Volume 2009, Article ID 193035,19pages doi:10.1155/2009/193035
Research Article
Fixed Points and Stability of a Generalized
Quadratic Functional Equation
Abbas Najati
1and Choonkil Park
21Department of Science, University of Mohaghegh Ardabili, Ardabil 51664, Iran 2Department of Mathematics, Hanyang University, Seoul 133-791, South Korea
Correspondence should be addressed to Choonkil Park,[email protected]
Received 26 November 2008; Revised 24 January 2009; Accepted 11 February 2009
Recommended by Patricia J. Y. Wong
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the generalized quadratic functional equationfrxsy r2fx s2fy rs/2fxy−fx−yin Banach modules, wherer, sare nonzero rational numbers withr2s2/1.
Copyrightq2009 A. Najati and C. Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The stability problem of functional equations originated from a question of Ulam 1 concerning the stability of group homomorphisms:letG1,∗be a group and letG2,, dbe a metric group with the metricd·,·. Given >0, does there existδ> 0such that if a mapping
h:G1 → G2satisfies the inequality
dhx∗y, hxhy< δ 1.1
for allx, y∈G1, then there is a homomorphismH:G1 → G2with
dhx, Hx< 1.2
for allx∈G1?
Hyers2gave a first affirmative answer to the question of Ulam for Banach spaces. LetXandYbe Banach spaces.Assume thatf :X → Y satisfies
for someε≥0and allx, y∈X. Then there exists a unique additive mappingT :X → Y such that
fx−Tx≤ε 1.4
for allx∈X.
Aoki 3 and Th. M. Rassias 4 provided a generalization of the Hyers’ theorem for additive and linear mappings, respectively, by allowing the Cauchy difference to be unbounded.
Theorem 1.1Th. M. Rassias4. Letf : E → Ebe a mapping from a normed vector spaceE
into a Banach spaceEsubject to the inequality
fxy−fx−fy≤ x p y p 1.5
for allx, y∈E, whereandpare constants with >0andp <1. Then the limit
Lx lim
n→ ∞
f2nx
2n 1.6
exists for allx∈EandL:E → Eis the unique additive mapping which satisfies
fx−Lx≤ 2
2−2p x
p 1.7
for allx∈E. Ifp <0, then the inequality1.5holds forx, y /0and1.7forx /0. Also, if for each
x∈Ethe mappingt→ftxis continuous int∈R, thenLisR- linear.
Theorem 1.2J. M. Rassias5–7. LetXbe a real normed linear space and letY be a real Banach space. Assume thatf :X → Yis a mapping for which there exist constantsθ≥0andp, q∈Rsuch thatrpq /1andfsatisfies the functional inequality
fxy−fx−fy≤θ x p y q 1.8
for allx, y∈X. Then there exists a unique additive mappingL:X → Y satisfying
fx−Lx≤ θ
2r−2 x
r 1.9
for allx ∈X. If, in addition,f :X → Y is a mapping such that the transformationt → ftxis continuous int∈Rfor each fixedx∈X,thenLis linear.
In 1994, a generalization of Theorems1.1and1.2was obtained by G˘avrut¸a8, who replaced the boundsε x p y pandθ x p y qby a general control functionϕx, y.
The functional equation
is called a quadratic functional equation. Quadratic functional equations were used to characterize inner product spaces 9–11. In particular, every solution of the quadratic equation1.10is said to be aquadratic mapping.It is well known that a mappingf between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive mappingBsuch thatfx Bx, xfor allxsee9,12. The biadditive mappingBis given by
Bx, y 1
4
fxy−fx−y. 1.11
The generalized Hyers-Ulam stability problem for the quadratic functional equation 1.10was proved by Skof for mappingsf : E1 → E2,whereE1 is a normed space andE2 is a Banach spacesee13. Cholewa14noticed that the theorem of Skof is still true if the relevant domainE1 is replaced by an Abelian group. J. M. Rassias 15and Czerwik16, proved the stability of the quadratic functional equation1.10. Grabiec17has generalized these results mentioned above. J. M. Rassias18introduced and investigated the stability problem of Ulam for the Euler-Lagrange quadratic mappings:
fa1x1a2x2
fa2x1−a1x2
a21a22fx1
fx2
. 1.12
In addition, J. M. Rassias 19 generalized the Euler-Lagrange quadratic mapping 1.12 and investigated its stability problem. The Euler-Lagrange quadratic mapping 1.12 has provided a lot of influence in the development of general Euler-Lagrange quadratic equations mappingswhich is now known as Euler-Lagrange-Rassias quadratic functional equations mappings.
