On the Stability of Generalized Quartic Mappings in Quasi-β-Normed Spaces

Volume 2010, Article ID 198098,11pages doi:10.1155/2010/198098

Research Article

On the Stability of Generalized Quartic Mappings in Quasi-β-Normed Spaces

Dongseung Kang

Department of Mathematical Education, Dankook University, 126 Jukjeon, Suji, Yongin, Gyeonggi 448-701, South Korea

Correspondence should be addressed to Dongseung Kang,[email protected] Received 28 August 2009; Revised 7 December 2009; Accepted 25 January 2010 Academic Editor: Patricia J. Y. Wong

Copyrightq2010 Dongseung Kang. 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.

We investigate the generalized Hyers-Ulam-Rassias stability problem in quasi-β-normed spaces and then the stability by using a subadditive function for the generalized quartic functionf:X → Ysuch thatfaxbyfax−by−2a²a²−b²fx ab²fxyfx−y−2b²a²−b²fy, wherea /0,b /0,a±b /0, for allx, y∈X.

1. Introduction

One of the interesting questions concerning the stability problems of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to the solution of the given functional equation? Such an idea was suggested in 1940 by Ulam1as follows. LetG1 be a group and letG2 be a metric group with the metric d·,·. Givenε > 0, does there exist aδ > 0 such that if a functionh : G1 → G2 satisfies the inequalitydhxy, hxhy< δfor allx, y∈G1, then there is a homomorphismH:G1 → G2

with dhx, Hx < εfor allx ∈ G1? In other words, we are looking for situations when the homomorphisms are stable; that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it. In 1941, Hyers2considered the case of approximately additive mappings in Banach spaces and satisfying the well-known weak Hyers inequality controlled by a positive constant. The famous Hyers stability result that appeared in 2 was generalized in the stability involving a sum of powers of norms by Aoki3. In 1978, Rassias4provided a generalization of Hyers Theorem which allows the Cauchy difference to be unbounded. During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors5–10. In particular,

Rassias11introduced the quartic functional equation f

x2y f

x−2y

6fx 4f xy

4f x−y

24f y

. 1.1

It is easy to see thatfx x⁴is a solution of1.1by virtue of the identity x2y₄

x−2y₄

x⁴4 xy₄

4 x−y₄

24y⁴. 1.2 For this reason,1.1is called a quartic functional equation. Also Chung and Sahoo 12determined the general solution of1.1without assuming any regularity conditions on the unknown function. In fact, they proved that the functionf :R → Ris a solution of1.1 if and only iffx Ax, x, x, x,where the functionA:R⁴ → Ris symmetric and additive in each variable. Lee and Chung13introduced a quartic functional equation as follows:

f axy

f ax−y

a²f xy

a²f x−y

2a² a²−1

fx−2 a²−1

f y

, 1.3

for fixed integerawitha /0,±1.

Letβbe a real number with 0< β≤1 and letKbe eitherRorC.We will consider the definition and some preliminary results of a quasi-β-norm on a linear space.

Definition 1.1. LetX be a linear space over a fieldK.A quasi-β-norm · is a real-valued function onXsatisfying the followings.

1x ≥0 for allx∈Xandx0 if and only ifx0.

2λx|λ|^β· xfor allλ∈Kand allx∈X.

3There is a constantK≥1 such thatxy ≤Kxyfor allx, y∈X.

The pairX, · is called a quasi-β-normed space if · is a quasi-β-norm onX.The smallest possibleKis called the modulus of concavity of·.A quasi-Banach space is a complete quasi-β-normed space.

A quasi-β-norm · is called aβ, p-norm0< p≤1ifxy^p≤ x^py^p,for all x, y∈X.In this case, a quasi-β-Banach space is called aβ, p-Banach space; see14–16.

In this paper, we consider the following the generalized quartic functional equation:

f

axby f

ax−by

−2a²

a²−b² fx ab²

f xy

f x−y

−2b²

a²−b² f

y ,

1.4

for fixed integersaandbsuch thata /0, b /0, a±b /0,for allx, y ∈X.We investigate the generalized Hyers-Ulam-Rassias stability problem in quasi-β-normed spaces and then the stability by using a subadditive function for the generalized quartic functionf : X → Y satisfying1.4.

For the same reason as 1.1 and 1.2, we call 1.4 generalized quartic functional equation.

