Hyperstability and Stability of a Logarithm type Functional Equation

(1)

Hyperstability and Stability of a Logarithm-type

Functional Equation

Young Whan Lee

1

_,

_{Gwang Hui Kim}

2,∗

1_Department_of_Computer_Hacking_and_Information_Security,_College_of_Engineering_Daejeon_University,_Daejeon,_34520,_Korea 2_Department_of_Mathematics,_Kangnam_University,_Yongin,_Gyeonggi,_16979,_Korea

Abstract

In 2001, Maksa and P´ales [12] introduced a new type’s stability: hyperstability for a class of linear functional equationf(x) +f(y) = _n1Pn

i=1f(xϕi(y)). Riedel and Sahoo [14] have generalized a functional equation associated with

the distance between the probability distributions, which isf(pr, qs) +f(ps, qr) = 2M(rs)f(p, q) + 2M(pq)f(r, s). Elfen etc. [7] obtained the solution of the functional equationf(pr, qs) +f(ps, qr) = 2f(p, q) + 2f(r, s)on semigroupG. The aim of this paper is to investigate the hyperstability and the Hyers-Ulam stability for the above Logarithm-type functional equation considered by Elfen, etc. Namely, iff is an approximative equation related to the above equation, then it is a solution of this equation which exists withinε−bound of a given approximative functionf.

Keywords

Information Measure, Distance Measure, Superstability, Multiplicative Function, Stability of Functional Equation

AMS Subject Classification:39B82, 39B52

1 Introduction

The following stability problem is well-known as Ulam’s stability problem [16]:

LetG1be a group and letG2be a metric group with a metricd(·,·). Given >0, does there exist a δ >0

such that if a mappingh:G1→G2satisfies the inequalityd(h(xy), h(x)h(y))< δfor allx,y∈G1, then there

exists a homomorphismH :G1→G2withd(h(x), H(x))< for allx∈G1?

In next year, Hyers [11] proved a first partial answer to Ulam’s problem for an additive mapping on a Banach space. D. G. Bourgin obtained many excellent results for the stability ([3], [4]).Hyers’ theorem was generalized by Aoki [1] for the case bounded by variables, and their results are improved by Rassias [13] to the case of the linear mapping and by Ger [9] . G˘avruta [8] proved a further generalization of the Rassias’ theorem by using a general control function.

The superstability phenomenon of the exponential equationf(x+y) =f(x)f(y)was discovered by Baker, Lawrence, and Zorzitto [2] in 1979. The superstability for asymptotic phenomenon of the exponential equation was discovered by Ger [9].

In 2001, Maksa and P´ales [12] proved a new type’s stability for a class of linear functional equation f(x) +f(y) = 1

n

X

i=1

f(xϕi(y)), (1)

wheref is a real-valued mapping defined on a semigroupS, and the mappingsϕ1, ϕ2,· · ·, ϕn : S → Sare pairwise distinct

automorphisms. That is as following:

Letε:S×S→Rbe a function such that there exists a sequenceukthat satisfies

lim

k→∞ε(uks, t) = 0 (s, t∈S).

Assume thatf :S →Xsatisfies the stability inequality

f(s) +f(t)−

1 n

n

X

i=1

f(sϕi(t))

(2)

whereX is a real normed space. Then,fis a solution of (1).

Such a phenomenon is called the hyperstability of the functional equation. Gselmann [10], Brazde¸k and Ciepli´nski [5] investigated the hyperstability of functional equations. A similar concept was introduced by Sirouni and Kabbaj [15].

