A Fixed Point Approach to the Stability of Quadratic Functional Equation with Involution

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 Jung1_{and Zoon-Hee Lee}2

1_{Mathematics Section, College of Science and Technology, Hong-Ik University,}

339-701 Chochiwon, South Korea

2_{Department 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 metric d·,·. Givenε > 0, does there exist aδ >0such 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

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 x for 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 fxσy2fx 2fy 1.2

is called the quadratic functional equation with involution. According to 11, Corollary 8,

a function f : E1→E2 is a solution of 1.2 if and only if there exists an additive function A : E1→E2and a biadditive symmetric functionB : E1×E1→E2such thatAσx Ax, Bσx, y −Bx, yandfx Bx, x Axfor allx∈E1.

Indeed, if we setσ Iin1.2, whereI:E1→E1denotes the identity function, then1.2 reduces to the additive functional equation

fxy fx fy. 1.3

On the other hand, if σ −I in 1.2, 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.

athe sequence{Λn_{x}converges to a fixed pointx}∗_ofΛ; bx∗is the unique fixed point ofΛin

X∗y∈X:dΛkx, y<∞; 2.1

cify∈X∗, then

dy, 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 equation see 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 with0< β≤1. Suppose a functionϕ:E1×E1→0,∞is given and there exists a constantL,0< L <1, such that

ϕ2x,2y≤ 4

β

2Lϕx, y,

ϕxσx, yσy≤ 4

β

2 Lϕx, y

3.1

for allx, y ∈E1. Furthermore, letf:E1→E2be a function satisfying the inequality

_{fxy f}

xσy−2fx−2fy_β≤ϕx, y 3.2

for all x, y ∈ E1, whereσ : E1→E1 is an involution of E1. Then there exists a unique solution T : E1→E2of 1.2such that

_fx−Tx β≤

1 4β

1

1−Lϕx, x 3.3

Proof. First, let us defineXto be the set of all functionsh:E1→E2and introduce a generalized metric onXas follows:

dg, h infC∈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 given ε > 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 m increase to infinity, it follows from3.5that for any ε > 0, there exists a positive integerNε withfnx−fxβ ≤ εϕx, xfor alln ≥ Nε and for allx ∈ E1, that is,

for anyε >0, there exists a positive integerNε such 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 hxσx 3.7

for allx∈E1.

First, we assert thatΛ is strictly contractive onX. Giveng, h ∈ X, let C ∈ 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−hxσx_β

≤ 1

4βg2x−h2xβ

1 4βg

xσx−hxσx_β

≤ C

4βϕ2x,2x

C 4βϕ

xσx, xσx≤LCϕx, x

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 puty xin3.2and we divide both sides by 4β_{, then we get}

_Λfx−fx

β

1₄f2x fxσx−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 thatdΛn_{f, T→0 asn→ ∞.}

By mathematical induction, we can easily showand hence we can omit to showthat

Λn_fx 1

22n

f2nx2n−1f2n−1x2n−1σx 3.13

for eachn∈N.

Since dΛn_{f, T→0 as n→ ∞, there exists a sequence {C}

n}such that Cn→0 as n→ ∞

anddΛnf, T≤Cnfor everyn∈N. Hence, it follows from the definition ofdthat _Λn_fx−Tx

β ≤Cnϕx, x 3.14

for allx∈E1. Thus, for eachfixedx∈E1, we have

lim

n→∞Λ

n_fx−Tx

β0. 3.15

Therefore

Tx lim

n→∞ 1 22n

f2nx2n−1f2n−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

2nx2ny2n−1f2n−1xy 2n−1σx σy

f2nx2nσy2n−1f2n−1xσy2n−1σx y

−2f2nx−22n−1f2n−1xσx

≤ lim

n→∞

1 4βnf

2nx2nyf2nx2nσy−2f2nx−2f2ny_β

lim

n→∞

2n−1β 4βn f

2n−1xσx2n−1yσyf2n−1xσx2n−1yσy

−2f2n−1xσx−2f2n−1yσy_β

≤ lim

n→∞

1 4βnϕ

2nx,2ny lim

n→∞

2n₋₁β

4βn ϕ

2n−1xσx,2n−1yσy

≤ lim

n→∞

1 4βn

4β

2 L

n

ϕx, y lim

n→∞

2n−1β 4βn

4β

2L

n

ϕx, y 0

3.17

for allx, y ∈E1, which implies thatTis a solution of1.2.

