On the Stability of a General Mixed Additive Cubic Functional Equation in Random Normed Spaces

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Volume 2010, Article ID 328473,16pages doi:10.1155/2010/328473

Research Article

On the Stability of a General Mixed Additive-Cubic

Functional Equation in Random Normed Spaces

Tian Zhou Xu,

1

_{John Michael Rassias,}

2

_{and Wan Xin Xu}

3

1_{Department of Mathematics, School of Science, Beijing Institute of Technology, Beijing 100081, China} 2_{Pedagogical Department E.E., Section of Mathematics and Informatics, National and}

Kapodistrian University of Athens, 4, Agamemnonos Str., Aghia Paraskevi, 15342 Athens, Greece

3_{School of Communication and Information Engineering, University of Electronic Science and}

Technology of China, Chengdu 611731, China

Correspondence should be addressed to Tian Zhou Xu,[email protected]

Received 6 June 2010; Revised 1 August 2010; Accepted 23 August 2010

Academic Editor: Radu Precup

Copyrightq2010 Tian Zhou Xu 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.

We prove the generalized Hyers-Ulam stability of the following additive-cubic equationfkx y fkx−y kfxy kfx−y 2fkx−2kfxin the setting of random normed spaces.

1. Introduction

A basic question in the theory of functional equations is as follows: when is it true that a function, which approximately satisfies a functional equation, must be close to an exact solution of the equation?

If the problem accepts a unique solution, we say the equation is stablesee1. The first stability problem concerning group homomorphisms was raised by Ulam2in 1940 and aﬃrmatively solved by Hyers3. The result of Hyers was generalized by Rassias4 for approximate linear mappings by allowing the Cauchy diﬀerence operatorCDfx, y fxy−fxfyto be controlled byxp_yp_{. In 1994, a generalization of Rassias’}

theorem was obtained by G˘avrut¸a 5, who replaced xp _yp _{by a general control}

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The theory of random normed spacesRN-spaces is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations. The RN-spaces may also provide us the appropriate tools to study the geometry of nuclear physics and have important application in quantum particle physicssee17and the references therein. The generalized Hyers-Ulam stability of diﬀerent functional equations in random normed spaces, fuzzy normed spaces, and non-Archimedean fuzzy normed spaces has been recently studied in14–28.

Najati and Eskandani29established the general solution and investigated the Ulam-Hyers stability of the following functional equation.

f2xyf2x−y2fxy2fx−y2f2x−4fx, 1.1

withf0 0 in the quasi-Banach spaces. It is easy to see that the mappingfx ax3_bx_is a solution of the functional equation1.1, which is called a mixed additive-cubic functional equation, and every solution of the mixed additive-cubic functional equation is said to be a mixed additive-cubic mapping.

In 14–16, we considered the following general mixed additive-cubic functional equation:

fkxyfkx−ykfxykfx−y2fkx−2kfx. 1.2

It is easy to show that the functionfx ax3_bx_{satisfies the functional equation}_1.2_{. We} observe that in casek21.2yields mixed additive-cubic equation1.1. Therefore,1.2is a generalized form of the mixed additive-cubic equation.

In the present paper, we first prove a theorem on stability of equation gax asgx a, s ∈ N, a ≥ 2in random normed spaces and derive from it results on stability of equationf4x 10f2x−16fx. Next, use those results to establish Ulam-Hyers stability for the general mixed additive-cubic functional equation 1.2 in the setting of random normed spaces. In this way some results will be obtained on stability of the linear functional equations also for the random normed spaces, which correspond, for example, to the papers

30–33.

2. Preliminaries

In the sequel we adopt the usual terminology, notations and conventions of the theory of random normed spaces, as in 17,28. Throughout this paper, the space of all probability distribution functions is denoted by

Δ_F_:_R_{∪ {−∞}_,_{∞} →} ₀_,₁_:_F _{is left-continuous and nondecreasing on}_R

and F0 0, F∞ 1}, 2.1

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ordering of functions, that is,F ≤ G if and only ifFt ≤ Gtfor all t ∈ R. The maximal element forΔin this order is the distribution function given by

ε0t

⎧ ⎨ ⎩

0, ift≤0,

1, ift >0. 2.2

Definition 2.1see17,28. A functionT :0,1×0,1 → 0,1is a continuous triangular normbriefly, a continuoust-normifTsatisfies the following conditions:

aTis commutative and associative;

bTis continuous;

cTa,1 afor alla∈0,1;

dTa, b≤ Tc, dwhenevera≤candb≤dfor alla, b, c, d∈0,1.

