A Fixed Point Approach to Stability of Functional Equations in Modular Spaces

MALAYSIANMATHEMATICAL

SCIENCESSOCIETY http://math.usm.my/bulletin

A Fixed Point Approach to Stability of Functional Equations in Modular Spaces

GHADIRSADEGHI

Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, P.O. Box 397, Iran

[email protected], [email protected]

Abstract. In this paper, we present a fixed point method to prove generalized Hyers–Ulam stability of the generalized Jensen functional equationf(rx+sy) =rg(x) +sh(x)in modular spaces.

2010 Mathematics Subject Classification: Primary 39B52; Secondary 39B72, 47H09 Keywords and phrases: Stability, Jensen’s functional equation, fixed point, modular space.

1. Introduction

The concept of stability for a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. Recall that the problem of stability of functional equations was motivated by a question of Ulam being asked in 1940 [24] and Hyers answer to it was published in [4]. Hyers’s theorem was generalized by Aoki [1] for additive mappings and by Rassias [21] for linear mappings by considering an unbounded Cauchy difference. During the past decades, a number of results concerning the stability of various functional equations have been obtained [3, 5, 7, 8].

The result on the stability of the classical Jensen functional equation was first given by Kominek [11]. The author who presumably investigated the stability problem on a restricted domain for the first time was Skof [22]. The stability of the Jensen equation and its gen- eralizations were studied by a number of mathematicians (cf., e.g., [2, 6, 9, 16, 18]). In this paper, by using some ideas of [9], we investigate the generalized Hyers–Ulame stability of a generalized Jensen functional equation for mappings from linear spaces into modular spaces. The theory of modulars on linear spaces and the corresponding theory of modular linear spaces were founded by Nakano [19] and were intensively developed by Amemiya, Koshi, Shimogaki, Yamamuro [12, 25] and others. Further and the most complete develop- ment of these theories are due to Orlicz, Mazur, Musielak, Luxemburg, Turpin [14, 17, 23]

and their collaborators. In the present time the theory of modulars and modular spaces is extensively applied, in particular, in the study of various Orlicz spaces [20] and interpola- tion theory [13, 15], which in their turn have broad applications [17]. The importance for

Communicated byV. Ravichandran.

Received:December 7, 2011;Revised:March 15, 2012.

applications consists in the richness of the structure of modular function spaces, that be- sides being Banach spaces (orF-spaces in more general setting) are equipped with modular equivalent of norm or metric notions.

Definition 1.1. LetX be an arbitrary vector space.

(a) A functionalρ:X →[0,∞]is called a modular if for arbitrary x,y∈X, (i) ρ(x) =0if and only if x=0,

(ii) ρ(αx) =ρ(x)for every scalerαwith|α|=1,

(iii) ρ(αx+βy)≤ρ(x) +ρ(y)if and only ifα+β =1andα,β≥0, (b) if (iii) is replaced by

(iii)⁰ ρ(αx+βy)≤α ρ(x) +β ρ(y)if and only ifα+β =1andα,β ≥0, then we say thatρ is a convex modular.

A modularρdefines a corresponding modular space, i.e., the vector spaceXρgiven by Xρ={x∈X :ρ(λx)→0 as λ→0}.

Letρbe a convex modular, the modular spaceXρ can be equipped with a norm called the Luxemburg norm, defined by

kxkρ=infn

λ>0 ; ρ x

λ ≤1o

.

A function modular is said to satisfy the∆2-condition if there existsκ >0 such that ρ(2x)≤κ ρ(x)for allx∈Xρ.

Definition 1.2. Let{x_n}and x be inXρ. Then

(i) the sequence{x_n}, with x_n∈Xρ, isρ-convergent to x and write x_n−→^ρ x ifρ(x_n− x)→0as n→∞.

(ii) The sequence{x_n}, with x_n∈Xρ, is calledρ-Cauchy ifρ(x_n−x_m)→0as n,m→

∞.

