The additive approximation on a four variate Jensen type operator equation

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THE ADDITIVE APPROXIMATION ON A FOUR-VARIATE

JENSEN-TYPE OPERATOR EQUATION

JIAN WANG

Received 27 October 2002

We study the Hyers-Ulam stability theory of a four-variate Jensen-type functional equation by considering the approximate remainderφand obtain the correspond-ing error formulas. We brcorrespond-ing to light the close relation between theβ-homogeneity of the norm onF∗-spaces and the approximate remainder φ, where we allow p, q, r, andsto be different in their Hyers-Ulam-Rassias stability.

2000 Mathematics Subject Classification: 47A62, 39B72.

1. Introduction. Throughout this paper, we denote byGa linear space and byEa real or complex Hausdorff topological vector space. ByNandRwe de-note the sets of positive integers and of reals, respectively. Letfbe a mapping fromGintoE. We refer to the equations

2f

_x+y

2

−f (x)−f (y)=θ, (1.1)

4f

_x+y+z+w

4

+2f

_x+y

2

+2f

_x+w

2

+2f

_y+z

2

+2f

_z+w

2

−3f

_x+y+z

3

−3f

_y+z+w

3

−3f

_z+w+x

3

−3f

_w+x+y

3

=θ

(1.2)

as a Jensen equation and a four-variate Jensen-type functional equation, re-spectively. The approximate remainderφis defined by

4f

_x+y+z+w

4

+2f

_x+y

2

+2f

_x+w

2

+2f

_y+z

2

+2f

_z+w

2

−3f

_x+y+z

3

−3f

_y+z+w

3

−3f

_z+w+x

3

−3f

_w+x+y

3

=φ(x, y, z, w)

(1.3)

In 1940, the following problem was proposed (see Ulam [11]): let G be a group and letE be a metric group with the metryd(·,·). Givenε >0, does there exist aδ >0 such that if a function h:G→E satisfies the inequality

d(h(xy), h(x)h(y)) < δfor allx, y∈G, then there exists a homomorphism

H:G→Ewithd(h(x), H(x)) < εfor allx∈G?

In 1941, Hyers [2] answered this question in the affirmative whenGandEare Banach spaces. In 1978, Rassias [6] generalized the result of Hyers. The result was further generalized by Rassias [7], Rassias and Šemrl [9], and G˘avru¸ta [1].

The stability problems of Jensen equations can be found in [3,4,5]. The author [12] considered Hyers-Ulam-Rassias stability of several func-tional equations under the assumption thatGandE are a power-associative groupoid and a sequentially complete topological vector space, respectively. In the following, we introduce [12, Theorem 4].

Theorem1.1. The approximate remainderφ:G×G→Eof Jensen equation

(1.1) satisfies

lim n→∞

φ3n_x,3n_y

3n =θ ∀x, y∈G, ∞

k=1

φ3k−1_x,−3k−1_x−φ−3k−1_x,3k_x

3k =η(x)∈E ∀x∈G

(1.4)

if and only if the limitT (x)=limn_→∞f (3n_x)/3n_{exists for allx∈G, andT is} additive, whereGis a real linear space andEis a real Hausdorff topological vector space. In addition,

T (x)−f (x)+f (θ)=η(x) ∀x∈G. (1.5)

Trif [10] investigated the Hyers-Ulam-Rassias stability of the three-variate Jensen-type functional equation

3f

_x+y+z

3

+f (x)+f (y)+f (z)

=2f

_x+y

2

+2f

_y+z

2

+2f

_z+x

2

(1.6)

under the assumption thatGandEare a real normed linear space and a real Banach space, respectively.

In this paper, we investigate the Hyers-Ulam stability of (1.2) by considering the approximate remainders under the assumption that G and E are a real linear space and a certain kind ofF∗-space, respectively. First we solve (1.2) in

2. Solutions of (1.2). From now we letGbe a real linear space andEa real Hausdorff topological vector space, unless otherwise specified. In this section, we claim that (1.2) is equivalent to (1.1). It is well known that ifGandEare real linear spaces, then a functionf:G→Esatisfyingf (θ)=θis a solution of (1.1) if and only if it is additive.