Typical examples of continuoust-norms areTPa, b ab,TMa, b mina, band TLa, b maxab−1,0 the Lukasiewiczt-norm.

Now, ifTis at-norm and{xi}is a given sequence of numbers in0,1, we define a

sequenceTn _{recursively by} _T1

i1x1 x1 andTin1xi TTin−11xi, xn for alln ≥ 2.T∞inxi is

defined asT∞_i₁xni.

Definition 2.2see17,28. A random normed spacebriefly, RN-spaceis a tripleX, μ,T, whereXis a vector space,Tis a continuoust-norm, andμis a mapping fromXintoDsuch that the following conditions hold:

RN1μxt ε0tfor allt >0 if and only ifx0;

RN2μαxt μxt/|α|for allxinX,α /0 and allt≥0;

RN3μxyts≥ Tμxt, μysfor allx, y∈Xand allt, s≥0.

Example 2.3. Let X, · be a normed space. For all x ∈ X and t > 0, consider μxt t/tx. ThenX, μ,TMis a random normed space, whereTMis the minimumt-norm.

This space is called the induced random normed space.

Definition 2.4. LetX, μ,Tbe an RN-space.

1A sequence{xn}inXis said to be convergent to a pointx∈Xif, for everyt >0 and

ε >0, there exists a positive integerNsuch thatμxn−xt>1−εwhenevern≥N.

2A sequence{xn}inXis called a Cauchy sequence if, for everyt >0 andε >0, there

exists a positive integerNsuch thatμxn−xmt>1−εwhenevern≥m≥N.

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3. On the Stability of a General Mixed Additive-Cubic Equation

in RN-Spaces

Theorem 3.1. Lets, a∈Nwitha≥2,Xbe a linear space,Y, μ,TMbe a complete RN-space, and

g:X → Ybe a mapping for which there isψ:X → Dsuch that

μgax−as_g_xt≥ψxt 3.1

for allx∈Xandt >0. If for some0< α < as_,

ψaxt≥ψx

t

α 3.2

for allx ∈ X andt > 0, then there exists a uniquely determined mappingG : X → Y such that

Gax as_G_x_and

μgx−Gxt≥ψx

as₋_α_t

2 3.3

for allx∈Xandt >0.

Proof. Replacingxbyai_x_in_3.1_{and using}_3.2_{, we get}

μgai1_x_/asi1₋_g_ai_x_/asi

αi_t

asi1

≥ψxt 3.4

for allx∈X, i∈N,andt >0. It follows that

μgan_x_/asn₋_g_am_x_/asm

_n₋₁

im

αi_t

asi1

μn−1

imgai1x/asi1−gaix/asi

_n₋₁

im

αit asi1

≥ψxt

3.5

for allx∈X, t >0 and all nonnegative integersnandmwithn > m. Hence

μgan_x_/asn₋_g_am_x_/asmt≥ψ_x

t/

n−1

im

αi

asi1

3.6

for allx∈X, t >0,andm, n∈Nwithn > m. As 0< α < as_and∞

i0αi/asi1<∞, the right hand side of the inequality tends to 1 asmtend to infinity. Then the sequence{gan_x_/asn_}

is a Cauchy sequence inY, μ,TM. SinceY, μ,TMis a complete RN-space, this sequence converges to some pointGx∈Y. Therefore, we may defineGx:limn→ ∞ganx/asnfor

allx∈X. Fixx∈X,and putm0 in3.6. Then we obtain

μgan_x_/asn₋_g_xt≥ψ_x

t/

n−1

i0

αi

asi1

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and so, byRN3, we have

μGx−gxt≥ TM

μGx−gan_x_/asn

t

2 , μganx/asn−gx

t

2

≥ TM

μGx−gan_x_/asn

t

2 , ψx

t/

n−1

i0 2αi

asi1

3.8

for everyt > 0. Taking the limit asn → ∞in3.8, byGx limn→ ∞ganx/asn, we get 3.3.

