Erratum to: A Note to Paper "On the Stability of Cubic Mappings and Quartic Mappings in Random Normed Spaces"

Volume 2009, Article ID 214530,6pages doi:10.1155/2009/214530

Erratum

A Note to Paper “On the Stability of

Cubic Mappings and Quartic Mappings in

Random Normed Spaces”

R. Saadati,

_{S. M. Vaezpour,}

_{and Y. J. Cho}

1_{Department of Mathematics and Computer Science, Amirkabir University of Technology,}

424 Hafez Avenue, Tehran 15914, Iran

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

Chinju 660-701, South Korea

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

Received 16 February 2009; Accepted 21 April 2009

Recently, Baktash et al.2008proved the stability of the cubic functional equationf2xyf2x−

y 2fxy 2fx−y 12fxand the quartic functional equationf2xy f2x−y

4fxy4fx−y24fx−6fyin random normed spaces. In this note, we correct the proofs. Copyrightq2009 R. Saadati 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

If inf{t >0 :Ft> a} ≤inf{t >0 :Gt> a}, in general we cannot conclude thatFt≥Gt. For example, letFt 3/4,Gt t/t1anda 1/2. We know that inf{t > 0 : 3/4 > 1/2}0 ≤inf{t >0 :t/t1>1/2}1 butF4 3/4< G4 4/5. This example shows that in1, inequalities2.13and3.13do not follow from inequalities2.12and3.12.

The functional equation

f2xyf2x−y2fxy2fx−y12fx 1.1

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 solved by Jun and Kim2and Lee3for mappings f : X → Y, whereX is 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 equation4. The functional equation

is said to be the quartic functional equation since the function fx cx4 _{is its solution.} Every solution of the quartic functional equation is said to be a quartic mapping. The stability problem for the quartic functional equation first was solved by Rassias5and Lee and Chung 6for mappingsf:X → Y, whereXis a real normed space andY is a Banach space.

In the sequel, we shall adopt the usual terminology, notations and conventions of the theory of random normed spaces as in 7–15. 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 alltinR. The maximal element forΔin this order is the distribution function given by

ε0t ⎧ ⎨ ⎩

0, ift≤0,

1, ift≤0. 1.4

Definition 1.1see13. 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− 1,0andTa, b mina, b.

Recall that, ifT is at-norm and{an}is a given sequence of numbers in0,1,T_in₁aiis

defined recursively byT1

i1aia1andT_in₁aiTT_in₁−1ai, anforn≥2.

Definition 1.2. Arandom normed space briefly, RN-space is a triple X, μ, T, where X is a vector space, T is a continuous t-norm andμ is a mapping fromX into D such that the following conditions hold:

PN1μxt ε0tfor allt >0 if and only ifx0;

PN2μαxt μxt/|α|for allxinX,α /0 andt≥0;

PN2μxyts≥Tμxt, μysfor allx, y∈Xandt, 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}inX is calledCauchy sequenceif, for everyt > 0 andε >0, there

3An RN-spaceX, μ, Tis said to becompleteif and only if every Cauchy sequence in Xis convergent to a point inX.

Theorem 1.4see13. IfX, μ, Tis an RN-space and{xn}is a sequence such thatxn → x, then

limn→ ∞μxnt μxt.

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

2. On the Stability of Cubic Mappings in RN-Spaces

Theorem 2.1. LetXbe a linear space,Z, μ,minbe an RN-space,ϕ:X×X → Zbe a 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 ∈X andt >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∈X, t >0, 2.2

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

μfx−Cxt≥μ_ϕx,028−αt. 2.3

Proof. Puttingy0 in2.2, we get

μf2x/8−fxt≥μϕx,016t, ∀x∈X. 2.4

Replacingxby 2n_{xin2.4and using2.1, we obtain}

μf2n1_x/8n1_−f2n_x/8nt≥μ_ϕ2n_x,016×8n

≥μ_ϕx,0

16×8n

αn .

2.5

It follows fromf2nx/8n−fx nk−01f2k1x/8k1−f2kx/8kand2.5that

μf2n_x/8n_−fx

t

n−1

k0 αk

16×8k

≥Tn−1

k0

that is,

μf2n_x/8n_−fxt≥μ_ϕx,0

t n−1

k0

αk_/16×8k

. 2.7

By replacingxwith 2m_{xin2.7, we observe that}

μf2nm_x/8nm_−f2m_x/8mt≥μ

ϕx,0

t _nm

km

αk_/16×8k

. 2.8

Asμ_ϕx,0t/_knm_mαk_/16×8k_{tends to 1 asm, ntend to∞, then{f2}n_x/8n_{}is a Cauchy}

sequence inY, μ,min. SinceY, μ,minis a complete RN-space, this sequence converges to some pointCx∈Y. Fixx∈Xand putm0 in2.8. Then we obtain

μf2n_x/8n_−fxt≥μ_ϕx,0

t _n−1

k0

αk_/16×8k

2.9

and so, for everyδ >0, we have

μCx−fxtδ≥T

μCx−f2n_x/8nδ, μ_f2n_x/8n_−fxt

≥T

μ_Cx−f2n_x/8nδ, μ_ϕx,0

t n−1

k0

αk_/16×8k

.