Theorem _2.1. _{A function f :G}→_{E satisfies (1.2) for all x, y, z, w} ∈_{G if}

and only if there exist a constant elementC∈Eand a unique additive mapping

T:G→Esuch that

f (x)=T (x)+C ∀x∈G. (2.1)

Proof. _{The proof of the sufficiency is straightforward, so we will show only}

the necessity. SetC=f (θ)andT (x)=f (x)−Cfor eachx∈G. ThenT (θ)=θ

and

4T

_x+y+z+w

4

+2T

_x+y

2

+2T

_x+w

2

+2T

_y+z

2

+2T

_z+w

2

=3T

_x+y+z

3

+3T

_y+z+w

3

+3T

_z+w+x

3

+3T

_w+x+y

3

(2.2)

for anyx, y, z, w∈G. We will show thatT is additive. Letx∈G. Puty=x

andz=w= −xin (2.2) to yield

T (x)+T (−x)=3 T _x 3 +T −x 3 . (2.3)

Takey= −xandz=w=θin (2.2) to get

2 T _x 2 +T −x 2 =3 T _x 3 +T −x 3 . (2.4)

From (2.3) and the last equality, we obtain

T (x)+T (−x)=2 T _x 2 +T −x 2 . (2.5)

Puttingy=x,z= −2x, andw=θin (2.2) gives

2T (x)+T (−x) +2 T _x 2 +T −x 2

=6T

From (2.5) and the last equality, we have

T (x)+T (−x)=2T

−x₃

+T

_2x

3

. (2.7)

Puty=z=xandw= −3xin (2.2) to conclude that

T (x)+4T (−x)=9T

−x₃

. (2.8)

Replacingxby−xin the above equality, we have

T (−x)+4T (x)=9T

_x

3

. (2.9)

Adding the last two formulas together produces

5T (x)+T (−x) =9

T

_x

3

+T

−x

3

. (2.10)

Hence, from (2.3) and the last equality, we conclude that

T (x)+T (−x)=θ, that is, T (−x)= −T (x). (2.11)

It follows from (2.7), (2.9), and (2.11) that

T

_x

3

=1

3T (x), T _2x

3

=2T

_x

3

. (2.12)

Replacingx/3 byxin the last equality, we obtain

T (2x)=2T (x), that is, T

_x

2

=1₂T (x), (2.13)

and so,T (x/4)=(1/4)T (x). Substituting

T

_x

2

=1

2T (x), T _x

3

=1

3T (x), T _x

4

=1

4T (x) (2.14)

into (2.2) supplies

T (x+y+z+w)+T (x+y)+T (x+w)+T (y+z)+T (z+w)

=T (x+y+z)+T (y+z+w)+T (z+w+x)+T (w+x+y). (2.15)

3. Hyers-Ulam-Rassias stability of (1.2). Next we are interested in the Hyers-Ulam stability of (1.2). For convenience, we setϕ(x, y)=φ(x, y, x, y)

for allx, y∈G, whereφis of (1.3).

Theorem3.1. The mapϕ:G×G→Esatisfies

lim n→∞

ϕ3n_x,3n_y

3n =θ ∀x, y∈G, (3.1)

1 2

∞

k=1

ϕ3k_x,−3k_x−ϕ−3k−1_(5x),3k−1_(7x)

3k+1 =η(x)∈E ∀x∈G (3.2)

if and only if the limitT (x)=limn→∞f (3nx)/3nexists for allx∈G, andT is additive. In this case (1.5) holds.

Proof. We omit the easy proof of sufficiency and, likeTheorem 2.1, we will

show the necessity only. Let anyx, y∈G. Puttingz=xandw=yin (1.3), we get

2f

_x+y

2

−f

_2x+y

3

−f

_x+2y

3

=1₆ϕ(x, y). (3.3)

Letu, v∈G,x=2u−v, andy= −u+2v. Thenu=(2x+y)/3,v=(x+ 2y)/3, andx+y=u+v, and so we have

2f

_u+v

2

−f (u)−f (v)=Φ(u, v), (3.4)

whereΦ(u, v)def= (1/6)ϕ(2u−v,−u+2v). On the one hand, clearly,

lim n→∞

Φ3n_u,3n_v

3n =

1 6nlim→∞

ϕ3n_(2u−v),3n_(−u+2v)

3n . (3.5)

This yields from assumption (3.1) that

lim n→∞

Φ3n_u,3n_v

3n =θ. (3.6)

On the other hand, using the definition ofΦ(u, v), we compute

Φ3k−1_u,3k−1_u=1 6ϕ

3k−1_u,−3k−1_u,

Φ−3k−1_u,3k_u=1 6ϕ

−3k−1_(5u),3k−1_(7u),

then we conclude from (3.2) that

∞

k=1

Φ3k−1_u,−3k−1_u−Φ−3k−1_u,3k_u 3k

=1 2

∞

k=1

ϕ3k_u,−3k_u−ϕ−3k−1_(5u),3k−1_(7u)

3k+1 =η(u)∈E.

