from Theo-
rem 5 that there exists a unique quadratic function Q : X “ Y such thatn
f x y Q x F 39« 17Ž . Ž . Ž .n n
for all x g X. Let l and m be integers satisfying m ) l ) 0. Obviously,
Ž .from 17 we obtain
f x y Q x F 39« F 39«Ž . Ž .m m l
for any x in X. Hence, the uniqueness of Q implies that Q s Q holdsn m l
Ž .for any l, m g N. By letting n “ ‘ in 17 , we conclude that f is
quadratic. The reverse assertion is trivial in view of Theorem 1.
ACKNOWLEDGMENT
The author would like to thank Professor Themistocles M. Rassias for his helpful advice.
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On the Hyers–Ulam–Rassias Stability of a Quadratic Functional Equation 

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On the Hyers–Ulam–Rassias Stability of a Quadratic Functional Equation

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