Riedel and Sahoo [14] solved a functional equation associated with the distance between the probability distributions. Let M : (0.1)→_Cbe a given multiplicative function. Then, iff : (0.1)2_→

Csatisfies the functional equation f(pr, qs) +f(ps, qr) = 2M(rs)f(p, q) + 2M(pq)f(r, s)

if and only if

f(p, q) =M(p)M(q)hL(p) +L(q) +l(p q,

p q)

i

,

whereM : (0.1)→_Cis an arbitrary logarithmic function andl: (0.1)2_→

Cis a bilogarithmic function. Thus, we will call it a logarithm-type functional equation

In addition, Elfen, Riedel and Sahoo [7] solved a functional equation

f(pr, qs) +f(sp, rq) = 2f(p, q) + 2f(r, s)

on semigroupG. Its solution type offon ¯G is given by

f(p, q) =A(p) +A(q) +ψ(pq−1, pq−1), whereA:G→_Cis a homomorphism andψ:G→_Cis a symmetric bi-homomorphism.

Now we consider the logarithm-type functional equation given by 1

2[f(pr, qs) +f(ps, qr)] =f(p, q) +f(r, s). (2) For example, iff(x, y) = lnxy, thenf is a solution of the equation (2). In this paper, we investigate the hyperstability and stability of the functional equation (2). Namely, we prove that iff satisfies a stability inequality for the equation (2), then it is also a solution of this equation and also we can find an another solution of it which has anε−error bound forf.

2 Hyperstability of the logarithm-type functional equation

In this section, we investigate the hyperstability of the equation (2). Throughout this section, let(G,·)denote a noncommu-tative semigroup,Xa real normed space, andRthe set of real numbers. And letR+denote the set of positive real numbers.

Theorem 1. Letε:G2×G2−→Rbe a function such that there exists a sequenceuk ∈Gthat satisfies

lim

k→∞ε(uk(p, q),(r, s)) = 0

for allp, q, r, s∈G. Assume thatf :G×G−→Xsatisfies the stability inequality

1 2

h

f(pr, qs) +f(ps, qr)i−f(p, q)−f(r, s)

≤ε((p, q),(r, s)) (3) for allp, q, r, s∈G.Then,

1 2

h

f(pr, qs) +f(ps, qr)i=f(p, q) +f(r, s).

Proof. Define a functionF :G2_×_G2_−→_X_by

F((p, q),(r, s)) =f(p, q) +f(r, s)−1

2

h

f(pr, qs) +f(ps, qr)i.

Then, for allp, q, r, s, v, w∈G, we have F((p, q),(r, s)) +1

2

h

F((pr, qs),(v, w)) +F((ps, qr),(v, w))i

=f(p, q) +f(r, s) +f(v, w)

−1

2

h

(3)

And also, for allp, q, r, s, v, w∈G, we have F((r, s),(v, w)) +1

2

h

F((p, q),(rv, sw)) +F((p, q),(rw, sv))i

=f(p, q) +f(r, s) +f(v, w)

−1

2

h

f(prv, qsw) +f(psw, qrv) +f(prw, qsv) +f(psv, qrw)i.

Thus,F satisfies the following functional equation F((p, q),(r, s)) +1

2

h

F((pr, qs),(v, w)) +F((ps, qr),(v, w))i (4) =F((r, s),(v, w)) +1

2

h

F((p, q),(rv, sw)) +F((p, q),(rw, sv))i.

By (3), we get

||F((p, q),(r, s))|| ≤ε((p, q),(r, s)),

and with the assumed sequence{uk}, we obtain

lim

k→∞F(uk(p, q),(r, s))≤klim→∞ε(uk(p, q),(r, s)) (5)

for allp, q, r, s∈G. The equation (4) implies

F((r, s),(v, w)) (6) =F((p, q),(r, s)) +1

2

h

F((pr, qs),(v, w)) +F((ps, qr),(v, w))i

−1

2

h

F((p, q),(rv, sw)) +F((p, q),(rw, sv))i.

Letr, s, v, w, p0, q0be fixed. Applying the norm and substitutingp=ukp0, q =ukq0in (6) , and ask→ ∞, respectively,

we obtain

|| lim

k→∞F(r, s),(v, w))||

=|| lim

k→∞

h

F(uk(p0, q0),(r, s)) +

1 2

h

F(uk(p0r, q0s),(v, w)) +F(uk(p0s, q0r),(v, w))

i

−1

2

h

F(uk(p0, q0),(rv, sw)) +F(uk(p0, q0),(rw, sv))

ii

||.