(3.8)

Thus, byTheorem 1.1, the limitT (u)=limn_→∞f (3n_u)/3n_{exists,Tis additive,} and the equalityT (u)−f (u)+f (θ)=η(u)holds for eachu∈G.

The proof is complete.

For abbreviation, we set

B(x,−x)=co{θ}∪ϕ3ix,−3ix∞i=1

∀x∈G,

B(−5x,7x)=co{θ}∪ϕ−3i−1_(5x),3i−1_(7x)∞ i=1

∀x∈G.

(3.9)

ByTheorem 3.1and [12, Corollary 6], we conclude the following corollary.

Corollary _3.2. _{Let E be sequentially complete and let (3.1) hold. If B}

(x,−x)andB(−5x,7x)are bounded for anyx∈G, then there exists a unique additive mappingT:G→Esuch that

T (x)−f (x)+f (θ)∈1₆Bs(x,−x)−Bs(−5x,7x) ∀x∈G, (3.10)

whereco(A)is the convex hull of a setA, andAsdenotes the sequential closure of setA. IfEis also locally convex, then the boundedness of{ϕ(3i_x,−3i_x)}∞

i=1 and {φ(−3i−1_(5x),3i−1_(7x))}∞

i=1 ensures the boundedness of B(x,−x) and B(−5x,7x), respectively.

Next we derive the Hyers-Ulam-Rassias stability of (1.2), which is an applica-tion ofTheorem 3.1. Note that it is close correlative with theβ-homogeneity of the norm onF∗-spaces. Simultaneously, we allowp, q, r, andsto be different. LetXbe a linear space. A nonnegative-valued function·defined onXis called anF-norm if it satisfies the following conditions:

(n1) x =0 if and only ifx=0; (n2) ax = xfor alla,|a| =1; (n3) x+y ≤ x+y; (n4) anx →0 providedan→0; (n5) axn →0 providedxn→0.

Corollary _3.3. _{Suppose that G is an F}∗_{-space and E a β-homogeneous}

F-space(0< β≤1). Givenε1, ε2, ε3, ε4, δ≥0and0≤p, q, r , s < β, ifφsatisfies φ(x, y, z, w)

≤δ+ε1xp+ε2yq+ε3zr+ε4ws ∀x, y, z, w∈G,

(3.11)

then there exists a unique additive mappingT:G→Esuch that T (x)−f (x)+f (θ)≤Aδ+ε1B1xp+ε2B2xq

+ε3B3xr+ε4B4xs

(3.12)

for allx∈G, where

Adef= ₆β₃2β₋₁, B1 def

=

3p₊₅p

6β₃β₋₃p, B2 def

=

3q₊₇q 6β₃β₋₃q,

B3 def

=

3r₊₅r

6β₃β₋₃r, B4 def

=

3s₊₇s 6β₃β₋₃s.

(3.13)

Proof. _{Let any x, y} ∈_{G. Firstly, put z}=_{x and w} =_{y in (3.11) to get}

according to the definition ofϕthat

ϕ(x, y)=φ(x, y, x, y)≤δ+ε1xp+ε2yq +ε3xr+ε4ys ∀x, y∈G.

(3.14)

It follows fromp, q, r , s < βthat

lim n→∞

ϕ3n₃x,n3ny ≤lim

n→∞ _δ

3n+

ε1 3n(β−p)x

p₊ ε2 3n(β−q)y

q

+ ε3 3n(β−r )x

r₊ ε4 3n(β−s)y

s

=0.