To prove the uniqueness of the mappingG, assume that there exists another mapping

H : X → Y which satisfies3.3andHax as_H_x_{for all}_x _∈ _X_{. Fix}_x _∈ _X_{. Clearly,}

Gan_x _asn_G_x_,_and_H_an_x _asn_H_x_{for all}_n_∈_N_{. It follows from}_3.2_and_3.3_that

μGx−Hxt≥ TM

μGan_x_/asn₋_g_an_x_/asn

t

2 , μganx/asn−Hanx/asn

t

2

≥ψx

as₋_α_asn_t

4αn .

3.9

Since limn→ ∞as−αasnt/4αn ∞, we get limn→ ∞ψxas−αasnt/4αn 1. Therefore,

it follows from3.9thatμGx−Hxt 1 for allt > 0,and soG H. This completes the

proof.

Corollary 3.2. Lets∈ {1,3}be fixed,Xbe a linear space,Y, μ,TMbe a complete RN-space, and

f:X → Ybe a mapping for which there isψ:X → Dsuch that

μf4x−10f2x16fxt≥ψxt 3.10

for allx∈Xandt >0. If for some0< α <2s_,_ψ

2xt≥ψxt/αfor allx∈Xandt >0, then there

exists a uniquely determined mappingFs:X → Ysuch thatFs2x 2sFsxand

μ_f₂_x₋₂3/s_f_x₋_F_s_xt≥ψ_x

2s₋_α_t

2 3.11

Theorem 3.3. LetX be a linear space,Z, μ,TMbe an RN-space,Y, μ,TMbe a complete RN-space, and f : X → Y be a mapping with f0 0 for which there isϕ : X×X → Z such that

μfkxyfkx−y−kfxy−kfx−y−2fkx2kfxt≥μ_ϕ_x,yt 3.12

for allx, y∈Xandt >0. If for some0< α <2,

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for allx, y∈Xandt >0, then there exists a unique additive mappingA:X → Y such that

μf2x−8fx−Axt≥ψx

2−αk3₋_k_t 2

3.14

for allx∈Xandt >0, where

ψxt: TM32_i₁

μ_ϕ_x/₂_,₂_k₁_x/₂

t

384k , μ

ϕx/2,2k−1x/2

t

384k ,

μ_ϕ_x/₂_,₃_kx/₂

t

384 , μ

ϕ0,3k−1x/2

k−1t

384k , μ

ϕx,x

t

96k2 ,

μ_ϕ_x/₂_,kx/₂

t

96 , μ

ϕ0,k1x/2

k−1t

96k , μ

ϕ0,k−1x

k−1t

96k2 ,

μ_ϕ₀_,kx

t

96k1 , μ

ϕx,k1x

t

128 , μ

ϕx,k−1x

t

128 ,

μ_ϕ₀_,x

k−1t

128 , μ

ϕ0,kx

k−1t

128k , μ

ϕ2x,x

t

32 , μ

ϕ2x,kx

t

16 ,

μ_ϕ_x,₂_k₋₁_x

t

56 , μ

ϕx,2k1x

t

56 , μ

3x,x

t

56 , μ

ϕx,x

t

56 ,

μ_ϕ₀_,_k₁_x

k−1t

56 , μ

ϕ0,k−1x

k−1t

56 , μ

ϕ0,2kx

k−1t

56k ,

μ_ϕ_x,₂_k₁_x

t

384k , μ

ϕx,2k−1x

t

384k , μ

ϕx,3kx

t

384 ,

μ_ϕ₀_,₃_k₋₁_x

k−1t

384k , μ

ϕ2x,2x

t

96k2 , μϕx,kx

t

96 ,

μ_ϕ₀_,_k₁_x

k−1t

96k , μ

ϕ0,2k−1x

k−1t

96k2 ,

μ_ϕ₀_,₂_kx

t

96k1 , μ

ϕ2x,2kx

t 16 . 3.15

Proof. Lettingx0 in3.12, we get

μfyf−yt≥μ_ϕ₀_,yk−1t 3.16

for ally∈Xandt >0. Puttingyxin3.12, we have

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for allx∈Xandt >0. Replacingxby 2xin3.17, we obtain