Jun and Lee 20 proved the generalized Hyers-Ulam stability of a pexiderized quadratic equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problemsee8,20–47. We also refer the readers to the books48–51.
LetEbe a set. A functiond : E×E → 0,∞is called ageneralized metric onEif d
satisfies
idx, y 0 if and only ifxy, iidx, y dy, xfor allx, y∈E,
iiidx, z≤dx, y dy, zfor allx, y, z∈E.
We recall the following theorem by Margolis and Diaz.
Theorem 1.3see52. LetE, dbe a complete generalized metric space and letJ :E → Ebe a strictly contractive mapping with Lipschitz constant0 < L <1. Then for each given elementx∈E, either
dJnx, Jn1x∞ 1.13
for all nonnegative integersnor there exists a nonnegative integern0such that 1dJnx, Jn1x<∞for alln≥n
0,
3y∗is the unique fixed point ofJin the setY {y∈E:dJn0x, y<∞},
4dy, y∗≤1/1−Ldy, Jyfor ally∈Y.
Throughout this paper, we assume thatr, sare nonzero rational numbers withr2
s2/1,and thatAis a unital Banach algebra with unite, norm| · |, andA1:{a∈A:|a|1}. Assume thatXis a normed leftA-module andY is aunit linkedBanach leftA-module. A quadratic mappingT : X → Y is calledA-quadraticifTax a2Txfor alla∈Aand all
x∈X.
In this paper, we investigate anA-quadratic mapping associated with the generalized quadratic functional equation
frxsy r2fx s2fy rs
2
fxy−fx−y, 1.14
and using the fixed point methodsee24,25,38,53–55, we prove the generalized Hyers-Ulam stability ofA-quadratic mappings in BanachA-modules associated with the functional equation1.14. In 1996, Isac and Th. M. Rassias56were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications.
For convenience, we use the following abbreviation for a givena∈Aand a mapping
f:X → Y:
Dafx, y:fraxsy−r2a2fx−s2fy−
rs
2
faxy−fax−y 1.15
for allx, y∈X.
2. Fixed Points and Stability of the Generalized Quadratic
Functional Equation
1.14
Proposition 2.1. A mappingf:X → Ysatisfies
D1fx, y 0 2.1
for allx, y∈Xif and only iffis quadratic.
Proof. Letf satisfy2.1. Sincer2s2/1,lettingxy 0 in2.1, we getf0 0. Letting
y0 in2.1, we get
frx r2fx 2.2
for allx∈X. It follows from2.1thatD1fx, y D1fx,−y 0 for allx, y∈X.Hence
for allx, y∈X.We decomposefinto the even part and the odd part by putting
fex
fx f−x
2 , fox
fx−f−x
2 2.4
for allx ∈ X.It is clear thatfx fex foxfor allx ∈ X.It is easy to show that the
mappingsfeandfosatisfy2.2and2.3. Thus we have
ferxsy ferx−sy 2r2fex 2s2fey, 2.5
forxsy forx−sy 2r2fox 2.6
for allx, y∈X.Lettingx0 in2.5, we get
fesy s2fey 2.7
for ally∈X. It follows from2.2,2.5, and2.7that
ferxsy ferx−sy 2ferx 2fesy 2.8
for allx, y∈X.Therefore,
fexy fex−y 2fex 2fey 2.9
for allx, y ∈ X. Sofe is quadratic. We claim thatfo ≡ 0.For this, it follows from2.2and
2.6that
forxsy forx−sy 2forx 2.10
for allx, y∈X. So
foxy fox−y 2fox 2.11
for allx, y∈X. Lettingyxin2.11, we getfo2x 2foxfor allx∈X. So it follows from
2.11that
foxy fox−y fo2x 2.12
for allx, y ∈ X. Replacingxbyxy/2 andybyx−y/2 in2.12, we infer that fois
additive. To complete the proof we have two cases.
Case 1r 1. Sincefois additive and satisfies2.1, lettingx 0 and replacingfobyf in
2.1, we gets2f
Case 2r /1. Sincefois additive and satisfies2.2, we haver2−rfox 0 for allx∈X.
Sincer /0,1, we getfo≡0.
Henceffeand this proves thatfis quadratic.
Conversely, let f be quadratic. Then there exists a unique symmetric biadditive mappingB:X×X → Y such thatfx Bx, xfor allx∈Xand
Bx, y 1
4
fxy−fx−y 2.13
for allx, y∈Xsee9,12. Hence
frxsy Brxsy, rxsy
r2Bx, x s2By, y 2rsBx, y
r2fx s2fy rs 2
fxy−fx−y
2.14
for allx, y∈X. Hencefsatisfies2.1.
