Volume 2008, Article ID 732086,11pages doi:10.1155/2008/732086
Research Article
A Fixed Point Approach to the Stability of
Quadratic Functional Equation with Involution
Soon-Mo Jung1and Zoon-Hee Lee2
1Mathematics Section, College of Science and Technology, Hong-Ik University, 339-701 Chochiwon, South Korea
2Department of Mathematics, Chungnam National University, 305-764 Deajeon, South Korea
Correspondence should be addressed to Soon-Mo Jung,[email protected] Received 27 September 2007; Accepted 26 November 2007
Recommended by Tomas Dom´ınguez Benavides
C˘adariu and Radu applied the fixed point method to the investigation of Cauchy and Jensen functional equations. In this paper, we will adopt the idea of C˘adariu and Radu to prove the Hyers- Ulam-Rassias stability of the quadratic functional equation with involution.
Copyrightq2008 S.-M. Jung and Z.-H. Lee. 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
In 1940, Ulam1gave a wide ranging talk before the Mathematics 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.
LetG1 be a group and letG2be a metric group with the metricd·,·. Givenε > 0, does there exist aδ >0 such that if a functionh:G1→G2satisfies the inequalitydhxy, hxhy< δfor all x, y∈G1, then there exists a homomorphismH :G1→G2withdhx, Hx< εfor allx∈G1?
The case of approximately additive functions was solved by Hyers 2 under the assumption that G1 and G2 are Banach spaces. Indeed, he proved that each solution of the inequalityfxy−fx−fy ≤ε, for allxandy, can be approximated by an exact solution, say an additive function. Rassias3attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows:
fxy−fx−fy≤ε
xpyp
, 1.1
and generalized the result of Hyers. Since then, the stability problems for several functional equations have been extensively investigated.
The terminology Hyers-Ulam-Rassias stability originates from these historical back- grounds. The terminology can also be applied to the case of other functional equations. For more detailed definitions of such terminologies, we can refer to4–9.
LetE1andE2be real vector spaces. If an additive functionσ:E1→E1satisfiesσσx xfor all x ∈ E1, then σ is called an involution of E1 see 10, 11. For a given involution σ:E1→E1, the functional equation
fxy f
xσy
2fx 2fy 1.2 is called the quadratic functional equation with involution. According to 11, Corollary 8, a functionf : E1→E2 is a solution of 1.2 if and only if there exists an additive function A:E1→E2 and a biadditive symmetric functionB :E1×E1→E2 such thatAσx Ax, Bσx, y −Bx, yandfx Bx, x Axfor allx∈E1.
Indeed, if we setσIin1.2, whereI :E1→E1denotes the identity function, then1.2 reduces to the additive functional equation
fxy fx fy. 1.3
On the other hand, ifσ −I in1.2, then1.2is transformed into the quadratic functional equation
fxy fx−y 2fx 2fy. 1.4
Recently, Belaid et al. have proved the Hyers-Ulam-Rassias stability of the quadratic functional equation with involution1.2 see10.
In this paper, we will apply the fixed point method to prove the Hyers-Ulam-Rassias stability of the functional equation1.2for a large class of functions from a vector space into a completeβ-normed space. We remark that Isac and Rassias12were the first to apply the Hyers-Ulam-Rassias stability approach for the proof of new fixed point theorems.
2. Preliminaries
LetXbe a set. A functiond:X×X→0,∞is called a generalized metric onXif and only if dsatisfies
M1dx, y 0,if and only ifxy;
M2dx, y dy, x,for allx, y∈X;
M3dx, z≤dx, y dy, z,for allx, y, z∈X.
Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity.
We now introduce one of fundamental results of fixed point theory. For the proof, refer to 13. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al.14.
Theorem 2.1. LetX, dbe a generalized complete metric space. Assume thatΛ:X→Xis a strictly contractive operator with the Lipschitz constant 0< L <1. If there exists a nonnegative integerksuch thatdΛk1x,Λkx<∞for somex∈X, then the followings are true:
athe sequence{Λnx}converges to a fixed pointx∗ofΛ;
bx∗is the unique fixed point ofΛin X∗
y∈X:d Λkx, y
<∞
; 2.1
cify∈X∗, then
d y, x∗
≤ 1
1−LdΛy, y. 2.2
Throughout this paper, we fix a real numberβwith 0 < β≤1 and letKdenote eitherR orC. SupposeEis a vector space overK. A function·β:E→0,∞is called aβ-norm if and only if it satisfies
N1xβ 0,if and only ifx0;
N2λxβ|λ|βxβ,for allλ∈Kand allx∈E;
N3xyβ≤ xβyβ,for allx, y∈E.