μf2k1xf2k−1x−kf4x−2f2kx2kf2xt≥μϕ2x,2xt 3.18

for allx∈Xandt >0. Lettingykxin3.12, we get

μf2kx−kfk1x−kf−k−1x−2fkx2kfxt≥μϕx,kxt 3.19

for allx∈Xandt >0. Lettingy k1xin3.12, we have

μf2k1xf−x−kfk2x−kf−kx−2fkx2kfxt≥μϕx,k1xt 3.20

for allx∈Xandt >0. Lettingy k−1xin3.12, we have

μf2k−1x−k2fkx−kf−k−2x2k1fxt≥μϕx,k−1xt 3.21

for allx∈Xandt >0. Replacingxandyby 2xandxin3.12, respectively, we get

μf2k1xf2k−1x−2f2kx−kf3x2kf2x−kfxt≥μϕ2x,xt 3.22

for allx∈Xandt >0. Replacingxandyby 3xandxin3.12, respectively, we get

μf3k1xf3k−1x−2f3kx−kf4x−kf2x2kf3xt≥μϕ3x,xt 3.23

for allx∈Xandt >0. Replacingxandyby 2xandkxin3.12, respectively, we have

μf3kxfkx−kfk2x−kf−k−2x−2f2kx2kf2xt≥μϕ2x,kxt 3.24

for allx∈Xandt >0. Settingy 2k1xin3.12, we have

μf3k1xf−k1x−kf2k1x−kf−2kx−2fkx2kfxt≥μϕx,2k1xt 3.25

for allx∈Xandt >0. Lettingy 2k−1xin3.12, we have

μf3k−1xf−k−1x−kf−2k−1x−kf2kx−2fkx2kfxt≥μϕx,2k−1xt 3.26

for allx∈Xandt >0. Lettingy3kxin3.12, we have

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for allx∈Xandt >0. By3.16,3.17,3.23,3.25, and3.26, we get

μkf2k1xkf−2k−1x6fkx−2f3kx−kf4x2kf3x−6kfxt

≥TM7

i1

μ_ϕ_x,₂_k₋₁_x

t

7 , μ

ϕx,2k1x

t

7 , μ

ϕ3x,x

t

7 , μ

ϕx,x

t

7 ,

μ_ϕ₀_,_k₁_x

k−1t

7 , μ

ϕ0,k−1x

k−1t

7 , μ

ϕ0,2kx

k−1t

7k

3.28

for allx∈Xandt >0. By3.16,3.20, and3.21, we have

μf2k1xf2k−1x−kfk2x−kf−k−2x−4fkx4kfxt

≥TM4

i1

μ_ϕ_x,_k₁_x

t

4 , μ

ϕx,k−1x

t

4 , μ

ϕ0,x

k−1t

4 , μ

ϕ0,kx

k−1t

4k

3.29

for allx∈Xandt >0. It follows from3.22and3.29that

μkfk2xkf−k−2x−2f2kx4fkx−kf3x2kf2x−5kfxt

≥TM5

i1

μ_ϕ_x,_k₁_x

t

8 , μ

ϕx,k−1x

t

8 ,

μ_ϕ₀_,x

k−1t

8 , μ

ϕ0,kx

k−1t

8k , μ

ϕ2x,x

t

2

3.30

for allx∈Xand t>0. By3.24and3.30, we have

μf3kx−4f2kx5fkx−kf3x4kf2x−5kfxt

≥TM6

i1

μ_ϕ_x,_k₁_x

t

16 , μ

ϕx,k−1x

t

16 , μ

ϕ0,x

k−1t

16 ,

μ_ϕ₀_,kx

k−1t

16k , μ

ϕ2x,x

t

4 , μ

ϕ2x,kx

t

2

3.31

for allx∈Xandt >0. By3.16and3.25–3.27, we have

μkf−k1x−kf−k−1x−k2_f₂_k₁_x_k2_f₋₂_k₋₁_x_k2_f₂_kx₋_k2₋₁_f₋₂_kx_f₄_kx₋₂_f_kx₂_kf_xt