Corollary 2.2. Letf :X → Y be a mapping satisfying
Dafx, y 0 2.15
for allx, y∈Xand alla∈A1.If for eachx∈Xthe mappingt→ftxis continuous int∈R, then
fisA-quadratic.
Proof. Leta e.ByProposition 2.1,f is quadratic. Thusf isQ-quadratic. Letα∈Rand let
{rn}nbe a sequence of rational numbers such that limn→ ∞rn α.SincefisQ-quadratic and
the mappingt→ftxis continuous int∈Rfor eachx∈X, we have
fαx lim
n→ ∞f
rnx
lim
n→ ∞r 2
nfx α2fx 2.16
for allx∈X. SofisR-quadratic. Lettingy0 in2.15, we get
fax a2fx 2.17
for allx ∈ X and alla ∈ A1.It is clear that2.17is also true fora 0.For each element
a∈Aa /0, a|a| ·a/|a|.SincefisR-quadratic andfbx b2fxfor allx∈Xand all
b∈A1,we have
fax f
|a| · a
|a|x |a|
2f
a
|a|x |a|
2· a2
|a|2 ·fx a
2fx 2.18
Now we prove the generalized Hyers-Ulam stability of A-quadratic mappings in BanachA-modules.
Theorem 2.3. Let f : X → Y be a mapping with f0 0 for which there exists a function
ϕ:X2 → 0,∞such that
Dafx, y≤ϕx, y 2.19
for allx, y∈Xand alla∈A1. Let0< L <1be a constant such thatr2ϕx, y≤Lϕrx, ryfor all
x, y∈X.If for eachx∈Xthe mappingt→ftxis continuous int∈R, then there exists a unique
A-quadratic mappingQ:X → Y satisfying
fx−Qx≤ L
r21−Lϕx,0 2.20
for allx∈X.
Proof. It follows fromr2ϕx, y≤Lϕrx, rythat
lim
n→ ∞r 2nϕ
x rn,
y
rn 0 2.21
for allx, y∈X.
Lettingy0 in2.19, we get
frax−r2a2fx≤ϕx,0 2.22
for allx∈Xand alla∈A1. Hence
fax−r2a2f
x r ≤ϕ
x r,0 ≤
L
r2ϕx,0 2.23
for allx ∈X and alla∈A1. LetE :{g : X → Y |g0 0}.We introduce a generalized metric onEas follows:
dg, h:infC∈0,∞:gx−hx≤Cϕx,0∀x∈X. 2.24
It is easy to show thatE, dis a generalized complete metric space24. Now we consider the mappingΛ:E → Edefined by
Λgx r2g
x
Letg, h∈Eand letC∈0,∞be an arbitrary constant withdg, h≤C. From the definition ofd, we have
gx−hx≤Cϕx,0 2.26
for allx∈X. By the assumption and the last inequality, we have
Λgx−Λhxr2g
x r −h
x r ≤r
2Cϕ
x
r,0 ≤CLϕx,0 2.27
for allx∈X. So
dΛg,Λh≤Ldg, h 2.28
for anyg, h∈E. It follows from2.23 by lettingaethatdΛf, f≤ L/r2. According to Theorem 1.3, the sequence{Λnf}converges to a fixed pointQofΛ, that is,
Q:X−→Y, Qx lim
n→ ∞
Λnfx lim n→ ∞r
2nf
x
rn , 2.29
andQrx r2Qxfor allx∈X. AlsoQis the unique fixed point ofΛin the setE∗ {g ∈
E:df, g<∞}and
dQ, f≤ 1
1−LdΛf, f≤ L
r21−L, 2.30
that is, the inequality2.20holds true for allx∈X. It follows from the definition ofQ,2.19, and2.21that
DaQx, y lim n→ ∞r
2nD af
x rn,
y
rn ≤nlim→ ∞r 2nϕ
x rn,
y
rn 0 2.31
for all x, y ∈ X and all a ∈ A1. By Proposition 2.1by letting a e, the mapping Q is quadratic. LetL:Y → Rbe a continuous linear functional. For anyx∈X, we consider the mappingψx:R → Rdefined by
ψxt:L
Qtx. 2.32
SinceQis quadratic andLis linear,
ψxuv ψxu−v L
Quxvx Qux−vx
L2Qux 2Qvx
2ψxu 2ψxv
for allu, v∈R.Soψxis quadratic. Alsoψxis measurable since it is the pointwise limit of the
sequence
ψn,xt:r2nL
f
tx
rn . 2.34
It follows from48, Corollary 10.2thatψxt t2ψx1for allt∈R.Then
LQtxψxt t2ψx1 t2L
QxLt2Qx 2.35
for allt∈R.HenceQtx t2Qxfor allt∈Rand allx∈X.ByCorollary 2.2, the mapping
QisA-quadratic.
