Superstability and Stability of the Pexiderized Multiplicative Functional Equation

Journal of Inequalities and Applications Volume 2010, Article ID 486325,15pages doi:10.1155/2010/486325

Research Article

Superstability and Stability of the Pexiderized

Multiplicative Functional Equation

Young Whan Lee

Department of Computer and Information Security, Daejeon University, Daejeon 300-716, South Korea

Correspondence should be addressed to Young Whan Lee,[email protected]

Received 14 August 2009; Accepted 28 December 2009

Academic Editor: Yeol Je Cho

Copyrightq2010 Young Whan 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.

We obtain the superstability of the Pexiderized multiplicative functional equation fxy gxhyand investigate the stability of this equation in the following form: 1/1ψx, y ≤ fxy/gxhy≤1ψx, y.

1. Introduction

The superstability of the functional equationfxy fxfy was studied by Baker

et al. 1. They proved that if f is a functional on a real vector space W satisfying|fx

y−fxfy| ≤ δfor some fixedδ > 0 and allx, y ∈W, thenf is either bounded or else

fxy fxfyfor all x, y ∈ W. This result was genealized with a simplified proof by

Baker2as follows.

Theorem 1.1Baker2. Letδ >0,Sbe a semigroup andf:S → Csatisfying

_f

xy−fxfy≤δ 1.1

for allx, y∈S. Putβ: 1√14δ/2.Then|fx| ≤βfor allx∈Sor elsefxy fxfy

for allx, y∈S.

A different generalization of the result of Baker et al. was given by Sz´ekelyhidi3.

Theorem 1.2Sz´ekelyhidi3. LetGbe a commutative group with identity1and letf, m:S → C

be functions such that there exist functionsM1, M2:G → 0,∞with

fxy−fxmy≤minM1x, M2

y 1.2

for allx, y∈G. Thenfis bounded ormis an exponential andff1m.

In this paper, we prove the superstability of the Pexiderized multiplicative functional

equationPMFE

fxygxhy. 1.3

That is, we prove that iff, g, hare functional on a semigroupS with identity 1 satisfying

g1 1 and

_f

xy−gxhy≤ϕx, y 1.4

for allx, y∈Sand for a functionϕ:S×S → 0,∞with some coditions, thengis bounded

or elsegis an exponential andhh1g. This is a generalization of the result of Sz´ekelyhidi.

Also we investigate the stability of the Pexiderized multipicative functional equation1.3in

the sense of Ger4.

2. Superstability of the PMFE

In this section, letS,·be a semigroup with identity 1 andϕ:S×S → 0,∞a function with

Φwx:

∞

k0

ϕw, xk2_{ϕwx, x}k1

2k <∞ 2.1

for allx, w∈Sand

lim k→ ∞ϕ

wx, zyk exists 2.2

for allx, y, z, w∈S.

Example 2.1. The following functions satisfy conditions2.1and2.2above.

aϕx, y δ,for everyx, y∈Randδ≥0.

bϕx, y |tx|, for everyx, y∈Sandtis a functional onS.

cϕx, y |x|1/1|y|, for everyx, y∈R.

Example 2.2. LetS,· 0,∞,and alsogx ex_{, hx e}xc_,

fx exc 1

1x 2.3

for allx ∈ S and for somec ∈ S. Let ϕx, y 1/1xy. Thenf, g, h, ϕ satisfy the

conditions1.4,2.1,2.2and

_f

xy−gxhy 1

1xy. 2.4

In particular, we know thatg0 1, gxy gxgy, hh0g, andfxy/fxfy.

Theorem 2.3. Let S,· be a semigroup with identity 1. Iff, g, h : S → C are functions with

|gm| ≥max{2,2Φ1m/|hm|}for somem∈Ssatisfyingg1 1and condition1.4, that is,

fxy−gxhy≤ϕx, y, 2.5

then

gxygxgy 2.6

for allx, y∈Sandhh1g.

