Superstability of Generalized Multiplicative Functionals

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Volume 2009, Article ID 486375,7pages doi:10.1155/2009/486375

Research Article

Superstability of Generalized Multiplicative

Functionals

Takeshi Miura,

1

_{Hiroyuki Takagi,}

2

_{Makoto Tsukada,}

3

and Sin-Ei Takahasi

1

1_{Department of Applied Mathematics and Physics, Graduate School of Science and Engineering,}

Yamagata University, Yonezawa 992-8510, Japan

2_{Department of Mathematical Sciences, Faculty of Science, Shinshu University,}

Matsumoto 390-8621, Japan

3_{Department of Information Sciences, Toho University, Funabashi, Chiba 274-8510, Japan}

Correspondence should be addressed to Takeshi Miura,[email protected]

Received 2 March 2009; Accepted 20 May 2009

Recommended by Radu Precup

LetXbe a set with a binary operation◦such that, for eachx, y, z∈X, eitherx◦y◦z x◦z◦y, orz◦x◦y x◦z◦y. We show the superstability of the functional equationgx◦y gxgy. More explicitly, ifε≥0 andf :X → Csatisfies|fx◦y−fxfy| ≤εfor eachx, y∈X, then

fx◦y fxfyfor allx, y∈X, or|fx| ≤1√14ε/2 for allx∈X. In the latter case, the constant1√14ε/2 is the best possible.

Copyrightq2009 Takeshi Miura et al. 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

It seems that the stability problem of functional equations had been first raised by S. M. Ulam

cf. 1, Chapter VI. “For what metric groupsG is it true that anε-automorphism ofG is necessarily near to a strict automorphism?Anε-automorphism ofGmeans a transformation

fofGinto itself such thatρfx·y, fx·fy< εfor allx, y∈G.” D. H. Hyers2gave an aﬃrmative answer to the problem: ifε≥0 andf :E1 → E2is a mapping between two real Banach spacesE1 andE2satisfyingfxy−fx−fy ≤εfor allx, y ∈E1, then there

exists a unique additive mappingT :E1 → E2 such thatfx−Tx ≤εfor allx∈E1. If,

in addition, the mappingR t→ftxis continuous for each fixedx∈E1, thenT is linear. This result is calledHyers-Ulam stabilityof theadditiveCauchy equationgxy gx

gy. J. A. Baker 3, Theorem 1considered stability of the multiplicative Cauchy equation

gxy gxgy: ifε ≥ 0 andf is a complex valued function on a semigroupSsuch that

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for allx∈S. This result is called superstability of the functional equationgxy gxgy. Recently, A. Najdecki 4, Theorem 1 proved the superstability of the functional equation

gxφy gxgy: ifε≥0,fis areal or complex valuedfunctional from a commutative semigroupX,◦,andφis a mapping fromXinto itself such that|fx◦φy−fxfy| ≤ε for allx, y∈X, thenfx◦φy fxfyholds for allx, y∈X, orfis bounded.

In this paper, we show that superstability of the functional equation gx ◦ y

gxgyholds for a setXwith a binary operation◦under an additional assumption.

2. Main Result

Theorem 2.1. Letε≥0andXa set with a binary operation◦such that, for eachx, y, z∈X, either

x◦y◦z x◦z◦y, or z◦x◦yx◦z◦y. 2.1

Iff:X → Csatisfies

_f

x◦y−fxfy≤ε ∀x, y∈X, 2.2

thenfx◦y fxfyfor allx, y∈X, or|fx| ≤1√14ε/2for allx∈X. In the latter case, the constant1√14ε/2is the best possible.

_f_x2_≤_ε_f_x_◦_x_≤_ε_M.

2.3

Thus,M2 _≤ _ε_M_{. Now it is easy to see that}_M _≤ ₁√₁₄_ε_/_{2. Consequently, if} _f _is

bounded, then|fx| ≤ 1√14ε/2 for allx ∈ X. The constant1√14ε/2 is the best possible sincegx 1√14ε/2 forx∈Xsatisfiesgxgy−gx◦y εfor each

x, y∈X. It should be mentioned that the above proof is essentially due to P. ˇSemrl5, Proof of Theorem 2.1 and Proposition 2.2 cf.6, Proposition 5.5.