≥TM4

i1

μ_ϕ_x,₂_k₁_x

t

4k , μ

ϕx,2k−1x

t

4k , μ

ϕx,3kx

t

4 , μ

ϕ0,3k−1x

k−1t

4k

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for allx∈Xandt >0. It follows from3.16,3.18,3.19, and3.32that

μf4kx−2f2kx−k3_f₄_x₂_k3_f₂_xt

≥TM9

i1

μ_ϕ_x,₂_k₁_x

t

24k , μ

ϕx,2k−1x

t

24k , μ

ϕx,3kx

t

24 , μ

ϕ0,3k−1x

k−1t

24k ,

μ_ϕ₂_x,₂_x

t

6k2 , μϕx,kx

t

6 , μ

ϕ0,k1x

k−1t

6k ,

μ_ϕ₀_,₂_k₋₁_x

k−1t

6k2 , μ

ϕ0,2kx

t

6k1

3.33

for allx∈Xandt >0. Hence

μf2kx−2fkx−k3_f₂_x₂_k3_f_xt

≥TM9

i1

μ_ϕ_x/₂_,₂_k₁_x/₂

t

24k , μ

ϕx/2,2k−1x/2

t

24k , μ

ϕx/2,3kx/2

t

24 ,

μ_ϕ₀_,₃_k₋₁_x/₂

k−1t

24k , μ

ϕx,x

t

6k2 , μϕx/2,kx/2

t

6 ,

μ_ϕ₀_,_k₁_x/₂

k−1t

6k , μ

ϕ0,k−1x

k−1t

6k2 , μ

ϕ0,kx

t

6k1

3.34

for allx∈Xandt >0. By3.19, we have

μf4kx−kf2k1x−kf−2k−1x−2f2kx2kf2xt≥μϕ2x,2kxt 3.35

for allx∈Xandt >0. From3.33and3.35, we have

μkf2k1xkf−2k−1x−k3_f₄_x₂_k3₋₂_k_f₂_xt

≥TM10

i1

μ_ϕ_x,₂_k₁_x

t

48k , μ

ϕx,2k−1x

t

48k , μ

ϕx,3kx

t

48 , μ

ϕ0,3k−1x

k−1t

48k ,

μ_ϕ₂_x,₂_x

t

12k2 , μ

ϕx,kx

t

12 , μ

ϕ0,k1x

k−1t

12k , μ

ϕ0,2k−1x

k−1t

12k2 ,

μ_ϕ₀_,₂_kx

t

12k1 , μ

ϕ2x,2kx

t

2

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for allx∈Xandt >0. Also, from3.28and3.36, we get

μ2f3kx−6fkxk−k3_f₄_x₋₂_kf₃_x₂_k3₋₂_k_f₂_x₆_kf_xt

≥TM17

i1

μ_ϕ_x,₂_k₋₁_x

t

14 , μ

ϕx,2k1x

t

14 ,

μ_ϕ₃_x,x

t

14 , μ

ϕx,x

t

14 , μ

ϕ0,k1x

k−1t

14 ,

μ_ϕ₀_,_k₋₁_x

k−1t

14 , μ

ϕ0,2kx

k−1t

14k , μ

ϕx,2k1x

t

96k ,

μ_ϕ_x,₂_k₋₁_x

t

96k , μ

ϕx,3kx

t

96 , μ

ϕ0,3k−1x

k−1t

96k ,

μ_ϕ₂_x,₂_x

t

24k2 , μ

ϕx,kx

t

24 , μ

ϕ0,k1x

k−1t

24k ,

μ_ϕ₀_,₂_k₋₁_x

k−1t

24k2 , μ

ϕ0,2kx

t

24k1 , μ

ϕ2x,2kx

t

4

3.37

for allx∈Xandt >0.