Corollary 2.4. Letp >0andθbe nonnegative real numbers such thatr2<|r|pand letf :X → Y be a mapping satisfying the inequality
Dafx, y≤θ
x p y p 2.36
for allx, y∈Xand alla∈A1. If for eachx∈Xthe mappingt→ftxis continuous int∈R, then there exists a uniqueA-quadratic mappingQ:X → Y such that
fx−Qx≤ θ
|r|p−r2 x
p 2.37
for allx∈X.
Proof. Lettinga eandx y 0 in2.36, we getf0 0.Now, the proof follows from Theorem 2.3by taking
ϕx, y:θ x p y p 2.38
for allx, y∈X. Then we can chooseL|r|2−pand we get the desired result.
Remark 2.5. Letf : X → Y be a mapping withf0 0 for which there exists a function Φ:X2 → 0,∞such that
Dafx, y≤Φx, y 2.39
for allx, y∈X and alla∈A1. Let 0 < L <1 be a constant such thatΦrx, ry≤r2LΦx, y for allx, y∈X. By a similar method to the proof ofTheorem 2.3, one can show that if for each
x∈Xthe mappingt→ftxis continuous int∈R, then there exists a uniqueA-quadratic mappingQ:X → Y satisfying
fx−Qx≤ 1
r21−LΦx,0 2.40
For the caseΦx, y:δθ x p y p whereδ, θare nonnegative real numbers and
p >0 with 1<|r|p< r2, there exists a uniqueA-quadratic mappingQ:X → Ysatisfying
fx−Qx≤ δ r2− |r|p
θ r2− |r|p x
p 2.41
for allx∈X.
Corollary 2.6. Letp, q > 0and letθbe nonnegative real numbers such thatr2/|r|pq and letf :
X → Y be a mapping satisfying the inequality
Dafx, y≤θ x p y q 2.42
for allx, y∈Xand alla∈A1. If for eachx∈Xthe mappingt→ftxis continuous int∈R, then
fisA-quadratic.
Theorem 2.7. Letf:X → Y be an even mapping for which there exists a functionϕ:X2 → 0,∞ satisfying2.19and
lim
n→ ∞4 nϕ
x
2n,
y
2n 0 2.43
for allx, y∈Xand alla∈A1. Let0< L <1be a constant such that the mapping
x−→φx:ϕ
x r,
x s ϕ
x r,
−x s 2ϕ
x
r,0 2ϕ
0,x
s 2.44
satisfying4φx≤Lφ2xfor allx∈X.If for eachx∈Xthe mappingt→ftxis continuous in
t∈R, then there exists a uniqueA-quadratic mappingQ:X → Y satisfying
fx−Qx≤ L
41−Lφx 2.45
for allx∈X.
Proof. Sinceϕ0,0 0,it follows from2.19thatf0 0 and
Dafx, y Dafx,−y−2Dafx,0−2Daf0, y
≤ϕx, y ϕx,−y 2ϕx,0 2ϕ0, y
2.46
for allx, y∈Xand alla∈A1. Therefore,
fraxsy frax−sy−2frax−2fsy
≤ϕx, y ϕx,−y 2ϕx,0 2ϕ0, y
for allx, y∈Xand alla∈A1. Lettingaeand replacingxbyx/randybyy/sin2.47, we get
fxy fx−y−2fx−2fy≤Φx, y 2.48
for allx, y∈X, where
Φx, y:ϕ
x r,
y s ϕ
x r,
−y s 2ϕ
x
r,0 2ϕ
0,y
s . 2.49
Lettingyxin2.48, we get
f2x−4fx≤φx 2.50
for allx∈X.Hence
4f
x
2 −fx ≤φ
x
2 ≤
L
4φx 2.51
for allx ∈ X.LetE :{g : X → Y | g0 0}.We introduce a generalized metric onEas follows:
dg, h:infC∈0,∞:gx−hx≤Cφx∀x∈X. 2.52
Now we consider the mappingΛ:E → Edefined by
Λgx 4g
x
2 , ∀g ∈E, x∈X. 2.53
Similar to the proof ofTheorem 2.3, we deduce that the sequence{Λnf}converges to a fixed
pointQofΛwhich isA-quadratic. AlsoQis the unique fixed point ofΛin the setE∗{g ∈ E:df, g<∞}and satisfies2.45.