(iii) A subset S of Xρ is called ρ-complete complete if and only if anyρ-Cauchy sequence isρ-convergent to an element ofS.

The modularρhas the Fatou property if and only ifρ(x)≤lim infn→∞ρ(x_n)whenever the sequence{x_n}isρ-convergent tox.

Remark 1.1. Note that ρ is an increasing function. Suppose 0<a<b, then property (iii) of Definition 1.1 withy=0 shows that ρ(ax) =ρ((a/b)bx)≤ρ(bx)for allx∈X. Moreover, if ρ is a convex modular onX and|α| ≤1, then ρ(αx)≤α ρ(x) and also ρ(x)≤(1/2)ρ(2x)for allx∈X.

A convex functionϕ defined on the interval[0,∞), nondecreasing and continuous for α≥0 and such thatϕ(0) =0,ϕ(α)>0 forα>0,ϕ(α)→∞asα→∞, is called an Orlicz function. The Orlicz functionϕ satisfies the∆2-condition if there existsκ>0 such that ϕ(2α)≤ϕ(α)for allα >0. Let(Ω,Σ,µ)be a measure space. Let us consider the space L⁰(µ)consisting of all measurable real-valued (or complex-valued) functions onΩ. Define for every f∈L⁰(µ)the Orlicz modularρϕ(f)by the formula

ρϕ(f) = Z

Ω

ϕ(|f|)dµ.

The associated modular function space with respect to this modular is called an Orlicz space, and will be denoted byL^ϕ(Ω,µ)or brieflyL^ϕ. In other words,

L^ϕ={f ∈L⁰(µ)|ρϕ(λf)→0 as λ →0}

or equivalently as

L^ϕ={f∈L⁰(µ)|ρϕ(λf)<∞ for some λ>0}.

It is known that the Orlicz spaceL^ϕ isρϕ-complete. Moreover,(L^ϕ,k.k_ρϕ)is a Banach space, where the Luxemburg normk.k_ρϕ is defined as follows

kfk_ρϕ =inf

λ>0 : Z

Ω

ϕ |f|

λ

dµ≤1

.

Moreover, ifLis the space of sequencesx={x_i}^∞_i=1with real or complex termsx_i,ϕ = {ϕ_i}^∞_i=1,ϕiare Orlicz functions andρϕ(x) =Σ^∞_i=1ϕi(|x_i|), we shall write`<sup>ϕ</sup>in place ofL<sup>ϕ</sup>. The space`^ϕ is called the generalized Orlicz sequence space. The motivation for the study of modular spaces (and Orlicz spaces) and many examples are detailed in [19, 17, 20, 15].

2. Stability of a generalized Jensen functional equation

Throughout this paper, we assume thatρis a convex modular onX with the Fatou property such that satisfies the∆₂-condition with 0<κ≤2. In addition, we assume thatr,sconstant positive integer numbers. In this section, we use some ideas from [9] and we establish the conditional stability of a generalized Jensen functional equation.

Theorem 2.1. LetE be a real or complex linear space and letXρbe aρ-complete modular space. Suppose f :E →Xρsatisfies the condition f(0) =0and an inequality of the form (2.1) ρ(f(x+y)−f(x)−f(y))≤φ(x,y)

for all x,y∈E, whereφ:E×E →[0,∞)is a given function such that φ(2x,2x)≤2Lφ(x,x)

for all x∈E and has the property

(2.2) lim

n→∞

φ(2ⁿx,2ⁿy)

2ⁿ =0

for all x,y∈E and a constant0<L<1. Then there exists a unique additive mapping j:E →Xρ such that

(2.3) ρ(j(x)−f(x))≤ 1

2(1−L)φ(x,x) for all x∈E.

Proof. We consider the set

M ={g:E →Xρ, g(0) =0}

and introduce the convex modularρeonM as follows,

ρe(g) =inf{c>0 :ρ(g(x))≤cφ(x,x)}.