Proof. If we replacexbymand alsoybymin1.4, we get

fm2 −gmhm≤ϕm, m. 2.7

Also we replacexby 1 in1.4, then we have

fy−hy≤ϕ1, y 2.8

for ally∈S. An induction argument implies that for alln≥2,

fmn_−gmn−1_{hm≤ϕm, m}n−1 n−2 k1

Indeed, if inequality2.9holds, using inequality1.4and2.8we have

fmn1 −gmnhm

≤fmmn−gmhmngmhmn−fmn

gmfmn−gmn−1hm

≤ϕm, mn gmϕ1, mn

gm

ϕm, mn−1

n−2

k1

gmkϕ1, mn−k ϕm, mn−k−1

ϕm, mn

n−1

k1

gmkϕ1, mn1−k ϕm, mn−k

2.10

for alln≥2. By2.9, we have

fmn

gmn−1hm−1

≤ 1

|hm|gm

ϕm, mn−1

_gmn−2

n−2

k1

1

gmn−k−2

ϕ1, mn−k ϕm, mn−k−1

≤ 1

|hm|gm

ϕ1, m2 ϕm, m 1

2

ϕ1, m3 ϕm, m2 · · ·

1

2n−3

ϕ1, mn−1 _{ϕm, m}n−2

1 2n−2ϕ

m, mn−1

≤ 1

|hm|gm

_n−3

k0

1 2k

ϕ1, mk2 _{ϕm, m}k1 1

2n−2ϕ

m, mn−1 _{ϕ1, m}n

≤ 1

|hm|gm

∞

k0

1 2k

ϕ1, mk2 ϕm, mk1

Φ1m

|hm|gm ≤

1 2.

2.11

Thus we can easily show that|fmn_{| → ∞from|gm}n−1_{hm| → ∞asn → ∞and thus}

|hmn_{| → ∞asn → ∞. By1.4,}

f_hxm_mnn−gx

≤ ϕ_|x, mn

and thus we have

gx lim n→ ∞

fxmn

hmn 2.13

for allx∈S. Then, by2.2,

gxy−gxgy lim n→ ∞

1

|hmn_|f

xymn−gxfymn

lim n→ ∞

1

|hmn_|f

xymn−gxhymngxhymn−fymn

≤ lim n→ ∞

1

|hmn_|

ϕx, ymngxϕ1, ymn0,

2.14

and so

gxygxgy 2.15

for allx, y∈S. Thus we have|gmn_||gmn_{| → ∞asn → ∞. Since}

f_gxm_mnn −hx

≤ ϕx, mn

gmn −→0 2.16

asn → ∞, we can definehby

hx lim n→ ∞

fxmn

gmn 2.17

for allx∈S. Then

h1gx lim n→ ∞

fmn

gmn·

fxmn

hmn g1hx hx 2.18

for allx∈S.

Corollary 2.4. LetS,be a semigroup with identity0andf, g, h :S → Cfunctions satisfying

the inequality

_f

xy−gxhy≤ϕx, y 2.19

Theorem 2.5. LetS,·be a semigroup with identity1and f, g, h : S → Cfunctions satisfying condition1.4, that is,

fxy−gxhy≤ϕx, y. 2.20

Ifg satisfies thatgs/0 for somes ∈ Sand |gsm| ≥ max{2|gs|,2Φsm/|hm|}for some

m∈S,then

gsxy 1

gsgsxg

sy 2.21

for allx, y∈Sandhx h1/gsgsx.

Proof. Letfx fsx, gx gsx/gsandhx gshxfor everyx∈Sandϕx, y ϕsx, y. Then

fxy−gxhy≤ϕx, y 2.22

for allx, y ∈ S,|gm| ≥ max{2,2Φ1m/|hm|}andg1 1 whereΦ1m Φsm. By

Theorem 2.3, we complete the proof.

Corollary 2.6. LetS,·be a semigroup with identity1. Iff, g, h : S → Care nonzero functions satisfying condition1.4, that is,

_f

xy−gxhy≤ϕx, y, 2.23

then eithergis bounded, or else

gsxy 1

gsgsxg

sy 2.24

for allx, y∈Sandhx h1/gsgsx.

Proof. Letgs/0 for somes∈S. Ifgis unbounded, then there existsmsuch that|gsm| ≥

max{2|gs|,2Φsm/|hm|}. ByTheorem 2.5, we complete the proof.