2. Quartic Functional Equations

LetX, Ybe real vector spaces. In this section, we will investigate that the functional equation 1.1is equivalent to the presented functional equation1.4.

Lemma 2.1. A mappingf:X → Ysatisfies the functional equation1.1if and only iffsatisfies

f xay

f x−ay

2 a²−1

fx a² f

xy f

x−y 2a²

a²−1 f

y , 2.1

wherea /0, a / ±1,for allx, y∈X.

Proof. We will show it by induction ona.Assume that it holds for all less than equala.Now, lettingxbexy in2.1,

f

x a1y f

x−a−1y 2

a²−1 f

xy a²

f x2y

fx 2a²

a²−1 f

y ,

2.2

and also replacingxbyx−yin2.1, f

x a−1y f

x−a1y 2

a²−1 f

x−y a²

fx f

x−2y 2a²

a²−1 f

y ,

2.3

for allx, y∈X.Adding2.2and2.3, we have f

x a1y f

x−a1y f

x a−1y f

x−a−1y 2

a²−1 f

xy f

x−y a²

f x2y

f

x−2y

2a²fx 4a² a²−1

f y

,

2.4

for allx, y∈X.By induction steps, we have f

x a1y f

x−a1y

−2

a−1²−1 fx a−1²

f xy

f x−y

2a−1²

a−1²−1 f

y 2

a²−1₂ f

xy f

x−y a²

−6fx 4 f

xy f

x−y 24f

y 2a²fx 4a²

a²−1 f

y .

2.5

Hence we have f

x a1y f

x−a1y 2

a1²−1 fx a1²

f xy

f x−y

2a1²

a1²−1 f

y ,

2.6

for allx, y∈X.Thus they are equivalent.

Theorem 2.2. If a mappingf : X → Y satisfies the functional equation1.4, thenf satisfies the functional equation2.1.

Proof. By lettingxy0 in2.1, we have 2a²a²−1f0 0.Sincea /0 anda /±1, f0 0.

Puttingx0 in2.1, f

ay f

−ay a²

f y

f

−y 2a²

a²−1 f

y

. 2.7

Now, replacingyby−yin2.7, f

ay f

−ay a²

f y

f

−y 2a²

a²−1 f

−y

. 2.8

By2.7and2.8, we have 2a²a²−1fy 2a²a²−1f−y,that is,fy f−y.Hence f is even. This implies that 2fay 2a²fy 2a²a²−1fy,that is,fay a⁴fy,for ally∈X.Now, we will show that2.1implies1.4. By lettingxbxin2.1, we have

f

bxay f

bx−ay 2

a²−1 fbx a²

f bxy

f

bx−y 2a²

a²−1 f

y .

2.9

Switchingxandyin the previous equation, f

axby f

ax−by 2

a²−1 f

by a²

f xby

f

x−by 2a²

a²−1 fx.

2.10

By2.1withb, the previous equation implies that f

axby f

ax−by 2b⁴

a²−1 f

y a²b²

f xy

f x−y

2a²b² b²−1

f y

−2a² b²−1

fx 2a² a²−1

fx.

2.11

Hence we have f

axby f

ax−by

−2a²

a²−b² fx ab²

f xy

f x−y

−2b²

a²−b² f

y ,

2.12

for allx, y∈X.

Corollary 2.3. If a mappingf :X → Y satisfies the functional equation1.1, thenfsatisfies the functional equation1.4.

3. Stabilities

Throughout this section, letXbe a quasi-β-normed space and letYbe a quasi-β-Banach space with a quasi-β-norm · _Y. LetK be the modulus of concavity of · _Y. We will investigate the generalized Hyers-Ulam-Rassias stability problem for the functional equation1.4. After then we will study the stability by using a subadditive function. For a given mappingf : X → Y and all fixed integersaandbwitha /0, a /0, a±b /0,let

Df x, y

:f

axby f

ax−by

−2a²

a²−b² fx 2b²

a²−b² f

y

−ab² f

xy f

x−y

, x, y∈X.