Corollary 2.8. Letp > 2 and letθ be nonnegative real numbers and letf : X → Y be an even mapping satisfying the inequality2.36for allx, y∈Xand alla∈A1. If for eachx∈Xthe mapping
t→ftxis continuous int∈R, then there exists a uniqueA-quadratic mappingQ:X → Y such that
fx−Qx≤ 4θ
|r|p|s|p
2p−4|rs|p x
p 2.54
Proof. Lettinga eandx y 0 in 2.36, we getf0 0.Now the proof follows from Theorem 2.7by taking
ϕx, y:θ x p y p 2.55
for allx, y∈X. Then we can chooseL 22−pand we get the desired result.
Remark 2.9. Letf:X → Ybe an even mapping withf0 0 for which there exists a function Φ:X2 → 0,∞such that
lim
n→ ∞ 1 4nΦ
2nx,2ny0, Dafx, y≤Φx, y 2.56
for allx, y∈Xand alla∈A1. Let 0< L <1 be a constant such that the mapping
x−→φx: Φ
x r,
x s Φ
x r,
−x s 2Φ
x
r,0 2Φ
0,x
s 2.57
satisfyingφ2x≤4Lφxfor allx∈X.By a similar method to the proof ofTheorem 2.7, one can show that if for eachx∈Xthe mappingt→ftxis continuous int∈R, then there exists a uniqueA-quadratic mappingQ:X → Y satisfying
fx−Qx≤ 1
41−Lφx 2.58
for allx∈X.
For the caseΦx, y : δθ x p y p whereδ, θare nonnegative real numbers
and 0< p <2, there exists a uniqueA-quadratic mappingQ:X → Y satisfying
fx−Qx≤ 6δ
4−2p
4θ|r|p|s|p
4−2p|rs|p x
p 2.59
for allx∈X.
Corollary 2.10. Let p, q > 0 and let θ be nonnegative real numbers such that pq /2 and let
f:X → Ybe an even mapping satisfying the inequality2.42for allx, y∈Xand alla∈A1. If for eachx∈Xthe mappingt→ftxis continuous int∈R, thenfisA-quadratic.
We may omit the evenness of the mappingfinTheorem 2.7.
Theorem 2.11. Letf : X → Y be a mapping for which there exists a functionϕ :X2 → 0,∞ satisfying2.19and2.43for allx, y∈Xand alla∈A1. Let0< L <1be a constant such that the mapping
x−→φx:ϕ
x r,
x s ϕ
x r,
−x s 2ϕ
x
r,0 2ϕ
0,x
satisfying4φx≤Lφ2xfor allx∈X.If for eachx∈Xthe mappingt→ftxis continuous in
t∈R, then there exists a uniqueA-quadratic mappingQ:X → Y satisfying
fx−Qx≤ L4−3L
81−L2−L
φx φ−x 2.61
for allx∈X.
Proof. Sinceϕ0,0 0,it follows from2.19thatf0 0.We decomposef into the even partfeand the odd partfo.It follows from2.19that
Dafex, y≤
1 2
ϕx, y ϕ−x,−y,
Dafox, y≤
1 2
ϕx, y ϕ−x,−y
2.62
for allx, y ∈ Xand alla∈A1. ByTheorem 2.7, there exists a uniqueA-quadratic mapping
Q:X → Ysatisfying
fex−Qx≤ L
81−L
φx φ−x 2.63
for allx∈X. We get from2.62that
Dafox, y Dafox,−y−2Dafox,0≤Ψx, y 2.64
for allx, y∈Xand alla∈A1, where
Ψx, y : 1
2
ϕx, y ϕ−x,−y ϕx,−y ϕ−x, y 2ϕx,0 2ϕ−x,0. 2.65
Hence
foxy fox−y−2fox≤Ψ
x r,
y
s 2.66
for allx, y∈X. Lettingyxin2.66, we get
fo2x−2fox≤Ψ
x r,
x
s 2.67
for allx∈X. Therefore, 2fo
x
2 −fox ≤ 1
2
φ
x
2 φ −
x
2 ≤
L
8
for allx ∈X. LetE: {g :X → Y | g0 0}.We introduce a generalized metric onEas follows:
dg, h:infC∈0,∞:gx−hx≤Cφx φ−x∀x∈X. 2.69
Now we consider the mappingΛ:E → Edefined by
Λgx 2g
x
2 , ∀g ∈E, x∈X. 2.70
Similar to the proof ofTheorem 2.3, we deduce that the sequence{Λnf
o}converges to a fixed
pointTofΛwhich is quadratic and
dT, fo
≤ 2
2−Ld
Λfo, fo
≤ 2L
16−8L. 2.71
AlsoT is odd sincefois odd. Therefore,T ≡ 0 sinceT is quadratic too. Now2.61follows
from2.63and2.71.