3. Stability of the PMFE

In 1940, Ulam gave a wide-ranging talk in the Mathematical Club of the University of

Wisconsin in which he discussed a number of important unsolved problems5. One of those

was the question concerning the stability of homomorphisms.

Let G1 be a group and let G2 be a metric group with a metric d·,·. Given > 0, does there exist aδ > 0 such that if a mapping h : G1 → G2 satisfies the inequality

dhxy, hxhy< δfor allx, y∈G1, then there exists a homomorphismH:G1 →

In the next year, Hyers6answered the Ulam’s question for the case of the additive

mapping on the Banach spacesG1, G2. Thereafter, the result of Hyers has been generalized

by Rassias7. Since then, the stability problems of various functional equations have been

investigated by many authorssee6,8–18.

Ger 4 suggested another type of stability for the exponential equation in the

following type:

fxy fxfy−1

≤δ. 3.1

In this section, the stability problem for the Pexiderized multiplicative functional equation in the following form:

1

1ψx, y ≤

fxy

gxhy ≤1ψ

x, y 3.2

will be investigated.

Throughout this section, we denote by S,· a commutative semigroup and by ψ :

S×S → 0,∞a function such that

Ψx, y, z, w

∞

n0

1 2nln

1ψxz2n, yw2n <∞ 3.3

for allx, y, z, w∈S. Also we let

ux, y:ln1ψx, y1ψy, x1ψx, x1ψy, y 3.4

for allx, y∈S.Inequality3.3implies that

afor allx, z∈S

∞

n0

1 2nu

xz2n, xz2n

∞

n0

1 2nln

1ψxz2n, xz2n 44Ψx, x, z, z<∞, 3.5

bfor allx, z∈S

∞

n0

1 2nu

x2, z2n

∞

n0

1 2nln

1ψx2, z2n 1ψz2n, x2

·1ψx2, x2 1ψz2n, z2n

cfor allx, z∈S

∞

n0

1 2nu

x4, z2n Ψx4,1,1, z Ψ1, x4, z,1 Ψx4, x4,1,1 Ψ1,1, z, z<∞, 3.7

dfor allx, y∈Sfor

lim n→ ∞

1 2nu

x2n, y2n

lim n→ ∞

1 2nln

1ψx2n, y2n 1ψy2n, x2n ·1ψx2n, x2n 1ψy2n, y2n 0,

3.8

because

Ψ1,1, x, y Ψ1,1, y, x Ψ1,1, x, x Ψ1,1, y, y<∞. 3.9

Example 3.1. The following functions satisfy condition3.3above.

aψx, y δ,for everyx, y∈Randδ≥0.

bψx, y 1/1|x||y|, for everyx, y∈R.

Example 3.2. LetS,· 0,∞,and alsogx ex_{, hx e}xc_,

fx exc

1 1

1x

3.10

for allx∈Sand for somec∈S. Letψx, y 1/1xy. Thenf, g, h, ψsatisfy condition

3.3and

1

1ψx, y ≤

fxy

gxhy ≤1ψ

x, y. 3.11

In particular, we know that if we letTx ex_then

e−c

2 ≤

Tx fx ≤e

−c_, Tx

gx 1,

Tx hx e

−c_{. 3.12}

Theorem 3.3. Iff, g, h:S → 0,∞are functions such that

1

1ψx, y ≤

fxy

gxhy ≤1ψ

for allx, y∈S, then there exists a functionT :S → 0,∞and there exists a constantMsuch that

Txy TxTyfor allx, y∈Sand

e−M≤ Tx fx ≤e

M _3.14

for allx∈S. Moreover, ifψis bounded, then

e−M1 ≤ Tx

gx ≤e

M1_,

e−M1 ≤ Tx

hx ≤e

M1

3.15

for allx∈Sand for some constantM1.