ByTheorem 2.1cand by3.12, we obtain

df, T≤ 1

1−LdΛf, f≤ 1

4β_1−L, 3.18

that is,3.3is true for allx∈E1.

Assume thatT1 : E1→E2is another solution of 1.2satisfying3.3.We know thatT1 is a fixed point of Λ.In view of3.3and the definition ofd, we can conclude that 3.18is 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 with0< 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,

ϕxσx, yσy≤2β_ϕ2x,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→E2of 1.2such 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

for allx∈E1. By induction, we can prove that

Λn_{fx 2}2n

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

4βg x 2 −1 2g x 4

σx 4

−h x 2 1 2h x 4

σx 4

β

≤4βg x 2

−h x 2

β

2βg 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 4

σx 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

According toTheorem 2.1athere exists a unique functionT :E1→E2, which is a fixed point ofΛ, such that

Tx lim

n→∞2

2n

f x

2n

1

2n −1

f x

2n1 σx

2n1

3.28

for all x ∈ E1. Analogously to the proof of Theorem 3.1, we can show that T 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 constant β with p1/2 < β ≤1. LetE1be a normed space overKand letE2be a completeβ-normed space overK. If a function f :E1→E2satisfies

_{fxy f}

xσy−2fx−2fy_β ≤εxpyp 3.30

and xσxp _{≤ 2}p_xp _{for all x ∈ E}

1 and for someε > 0, then there exists a unique solution T :E1→E2of 1.2such that

_fx−Tx β ≤

2ε 4β₋₂p1x

p _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

ϕ2x,2y2pεxpyp 4

β

2 Lϕx, y 3.32

for allx, y ∈E1. Furthermore, we get

ϕxσx, yσy≤ 4

β

2 Lϕx, y 3.33

for anyx, y ∈E1.

Remark 3.4. It may be remarked that if we setp 0 andβ1 inCorollary 3.3, then it reduces to10, Theorem 2.1.

If we set σx −x in Corollary 3.3, then xσxp ₀p _{≤ 2}p_xp _{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, thenxσxp₂p_xp _{holds 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 than1and choose a constantβwith0 < β <p−1/2. LetE1 be a normed space overKand letE2be a complete β-normed space overK. If a functionf : E1→E2 satisfies3.30andxσxp _{≤ 2}pβ_xp _{for allx, y ∈ E}

1and for someε > 0, then there exists a unique solutionT :E1→E2of 1.2such that

_fx−Tx β ≤

2ε 2p−1₋₄βx

p _3.38

for anyx∈E1.

Proof. If we setϕx, y εxp_yp_{for allx, y∈E}

1and if we setL4β/2p−1, then we have 0< L <1 and

ϕx, y εxpyp L 2·4βϕ

2x,2y 3.39

for allx, y ∈E1. Furthermore, we get

ϕxσx, yσy≤2βϕ2x,2y 3.40

for anyx, y ∈E1.

Remark 3.6. If σx −x in Corollary 3.5, then xσxp ₀p _{≤ 2}pβ_xp _{is true for all}

x∈E1. In this case,3.30reduces to

_{fxy fx−y−2fx−2fy} β ≤ε

xpyp, 3.41

and the quadratic functionTis defined by

Tx lim

n→∞2 2n_f 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 for3.41whenp 2 and the range spaceE2of the relevant functionsf is a Banach spacei.e.,E2 is 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 xin Corollary 3.5, thenxσxp₂p_xp _{≤ 2}pβ_xp _{for allx ∈ E}

1, 3.30reduces to

_{fxy−fx−fy} β ≤

ε 2β

xpyp, 3.43

and the additive functionTis given by

Tx lim

n→∞2

n_f 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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