(3.15)

Secondly, in light of the triangle inequality ofF-norm andp, q, r , s≥0, we have, for anyi∈N,

_ϕ

3ix,−3ix≤δ+ε13ipxp+ε23iqxq+ε33irxr+ε43isxs, ϕ−3i−1(5x),3i−1(7x)≤δ+ε13(i−1)p5pxp+ε23(i−1)q7qxq

+ε33(i−1)p5rxr+ε43(i−1)q7sxs.

(3.16)

As in the proof of [12, Theorem 3], we infer from (3.4) that

1 3nf

3n_{x−f (x)=} n

k=1

Ψ3k−1_x

3k (3.17)

Consequently, for anyn∈N,

1 3nf

3nx−f (x)+2

n k=1 f (θ) 3k = n k=1

Φ3k−1_x,−3k−1_x−Φ−3k−1_x,3k_x 3k =1 2 n k=1

ϕ3k_x,−3k_x−ϕ−3k−1_(5x),3k−1_(7x)

3k+1 .

(3.18)

It is easy to see that

1 2

∞

k=1

ϕ3k_x,−3k_x−ϕ−3k−1_(5x),3k−1_(7x)

3k+1 (3.19)

exists for everyx∈G. Indeed, from the above, we conclude that

f3m_x 3m −

f3n_x 3n

=₃1n _f

3m−n₃n_x 3m−n −f

3n_x

= 1 3n

m−n

k=1

Ψ3n+k−1_x

3k =

m

k=n+1

Ψ3k−1_x 3k

=1₂ m

k=n+1

ϕ3k_x,−3k_x−ϕ−3k−1_(5x),3k−1_(7x)

3k+1 −2

m

k=n+1 f (θ)

3k

(3.20)

for anym > n, wherem, n∈N, and so

f3m_x 3m −

f3n_x 3n

≤₂1_β m

k=n+1 2δ+ε1

3kp₊₃(k−1)p₅p_xp_+ε 2

3kq₊₃(k−1)q₇q_xq 3(k+1)β

+ 1 2β

m

k=n+1

ε33kr+3(k−1)r5rxr+ε43ks+3(k−1)s7sxs 3(k+1)β

+2f (θ)

m

k=n+1 1 3k

≤ m

k=n+1 21−β_δ 3(k+1)β+

ε1 2β

m

k=n+1 ₁

3β3

k(p−β)₊ 5p 32β3

(k−1)(p−β)_xp

+ε2 2β

m

k=n+1 ₁

3β3

k(q−β)₊ 7q 32β3

(k−1)(q−β)

+ε3 2β

m

k=n+1 ₁

3β3

k(r−β)₊ 5r 32β3

(k−1)(r−β)_xr

+ε4 2β

m

k=n+1 ₁

3β3

k(s−β)₊ 7s 32β3

(k−1)(s−β)

xs_{+2f (θ)} m

k=n+1 1 3k

(3.21)

for anym > n, wherem, n∈N. Sincep, q, r , s < β,{f (3n_x)/3n_{}is a Cauchy} sequence ofE. By the completeness ofE,{f (3n_x)/3n_{}converges to an element} ofE.

Thus, byTheorem 3.1,T (x)=limn_→∞(f (3n_x)/3n_{)and it is additive. In} ad-dition, from (3.18), inequality (3.12) holds for allx∈G.

In order to prove the uniqueness ofT, suppose thatU:G→E is another additive mapping which satisfies

U (x)−f (x)+f (θ)≤Aδ+ε1B1xp+ε2B2xq +ε3B3xr+ε4B4xs

(3.22)

for allx∈G. On account of the last two inequalities, we conclude that, for all

x∈G,

U (x)−T (x)

=_n1βU (nx)−T (nx)

= 1

nβU (nx)−f (nx)+f (θ)−T (nx)+f (nx)−f (θ)

≤ 1

nβ

U (nx)−f (nx)+f (θ)+T (nx)−f (nx)+f (θ)

≤_n2β

Aδ+ε1B1nxp+ε2B2nxq+ε3B3nxr+ε4B4nxs

=2 _Aδ

nβ+

ε1B1 nβ−px

p₊ε2B2 nβ−qx

q₊ε3B3 nβ−rx

r₊ε4B4 nβ−rx

s

,

(3.23)

and so, U (x)−T (x) →0 asn→ ∞ sincep, q, r , s < β. As a consequence,

U (x)=T (x)for allx∈G. Therefore, the result holds.