Proof. If we define functionsF, G, H:S → Rby

Fx lnfx, Gx lngx, Hx lnhx 3.16

for allx∈S, then equality3.13may be transformed into

ln 1

1ψx, y ≤F

xy−Gx−Hy≤ln1ψx, y, 3.17

and thus

Fxy−Gx−Hy≤ln1ψx, y, 3.18

for allx, y∈S. For the case ofxy, the above inequality implies

Fx2 −Gx−Hx≤ln1ψx, x 3.19

and so

2Fxy−Fx2 _−Fy2

≤Fxy−Gx−HyFxy−Gy−Hx

Gx Hx−Fx2 GyHy−Fy2

≤ln1ψx, y1ψy, x·1ψx, x1ψy, y:ux, y

3.20

for allx, y∈S. Puttingxzinstead ofxandyzinstead ofyin3.20, respectively, we get

Lettingxbyx2_andybyz2_{in3.20, we have}

Fx2z2 −1

2F

x4 −1

2F

z4 ≤ 1

2u

x2, z2 , 3.22

and also

Fy2z2 − 1

2F

y4 −1

2F

z4 ≤ 1

2u

y2, z2 . 3.23

From3.21,3.22and3.23,

2Fxz2y −1

2F

x4 −1

2F

y4 −Fz4 ≤uxz, yz 1

2u

x2, z2 1

2u

y2, z2 3.24

for allx, y, z∈S. Now replacingxbyxzandybyyz, respectively, we have

2Fxz4y − 1

2F

x4y4 −1

2F

y4z4 −Fz4

≤uxz2, yz4 1

2u

x2z2, z2 1

2u

y2z2, z2

3.25

for allx, y, z∈S. Replacingxbyxzandybyyzin3.21,3.22, and3.23, respectively, one

obtains

2Fxz4y −Fx2z4 −Fy2z4 ≤uxz2, yz2 ,

Fx2z4 −1

2F

x4z4 −1

2F

z4 ≤ 1

2u

x2z2, z2 ,

Fy2z4 − 1

2F

y4z4 −1

2F

z4 ≤ 1

2u

y2z2, z2

3.26

for allx, y, z∈S. Also from3.22and3.23, we have

1₂Fx4z4 − 1

4F

x8 −1

4F

z8 ≤ 1

4u

x4, z4 ,

1₂Fy4z4 −1

4F

y8 −1

4F

z8 ≤ 1

4u

y4, z4

for allx, y, z∈S. Thus we have

2Fxz4y −1

4F

x8 −1

4F

y8 −1

2F

z8 −Fz4

≤2Fxz4y −Fx2z4 −Fy2z4 Fx2z4 −1

2F

x4z4 − 1

2F

z4

Fy2z4 −1

2F

y4z4 −1

2F

z4 1

2F

x4z4 −1

4F

x8 − 1

4F z8 1 2F

y4z4 −1

4F

y8 −1

4F

z8

≤uxz2_{, yz}2 1

2u

x2_z2_{, z}2 1

2u

y2_z2_{, z}2 1

4u

x4_{, z}4 1

4u

y4_{, z}4 _,

3.28

for allx, y, z∈S. For arbitrary positive integern, puttingz2n instead ofzin3.24andz2n−1