By applying of (5) and the triangle inequalities, we obtain

||F(r, s),(v, w))||

≤

lim

k→∞

h

ε(uk(p0, q0),(r, s)) +

1 2

h

ε(uk(p0r, q0s),(v, w)) +ε(uk(p0s, q0r),(v, w))

ii

+ lim

k→∞

1 2

h

ε(uk(p0, q0),(rv, sw)) +ε(uk(p0, q0),(rw, sv))

i

.

Hence, we obtain from the assumed sequence{uk}the required result

F((r, s),(v, w)) = 0

for anyr, s, v, w∈G.

Corollary 2. Assume thatf :R+×R+−→Xsatisfies the stability inequality

1 2

h

≤

rs

pq or pqrs (7)

for allp, q, r, s∈R+. Then,

1 2

h

f(pr, qs) +f(ps, qr)i=f(p, q) +f(r, s).

Proof. Letε((p, q),(r, s)) = rs_pqanduk =ak fora >1, orε((p, q),(r, s)) =pqrsanduk =ak for0< a <1. Then, we

obtain

lim

k→∞ε(uk(p, q),(r, s)) = 0,

(4)

3 Stability of the logarithm-type functional equation

In this section, we investigate the stability of the equation (2). Throughout this section, let(G,·) denote a commutative semigroup,N the set of natural numbers, andXa Banach space.

Theorem 3. Letε >0.Assume thatf :G×G−→Xsatisfies the stability inequality

1 2

h

≤ε (8)

for allp, q, r, s∈G.Then there exists a functionF :G×G−→X such that

1 2

h

F(pr, qs) +F(ps, qr)i=F(p, q) +F(r, s)

and||F(p, q)−f(p, q)|| ≤ 39ε

40 for anyp, q∈G, whereFis defined by

F(p, q) := lim

n→∞

h 1

22nf(p 2n_{, q}2n₎

+

n−2

X

i=0

1

2i_·₂2n−i +

1 2n+1

f((pq)2n−1,(pq)2n−1)i

for anyp, q, r, s∈G.

Proof. Lettingr=p, s=qin (8) and dividing it by2, we have

1 22

h

f(p2, q2) +f(pq, pq)i−f(p, q)

≤

ε

2. (9)

And also, lettingp=q=r=sin (8) and dividing it by2, we have

1 2f(p

2_{, p}2₎₋_f_{(p, p)}

≤

ε

2. (10)

Let us show that the following inequality holds for everyn∈N:

1 2f(p

2n_{, p}2n₎₋_f_(p2n−1_{, p}2n−1₎

≤

ε

2. (11)

Replacingpbyp2_and_q_by_q2_{in (9) respectively, and dividing}₂2_{, we have}

1

222

h

f(p22, q22) +f((pq)2,(pq)2)i−f(p

2_{, q}2₎

22

≤

ε

2·22. (12)

Thus by (9),(10), and (12), we have

1

222f(p

22_{, q}22_{) + (} 1

222+

1 22

1 2)f((pq)

2_,_(pq)2₎₋_f_{(p, q)}

≤

1

222

h

f(p22, q22) +f((pq)2,(pq)2)i−f(p

2_{, q}2₎

22

+

1 22

h

(13)

+ 1 22

f((pq)2,(pq)2)

2 −f(pq, pq)

≤ ε

2·22 +

ε 2 +

ε 2·22 =

ε 2+

ε 22.