In order to show thatCorollary 3.3is valid in the case thatp, q, r , s >1/β, we need the following theorem, which can be proved in the same manner as

Theorem_3.4. _{The approximate remainderφ:G}×_G→_{Eof (1.1) satisfies}

lim n→∞3

n_φ3−n_x,3−n_{y=θ ∀x, y∈G,}

∞

k=1

3k−1_φ3−k_x,−3−k_x−φ−3−k_x,3−k+1_{x =η(x)∈E ∀x∈G} (3.24)

if and only if the limitT (x)=limn→∞3n[f (3−nx)−f (θ)]exists for allx∈G, andT is additive. In this case (1.5) holds.

Proof. Note that if setg(x)=f (x)−f (θ)for anyx∈G, theng(θ)=θ

and the approximate remaindersφg andφf of (1.1) with respect togandf, respectively, are equal. We still write it asφ. As in the proof ofTheorem 1.1, we can conclude that, for everyxinGwithx≠0 and everyninN,

g(x)−3n₃−n_x= n

k=1

φ3−k_x,−3−k_x−φ−3−k_x,3−k+1_{x . (3.25)}

We may see that it is possible thatT (x)=limn→∞3n[f (3−nx)−f (θ)]exists, in particular, iffis differentiable atθinG.

Corollary3.5. Suppose thatGis aβ-homogeneousF∗-space(0< β≤1)

andE anF-space with a nondecreasingF-norm. Givenε1, ε2, ε3, ε4∈[0,+∞) andp, q, r , s∈(1/β,+∞), ifφsatisfies

φ(x, y, z, w)≤ε1xp+ε2yq

+ε3zr+ε4ws ∀x, y, z, w∈G,

(3.26)

then there exists a unique additive mappingT:G→Esuch that

T (x)−f (x)+f (θ)≤ε1B1xp+ε2B2xq

+ε3B3xr+ε4B4xs ∀x∈G,

(3.27)

where

B1 def

=

3pβ₊₅pβ

3pβ₋₃ , B2 def

=

3qβ₊₇qβ

3qβ₋₃ ,

B3def=

3r β₊₅r β

3r β₋₃ , B4 def

=

3sβ₊₇sβ

3sβ₋₃ .

Proof. _Letg(x)=_{f (x)}−_{f (θ)for anyx}∈_{G. UsingTheorem 3.4, as in the}

proofs ofTheorem 3.1andCorollary 3.3, we can achieve that

1 6

∞

k=1

3k−1_ϕ3−k+1_x,−3−k+1_x−ϕ−3−k_(5x),3−k_{(7x) (3.29)}

exists for everyx∈Gand

g(x)−3n_g3−n_x

=1 6

n

k=1

3k−1ϕ3−k+1x,−3−k+1x−ϕ−3−k(5x),3−k(7x) . (3.30)

Finally, we can evaluate the error formula.

We may also deal with the Hyers-Ulam stability of (1.2) as usual.

Theorem_3.6. _{The approximate remainderφsatisfies}

lim n→∞

φ3n_x,3n_y,3n_z,3n_w

3n =θ ∀x, y, z, w∈G, (3.31)

∞

k=1

ψ3k_x

3k =η(x)∈E ∀x∈G (3.32)

if and only if the limitT (x)=limn→∞f (3nx)/3nexists for allx∈G, andT is additive. Moreover, (1.5) holds, where

ψ(x)def= 1

4φ(x, x,−x,−x)+ 1 6

φ(−x,−x,−x,3x)−φ(x, x, x,−3x) .

(3.33)

Proof. It is enough to show the necessity. Definegas above.

Let anyx∈G. Puty=xandz=w= −xin (1.3) to yield

g(x)+g(−x)−3

g

_x

3

+g

−x

3

=1

2φ(x, x,−x,−x). (3.34)

Puty=z=xandw= −3xin (1.3) to give

g(x)+4g(−x)−9g

−x

3

Replacingxby−xin the above equality, we have

g(−x)+4g(x)−9g

_x

3

=φ(−x,−x,−x,3x). (3.36)

Adding the last two formulas together, we conclude that

5g(x)+g(−x) −9

g

_x

3

+g

−x₃

=φ(x, x, x,−3x)+φ(−x,−x,−x,3x).