2.10

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

μCx−fxtδ≥μϕx,02t8−α. 2.11

Sinceδwas arbitrary, by takingδ → 0 in2.11, we get

μCx−fxt≥μ_ϕx,02t8−α. 2.12

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

μ_f2n_2xy/8n_f2n_2x−y/8n_−2f2n_xy/8n_−2f2n_x−y/8n_−12f2n_x/8nt

≥μ_ϕ2n_x,2n_y 8

n_t 2.13

for all x, y ∈ X and for all t > 0. Since limn→ ∞μϕ2n_x,2n_y8nt 1, we conclude that C

cubic mappingD : X → Y which satisfies2.3. Fixx ∈ X. Clearly,C2n_{x 8}n_Cxand

D2n_{x 8}n_{Dxfor 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 , μD2nx/8n−f2nx/8n

t 2

≥μ_ϕ2n_x,08n8−αt

≥μ_ϕx,0

8n8−αt αn .

2.14

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.

3. On the Stability of Quartic Mappings in RN-Spaces

Theorem 3.1. LetXbe a linear space,Z, μ,minbe an RN-space,ϕ:X×X → Zbe a function such that for some0< α <16,

μ_ϕ2x,0t≥μ_αϕx,0t, ∀x∈X, t >0, 3.1

f0 0andlimn→ ∞μϕ2n_x,2n_y16nt 1for allx, y ∈X andt >0. LetY, μ,minbe a complete RN-space. Iff:X → Y is a mapping such that

μ_{f2xyf2x−y−4fxy−4fx−y−24fx6fy}t≥μ_ϕx,yt, ∀x∈X, t >0, 3.2

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

μfx−Qxt≥μϕx,0216−αt. 3.3

Proof. The proof is the same asTheorem 2.1.

Acknowledgments

References

1 E. Baktash, Y. J. Cho, M. Jalili, R. Saadati, and S. M. Vaezpour, “On the stability of cubic mappings and quadratic mappings in random normed spaces,”Journal of Inequalities and Applications, vol. 2008, Article ID 902187, 11 pages, 2008.

2 K.-W. Jun and H.-M. Kim, “The generalized Hyers-Ulam-Rassias stability of a cubic functional equation,”Journal of Mathematical Analysis and Applications, vol. 274, no. 2, pp. 867–878, 2002. 3 Y.-S. Lee, “Stability of a quadratic functional equation in the spaces of generalized functions,”Journal

of Inequalities and Applications, vol. 2008, Article ID 210615, 12 pages, 2008.

4 K.-W. Jun, H.-M. Kim, and I.-S. Chang, “On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation,”Journal of Computational Analysis and Applications, vol. 7, no. 1, pp. 21–33, 2005. 5 J. M. Rassias, “Solution of the Ulam stability problem for quartic mappings,”Glasnik Matematiˇcki, vol.

34, no. 2, pp. 243–252, 1999.

6 Y.-S. Lee and S.-Y. Chung, “Stability of cubic functional equation in the spaces of generalized functions,”Journal of Inequalities and Applications, vol. 2007, Article ID 79893, 13 pages, 2007.

7 S. Chang, Y. J. Cho, and S. M. Kang,Nonlinear Operator Theory in Probabilistic Metric Spaces, Nova Science, Huntington, NY, USA, 2001.

8 O. Hadˇzi´c and E. Pap,Fixed Point Theory in Probabilistic Metric Spaces, vol. 536 ofMathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.

9 D. Mihet¸ and V. Radu, “On the stability of the additive Cauchy functional equation in random normed spaces,”Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 567–572, 2008.

10 D. Mihet¸, R. Saadati, and S. M. Vaezpour, “The stability of the quartic functional equation in random normed spaces,”Acta Applicandae Mathematicae, 2009.

11 D. Mihet¸, R. Saadati, and S. M. Vaezpour, “The stability of an additive functional equation in Menger probabilisticϕ-normed spaces,”Mathematica Slovaca. In press.

12 R. Saadati and S. M. Vaezpour, “Linear operators in finite dimensional probabilistic normed spaces,”

Journal of Mathematical Analysis and Applications, vol. 346, no. 2, pp. 446–450, 2008.

13 B. Schweizer and A. Sklar,Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland, New York, NY, USA, 1983.

14 A. N. ˇSerstnev, “On the concept of a stochastic normalized space,”Doklady Akademii Nauk SSSR, vol. 149, pp. 280–283, 1963Russian.