On the Stability of Cubic Mappings and Quadratic Mappings in Random Normed Spaces

Journal of Inequalities and Applications Volume 2008, Article ID 902187,11pages doi:10.1155/2008/902187

Research Article

On the Stability of Cubic Mappings and Quadratic

Mappings in Random Normed Spaces

E. Baktash,1 _{Y. J. Cho,}2_{M. Jalili,}3_{R. Saadati,}4, 5_{and S. M. Vaezpour}4

1_{Youngs Researchers Club and Department of Basic Sciences, Islamic Azad University,}

Ayatollah Amoli Branch, P.O. Box 678, Amol, Iran

2_{Department of Mathematics Education and the RINS, Gyeongsang National University,}

Chinju 660-701, South Korea

3_{Department of Mechanical Engineering, Islamic Azad University, Ayatollah Amoli Branch,}

P.O. Box 678, Amol, Iran

4_{Department of Mathematics and Computer Science, Amirkabir University of Technology,}

424 Hafez Avenue, Tehran 15914, Iran

5_{Faculty of Sciences, University of Shomal, P.O. Box 731, Amol, Iran}

Correspondence should be addressed to Y. J. Cho,[email protected]

Received 27 August 2008; Revised 23 October 2008; Accepted 24 November 2008

Recommended by Wing-Sum Cheung

Recently, the stability of the cubic functional equationf2xy f2x−y 2fxy 2fx− y 12fxin fuzzy normed spaces was proved in earlier work; and the stability of the additive functional equationsfxy fx fy, 2fxy/2 fx fyin random normed spaces was proved as well. In this paper, we prove the stability of the cubic functional equation

f2xy f2x−y 2fxy 2fx−y 12fxin random normed spaces by an alternative proof which provides a better estimation. Finally, we prove the stability of the quartic functional equationf2xy f2x−y 4fxy 4fx−y 24fx−6fyin random normed spaces.

Copyrightq2008 E. Baktash 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.

1. Introduction and preliminaries

The study of stability problems for functional equations is related to a question of Ulam

1 concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers2. Subsequently, the result of Hyers was generalized by Aoki3

for additive mappings and by Th. M. Rassias 4 for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias has provided a lot of influence in the development of what we now call Hyers-Ulam-Rassias stability of functional equations. We refer the interested readers for more information on such problems to the papers5–9. The functional equation

is said to be the cubic functional equation since the functionfx cx3_{is its solution. Every} solution of the cubic functional equation is said to be a cubic mapping. The stability problem for the cubic functional equation was proved by Jun and Kim10for mappingsf :X → Y, whereXis a real normed space andY is a Banach space. Later, a number of mathematicians have worked on the stability of some types of the cubic equation11. The functional equation

f2xy f2x−y 4fxy 4fx−y 24fx−6fy 1.2

is said to be the quadratic functional equation since the functionfx cx4 _{is its solution.} Every solution of the quadratic functional equation is said to be a quadratic mapping. The stability problem for the quadratic functional equation first was proved by J. M. Rassias

12for mappingsf : X → Y, where X is a real normed space andY is a Banach space. In addition, Mirmostafaee et al.13–15, Alsina16, Mihet¸ and Radu17investigated the stability in the settings of fuzzy, probabilistic, and random normed spaces.

In the sequel, we will adopt the usual terminology, notations, and conventions of the theory of random normed spaces as in17–21. Throughout this paper, the space of all probability distribution functions is denoted by

Δ _{F:R∪ {−∞,∞} −→0,1:F is left-continuous}

and nondecreasing onRandF0 0, F∞ 1, 1.3

and the subsetD ⊆ Δis the setD {F ∈ Δ : l−F∞ 1}, wherel−fxdenotes the left limit of the functionfat the pointx. The spaceΔis partially ordered by the usual point-wise ordering of functions, that is,F≤Gif and only ifFt≤Gtfor allt∈R. The maximal element forΔin this order is the distribution function given by

ε0t

⎧ ⎨ ⎩

0, ift≤0,

1, ift >0. 1.4

Definition 1.1see17. A functionT :0,1×0,1 → 0,1is a continuous triangular norm

briefly, at-normifTsatisfies the following conditions:

aT is commutative and associative;

bT is continuous;

cTa,1 afor alla∈0,1;

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

Three typical examples of continuoust-norms areTa, b ab,Ta, b maxab−

1,0, andTa, b mina, b.

the following conditions hold:

PN1μxt ε0tfor allt >0 if and only ifx0;

PN2μαxt μxt/|α|for allxinX,α /0 and allt≥0;

PN3μxyts≥Tμxt, μysfor allx, y∈Xand allt, s≥0.

Definition 1.3. LetX, μ, Tbe an RN-space.