(3.37)

Hence, from (3.34) and the above equality, we know that

g(x)+g(−x)=1 2

φ(x, x, x,−3x)+φ(−x,−x,−x,3x)

−3

4φ(x, x,−x,−x).

(3.38)

It follows from (3.36) and (3.38) that

g(x)−3g

_x

3

=1

4φ(x, x,−x,−x)

+1 6

φ(−x,−x,−x,3x)−φ(x, x, x,−3x)

=ψ(x).

(3.39)

With 3xin place ofxin the above equality and dividing by 3, we obtain

1

3g(3x)−g(x)= 1

3ψ(3x). (3.40)

We will prove by induction that

1 3ng

3nx−g(x)= n

k=1

ψ3k_x

3k ∀n∈N. (3.41)

Forn=1 this is trivial according to (3.40). Suppose that (3.41) holds for a certainm−1. Then (3.40) and the induction hypothesis imply that

1 3mg

3mx−g(x)=1_{3 1}

3m−1g

3m−1(3x)−g(3x)

+1₃g(3x)−g(x)

=1₃ m−1

k=1

ψ3k_(3x)

3k +

1

3ψ(3x)= m

k=1

ψ3k_x 3k ,

(3.42)

We defineT (x)=limn→∞g(3nx)/3n. Obviously,T (x)=limn→∞f (3nx)/3n, and so, by (3.32) and (3.41),T (x)exists and

T (x)−g(x)=η(x). (3.43)

Substituting the definition ofginto the last equality implies that

T (x)−f (x)+f (θ)=η(x). (3.44)

Finally, we verify thatT is additive. Indeed, the definition ofT implies that

T (θ)=lim n→∞

g3n_θ

3n =θ. (3.45)

Because of (3.31),Tis a solution of (1.2). HenceT (x)=T∗(x)+T (θ)=T∗(x)

byTheorem 2.1, whereT∗is additive. It follows thatT is additive.

To show the following corollary, we may use a manner analogous to that used inCorollary 3.3.

Corollary_3.7. _{Keeping all the hypotheses ofCorollary 3.3, there exists a}

unique additive mappingT:G→Esuch that (3.12) holds, where

Adef= ₁₂3β_β+2β+1

3β₋₁, B1 def

= 3p

3β₊₂β+1 12β₃β₋₃p, B2

def = 3q

3β₊₂β+1 12β₃β₋₃q,

B3 def

= 3r

3β₊₂β+1 12β₃β₋₃r, B4

def = 3s

3β₊₂β+1₃s 12β₃β₋₃s .

(3.46)

If there exists at least one ofp,q,r, ands such that it is strictly less than0, it is supposed that (3.11) holds for allx, y, z, w∈G\{θ}. Then the domain ofT is

G\{θ}instead ofG.

As earlier, we consider the case ofp, q, r , s >1/β.

Theorem_3.8. _{The approximate remainderφsatisfies}

lim n→∞3

n_φ3−n_x,3−n_y,3−n_z,3−n_{w=θ ∀x, y, z, w∈G,}

∞

k=1

3k−1_ψ3−(k−1)_{x=η(x)∈E ∀x∈G}

if and only if the limitT (x)=limn→∞3n[f (3−nx)−f (θ)]exists for allx∈G, andT is additive, whereψis as above. Moreover,

T (x)−f (x)+f (θ)=η(x) ∀x∈G. (3.48)

Proof. Letg(x)=f (x)−f (θ). Note that, by virtue of (3.39), we conclude

by induction that

g(x)−3n_g3−n_x= n

k=1

3k−1_ψ3−(k−1)_{x ∀x∈G, n∈N. (3.49)}

Corollary3.9. Keeping all the hypotheses ofCorollary 3.5, then there

ex-ists a unique additive mappingT:G→Esuch that

T (x)−f (x)≤ε1B1xp+ε2B2xq

+ε3B3xr+ε4B4xs ∀x∈G,

(3.50)

where

B1 def

= 3pβ+1

3pβ₋₃, B2 def

= 3qβ+1 3qβ₋₃,

B3 def

= 3r β+1

3r β₋₃, B4 def

= 3sβ

1+23sβ

3sβ₋₃ .

(3.51)

We still mention the following immediate consequence ofCorollary 3.3.