Suppose thatf :X → Cis an unbounded functional satisfying the inequality2.2. Sincefis unbounded, there exists a sequence{zk}k∈N⊂Xsuch that limk→ ∞|fzk|∞. Take

x, y∈Xarbitrarily. Set

N1

k∈N:x◦y◦zk x◦zk◦y,

N2

k∈N:zk◦x◦yx◦zk◦y.

2.4

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assumed to be infinite, for eachm >2 there existskm∈N1such thatkm−1 < km. Then{zkm}m∈N

is a subsequence of{zk}k∈Nwithkm∈N1for everym∈N. By the choice of{zk}k∈N, we have

lim

m→ ∞fzkm∞. 2.5

Thus we may and do assume that fzkm/0 for everym ∈ N. By2.2we have, for each w∈Xandm∈N,|fw◦zkm−fwfzkm| ≤ε. According to2.5, we have

fw◦zkm fzkm

−fw≤ ε

fzkm

−→0 asm → ∞. 2.6

Consequently, we have, for eachw∈X,

fw lim m→ ∞

fw◦zkm fzkm

. 2.7

Sincekm∈N1, we havex◦y◦zkm x◦zkm◦yfor everym∈N. Applying2.7, we have

fx◦y lim m→ ∞

fx◦y◦zkm

fzkm

lim m→ ∞

fx◦zkm◦y

fzkm

lim m→ ∞

fx◦zkm◦y−fx◦zkmf

y fzkm

lim m→ ∞

fx◦zkmf

y fzkm

.

2.8

By2.2and2.5, we have

lim m→ ∞

fx◦zkm◦y

−fx◦zkmf

y fzkm

≤mlim→ ∞

ε

_f_z_k_m0. 2.9

Consequently, we have by2.8and2.7

fx◦y lim m→ ∞

fx◦zkmf

y fzkm

lim m→ ∞

fx◦zkm fzkm

fyfxfy. 2.10

Next we consider the case whenN2is infinite. By a quite similar argument as in the

case whenN1is infinite, we see that there exists a subsequence{zkn}n∈N⊂ {zk}k∈Nsuch that kn∈N2for everyn∈N. Then

lim

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In the same way as in the proof of2.7, we have

fw lim n→ ∞

fzkn ◦w fzkn

, 2.12

for everyw∈X. According to2.2and2.11, we have

lim n→ ∞

fx◦zkn◦y

−fxfzkn◦y

fzkn

≤nlim→ ∞

ε

fzkn

0. 2.13

Sincezkn ◦x◦y x◦zkn◦yfor everyn∈N,2.11and2.12show that

fx◦y lim n→ ∞

fzkn◦

x◦y fzkn

lim n→ ∞

fx◦zkn ◦y

fzkn

lim n→ ∞

fx◦zkn ◦y

−fxfzkn ◦y

fzkn

lim n→ ∞

fxfzkn◦y

fzkn

lim n→ ∞

fxfzkn◦y

fzkn

fxlim n→ ∞

fzkn◦y

fzkn

fxfy.

2.14

Consequently, iffis unbounded, thenfx◦y fxfyfor allx, y∈X.

Remark 2.2. Letφbe a mapping from a commutative semigroupX into itself. We define the binary operation◦byx◦yxφyfor eachx, y∈X. Then◦satisfies2.1since

x◦y◦zxφyφz xφzφy x◦z◦y, 2.15

for allx, y, z∈X. Therefore,Theorem 2.1is a generalization of Najdecki4, Theorem 1and Baker3, Theorem 1.

Remark 2.3. LetXbe a set, andf :X → C. Suppose thatXhas a binary operation◦such that, for eachx, y, z∈X, either

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Iffsatisfies2.2for someε≥0, then by quite similar arguments to the proof ofTheorem 2.1, we can prove thatfx◦y fxfyfor allx, y ∈ X, or|fx| ≤ 1√14ε/2 for all

x∈X. Thus,Theorem 2.1is still true under the weaker condition2.16instead of2.2. This was pointed out by the referee of this paper. The condition2.16is related to that introduced by Kannappan7.