3.1

Theorem 3.1. Suppose that there exists a mappingφ : X² → R : 0,∞for which a mapping f:X → Ysatisfiesf0 0,

Dfx, y

Y ≤φ x, y

, 3.2

and the series ^∞_j0K/a^4βjφa^jx, a^jy converges for all x, y ∈ X. Then there exists a unique generalized quartic mappingQ:X → Y which satisfies1.4and the inequality

fx−Qx

Y ≤ K

2^βa^4β ∞

j0

K a^4β

_j φ

a^jx,0

, 3.3

for allx∈X.

Proof. By lettingy0 in the inequality3.2, sincef0 0,we have Dfx,0

Y 2fax−2a²a²−b²fx−2ab²fx

Y

2fax−2a⁴fx

Y

2a⁴β

fx− 1

a⁴fax Y

≤φx,0,

3.4

that is,

fx− 1

a⁴fax Y

≤ 1

2^βa^4βφx,0, 3.5

for allx∈X.Now, puttingxaxand multiplying 1/a^4β in the inequality3.5, we get 1

a^4β

fax− 1

a⁴fa²x Y

≤ 1 2^β

1 a^4β

₂

φax,0, 3.6

for allx∈X.Combining3.5and3.6, we have

fx− 1

a⁴ ₂

fa²x

Y

≤ K 2^βa^4β

φx,0 1

a^4βφax,0

, 3.7

for allx∈X.Inductively, sinceK≥1,we have fx− 1

a^4sfa^sx Y

≤ K 2^βa^4β

s−1 j0

K a^4β

_j φ

a^jx,0

, 3.8

for allx ∈ X, s ∈ N.For allsand dwiths < dand switchingxand a^sxand multiplying 1/a^4βsin the inequality3.5, inductively,

1 a⁴

_s

fa^sx− 1

a⁴ _d

fa^dx

Y

≤ K 2^βa^4β

d−1

js

K a^4β

_j φ

a^jx,0

, 3.9

for allx∈X.Since the right-hand side of the previous inequality tends to 0 asd → ∞,hence {1/a^4sfa^sx}is a Cauchy sequence in the quasi-β-Banach spaceY.Thus we may define

Qx lim

s→ ∞

1 a⁴

s

fa^sx, 3.10

for allx ∈ X.SinceK ≥ 1,replacingxandybya^sxanda^sy, respectively, and dividing by a^4βs in the inequality3.2, we have

1 a^4β

s

Dfa^sx, a^sy

Y

1

a^4β _s

a^s

axby f

a^s

ax−by

−2a²

a²−b² fa^sx 2b²a²−b²fa^sy−ab²fa^sxy−fa^sx−y

Y

≤ K

a^4β _s

φ

a^sx, a^sy ,

3.11

for allx, y ∈ X.By takings → ∞,the definition ofQ implies thatQ satisfies1.4for all x, y∈X; that is,Qis the generalized quartic mapping. Also, the inequality3.8implies the inequality3.3. Now, it remains to show the uniqueness. Assume that there existsT:X → Y satisfying1.4and3.3. It is easy to show that for allx∈X, Ta^sx a^4sTxandQa^sx a^4sQx,as in the proof ofTheorem 2.2. Then

Tx−Qx_Y 1

a^4β _s

Ta^sx−Qa^sx_Y

≤ 1

a^4β _s

KTa^sx−fa^sx

Y fa^sx−Qa^sx

Y

≤ 2K² 2^βa^4β

∞ j0

K a^4β

_sj φ

a^sjx,0 ,

3.12

for allx∈X.By lettings → ∞,we immediately have the uniqueness ofQ.

Theorem 3.2. Suppose that there exists a mappingφ : X² → R : 0,∞for which a mapping f:X → Ysatisfiesf0 0,

Dfx, y

Y ≤φ x, y

, 3.13

and the series ^∞_j1a^4βK^jφa^−jx, a^−jyconverges for all x, y ∈ X. Then there exists a unique generalized quartic mappingQ:X → Y which satisfies2.1and the inequality

fx−Qx

Y ≤ 1

2^βa^4β ∞

j1

a^4βK_j

φ

a^−jx,0

, 3.14

for allx∈X.

Proof. Ifxis replaced by1/axin the inequality3.5, then the proof follows from the proof ofTheorem 3.1.

Now we will recall a subadditive function and then investigate the stability under the condition that the spaceY is aβ, p-Banach space. The basic definitions of subadditive functions follow from16.