It is sufficient to show thatρesatisfies the following condition ρ(αe g+βh)≤αρe(g) +βρ(h)e

ifα+β=1 andα,β≥0. Letε>0 be given. Then there existc₁>0 andc₂>0 such that c₁≤ρ(g) +e ε;ρ(g(x))≤c₁φ(x,x)

and

c₂≤ρ(h) +e ε;ρ(h(x))≤c₂φ(x,x).

Ifα+β=1 andα,β ≥0, then we get

ρ(αg(x) +βh(x))≤α ρ(g(x)) +β ρ(h(x))≤(αc₁+βc₂)φ(x,x), whence

ρe(αg+βh)≤αeρ(g) +βρ(h) + (αe +β)ε.

Hence, we have

ρ(αe g+βh)≤αρe(g) +βρ(h).e Moreover,ρesatisfies the∆2-condition with 0<κ≤2.

Let{g_n}be aρ-Cauchy sequence ine Mρ_eand letε>0 be given. There exists a positive integern₀∈Nsuch thatρ(ge _n−g_m)≤εfor alln,m≥n₀. Now by considering the definition of the modularρ, we see thate

(2.4) ρ(g_n(x)−g_m(x))≤ε φ(x,x)

for all x∈E andn,m≥n0. Ifxis any given point ofE, (2.4) implies that{g_n(x)} is a ρ-Cauchy sequence inXρ. SinceXρisρ-complete, so{g_n(x)}isρ-convergent inXρ, for eachx∈E. Hence, we can define a functiong:E →Xρ by

g(x) =lim

n→∞g_n(x)

for anyx∈E. Letmincrease to infinity, then (2.4) implies that ρe(g_n−g)≤ε

for alln≥n₀, sinceρ has the Fatou property. Thus{g_n}isρ-convergent sequence ine M_eρ. ThereforeM_eρisρ-complete.e

Now, we consider the functionT :M_eρ→Mρ_edefined by Tg(x):=1

2g(2x)

for allg,h∈Mρ_e. Letg,h∈Mρ_eand letc∈[0,∞]be an arbitrary constant withρ(ge −h)≤c.

From the definition ofρe, we have

ρ(g(x)−h(x))≤cφ(x,x) for allx∈E. By the assumption and the last inequality, we get

ρ g(2x)

2 −h(2x) 2

≤1

2ρ(g(2x)−h(2x))≤1

2cφ(2x,2x)≤Lcφ(x,x)

for allx∈E. Hence,ρe(Tg−Th)≤Lρ(ge −h), for allg,h∈M_ρethat is,T is aρ-stricte contraction. We show that theρe-strict mappingT satisfies the conditions of Theorem 3.4 of [10].

Lettingx=yin (2.1), we get

(2.5) ρ(f(2x)−2f(x))≤φ(x,x)

for allx∈E. If we replacexby 2xin (2.5) we get

ρ(f(4x)−2f(2x))≤φ(2x,2x)

for allx∈E. Sinceρis convex modular and satisfies the∆₂-condition, we obtain ρ

f(4x)

2 −2f(x)

≤1

2ρ(f(4x)−2f(2x)) +1

2ρ(2f(2x)−4f(x))

≤1

2φ(2x,2x) +κ 2φ(x,x) for allx∈E. Moreover,

ρ

f(2²x) 2² −f(x)

≤1 2ρ

2f(4x)

2² −2f(x)

≤ 1

2²φ(2x,2x) + κ 2²φ(x,x).

for allx∈E. By mathematical induction, we can easily see that ρ

f(2ⁿx) 2ⁿ −f(x)

≤ 1 2ⁿ

n

∑

i=1

κⁿ⁻ⁱφ(2ⁱ⁻¹x,2ⁱ⁻¹x)≤ 1

2(1−L)φ(x,x) (2.6)

for allx∈E. Next, we assert thatδ

ρe(f) =sup{ρe(Tⁿ(f)−T^m(f));n,m∈N)}<∞. It follows from inequality (2.6) that

ρ

f(2ⁿx)