Remark_3.10. _{LetEbe aβ-homogeneousF-space(0< β}≤_{1). Ifφsatisfies}

the property that there existsδ∈[0+ ∞)such thatφ(x, y, z, w) ≤δfor anyx, y, z, w∈G, then there exists a unique additive mappingT:G→Esuch that

T (x)−f (x)+f (θ)≤ 2δ

6β₃β₋₁ ∀x∈G. (3.52)

Theorem_3.11. _{The functionf:}R→R_{defined byf (x)}def= _xlog2(1+ |_x|₎

satisfies the inequality

φ(x, y, z, w)≤14|x|+|y|+|z|+|w| ∀x, y, z, w∈R, (3.53)

but

supf (x)−T (x)

x

:x∈R\{0}

= ∞ (3.54)

for each additive mappingT:R→R.

Proof. _{For allx, y, z, w} ∈R_{, it follows from} |_{f (x}+_y)−_{f (x)}−_{f (y)}| ≤

|x|+|y|in [8] and|f (x+y+z)−f (x)−f (y)−f (z)| ≤(5/3)(|x|+|y|+|z|)

in [10] that

φ(x, y, z, w)=

4f

_x+y+z+w

4

−f (x+y+z+w)

+f (x+y+z+w)−f (x+z)−f (y+w)

+

f (x+z)−2f

_x+z

2

+

f (y+w)−2f

_y+w

2

−3f

_x+y+z

3

−f (x+y+z)

−

f (x+y+z)−f

_x+y

2

−f

_y+z

2

−f

_z+x

2

−

3f

_y+z+w

3

−f (y+z+w)

−

f (y+z+w)−f

_y+z

2

−f

_z+w

2

−f

_w+y

2

−

3f

_z+w+x

3

−f (z+w+x)

−f (z+w+x)−f

_z+w

2

−f

_w+x

2

−f

_x+z

2

−

3f

_w+x+y

3

−f (w+x+y)

−f (w+x+y)−f

_w+x

2

−f

_x+y

2

−f

_y+w

2

.

Furthermore, we evaluate that

φ(x, y, z, w)≤8x+y+z+w 4

+|x+z|+|y+w|+2x+z 2

+2y+w 2

+5

33

x+y+z

3

+5₃x+₂y+y+₂zz+₂x

+15₃ y+z₃+w+5₃y₂+z+z+₂ww+₂y

+15 3

z+w₃+x+5 3

z+₂w+w+x 2

x+₂z

+15₃ w+₃x+y+5₃w₂+x+x+₂yy+₂w

≤14|x|+|y|+|z|+|w|

(3.56)

for allx, y, z, w∈R. The rest of the proof has been proved in [10].

Remark3.12. Letfbe as inTheorem 3.11.

(i) IfG=(R,·1)with the Euclidean metric·1= |·|, andE=(R,·2) with theβ-homogeneous norm·2= |·|β, then

φ(x, y, z, w)₂

≤14βxβ1+y β 1+z

β 1+w

β 1

∀x, y, z, w∈G, (3.57)

but

sup f (x)−T (x)2 xβ1

:x∈R\{0}

= ∞ (3.58)

for each additive mappingT:G→E.

(ii) IfG=(R, · 1)with the β-homogeneous norm · 1= | · |β, andE= (R,·2)with the Euclidean metric·2= |·|, then

φ(x, y, z, w)2 ≤14x1/β

1 +y 1/β 1 +z

1/β 1 +w

1/β 1

∀x, y, z, w∈G, (3.59)

but

sup f (x)−T (x)2 x1/β

1

:x∈R\{0}

= ∞ (3.60)

for each additive mappingT:G→E.

but

supf (x)−T (x)

x

:x∈R\{0}

= ∞ (3.62)

for each additive mappingT:G→E.

Acknowledgments. I would like to express my deep gratitude to

Pro-fessor Themistocles M. Rassias for drawing my attention to these kinds of problems and the referee for his many valuable suggestions. This work was supported by the National Science Foundation of China, Grant 10171014, and the Foundation of Fujian Educational Committee, Grant JA02166. This work belongs to the Doctoral Programme Foundation of Institution of Higher Educa-tion, Grant 20010055013, and the Programme of National Science Foundation of China, Grant 10271060.

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Jian Wang: Department of Mathematics, Nanjing University, Nanjing 210093, China Current address: Department of Mathematics, Fujian Normal University, Fuzhou 350007, China