In addition, by lettingpbyp2n−1

andqbyq2n−1

in (9), and dividing22n−1

, the following inequality holds for everyn∈N:

1 22n

h

f(p2n, q2n) +f((pq)2n−1,(pq)2n−1)i−f(p

2n−1_{, q}2n−1₎

22n−1

≤ ε

(5)

By (10),(12), and (14), we have 1

223f(p

23_{, q}23_{) +} 1

223 + (

1

222+

1 23)

1 2

f((pq)22,(pq)22)−f(p, q) ≤ 1 223 h

f(p23, q23) +f((pq)22,(pq)22)i−f(p

22_{, q}22₎

222 + 1

222f(p

22_{, q}22_{) + (} 1

222 +

1

23)f((pq)

2_,_(pq)2₎₋_f_{(p, q)}

+ ( 1 222+

1 23)

f((pq)22_,_(pq)22₎

2 −f((pq)

2_,_(pq)2₎

(15)

≤ ε

2·222+

ε 2 +

ε 22 +

ε

222+

ε 23 1 2 =ε 2 + ε 22 +

ε

222 +

ε 24

=ε 2 +

ε 22 +

ε 23.

Note that

n

X

j=4

j−2

X

i=2

ε

2i_·₂2j−i +

n

X

i=1

ε

22i−1 +

n

X

i=3

ε 2i+1

≤ ε

22

1

222+

1

223 +

1

224 +· · ·

+ ε 23

1

222+

1

223 +

1

224 +· · ·

+· · ·+ ε 2n−1

1

222 +

1

223 +

1

224+· · ·

(16) +ε

2+ ε 22+ε

1

222 +

1

223 +

1

224+· · ·

+ ε 24

1 + 1 2+

1 22 +

1 23+· · ·

=ε 1 222+

1

223+

1

224 +· · ·

1 + 1 22+

1 23+

1 24+· · ·

+ε 2+

ε 22+

ε 24

1 +1 2 +

1 22 +

1 23 +· · ·

=ε· 1

15· 3 2+ ε 2+ ε 22 +

ε 23

=39ε 40.

Suppose that the following inequality holds forn≥4and for anyp, q∈G:

1 22nf(p

2n_{, q}2n_{) +}

n−2

X

i=0

1

2i_·₂2n−i +

1 2n+1

f((pq)2n−1,(pq)2n−1)−f(p, q)

≤ n X j=4

j−2

X

i=2

ε

2i_·₂2j−i +

n

X

i=1

ε

22i−1 +

n

X

i=3

ε

2i+1. (17)

Note that

n−1

X

i=0

1

2i_·₂2n+1−i =

1

22n+1 +

1 2

n−2

X

i=0

1

(6)

for alln∈N. Then, for anyp, q∈G, based on (14) and (18), we obtain 1

22n+1f(p 2n+1

, q2n+1)

+

n−1

X

i=0

1

2i_·₂2n+1−i +

1 2n+2

f((pq)2n,(pq)2n)−f(p, q)

≤ 1

22n+1

h

f(p2n+1, q2n+1) +f((pq)2n,(pq)2n)i−f(p

2n_{, q}2n₎

22n

+ 1 22nf(p

2n_{, q}2n₎ ₍₁₉₎

+

n−2

X

i=0

1

2i_·₂2n−i +

1 2n+1

f((pq)2n−1,(pq)2n−1)−f(p, q)

+

n−2

X

i=0

1

2i_·₂2n−i +

1 2n+1

f((pq)2n_,_(pq)2n₎

2 −f((pq)

2n−1_,_(pq)2n−1₎

= ε

2·22n +

n

X

j=4

j−2

X

i=2

ε

2i_·₂2j−i +

n

X

i=1

ε

22i−1

+

n

X

i=3

ε 2i+1 +

nX−2

i=0

1

2i_·₂2n−i +

1 2n+1

ε 2 = n X j=4

j−2

X

i=2

ε

2i_·₂2j−i +

n−1

X

i=2

ε

2i_·₂2n+1−i

+ ε 22n +

n

X

i=1

ε

22i−1

+ n X i=3 ε 2i+1 +

ε 2n+2

=

n+1

X

j=4

j−2

X

i=2

ε

2i_·₂2j−i +

n+1

X

i=1

ε

22i−1 +

n+1

X

i=3

ε 2i+1.