2ⁿ −f(2^mx) 2^m

≤1 2ρ

2f(2ⁿx)

2ⁿ −2f(x)

+1 2ρ

2f(2^mx)

2^m −2f(x)

≤κ 2ρ

f(2ⁿx) 2ⁿ −f(x)

+κ

2ρ

f(2^mx) 2^m −f(x)

≤ 1

1−Lφ(x,x), for everyx∈E andn,m∈N, which implies that

ρe(Tⁿ(f)−T^m(f))≤ 1 1−L, for alln,m∈N. By the definition ofδ

eρ(f), we haveδ

ρe(f)<∞. Lemma 3.3 of [10] shows that{Tⁿ(f)}isρ-converges toe j∈Mρ_e. Sinceρ has the Fatou property inequality (2.6), givesρ(e T j−f)<∞.

If we replacexby 2ⁿxin inequality (2.5), then we obtain ρe f(2ⁿ⁺¹x)−2f(2ⁿx)

≤φ(2ⁿx,2ⁿx), for allx∈E. Whence

ρ

f(2ⁿ⁺¹x)

2ⁿ⁺¹ − f(2ⁿx) 2ⁿ

≤ 1

2ⁿ⁺¹ρ f(2ⁿ⁺¹x)−2f(2ⁿx)

≤ 1

2ⁿ⁺¹φ(2ⁿ,2ⁿx)

≤ 1

2ⁿ⁺¹2ⁿLⁿφ(x,x)≤Lⁿ

2 ϕ(x,x)≤φ(x,x)

for allx∈E. Thereforeρ(e T(j)−j)<∞. It follows from [10, Theorem 3.4] thatρ-limite of{Tⁿ(f)}i.e., j∈M_eρ is fixed point of mapT. If we replacexby 2ⁿxandyby 2ⁿyin inequality (2.1), then we obtain

ρ(f(2ⁿ(x+y))−f(2ⁿx)−f(2ⁿy))≤φ(2^x,2ⁿy)

for allx,y∈E. Hence, ρ

f(2ⁿ(x+y))

2ⁿ − f(2ⁿx)

2ⁿ − f(2ⁿy) 2ⁿ

≤ 1

2ⁿρ(f(2ⁿ(x+y))−f(2ⁿx)−f(2ⁿy))

≤φ(2^x,2ⁿy) 2ⁿ

for allx,y∈E. Taking the limit, we deduce that j(x+y) = j(x) +j(y)for allx,y∈E. It follows from inequality (2.6) that

ρ(e j−f)≤ 1 2(1−L). If j^∗is another fixed point ofT, then

ρ(e j−j^∗)≤1

2ρ(2e T(j)−2f) +1

2ρ(2e T(j^∗)−2f)

≤κ

2eρ(T(j)−f) +κ

2ρ(e T(j^∗)−f)≤ κ

2(1−L)<∞.

SinceT isρ-strict contraction, we gete

ρ(e j−j^∗) =ρe(T(j)−T(j^∗))≤Lρ(e j−j^∗),

which implies thatρ(e j−j^∗) =0 or j= j^∗, sinceρe(j−j^∗)<∞. This prove the uniqueness of j.

Corollary 2.1. LetE be a normed space and letFbe a Banach space. Suppose f :E →F is a mapping with f(0) =0and there exist constantsε,θ≥0and p∈[0,1)such that

kf(x+y)−f(x)−f(y)k ≤ε+θ(kxk^p+kyk^p),

for all x,y∈E. Then there exists a unique additive mapping j:E →Fsuch that kf(x)−j(x)k ≤ ε

2−2^p+ 2θ 2−2^pkxk^p for all x,y∈E.

Proof. It is known that every normed space is modular space with the modularρ(x) =kxk andκ=2. Defineφ(x,y) =ε+θ(kxk^p+kyk^p)and apply Theorem 2.1.