Corollary 2.12. Letp >2and letθbe nonnegative real numbers and letf :X → Y be a mapping satisfying the inequality 2.36for all x, y ∈ X and alla ∈ A1. If for each x ∈ X the mapping
t→ftxis continuous int∈R, then there exists a uniqueA-quadratic mappingQ:X → Y such that
fx−Qx≤ 8θ
2p−3|r|p|s|p
2p−22p−4|rs|p x
p 2.72
for allx∈X.
Proof. Lettinga eandx y 0 in 2.36, we getf0 0.Now the proof follows from Theorem 2.11by taking
ϕx, y:θ x p y p 2.73
for allx, y∈X. Then we can chooseL 22−pand we get the desired result.
Remark 2.13. Letf : X → Y be a mapping withf0 0 for which there exists a function Φ:X2 → 0,∞such that
lim
n→ ∞
1 2nΦ
2nx,2ny0, D
afx, y≤Φx, y 2.74
for allx, y∈Xand alla∈A1. Let 0< L <1/2 be a constant such that the mapping
x−→φx: Φ
x r,
x s Φ
x r,
−x s 2Φ
x
r,0 2Φ
0,x
satisfyingφ2x ≤ 4Lφxfor allx ∈ X.By a similar method to the proof ofTheorem 2.11, one can show that if for eachx∈Xthe mappingt→ftxis continuous int∈R, then there exists a uniqueA-quadratic mappingQ:X → Y satisfying
fex−Qx≤
1 81−L
φx φ−x,
fox≤
1 41−2L
φx φ−x
2.76
for allx∈X. Hence
fx−Qx≤ 3−4L
81−L1−2L
φx φ−x 2.77
for allx∈X.
For the caseΦx, y : δθ x p y p whereδ, θare nonnegative real numbers
and 0< p <1, there exists a uniqueA-quadratic mappingQ:X → Y satisfying
fx−Qx≤ 12δ
3−2p
2−2p4−2p
8θ3−2p|r|p|s|p
2−2p4−2p|rs|p x
p 2.78
for allx∈X.
For the casep 2, we have the following counterexample which is a modification of the example of Czerwik16.
Example 2.14. Letφ:R → Rbe defined by
φx:
⎧ ⎨ ⎩
μx2 for|x|<1,
μ for|x| ≥1, 2.79
whereμis a positive real number. Consider the functionf:R → Rby the formula
fx:
∞
n0
α−2nφαnx, 2.80
whereα 1r2s2|rs|.It is clear thatf is continuous and bounded byα2/α2−1μ onR. We prove that
frxsy−r2fx−s2fy−rs
2
fxy−fx−y≤ α
10
α2−1μ
for allx, y∈R.To see this, ifx2y20 orx2y2 ≥α−4,then
frxsy−r2fx−s2fy−rs
2
fxy−fx−y
≤α2μ
∞
n0
α−2n≤ α
8
α2−1μ
x2y2.
2.82
Now suppose thatx2y2< α−4.Then there exists a nonnegative integerksuch that
α−4k2≤x2y2< α−4k1. 2.83
Therefore,
α2k|x|, α2k|y|, α2k|rxsy|, α2k|x±y| ∈−1,1. 2.84
Hence
α2m|x|, α2m|y|, α2m|rxsy|, α2m|x±y| ∈−1,1 2.85
for allm0,1, . . . ,2k.From the definition offand2.83, we have
frxsy−r2fx−s2fy−rs
2
fxy−fx−y
≤α2μ
∞
n2k1
α−2n≤ α
10
α2−1μ
x2y2.
2.86
Therefore,fsatisfies2.81. LetQ:R → Rbe a quadratic function such that
fx−Qx≤βx2 2.87
for allx∈R.Then there exists a constantc∈Rsuch thatQx cx2for allx∈Rsee57. So we have
fx≤β|c|x2 2.88
for allx ∈ R. Letm ∈ Nwith mμ > β|c|.If x ∈ 0, α1−m, then αnx ∈ 0,1for alln
0,1, . . . , m−1.So
fx≥
m−1
n0
α−2nφαnxmμx2>β|c|x2, 2.89
Corollary 2.15. Letp, q >0and letθbe nonnegative real numbers such thatpq >2pq <1
and letf :X → Y be a mapping satisfying the inequality2.42for allx, y ∈Xand alla∈A1. If for eachx∈Xthe mappingt→ftxis continuous int∈R, thenfisA-quadratic.