1A sequence{xn}inX is said to beconvergenttoxinXif, for everyt >0 andε >0,

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

2A sequence{xn}inXis called aCauchy sequenceif, for everyt >0 andε >0, there

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

3An RN-spaceX, μ, Tis said to becompleteif and only if every Cauchy sequence in

Xis convergent to a point inX.

Theorem 1.4see20. IfX, μ, Tis an RN-space and{xn}is a sequence such thatxn → x, then

limn→ ∞μxnt μxt.

Lemma 1.5. LetX, μ,minbe an RN-space and defineEλ,μ:X → R∪ {0}by

Eλ,μx inf{t >0 :μxt>1−λ}, ∀λ∈0,1, x∈X. 1.5

Then, one has

Eλ,μx1−xn≤Eλ,μx1−x2 · · ·Eλ,μxn−1−xn, 1.6

for allx1, . . . , xn∈Xand the sequence{xn}is convergent toxwith respect to random normμif and

only ifEλ,μxn−x → 0asn → ∞. Also, the sequence{xn}is a Cauchy sequence with respect to

random normμif and only if it is a Cauchy sequence withEλ,μ.

Proof. By the triangular inequality, we have

μx1−xn

Eλ, μx1−x2 · · ·Eλ, μxn−1−xn n−1δ

≥minμx1−x2

Eλ, μx1−x2 δ

, . . . , μxn−1−xn

Eλ, μxn−1−xn δ

>min1−λ, . . . ,1−λ 1−λ, ∀δ >0,

1.7

which implies that

Eλ,μx1−xn≤Eλ,μx1−x2 Eλ,μx2−x3 · · ·Eλ,μxn−1−xn n−1δ. 1.8

Sinceδ >0 is arbitrary, we have

Next, we haveμxn−xη>1−λ⇔Eλ,μxn−x< ηfor everyη >0. This completes the

proof.

In this paper, we establish the stability of the cubic and quadratic functional equations in the setting of random normed spaces.

2. On the stability of cubic mappings in RN-spaces

Theorem 2.1. LetX be a linear space,Z, μ,minan RN-space, andϕ : X×X → Za function such that for some0< α <8,

μ_ϕ2x,0t≥μ_αϕx,0t, ∀x∈X, t >0, 2.1

f0 0andlimn→ ∞μϕ2n_x,2n_y8nt 1for allx, y∈Xand allt >0. LetY, μ,minbe a complete

RN-space. Iff:X → Y is a mapping such that

μf2xyf2x−y−2fxy−2fx−y−12fxt≥μϕx,yt, ∀x, y∈X, t >0, 2.2

then there exists a unique cubic mappingC:X → Y such that

μfx−Cxt≥μϕx,028−αt. 2.3

Proof. From2.2, it follows that

Eλ, μ

f2xy f2x−y−2fxy−2fx−y−12fx inft >0 :μf2xyf2x−y−2fxy−2fx−y−12fxt>1−λ

≤inft >0 :μ_{ϕx, y}t>1−λ

Eλ, μϕx, y, ∀x, y∈X, λ∈0,1.

2.4

Puttingy0 in2.4, we get

Eλ, μ f

2x

8 −fx

≤ 1

16Eλ, μϕx,0, ∀x∈X. 2.5

Replacingxby 2n_{xin2.5and using2.1, we obtain}

Eλ, μ

f2n1x

8n1 −

f2nx

8n

≤ 1

16×8nEλ, μ

ϕ2nx,0

≤ αn

16×8nEλ, μϕx,0.

It follows fromf2nx/8n_−fx n−1

k0f2k1x/8k1−f2kx/8kand2.6that

Eλ, μ

f2nx

8n −fx

Eλ, μ n−1

k0

f2k1x

8k1 −

f2kx

8k

≤n−1

k0

Eλ, μ f

2k1x

8k1 −

f2kx

8k

≤n−1

k0 1 16×8kEλ, μ

ϕ2kx,0

≤n−1

k0

αk

16×8kEλ, μ

ϕx,0.

2.7

Replacingxwith 2m_{xin2.7, we observe that}

Eλ, μ

f2nmx

8nm −

f2mx

8m

≤n−1

k0

αk

16×8kmEλ, μ

ϕ2mx,0

≤n−1

k0

αkm

16×8kmEλ, μϕx,0

≤mn−1

km

αk

16×8kEλ, μϕx,0

Eλ, μϕx,0

16

mn−1

km α 8 k . 2.8

Then{f2nx/8n_{} is a Cauchy sequence inY, μ,min. SinceY, μ,minis a complete}

RN-space, this sequence converges to some pointCx ∈ Y. Fixx ∈ X and putm 0 in2.8. Then we obtain