On the other hand, it follows from3.31and3.37that

μ8f2kx−16fkxk−k3_f₄_x₂_k3₋₁₀_k_f₂_x₁₆_kf_xt

≥TM23

i1

μ_ϕ_x,_k₁_x

t

64 , μ

ϕx,k−1x

t

64 , μ

ϕ0,x

k−1t

64 , μ

ϕ0,kx

k−1t

64k ,

μ_ϕ₂_x,x

t

16 , μ

ϕ2x,kx

t

8 , μ

ϕx,2k−1x

t

28 ,

μ_ϕ_x,₂_k₁_x

t

28 , μ

ϕ3x,x

t

28 , μ

ϕx,x

t

28 , μ

ϕ0,k1x

k−1t

28 ,

μ_ϕ₀_,_k₋₁_x

k−1t

28 , μ

ϕ0,2kx

k−1t

28k , μ

ϕx,2k1x

t

192k ,

μ_ϕ_x,₂_k₋₁_x

t

192k , μ

ϕx,3kx

t

192 , μ

ϕ0,3k−1x

k−1t

192k ,

μ_ϕ₂_x,₂_x

t

48k2 , μ

ϕx,kx

t

48 , μ

ϕ0,k1x

k−1t

48k

μ_ϕ₀_,₂_k₋₁_x

k−1t

48k2 , μϕ0,2kx

t

48k1 , μ

ϕ2x,2kx

t

8

3.38

for allx∈Xandt >0. Therefore by3.34and3.38, we get

μf4x−10f2x16fx

t

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for allx∈Xandt >0. ByCorollary 3.2, there exists a unique mappingA:X → Y such that

A2x 2Axandμf2x−8fx−Axt≥ψx2−αk3−kt/2for allx∈Xandt >0.

It remains to show thatAis an additive map. Replacingx, yby 2n_x,₂n_y_in_3.12_we

get

μ1/2n_f_k₂n_x₂n_y_f_k₂n_x₋₂n_y₋_kf₂n_x₂n_y₋_kf₂n_x₋₂n_y₋₂_f_k₂n_x₂_kf₂n_xt

≥μ

ϕx,y

2n_t

αn

3.40

for allx, y∈Xandt >0. Hence

μfkxyfkx−y−kfxy−kfx−y−2fkx2kfxt

≥TM8

i1

μAkxy−g2n_kx_y_/₂n

t

8 , μAkx−y−g2nkx−y/2n

t

8 ,

μAxy−g2n_x_y_/₂n

t

8k , μAx−y−g2nx−y/2n

t

8k ,

μAkx−g2n_kx_/₂n

t

16 , μAx−g2nx/2n

t

16k ,

μ_ϕ_x,y

2n_t

8αn1 , μ

ϕx,y

2n_t

64αn

3.41

for allx, y ∈ X andt > 0. Taking the limit asn → ∞in3.41, we conclude thatAfulfills

1.2, and so by16, Lemma 3.1, we see that the mappingx → A2x−8Axis additive, which implies that the mappingAis additive. This completes the proof.

Similar toTheorem 3.3, one can prove the following result.

for allx, y ∈Xandt >0. If for some0< α <8,μ_ϕ₂_x,₂_yt≥μ_αϕ_x,ytfor allx, y∈Xandt >0, then there exists a unique cubic mappingC:X → Y such that

μf2x−2fx−Cxt≥ψx

8−αk3₋_k_t 2

3.43

for allx∈Xandt >0, whereψxtis defined as inTheorem 3.3.

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for allx, y∈Xandt >0. If for some0< α <2,

μ_ϕ₂_x,₂_yt≥μ_αϕ_x,yt _3.45

for allx, y∈Xandt >0, then there exist a unique additive mappingA:X → Yand a unique cubic mappingC:X → Y such that

μfx−Ax−Cxt≥ψx

32−αk3₋_k_t 2

3.46

for allx∈Xandt >0, whereψxtis defined as inTheorem 3.3.

Proof. By Theorems3.3and3.4, there exist an additive mappingA1 : X → Y and a cubic mappingC1 :X → Ysuch that

μf2x−8fx−A1xt≥ψx

2−αk3₋_k_t 2

, 3.47

μf2x−2fx−C1xt≥ψx

8−αk3₋_k_t 2

3.48

for allx∈Xandt >0. Therefore from3.47and3.48, we get

μ

fx1

6A1x− 1 6C1x

t≥ψx

32−αk3₋_k_t 2

3.49

for allx∈Xandt >0. LettingAx −1/6A1xandCx 1/6C1xfor allx∈X, it follows from3.49that

32−αk3₋_k_t 2

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for allx∈ Xandt >0. To prove the uniqueness ofAandC, letA, C: X → Y be another additive and cubic mapping satisfying3.46. SetAA−AandCC−C. So