Thus, by induction, inequality (17) holds for alln≥4and for anyp, q∈G. Now forn≥4, we have

1 22nf(p

2n_{, q}2n_{) +}

n−2

X

i=0

1

2i_·₂2n−i +

1 2n+1

f((pq)2n−1,(pq)2n−1)

− 1

22n−1f(p 2n−1

, q2n−1)

−

n−3

X

i=0

1

2i_·₂2n−1−i +

1 2n

f((pq)2n−2,(pq)2n−2)

(20) ≤ 1 22n

h

f(p2n, q2n) +f((pq)2n−1,(pq)2n−1)i− 1

22n−1f(p

2n−1_{, q}2n−1₎

+

n−3

X

i=0

1

2i_·₂2n−1−i

f((pq)2n−1_,_(pq)2n−1₎

2 −f((pq)

2n−2

,(pq)2n−2) + 1

2n+1f((pq)

2n−1_,_(pq)2n−1₎₋ 1

2nf((pq)

2n−2_,_(pq)2n−2₎

≤ ε

2·22n−1 +

n−3

X

i=0

1

2i_·₂2n−1−i ·

ε 2 +

ε 2·2n

≤ ε

2·22n−1 +

1 22n−1

n−3

X

i=0

1 2i+1 ·

ε 2 +

ε

2·2n −→ 0

asn→ ∞. Thus, if we let

Yn =

1 22nf(p

2n_{, q}2n_{) +}

n−2

X

i=0

1

2i_·₂2n−i +

1 2n+1

(7)

then{Yn}is a Cauchy sequence due to (20), and so we can define a functionF :G×G−→Xby

F(p, q) := lim

n→∞

h 1

22nf(p 2n_{, q}2n₎

+

n−2

X

i=0

1

2i_·₂2n−i +

1 2n+1

f((pq)2n−1,(pq)2n−1)i. (21) Then, due to (16), (17), and (21), we have

||F(p, q)−f(p, q)|| ≤ 39ε

40 ∀p, q∈G. Finally, the functionFdefined in (21) holds the required equation (2) as follows:

1 2

h

F(pr, qs) +F(ps, qr)i−F(p, q)−F(r, s)

≤ lim

n→∞

1 2

1

22nf((pr)

2n_,_(qs)2n₎

+1 2

nX−2

i=0

1

2i_·₂2n−i +

1 2n+1

f((pqrs)2n−1,(pqrs)2n−1)

+1 2

1 22nf((ps)

2n_,_(qr)2n₎

+1 2

nX−2

i=0

1

2i_·₂2n−i +

1 2n+1

f((pqrs)2n−1,(pqrs)2n−1)

− 1

22nf((p)

2n_,_(q)2n₎

−

n−2

X

i=0

1

2i_·₂2n−i +

1 2n+1

f((pq)2n−1,(pq)2n−1)

− 1

22nf((r)

2n_,_(s)2n₎

−

n−2

X

i=0

1

2i_·₂2n−i +

1 2n+1

f((rs)2n−1,(rs)2n−1)

≤ lim

n→∞

1 22n

1 2

h

f((pr)2n,(qs)2n) +f((ps)2n,(qr)2n)i

−f(p2n, q2n)−f(r2n, s2n)

+ lim

n→∞

nX−2

i=0

1

2i_·₂2n−i +

1 2n+1

×

1 2

h

f((pqrs)2n−1,(pqrs)2n−1) +f((pqrs)2n−1,(pqrs)2n−1)i

−f((pq)2n−1,(pq)2n−1)−f((rs)2n−1,(rs)2n−1)

≤ lim

n→∞

ε

22n + lim_n_→∞

nX−2

i=0

1

2i_·₂2n−i +

1 2n+1

ε

= 0.

Corollary 4. Letε >0.Assume thatf :G×G−→Xsatisfies the stability inequality

1 22

h

≤ε (22)

for allp, q∈G.Then there exists a functionF :G×G−→Xsuch that

1 22

h

F(p2, q2) +F(pq, pq)i=F(p, q)

and||F(p, q)−f(p, q)|| ≤ 39ε

(8)

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