Recently, C˘adariu and Radu15applied the fixed point method to the investigation of the Cauchy additive functional equationsee 16,17. Using such a clever idea, they could present a short, simple proof for the Hyers-Ulam stability of Cauchy and Jensen functional equations.
3. Main results
In this section, by using an idea of C˘adariu and Radusee15,16, we will prove the Hyers- Ulam-Rassias stability of the quadratic functional equation with involution1.2.
Theorem 3.1. LetE1be a vector space overKand letE2be a completeβ-normed space overK, whereβ is a fixed real number with 0< β≤1. Suppose a functionϕ:E1×E1→0,∞is given and there exists a constantL, 0< L <1, such that
ϕ2x,2y≤4β
2Lϕx, y, ϕ
xσx, yσy
≤ 4β
2Lϕx, y
3.1
for allx, y∈E1. Furthermore, letf:E1→E2be a function satisfying the inequality fxy f
xσy
−2fx−2fyβ≤ϕx, y 3.2
for allx, y ∈ E1, whereσ : E1→E1 is an involution ofE1. Then there exists a unique solutionT : E1→E2of 1.2such that
fx−Txβ ≤ 1 4β
1
1−Lϕx, x 3.3
for allx∈E1.
Proof. First, let us defineXto be the set of all functionsh:E1→E2and introduce a generalized metric onXas follows:
dg, h inf
C∈0,∞:gx−hx
β≤Cϕx, x∀x∈E1
. 3.4
Let {fn} be a Cauchy sequence in X, d. According to the definition of Cauchy sequences, there exists, for any givenε > 0, a positive integer Nε such that dfm, fn ≤ ε for allm, n≥Nε. By considering the definition of the generalized metricd, we see that
∀ε >0∃Nε∈N ∀m, n≥Nε ∀x∈E1:fmx−fnx
β≤εϕx, x. 3.5 Ifxis any given point ofE1,3.5implies that{fnx}is a Cauchy sequence inE2. Since E2 is complete, {fnx} converges in E2 for each x ∈ E1. Hence, we can define a function f:E1→E2by
fx lim
n→∞fnx 3.6
for anyx∈E1.
If we let mincrease to infinity, it follows from3.5that for any ε > 0, there exists a positive integerNεwithfnx−fxβ ≤ εϕx, xfor alln≥ Nεand for allx ∈E1, that is, for anyε >0, there exists a positive integerNεsuch thatdfn, f≤εfor anyn≥Nε. This fact leads us to a conclusion that{fn}converges inX, d. Hence,X, dis a complete spacecf. the proof of15, Theorem 2.5.
We now define an operatorΛ:X→Xby Λhx 1
4
h2x h
xσx
3.7 for allx∈E1.
First, we assert thatΛis strictly contractive onX. Giveng, h ∈X, letC∈ 0,∞be an arbitrary constant withdg, h≤C, that is,
gx−hx
β ≤Cϕx, x 3.8
for allx∈E1. If we replaceybyxin3.2, then we obtain f2x f
xσx
−4fx
β≤ϕx, x 3.9
for everyx∈E1. It follows from3.1and3.8that Λgx−Λhx
β 1
4βg2x g
xσx
−h2x−h
xσx
β
≤ 1
4βg2x−h2x
β 1 4βg
xσx
−h
xσx
β
≤ C
4βϕ2x,2x C 4βϕ
xσx, xσx
≤LCϕx, x
3.10
for allx ∈E1, that is,dΛg,Λh ≤LC. We hence conclude thatdΛg,Λh≤ Ldg, hfor any g, h∈X. Therefore,Λis strictly contractive becauseLis a constant with 0< L <1.
Next, we assert thatdΛf, f<∞. If we putyxin3.2and we divide both sides by 4β, then we get
Λfx−fx
β
1 4
f2x f
xσx
−fx β
≤ 1
4βϕx, x 3.11 for anyx∈E1, that is,
dΛf, f≤ 1
4β <∞. 3.12
Now, it follows fromTheorem 2.1athat there exists a functionT : E1→E2 which is a fixed point ofΛ, such thatdΛnf, T→0 asn→ ∞.
By mathematical induction, we can easily showand hence we can omit to showthat Λnf
x 1 22n
f 2nx
2n−1
f
2n−1x2n−1σx
3.13 for eachn∈N.