Acknowledgments
The authors would like to thank the referees for bringing some useful references to their attention. The second author was supported by Korea Research Foundation Grant KRF-2008-313-C00041.
References
1 S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960.
2 D. H. Hyers, “On the stability of the linear functional equation,”Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222–224, 1941.
3 T. Aoki, “On the stability of the linear transformation in Banach spaces,”Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.
4 Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,”Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
5 J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”Journal of Functional Analysis, vol. 46, no. 1, pp. 126–130, 1982.
6 J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”Bulletin des Sciences Math´ematiques, vol. 108, no. 4, pp. 445–446, 1984.
7 J. M. Rassias, “Solution of a problem of Ulam,”Journal of Approximation Theory, vol. 57, no. 3, pp. 268–273, 1989.
8 P. G˘avrut¸a, “On the stability of some functional equations,” inStability of Mappings of Hyers-Ulam Type, Hadronic Press Collection of Original Articles, pp. 93–98, Hadronic Press, Palm Harbor, Fla, USA, 1994.
9 J. Acz´el and J. Dhombres,Functional Equations in Several Variables with Applications to Mathematics, Information Theory and to the Natural and Social Sciences, vol. 31 ofEncyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1989.
10 D. Amir,Characterizations of Inner Product Spaces, vol. 20 ofOperator Theory: Advances and Applications, Birkh¨auser, Basel, Switzerland, 1986.
11 P. Jordan and J. von Neumann, “On inner products in linear, metric spaces,”Annals of Mathematics, vol. 36, no. 3, pp. 719–723, 1935.
12 P. Kannappan, “Quadratic functional equation and inner product spaces,”Results in Mathematics, vol. 27, no. 3-4, pp. 368–372, 1995.
13 F. Skof, “Local properties and approximation of operators,”Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, no. 1, pp. 113–129, 1983.
14 P. W. Cholewa, “Remarks on the stability of functional equations,”Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76–86, 1984.
15 J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,” Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185–190, 1992.
16 S. Czerwik, “On the stability of the quadratic mapping in normed spaces,”Abhandlungen aus dem Mathematischen Seminar der Universit¨at Hamburg, vol. 62, no. 1, pp. 59–64, 1992.
17 A. Grabiec, “The generalized Hyers-Ulam stability of a class of functional equations,”Publicationes Mathematicae Debrecen, vol. 48, no. 3-4, pp. 217–235, 1996.
18 J. M. Rassias, “On the stability of the non-linear Euler-Lagrange functional equation in real normed linear spaces,”Journal of Mathematical and Physical Sciences, vol. 28, no. 5, pp. 231–235, 1994.
19 J. M. Rassias, “On the stability of the general Euler-Lagrange functional equation,”Demonstratio Mathematica, vol. 29, no. 4, pp. 755–766, 1996.
21 D. G. Bourgin, “Classes of transformations and bordering transformations,”Bulletin of the American Mathematical Society, vol. 57, pp. 223–237, 1951.
22 L. C˘adariu and V. Radu, “Fixed point methods for the generalized stability of functional equations in a single variable,”Fixed Point Theory and Applications, vol. 2008, Article ID 749392, 15 pages, 2008.
23 L. C˘adariu and V. Radu, “Remarks on the stability of monomial functional equations,”Fixed Point Theory, vol. 8, no. 2, pp. 201–218, 2007.
24 L. C˘adariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” inIteration Theory (ECIT ’02), vol. 346 of Grazer Mathematische Berichte, pp. 43–52, Karl-Franzens-Universit¨at Graz, Graz, Austria, 2004.
25 L. C˘adariu and V. Radu, “Fixed points and the stability of Jensen’s functional equation,”Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 4, pp. 1–7, 2003.
26 G. L. Forti, “On an alternative functional equation related to the Cauchy equation,” Aequationes Mathematicae, vol. 24, no. 2-3, pp. 195–206, 1982.
27 G. L. Forti, “The stability of homomorphisms and amenability, with applications to functional equations,”Abhandlungen aus dem Mathematischen Seminar der Universit¨at Hamburg, vol. 57, no. 1, pp. 215–226, 1987.