Eλ,μ

f2nx

8n −fx

≤ Eλ,μϕx,0

16

n−1

k0

α

8

k

, 2.9

and so

Eλ,μCx−fx≤Eλ,μ Cx− f

2nx

8n

Eλ,μ f

2nx

8n −fx

≤Eλ,μ Cx− f

2nx

8n

Eλ,μϕx,0

16

n−1

k0

Taking the limit asn → ∞and using2.10, we get

Eλ,μCx−fx≤

Eλ,μϕx,0

16−2α , 2.11

that is,

inf{t >0 :μCx−fxt>1−λ} ≤inf{t >0 :μ_ϕx,02t8−α>1−λ}. 2.12

Then, we have

μCx−fxt≥μϕx,02t8−α. 2.13

Replacingxandyby 2n_{xand 2}n_{yin2.2, respectively, we get}

μf2n_2xy/8n_f2n_2x−y/8n_−2f2n_xy/8n_−2f2n_x−y/8n_−12f2n_x/8nt

≥μ_ϕ2n_x,2n_y8nt, ∀x, y∈X, t >0.

2.14

Since limn→ ∞μϕ2n_x,2n_y8nt 1, we conclude thatCfulfills1.1.

To prove the uniqueness of the cubic mappingC, assume that there exists a cubic mapping D : X → Y which satisfies 2.3. Fix x ∈ X. Clearly, C2nx 8nCx and

D2nx 8nDxfor alln∈N. It follows from2.3that

μCx−Dxt lim

n→ ∞μC2nx/8n−D2nx/8nt,

μC2n_x/8n_−D2n_x/8nt≥min

μC2n_x/8n_−f2n_x/8n

t

2

, μD2n_x/8n_−f2n_x/8n

t

2

≥μ_ϕ2n_x,08n8−αt

≥μ_ϕx,0

8n_8−αt

αn

.

2.15

Since limn→ ∞8n8−αt/αn ∞, we get limn→ ∞μ_ϕx,08n8−αt/αn 1. Therefore, it

follows thatμCx−Dxt 1 for allt >0 and soCx Dx. This completes the proof.

Corollary 2.2. LetX be a linear space,Z, μ,minan RN-space, andY, μ,mina complete RN-space. Letp, qbe nonnegative real numbers and letz0 ∈Z. Iff :X → Y is a mapping such that

μf2xyf2x−y−2fxy−2fx−y−12fxt≥μxp_yq_z

0t, ∀x, y∈X, t >0, 2.16

f0 0andp, q <3, then there exists a unique cubic mappingC:X → Y such that

μfx−Cxt≥μxp_z

0

Proof. Letϕ : X×X → Zbe defined byϕx, y xp_yq_{z0. Then the proof follows}

fromTheorem 2.1byα2p_.

Corollary 2.3. LetX be a linear space,Z, μ,minan RN-space, andY, μ,mina complete RN-space. Letz0∈Z. Iff:X → Yis a mapping such that

μf2xyf2x−y−2fxy−2fx−y−12fxt≥μεz0t, ∀x, y∈X, t >0, 2.18

andf0 0, then there exists a unique cubic mappingC:X → Y such that

μfx−Cxt≥μεz014t, ∀x∈X, t >0. 2.19

Proof. Let ϕ : X ×X → Z be defined by ϕx, y εz0. Then, the proof follows from

Theorem 2.1byα1.

3. On the stability of quadratic mappings in RN-spaces

Theorem 3.1. LetX be a linear space, (Z, μ,min) an RN-space, andϕ :X×X → Za function such that for some0< α <16,

μ_ϕ2x,0t≥μ_αϕx,0t, ∀x∈X, t >0, 3.1

f0 0 and limn→ ∞μ_ϕ2n_x,2n_y16nt 1 for allx, y ∈ X and allt > 0. Let Y, μ,min be a

complete RN-space. Iff:X → Yis a mapping such that

μf2xyf2x−y−4fxy−4fx−y−24fx6fyt≥μϕx,yt, ∀x, y∈X, t >0, 3.2

then there exists a unique quadratic mappingQ:X → Ysuch that

μfx−Qxt≥μ_ϕx,0216−αt. 3.3

Proof. From3.2, it follows that

Eλ, μ

f2xy f2x−y−4fxy−4fx−y−24fx 6fy inft >0 :μf2xyf2x−y−4fxy−4fx−y−24fx6fyt>1−λ

≤inft >0 :μ_{ϕx, y}t>1−λ

Eλ, μϕx, y, ∀x, y∈X, λ∈0,1.

3.4

Puttingy0 in3.4, we get

Eλ,μ

f2x

16 −fx

≤ 1

Replacingxby 2n_{xin3.5and using3.1, we obtain}

Eλ, μ

f2n1x

16n1 −

f2nx

16n

≤ 1

32×16nEλ, μ

ϕ2nx,0

≤ αn

32×16nEλ, μϕx,0.