Example 2.4. Let ϕ and ψ be mappings from a semigroupX into itself with the following properties.

aϕxy ϕxϕyfor everyx, y∈X.

bψX⊂ {x∈X:ϕx x}.

cψxψy ψyψxfor everyx, y∈X.

If we definex◦yϕxψyfor eachx, y∈X, then we havex◦y◦z x◦z◦yfor every

x, y, z∈X. In fact, ifx, y, z∈X, then we have

x◦y◦zϕx◦yψz

ϕϕxψyψz

bya

ϕ2_x_ϕ_ψ_y_ψ_z

byb

ϕ2_x_ψ_y_ψ_z

byc

ϕ2_x_ψ_z_ψ_y

byb

ϕ2_x_ϕ_ψ_z_ψ_y

bya

ϕϕxψzψy

ϕx◦zψy

x◦z◦y

2.17

as claimed.

Letϕbe a ring homomorphism fromCinto itself, that is,ϕzw ϕz ϕwand

ϕzw ϕzϕwfor eachz, w ∈C. It is well known that there exist infinitely many such homomorphisms onCcf.8,9. Ifϕis not identically 0, then we see thatϕq qfor every

q∈ Q, the field of all rational real numbers. Thus, if we consider the case whenX C,ϕa nonzero ring homomorphism, andψ :X → Q, thenX, ϕ, ψsatisfies the conditionsa,b, andc.

If we definex∗yy◦xfor eachx, y∈X, thenz∗x∗y x∗z∗yholds for every

x, y, z∈X. In fact,

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Example 2.5. LetXC× {0,1},and, letϕ, ψ:C → C. We define the binary operation◦by

x, a◦y, b ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

xψy,0, if ab0,

ϕxy,1, if ab1,

0,0, if a /b,

2.19

for eachx, a,y, b ∈ X. Then◦ satisfies the condition2.1. In fact, letx, a,y, b,z, c

∈X.

aIfabc0, then we have

x, a◦y, b◦z, c xψyψz,0 x, a◦z, c◦y, b. 2.20

bIfabc1, then

z, c◦x, a◦y, bϕzϕxy,1

x, a◦z, c◦y, b. 2.21

cIfab0 andc1, then

x, a◦y, b◦z, c 0,0 x, a◦z, c◦y, b. 2.22

dIfab1 andc0, then

z, c◦x, a◦y, b 0,0 x, a◦z, c◦y, b,

x, a◦y, b◦z, c 0,0 x, a◦z, c◦y, b.

2.23

eIfa /b, then we have

x, a◦y, b◦z, c 0,0 x, a◦z, c◦y, b. 2.24

Therefore,◦satisfies the condition2.1. On the other hand, ifabc0, then

z, c◦x, a◦y, bzψxψy,0,

x, a◦z, c◦y, bxψzψy,0.

2.25

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Acknowledgments

The authors would like to thank the referees for valuable suggestions and comments to improve the manuscript. The first and fourth authors were partly supported by the Grant-in-Aid for Scientific Research.

References

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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, pp. 222–224, 1941.

3 J. A. Baker, “The stability of the cosine equation,”Proceedings of the American Mathematical Society, vol. 80, no. 3, pp. 411–416, 1980.

4 A. Najdecki, “On stability of a functional equation connected with the Reynolds operator,”Journal of Inequalities and Applications, vol. 2007, Article ID 79816, 3 pages, 2007.

5 P. ˇSemrl, “Non linear perturbations of homomorphisms onCX,”The Quarterly Journal of Mathematics. Series 2, vol. 50, pp. 87–109, 1999.

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7 Pl. Kannappan, “On quadratic functional equation,”International Journal of Mathematical and Statistical Sciences, vol. 9, no. 1, pp. 35–60, 2000.

8 A. Charnow, “The automorphisms of an algebraically closed field,”Canadian Mathematical Bulletin, vol. 13, pp. 95–97, 1970.