instead ofzin3.28, respectively, we see that

2Fxz2n1y −Fz2n2

≤uxz2n, yz2n 1

2u

x2, z2n1 1

2u

y2, z2n1 1

2

Fx4 1

2

Fy4 ,

2Fxz2n1

y −1

2F

z2n2

−Fz2n1

≤uxz2n, yz2n 1

2u

x2z2n, z2n 1

2u

y2z2n, z2n

1

4u

x4, z2n1 1

4u

y4, z2n1 1

4

Fx8 1

4

Fy8

3.29

for allx, y, z∈S. By3.29withxy,

Fz2n2

2n2 −

Fz2n1

2n1

≤ 1

2n1

1₂Fz2n2

−Fz2n1

≤ 1

2n1

Fz2n2 −2Fxz2n1x 2Fxz2n1x −1

2F

z2n2 −Fz2n1

≤ 1

2n1

2uxz2n, xz2n ux2, z2n1 ux2z2n, z2n

1

2u

x4, z2n1 Fx4 1

2

Fx8

for allx, z∈S. By3.30, for every positive integerk, mwithk≥m≥2, we have

Fz2mk

2mk −

Fz2m

2m

≤k

i1

Fz2mi

2mi −

Fz2mi−1

2mi−1

≤∞

i1

1 2mi−1

2uxz2mi−2, xz2mi−2 ux2, z2mi−1 ux2z2mi−2, z2mi−2

1

2u

x4, z2mi−1 Fx4 1

2

Fx8

≤ ∞

im−1

1 2iu

xz2i, xz2i

∞

im 1 2iu

x2, xz2i

∞

im 1 2iu

x2z2i−1, z2i

1

2

∞

im 1 2iu

x4_{, z}2i−1

Fx4 1

2

Fx8

_∞

im 1 2i −→0,

3.31

asm → ∞. This proves that{Fz2n

/2n_{}is a Cauchy sequence inR. Thus we can define a}

functionL:S → Rby

Lz lim n→ ∞

Fz2n

2n 3.32

for allz∈S. Then, by3.20and3.31, we have

Lxy−Lx−Ly≤ lim n→ ∞

Fx2n

y2n

2n −

Fx2n

2n −

Fy2n

2n

≤ lim n→ ∞

1 2n1

2Fx2ny2n −Fx2n1 −Fy2n1

lim n→ ∞

Fx2n1

2n1 −

Fx2n

2n

nlim→ ∞

Fy2n1

2n1 −

Fy2n

2n ≤ lim n→ ∞

1 2n1u

x2n, y2n 000

3.33

for allx, y∈S. Thus

for allx, y∈S. Now replacingxbyxzand thenybyyzin3.20, respectively, we obtain

2Fxyz−Fx2_z2 _−Fy2 _{≤uxz, y,}

2Fxyz−Fx2 −Fy2z2 ≤ux, yz

3.35

and so

Fx2 −Fy2 Fy2z2 −Fx2z2 ≤uxz, yux, yz 3.36

for allx, y, z∈S. By3.22,3.23, and3.36, we have

Fx2 −Fy2 −1

2F

x4 1

2F

y4 ≤uxz, yux, yz1

2u

x2, z2 1

2u

y2, z2 ,

3.37

and thus

Fx4 −2Fx2 ≤2uxz, y2ux, yzux2, z2 uy2, z2 Fy4 −2Fy2 <∞

3.38

for allx∈Sand for fixedy, z∈S. By3.3and3.30,

Lz4 −Fz4 4Lz−F

z4

4

4 limn→ ∞

Fz2n

2n −

Fz4

4

4 lim

n→ ∞ n−2

i1

Fz2i2

2i2 −

Fz2i1

2i1

<∞

3.39

for alls∈S. By3.20,3.38and3.39, for allz ∈Swithxzw, there exits a constantM

such that

|Lx−Fx||Lzw−Fzw|

Fzw−1

2F

z2 ₋1

2F

w2 1

4

Lz4 _−Fz4

1

4

Lw4 −Fw4 1

4

Fw4 −2Fw2

1

4

Fz4 −2Fz2

≤M.

Now we define a functionT :S → 0,∞by

Tx:eLx 3.41

for allx∈S. Then

TxyeLxyeLxLyTxTy 3.42

for allx, y∈S. By3.40, we have

−M≤Lx−lnfx≤M, 3.43

and thus for allx∈S

e−M≤ Tx fx ≤e

M_{. 3.44}

Ifψis bounded, there exist constantsM0, M1such that

|Gx−Hx| ≤Gx Hy0

−Fxy0F

xy0

−Gy0

−HxGy0

−Hy0

≤ln1ψx, y0

ln1ψy0, x

Gy0

−Hy0≤M0,

3.45

and so

|Lx−Gx| ≤ 1

2

Lx2 −Fx2 1

2

Fx2 −Hx−Gx1

2|Hx−Gx|

≤ 1

2

Mln1ψx, xM0

≤M1,

3.46

and by the same method above, we have

|Lx−Hx| ≤M1 3.47

for allx∈S. Therefore, we have

e−M1 ≤ Tx

gx ≤e

M1_{, e}−M1≤ Tx

hx ≤e

M1 _3.48

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