Now, we are ready to prove stability the functional equation f(rx+sy) =rg(x) +sh(y).

Theorem 2.2. Let f,g,h:E →Xρbe mappings with f(0) =g(0) =h(0) =0satisfying (2.7) ρ(f(rx+sy)−rg(x)−sh(y))≤φ(x,y)

for all x,y∈E, whereφ:E ×E →[0,∞)is given function. If there exists0<L<1such that

φ(2x,2x)≤2Lφ(x,x) and has the property

(2.8) lim

n→∞

φ(2ⁿx,2ⁿy)

2ⁿ =0

for all x,y∈E. Then there exists a unique additive mappingA :E →Xρsuch that ρ(f(x)−A(x))≤ψ(x,x)

ρ(g(x)−A(x))≤ κ

2r(φ(x,0) +ψ(rx,rx)) ρ(h(x)−A(x))≤ κ

2s(φ(0,x) +ψ(sx,sx)) for all x∈E, where ψ(x,x) =1/(2(1−L))

(κ/2)φ(x/r,x/r) + (κ²/4)(φ(x/r,0) +φ(0, x/s))

.

Proof. Lettingy=0 in (2.7) we get

ρ(f(rx)−rg(x)≤φ(x,0) for allx∈E. Lettingx=0 in (2.7) we get

ρ(f(sy)−sh(y))≤φ(0,y) for ally∈E. Then

ρ(f(rx+sy)−f(rx)−f(sy))

≤1

2ρ(2(f(rx+sy)−rg(x)−sh(y)) +1

2ρ(2(rg(x)−f(rx)−f(sy) +sh(y))

≤κ

2ρ(f(rx+sy)−rg(x)−sh(y)) +κ

2ρ(rg(x)−f(rx)−f(sy) +sh(y))

≤κ

2φ(x,y) +κ²

4 (φ(x,0) +φ(0,y)).

Replacingxby(1/r)xandyby(1/s)yin the above inequality, we obtain ρ(f(x+y)−f(x)−f(y))≤κ

2φ x

r,y s

+κ² 4

h φ

x r,0

+φ 0,y

s i

for allx,y∈E. By Theorem 2.1, there exists a unique additive mappingA :E →Xρgiven byA(x) =limn→∞(f(2ⁿx)/2ⁿ)such that

ρ(f(x)−A(x))≤ψ(x,x) (2.9)

for allx∈E. SinceA is a additive, we haveA(qx) =qA(x)for all rational numbersq andx∈E. It follows from inequalities (2.7) and (2.9) that

ρ(g(x)−A(x))≤1 2ρ

2

g(x)−1

rf(rx)

+1 2ρ

2

1

rf(rx)−A(x)

≤ κ

2r(ρ(rg(x)−f(rx))) + κ

2r(ρ(f(rx)−A(rx))

≤ κ

2rφ(x,0) + κ

2rψ(rx,rx) for allx∈E. Similarly, we obtain the following inequality

ρ(h(x)−A(x))≤ κ

2sφ(x,0) + κ

2sψ(sx,sx) for allx∈E.

Corollary 2.2. LetE be a normed space and letXρbe aρ-complete. Suppose f:E →Xρ

is a mapping with f(0) =0and there exist constantsε,θ≥0and p∈[0,1)such that ρ(f(rx+sy)−r f(x)−s f(y))≤ε+θ(kxk^p+kyk^p),

for all x,y∈E. Then there exists a unique additive mappingA :E →Xρsuch that ρ(f(x)−A(x))≤ κ+κ²

2(2−2^p)ε+ κ

r^p+ κ² 2r^p+ κ²

2s^p

θkxk^p 2−2^p. Proof. Defineφ(x,y) =ε+θ(kxk^p+kyk^p)and apply Theorem 2.2.