A functionφ:A → B having a domainAand a codomainB,≤that are both closed under addition is called

1a subadditive function ifφxy≤φx φy,

2a contractively subadditive function if there exists a constantLwith 0 < L < 1 such thatφxy≤Lφx φy,

3an expansively superadditive function if there exists a constantLwith 0< L <1 such thatφxy≥1/Lφx φy,

for allx, y∈A.

Theorem 3.3. Suppose that there exists a mappingφ : X² → R : 0,∞for which a mapping f:X → Ysatisfiesf0 0,

Dfx, y

Y ≤φ x, y

, 3.15

for allx, y∈X and the mapφis contractively subadditive with a constantLsuch thata^1−4βL <1.

Then there exists a unique generalized quartic mappingQ : X → Y which satisfies1.4and the inequality

fx−Qx

Y ≤ φx,0

2^βp

a^4βp−aL^p, 3.16

for allx∈X.

Proof. By the inequalities3.5and3.9of the proof ofTheorem 3.1, we have 1

a^4sfa^sx− 1

a^4dfa^dx ^p

Y

≤^d−1

js

1 a^4β

jp

fa^jx− 1

a⁴fa^j1x ^p

Y

≤ 1 2^βpa^4βp

d−1 js

1 a^4β

_jp φ

a^jx,0_p

≤ 1 2^βpa^4βp

d−1 js

1 a^4β

jp

aL^jpφx,0^p

φx,0^p 2^βpa^4βp

d−1

js

a^1−4βLjp

,

3.17

that is,

1 a⁴

s

fa^sx− 1

a⁴ d

fa^dx

p

Y

≤ φx,0^p 2^βpa^4βp

d−1

js

a^1−4βL_jp

, 3.18

for allx∈X,and for allsanddwiths < d.Hence{1/a^4sfa^sx}is a Cauchy sequence in the spaceY.Thus we may define

Qx lim

s→ ∞

1

a^4sfa^sx, 3.19

for allx∈X.Now, we will show that the mapQ:X → Yis a generalized quartic mapping.

Then

DQx, y^p

Y lim

s→ ∞

Dfa^sx, a^sy^p

Y

a^4βps

≤ lim

s→ ∞

φ

a^sx, a^sy_p a^4βps

≤ lim

s→ ∞φ x, y_p

a^1−4βL_ps 0,

3.20

for allx∈X.Hence the mappingQis a generalized quartic mapping. Note that the inequality 3.18implies the inequality3.16by lettings 0 and taking d → ∞.Assume that there existsT :X → Y satisfying1.4and3.16. We know thatTa^sx a^4sTx,for allx ∈X.

Then

Tx− 1

a⁴ _s

fa^sx ^p

Y

1

a^4β _ps

Ta^sx−fa^sx^p

Y

≤ 1

a^4β

_ps φa^sx,0^p 2^βp

a^4βp−aL^p

≤

a^1−4βLps φx,0^p 2^βp

a^4βp−aL^p,

3.21

that is,

Tx− 1

a⁴ _s

fa^sx Y

≤

a^1−4βLs φx,0 2^{β p}

a^4βp−aL^p, 3.22

for allx∈X.By lettings → ∞,we immediately have the uniqueness ofQ.

Theorem 3.4. Suppose that there exists a mappingφ : X² → R : 0,∞for which a mapping f:X → Ysatisfiesf0 0,

Dfx, y

Y ≤φ x, y

, 3.23

for allx, y ∈X and the mapφis expansively superadditive with a constantLsuch thata^4β−1L <1.

Then there exists a unique generalized quartic mappingQ : X → Y which satisfies1.4and the inequality

fx−Qx

Y ≤ φx,0

2^βL ^p

a^p−

a^4βL_p, 3.24

for allx∈X.

Proof. By lettingy0 in3.23, we have

2fax−2a⁴fx

Y ≤φx,0, 3.25

and then replacingxbyx/a,

fx−a⁴fx a

Y ≤ 1 2^βφx

a,0

, 3.26

for allx∈X.For allsanddwiths < d,inductively we have a^4sfx

a^s

−a^4df x

a^{d p}

Y

≤ φx,0^p 2^βpaL^p

d−1 js

a^4β−1L_jp

, 3.27

for allx∈X.The remains follow from the proof ofTheorem 3.3.

Acknowledgments

The author would like to thank referees for their valuable suggestions and comments. The present research was conducted by the research fund of Dankook University in 2009.

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