μ_A_x_C_xt≥ TM

μAxCx−fx

t

2 , μfx−Ax−Cx

t

2

≥ψx

32−αk3₋_k_t 4

3.51

for allx∈Xandt >0. ByA2x 2Ax,C2x 8Cx, and3.51, we get

μCxt≥ TM

μA2n_x_C₂n_x

8nt

2 , μA2n_x

8nt

2

≥ TM

ψx

32−αk3₋_k₈n_t

4αn

, μAx

4n_t

2

3.52

for allx ∈ X and t > 0. Since the right hand side of the inequality tends to 1 asntend to infinity, we find thatCx 0. ThereforeC0, and thenA0. This completes the proof.

Remark 3.7. We can formulate similar statements toTheorem 3.6forα >8.

Corollary 3.8. LetX, · be a normed space,Z, μ,TMbe an RN-space, andY, μ,TMbe a complete RN-space. Letpbe a non-negative real number such thatp∈0,1∪1,3∪3,∞,and let

z0∈Z. Iff:X → Yis a mapping withf0 0such that

μfkxyfkx−y−kfxy−kfx−y−2fkx2kfxt≥μ_xp_yp_z

0t 3.53

μfx−Ax−Cxt≥ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

μ_xp_z 0

2−2p_k2₋₁_t 25613k

, p∈0,1,

μ_xp_z 0

⎛ ⎜

⎝2p−2

k2₋₁_t 2561 3k3

⎞ ⎟

⎠, p∈1,log₂5,

μ_xp_z 0

⎛ ⎜

⎝8−2p

k2−1t

2561 3k3

⎞ ⎟

⎠, p∈log₂5,3,

μ_xp_z 0

2p₋₈_k2₋₁_t 2561 3kp

, p∈3,∞,

3.54

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Corollary 3.9. LetX, · be a normed space,Z, μ,TMbe an RN-space, andY, μ,TMbe a complete RN-space. Letr, sbe non-negative real numbers such thatλ:rs∈0,1∪1,3∪3,∞,

and letz0∈Z. Iff:X → Y be a mapping withf0 0such that

μfkxyfkx−y−kfxy−kfx−y−2fkx2kfxt≥μ_xr_ys_xrs_yrs_z

0t 3.55

μfx−Ax−Cxt≥ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

μ

xλ_z 0

2−2λ_k2₋₁_t 25616k

, λ∈0,1,

μ

xλ_z 0

⎛ ⎜ ⎝

2λ₋₂_k2₋₁_t 256123k3

⎞ ⎟

⎠, λ∈1,log₂5,

μ

xλ_z 0

⎛ ⎜ ⎝

8−2λk2−1t

256123k3

⎞ ⎟

⎠, λ∈log₂5,3,

μ

xλ_z 0

⎛ ⎜ ⎝

2λ₋₈_k2₋₁_t 256123kλ

⎞ ⎟

⎠, λ∈3,∞,

3.56

Now, we give one example to illustrate the main results ofTheorem 3.6. This example is a modification of the example of Zhang et al.34.

Example 3.10. LetX, · be a Banach algebra,x0be a unit vector inX,andμxtis defined as inExample 2.3. It is easy to see thatX, μ,TMis a completeRN-space.

Definef :X → Xbyfx x3_xp_x

0forx∈X. For 0< p <1, define

ϕx, y8kxpypx0, x, y∈X. 3.57

Since 0< p <1, the inequalityabp≤apbpholds whena≥0 andb≥0. A straightforward computation shows that

_f

kxyfkx−y−kfxy−kfx−y−2fkx 2kfx≤8kxpyp

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for allx, y ∈ X. Therefore, all the conditions ofTheorem 3.6hold, and there exist a unique additive mappingA:X → Xand a unique cubic mappingC:X → Xsuch that

32−αk3₋_k_t 2

3.59

for allx∈Xandt >0, whereψxtis defined as inTheorem 3.3.

Acknowledgments

The authors would like to thank the area editor professor Radu Precup and two anonymous referees for their valuable comments and suggestions. T. Z. Xu was supported by the National Natural Science Foundation of China10671013.

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