Corollary 2.3. LetE be a normed space and letFbe a Banach space. Suppose f :E →F is a mapping with f(0) =0and there exist constantsε,θ≥0and p∈[0,1)such that

kf(rx+sy)−r f(x)−s f(y)k ≤ε+θ(kxk^p+kyk^p),

for all x,y∈E. Then there exists a unique additive mappingA :E →Fsuch that kf(x)−A(x)k ≤ 3

2−2^pε+ 2

r^p+ 1 s^p

θkxk^p 2−2^p.

The following example shows that our results in this paper differ form some results of [9].

Example 2.1. Letϕbe an Orlicz function and satisfy the∆2-condition with 0<κ≤2. Let f,g,h:E →L^ϕbe mappings with f(0) =g(0) =h(0) =0 satisfying

(2.10)

Z

Ω

φ(|f(rx+sy)−rg(x)−sh(y)|)dµ≤φ(x,y)

for allx,y∈E, whereφ:E ×E →[0,∞)is given function. If there exists 0<L<1 such that

φ(2x,2x)≤2Lφ(x,x) and has the property

(2.11) lim

n→∞

φ(2ⁿx,2ⁿy)

2ⁿ =0

for allx,y∈E. Then there exists a unique additive mappingA :E →Xρsuch that Z

Ω

ϕ(|f(x)−A(x)|)dµ≤ψ(x,x) Z

Ω

ϕ(|g(x)−A(x)|)dµ≤ κ

2r(φ(x,0) +ψ(rx,rx)) Z

Ω

ϕ(|h(x)−A(x)|)dµ≤ κ

2s(φ(0,x) +ψ(sx,sx)) for allx∈E, where

ψ(x,x) = 1 2(1−L)

κ 2

φ x

r,x r

+ κ²

4

φ x

r,0 +φ

0,x s

.

3. Stability of generalized Jensen functional equation on restricted domains

In this section, we investigate the stability of our generalized Jensen equation on restricted domains. The idea and methods used in this section is taken from the paper by Junget al.

[9].

Theorem 3.1. Let(E,ρ)be a modular space and letXρ be aρ-complete modular space.

Let d>0,ε>0, and f:E →Xρwith f(0) =0such that (3.1) ρ(f(rx+sy)−r f(x)−s f(y))≤ε

for all x,y∈E withρ(x) +ρ(y)≥d. Then there exists a unique additive mappingA :E → Xρ such that

ρ(f(x)−A(x))≤κ+κ² 2

κ 2 +κ²

2²+κ³ 2³+κ⁴

2⁴+κ⁴ 2⁴

ε

for all x∈E.

Proof. Letx,y∈E withρ(x) +ρ(y)<d. Suppose that, forx=y=0,zis an element ofE withρ(z)≥d. Furthermore, forx6=0 ory6=0, let

z:=







1+ ^d

ρ(x)

x ifρ(x)≥ρ(y)

1+_ρ(y)^d

y ifρ(x)≤ρ(y).

Clearly, we see that ρ

h 2+s

r i

z+s ry

+ρ

1+2r

s

z−r sx

≥d, ρ(x) +ρ(z)≥d, ρ

2h

1+s r i

z

+ρ(y)≥d,

ρ

2h 1+s

r i

z +ρ

1+2r

s

z−a bx

≥d,

ρ h

2+s r i

z+s ry

+ρ(z)≥d.

Next, we show that the first inequality holds and other inequalities are trivial. To this end, letρ(x)≥ρ(y), we putα = [2+s/r]z+ (s/r)y,β = [1+ (2r/s)]z−(r/s)x,γ =−(s/r)y andη=−(r/s)x. Then we obtain

ρ(α) +ρ(β)≥2ρ α+γ

2

−ρ(γ) +2ρ

β+η 2

−ρ(η)

=2ρ 1

2 h

2+s r i

z

−ρ s

ry +2ρ

1 2

1+2r s

z

−ρ r

sx

≥ρ(z) +ρ s

2rz −ρ

s ry

+ρ

z 2

+ρ r

sz −ρ

r sx

≥ρ(z) +ρ s

rx

−ρs ry

+ρz 2

+ρ r

sx

−ρ r

sx

≥d. (3.2)