3.6

It follows fromf2nx/16n−fx nk−10f2k1x/16k1−f2kx/8kand3.6that

Eλ,μ

f2nx

16n −fx

Eλ,μ n−1

k0

f2k1x

16k1 −

f2kx

16k

≤n−1

k0

Eλ,μ

f2k1x

16k1 −

f2kx

16k

≤n−1

k0 1

32×16kEλ,μϕ2

k_x,0

≤n−1

k0

αk

32×16kEλ,μϕx,0.

3.7

Replacingxwith 2m_{xin3.7, we observe that}

Eλ, μ

f2nmx

16nm −

f2mx

16m

≤n−1

k0

αk

32×16kmEλ, μ

ϕ2mx,0

≤n−1

k0

αkm

32×16kmEλ, μϕx,0

≤mn−1

km

αk

32×16kEλ, μϕx,0

Eλ, μϕx,0

32

mn−1

km α 16 k . 3.8

Then{f2nx/16n}is a Cauchy sequence inY, μ,min. SinceY, μ,minis a complete RN-space, this sequence converges to some pointQx ∈ Y. Fixx ∈ Xand putm 0 in3.8. Then we obtain

Eλ,μ

f2nx

16n −fx

≤ Eλ,μϕx,0

32

n−1

k0

α

16

k

and so

Eλ,μQx−fx≤Eλ,μ

Qx−f2

n

x

16n

Eλ,μ

f2nx

16n −fx

≤Eλ,μ

Qx−f2

n_x

16n

Eλ,μϕx,0

32

n−1

k0

α

16

k

.

3.10

Taking the limit asn → ∞and using3.10, we get

Eλ,μQx−fx≤

Eλ,μϕx,0

32−2α , 3.11

that is,

inf{t >0 :μQx−fxt>1−λ} ≤inf{t >0 :μϕx,02t16−α>1−λ}. 3.12

Then, we have

μQx−fxt≥μϕx,02t16−α. 3.13

Replacingxandyby 2n_{xand 2}n_{yin3.2, respectively, we get}

μf2n_2xy/16n_f2n_2x−y/16n_−4f2n_xy/16n_−4f2n_x−y/16n_−24f2n_x/16n_6f2n_y/16nt ≥μ_ϕ2n_x,2n_y16nt, ∀x, y∈X, t >0.

3.14

Since limn→ ∞μϕ2n_x,2n_y16nt 1, we conclude thatQfulfills1.2.

To prove the uniqueness of the quadratic mapping Q, assume that there exists a quadratic mappingD : X → Y which satisfies3.3. Fixx ∈X. Clearly,Q2nx 16nQx

andD2nx 16nDxfor alln∈N. It follows from3.3that

μQx−Dxt _nlim_{→ ∞}μQ2nx/16n_−D2n_x/16nt,

μQ2n_x/16n_−D2n_x/16nt≥min

μQ2n_x/16n_−f2n_x/16n

t

2

, μD2n_x/8n_−f2n_x/8n

t

2

≥μ_ϕ2n_x,0

16n16−αt

≥μ_ϕx,0

16n16−αt αn

.

3.15

Since limn→ ∞16n16−αt/αn ∞, we get limn→ ∞μ_ϕx,016n16−αt/αn 1. Therefore, it

Corollary 3.2. LetX be a linear space,Z, μ,minan RN-space, andY, μ,mina complete RN-space. Letp, qbe nonnegative real numbers and letz0 ∈Z. Iff :X → Y is a mapping such that

μf2xyf2x−y−4fxy−4fx−y−24fx6fyt≥μxp_yq_z

0t, ∀x, y∈X, t >0, 3.16

f0 0andp, q <4, then there exists a unique quadratic mappingQ:X → Ysuch that

μfx−Qxt≥μ_xp_z

0

216−2pt, ∀x∈X, t >0. 3.17

Proof. Letϕ :X×X → Zbe defined byϕx, y xp_yq_z

0. Then, the proof follows fromTheorem 3.1byα2p.

Corollary 3.3. LetX be a linear space,Z, μ,minan RN-space, andY, μ,mina complete RN-space. Letz0∈Z. Iff:X → Yis a mapping such that

μf2xyf2x−y−4fxy−4fx−y−24fx6fyt≥μεz0t, ∀x, y∈X, t >0, 3.18

andf0 0, then there exists a unique quadratic mappingQ:X → Ysuch that

μfx−Qxt≥μεz030t, ∀x∈X, t >0. 3.19

Proof. Let ϕ : X ×X → Z be defined by ϕx, y εz0. Then, the proof follows from

Theorem 3.1byα1.

Acknowledgments

The authors would like to thank the referees for giving useful comments and suggestions for the improvement of this paper. The second author was supported by the Korea Research Foundation Grant funded by the Korean GovernmentMOEHRD KRF-2007-313-C00040.

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