SincedΛnf, T→0 as n→ ∞, there exists a sequence{Cn}such thatCn→0 as n→ ∞ anddΛnf, T≤Cnfor everyn∈N. Hence, it follows from the definition ofdthat
Λnf
x−Tx
β≤Cnϕx, x 3.14
for allx∈E1. Thus, for eachfixedx∈E1, we have
n→∞limΛnf
x−Tx
β0. 3.15
Therefore
Tx lim
n→∞
1 22n
f 2nx
2n−1
f
2n−1x2n−1σx
3.16 for allx∈E1. It follows from3.1,3.2, and3.16that
Txy T
xσy
−2Tx−2Ty
β
lim
n→∞
1 22βnf
2nx2ny
2n−1 f
2n−1xy 2n−1
σx σy f
2nx2nσy
2n−1 f
2n−1
xσy 2n−1
σx y
−2f 2nx
−22n−1 f
2n−1
xσx
−2f 2ny
−22n−1 f
2n−1
yσy
β,
≤ lim
n→∞
1 4βnf
2nx2ny f
2nx2nσy
−2f 2nx
−2f 2ny
β
lim
n→∞
2n−1β
4βn f 2n−1
xσx 2n−1
yσy f
2n−1
xσx 2n−1
yσy
−2f 2n−1
xσx
−2f 2n−1
yσy
β
≤ lim
n→∞
1 4βnϕ
2nx,2ny lim
n→∞
2n−1β 4βn ϕ
2n−1
xσx ,2n−1
yσy
≤ lim
n→∞
1 4βn
4β 2L
n
ϕx, y lim
n→∞
2n−1β
4βn 4β
2L n
ϕx, y 0
3.17 for allx, y∈E1, which implies thatT is a solution of1.2.
ByTheorem 2.1cand by3.12, we obtain df, T≤ 1
1−LdΛf, f≤ 1
4β1−L, 3.18
that is,3.3is true for allx∈E1.
Assume thatT1 :E1→E2is another solution of1.2satisfying3.3.We know thatT1 is a fixed point ofΛ.In view of3.3and the definition ofd, we can conclude that3.18is true withT1in place ofT. Due toTheorem 2.1b, we getT T1. This proves the uniqueness ofT.
In a similar way, by applyingTheorem 2.1, we can prove the following theorem.
Theorem 3.2. LetE1be a vector space overKand letE2be a completeβ-normed space overK, where βis a fixed real number with 0< β≤1. Assume that a functionϕ:E1×E1→0,∞is given and there exists a constantL, 0< L <1, such that
ϕx, y≤ L
2·4βϕ2x,2y, ϕ
xσx, yσy
≤2βϕ2x,2y
3.19
for allx, y ∈E1. Furthermore, letf : E1→E2 be a function satisfying3.2for allx, y ∈E1, where σ:E1→E1is an involution ofE1. Then there exists a unique solutionT :E1→E2of1.2such that
fx−Tx
β ≤ 1 4β
L
1−Lϕx, x 3.20
for allx∈E1.
Proof. We use the same definitions forX anddas in the proof ofTheorem 3.1. Then, we can similarly prove thatX, dis complete. Let us define an operatorΛ:X→Xby
Λhx 4
h x 2
−1 2h x
4 σx
4
3.21
for allx∈E1. By induction, we can prove that Λnf
x 22n
f x 2n
1
2n−1
f x
2n1 σx 2n1
3.22 for allx∈E1and for everyn∈N.
We apply the same argument as in the proof ofTheorem 3.1and prove thatΛis a strictly contractive operator. Giveng, h∈X, letC∈0,∞be an arbitrary constant withdg, h≤C, that is,gx−hxβ ≤Cϕx, xfor allx∈E1. It then follows from3.19and3.21that
Λgx−Λhx
β
4β g x
2
−1 2g x
4 σx
4
−h x 2
1 2h x
4 σx
4
β
≤4β g x
2
−h x 2
β
2β g x
4 σx
4
−h x
4 σx
4
β
≤4βCϕ x 2,x
2
2βCϕ x
4 σx
4 ,x
4 σx
4
≤LCϕx, x
3.23
for allx∈E1, that is,dΛg,Λh≤Ldg, h.
If we replacex/2, respectively,x/4σx/4, forxandyin3.2, then we obtain fx f x
2 σx 2
−4f x 2
β≤ϕ x 2,x
2
, 3.24
respectively,
f x 2 σx
2
−2f x 4 σx
4
β≤ 1 2βϕ x
4σx 4 ,x
4 σx
4
. 3.25
Therefore, it follows from3.19,3.21,3.24, and3.25that fx−Λfx
β
fx−4
f x 2
−1 2f x
4 σx
4
β
≤
fx f x
2 σx
2
−4f x 2
β
2f x
4 σx 4
−f x
2 σx
2
β
≤ϕ x 2,x
2
1 2βϕ x
4 σx
4 ,x 4 σx
4
≤ 1
4βLϕx, x
3.26
for allx∈E1. This means that
dΛf, f≤ 1
4βL. 3.27
According toTheorem 2.1athere exists a unique functionT :E1→E2, which is a fixed point ofΛ, such that
Tx lim
n→∞22n
f x 2n
1
2n−1
f x
2n1 σx 2n1
3.28
for allx ∈ E1. Analogously to the proof ofTheorem 3.1, we can show thatT is a solution of 1.2.