Now, we set

θ= f(rx+sy)−r fh 2+s

r i

z+s ry

−s f

1+2r s

z−r

sx

,

λ=f(rx+sz)−r f(x)−s f(z), µ=f(2(r+s)z+sy)−r f

2h 1+s

r i

z

−s f(y),

ν=−f(rx+sz) +r f 2h

1+s r i

z +s f

1+2r

s

z−r sx

and

ϑ=−f(2(r+s)z+sy) +r f h

2+s r i

z+s ry

+s f(z).

It follows from (3.1) and (3.2) that

ρ(f(rx+sy)−r f(x)−s f(y)) =ρ(θ+λ+µ+ν+ϑ)≤1

2ρ(2θ) +1

2ρ(2(λ+µ+ν+ϑ))

≤κ

2ρ(θ) +κ

2ρ(λ+µ+ν+ϑ) ...

≤κ

2ρ(θ) +κ²

2²ρ(λ) +κ³

2³ρ(µ) +κ⁴

2⁴ρ(ν) +κ⁴ 2⁴ρ(θ)

≤ κ

2 +κ² 2²+κ³

2³+κ⁴ 2⁴+κ⁴

2⁴

ε

for allx,y∈E. We thus obtain

ρ(f(rx+sy)−r f(x)−s f(y))≤ κ

2 +κ² 2²+κ³

2³+κ⁴ 2⁴+κ⁴

2⁴

ε

for allx,y∈E. Now the result asserted by the above Theorem can be deduced fairly easily from Corollary 2.2 withθ=p=0.

Corollary 3.1. LetE be a normed space and letFbe a Banach space. Let d>0,ε>0, and f:E →Fbe a mapping with f(0) =0such that

kf(rx+sy)−r f(x)−s f(y)k ≤ε

for all x,y∈E withkxk+kyk ≥d. Then there exists a unique additive mappingA :E →F such that

kf(x)−A(x)k ≤15ε for all x∈E.

Corollary 3.2. Let(E,ρ)be a modular space and letXρbe aρ-complete modular space.

Let f :E →Xρ be a mapping with f(0) =0. Then f is additive if and only if ρ(f(rx+sy)−r f(x)−s f(y))→0 as ρ(x) +ρ(y)→∞.

(3.3)

Proof. The proof of this corollary is similar to [9, Corollary 3.2].

Example 3.1. Let ˆϕ={ϕ_i}be a sequence of Orlicz functions and let(`<sup>ϕˆ</sup>,ρ<sub>ϕˆ</sub>)be a gener- alized Orlicz sequence space associated to ˆϕ={ϕ<sub>i</sub>}. Let(L<sup>ϕ</sup>,ρϕ)be an Orlicz space and ϕ satisfy the∆2-condition with 0<κ≤2. Supposed>0,ε>0 and f :`^ϕˆ →L^ϕ with

f(0) =0 such that

Z

Ω

ϕ(|f(rx+sy)−r f(x)−s f(y)|)≤ε

for allx,y∈`<sup>ϕˆ</sup> withρϕ(x) +ρϕ(y) =Σ<sup>∞</sup><sub>i=1</sub>ϕ<sub>i</sub>(|x<sub>i</sub>|) +Σ<sup>∞</sup><sub>i=1</sub>ϕ<sub>i</sub>(|y<sub>i</sub>|)≥d. Then there exists a unique additive mappingA :`^ϕˆ→L^ϕ such that

Z

Ω

ϕ(|f(x)−A(x)|)≤κ+κ² 2

κ 2+κ²

2²+κ³ 2³+κ⁴

2⁴+κ⁴ 2⁴

ε

for allx∈`^ϕˆ.

Acknowledgement.The author is grateful to the referees for their valuable suggestions.

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