28 G. L. Forti, “Hyers-Ulam stability of functional equations in several variables,” Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143–190, 1995.
29 P. G˘avrut¸a, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,”Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994.
30 P. G˘avrut¸a, “On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings,”Journal of Mathematical Analysis and Applications, vol. 261, no. 2, pp. 543–553, 2001.
31 P. G˘avrut¸a, “On the Hyers-Ulam-Rassias stability of the quadratic mappings,”Nonlinear Functional Analysis and Applications, vol. 9, no. 3, pp. 415–428, 2004.
32 K.-W. Jun and H.-M. Kim, “On the Hyers-Ulam stability of a difference equation,” Journal of Computational Analysis and Applications, vol. 7, no. 4, pp. 397–407, 2005.
33 K.-W. Jun and H.-M. Kim, “Stability problem of Ulam for generalized forms of Cauchy functional equation,”Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 535–547, 2005.
34 K.-W. Jun and H.-M. Kim, “Stability problem for Jensen-type functional equations of cubic mappings,”Acta Mathematica Sinica, vol. 22, no. 6, pp. 1781–1788, 2006.
35 K.-W. Jun and H.-M. Kim, “Ulam stability problem for a mixed type of cubic and additive functional equation,”Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 13, no. 2, pp. 271–285, 2006.
36 K.-W. Jun, H.-M. Kim, and J. M. Rassias, “Extended Hyers-Ulam stability for Cauchy-Jensen mappings,”Journal of Difference Equations and Applications, vol. 13, no. 12, pp. 1139–1153, 2007.
37 S.-M. Jung and T.-S. Kim, “A fixed point approach to the stability of the cubic functional equation,” Bolet´ın de la Sociedad Matem´atica Mexicana, vol. 12, no. 1, pp. 51–57, 2006.
38 S.-M. Jung and J. M. Rassias, “A fixed point approach to the stability of a functional equation of the spiral of Theodorus,”Fixed Point Theory and Applications, vol. 2008, Article ID 945010, 7 pages, 2008.
39 M. Mirzavaziri and M. S. Moslehian, “A fixed point approach to stability of a quadratic equation,” Bulletin of the Brazilian Mathematical Society, vol. 37, no. 3, pp. 361–376, 2006.
40 M. S. Moslehian and Th. M. Rassias, “Stability of functional equations in non-Archimedean spaces,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 2, pp. 325–334, 2007.
41 A. Najati, “Hyers-Ulam stability of ann-Apollonius type quadratic mapping,”Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 14, no. 4, pp. 755–774, 2007.
42 A. Najati, “On the stability of a quartic functional equation,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 569–574, 2008.
43 A. Najati and M. B. Moghimi, “Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces,”Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 399–415, 2008.
44 A. Najati and C. Park, “Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 763–778, 2007.
45 A. Najati and C. Park, “The Pexiderized Apollonius-Jensen type additive mapping and isomorphisms betweenC∗-algebras,”Journal of Difference Equations and Applications, vol. 14, no. 5, pp. 459–479, 2008.
47 K. Ravi, M. Arunkumar, and J. M. Rassias, “Ulam stability for the orthogonally general Euler-Lagrange type functional equation,”International Journal of Mathematics and Statistics, vol. 3, no. A08, pp. 36–46, 2008.
48 S. Czerwik,Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002.
49 D. H. Hyers, G. Isac, and Th. M. Rassias,Stability of Functional Equations in Several Variables, vol. 34 of Progress in Nonlinear Differential Equations and Their Applications, Birkh¨auser, Boston, Mass, USA, 1998.
50 S.-M. Jung,Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001.
51 Th. M. Rassias, Ed.,Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.
52 J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative, for contractions on a generalized complete metric space,”Bulletin of the American Mathematical Society, vol. 74, pp. 305–309, 1968.
53 V. Radu, “The fixed point alternative and the stability of functional equations,”Fixed Point Theory, vol. 4, no. 1, pp. 91–96, 2003.
54 J. M. Rassias, “Alternative contraction principle and Ulam stability problem,”Mathematical Sciences Research Journal, vol. 9, no. 7, pp. 190–199, 2005.
55 J. M. Rassias, “Alternative contraction principle and alternative Jensen and Jensen type mappings,” International Journal of Applied Mathematics & Statistics, vol. 4, no. M06, pp. 1–10, 2006.
56 G. Isac and Th. M. Rassias, “Stability ofΨ-additive mappings: applications to nonlinear analysis,” International Journal of Mathematics and Mathematical Sciences, vol. 19, no. 2, pp. 219–228, 1996.