UsingTheorem 2.1cand3.27, we get df, T≤ 1
4β L
1−L, 3.29
which implies the validity of3.20.
In the following corollaries, we will investigate some special cases of Theorems 3.1 and3.2.
Corollary 3.3. Fix a nonnegative numberp less than 1 and choose a constantβwithp1/2 <
β≤1. LetE1be a normed space overKand letE2be a completeβ-normed space overK. If a function f:E1→E2satisfies
fxy f
xσy
−2fx−2fyβ≤ε
xpyp
3.30 andxσxp ≤ 2pxpfor all x ∈ E1 and for some ε > 0, then there exists a unique solution T :E1→E2of1.2such that
fx−Txβ ≤ 2ε
4β−2p1xp 3.31
for anyx∈E1.
Proof. If we setϕx, y εxpypfor allx, y∈E1and if we setL2p1/4β, then we have 0< L <1 and
ϕ 2x,2y
2pε
xpyp 4β
2Lϕx, y 3.32
for allx, y∈E1. Furthermore, we get
ϕ
xσx, yσy
≤ 4β
2Lϕx, y 3.33
for anyx, y∈E1.
According toTheorem 3.1, there exists a unique solutionT : E1→E2 of1.2such that 3.31holds for everyx∈E1.
Remark 3.4. It may be remarked that if we setp0 andβ1 inCorollary 3.3, then it reduces to10, Theorem 2.1.
If we setσx −xinCorollary 3.3, thenxσxp 0p ≤ 2pxp is true for all x∈E1. In this case,3.30reduces to
fxy fx−y−2fx−2fy
β≤ε
xpyp
, 3.34
and the quadratic functionTis defined by Tx lim
n→∞
1 22nf
2nx
. 3.35
For the case whenσx −xandβ1,Corollary 3.3reduces to10, Corollary 3.3.
If we letσx xinCorollary 3.3, thenxσxp2pxpholds for allx∈E1,3.30 reduces to
fxy−fx−fy
β≤ ε 2β
xpyp
, 3.36
and the additive functionTis given by
Tx lim
n→∞
1 2nf
2nx
. 3.37
If we setσx xandβ1, then the upper bound of3.31is smaller than that of10, Corollary 3.2.
Corollary 3.5. Fix a numberplarger than 1 and choose a constantβwith 0< β <p−1/2. LetE1
be a normed space overKand letE2be a completeβ-normed space overK. If a functionf :E1→E2 satisfies3.30andxσxp ≤ 2pβxpfor allx, y ∈E1 and for someε > 0, then there exists a unique solutionT:E1→E2of 1.2such that
fx−Tx
β ≤ 2ε
2p−1−4βxp 3.38
for anyx∈E1.
Proof. If we setϕx, y εxpypfor allx, y∈E1and if we setL4β/2p−1, then we have 0< L <1 and
ϕx, y ε
xpyp L
2·4βϕ 2x,2y
3.39 for allx, y∈E1. Furthermore, we get
ϕ
xσx, yσy
≤2βϕ 2x,2y
3.40 for anyx, y∈E1.
According toTheorem 3.2, there exists a unique solutionT : E1→E2 of1.2such that 3.38holds for everyx∈E1.
Remark 3.6. Ifσx −xinCorollary 3.5, thenxσxp 0p ≤ 2pβxp is true for all x∈E1. In this case,3.30reduces to
fxy fx−y−2fx−2fyβ≤ε
xpyp
, 3.41
and the quadratic functionTis defined by Tx lim
n→∞22nf x 2n
3.42 for allx ∈E1. If we letσx −x,p >3 andβ 1 inCorollary 3.5, then the upper bound of 3.38is smaller than that of10, Corollary 4.3.
We cannot expect the Hyers-Ulam-Rassias stability for3.41whenp2 and the range spaceE2of the relevant functionsfis a Banach spacei.e.,E2is a complete 1-normed space see18. However, ifE2is a completeβ-normed space overK, whereβis a fixed real number with 0< β <1/2, then3.41is stable in the sense of Hyers, Ulam, and Rassias in spite ofp2.
If we setσx xinCorollary 3.5, thenxσxp2pxp ≤2pβxpfor allx∈ E1, 3.30reduces to
fxy−fx−fyβ≤ ε 2β
xpyp
, 3.43
and the additive functionTis given by
Tx lim
n→∞2nf x 2n
. 3.44
Unfortunately, if we setσx x,p > 3 andβ 1 inCorollary 3.5, then the upper bound of 3.38is larger than that of10, Corollary 4.2.
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