Volume 2009, Article ID 868423,14pages doi:10.1155/2009/868423
Research Article
Quadratic-Quartic Functional Equations in
RN-Spaces
M. Eshaghi Gordji,
1M. Bavand Savadkouhi,
1and Choonkil Park
2 1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran2Department of Mathematics, Hanyang University, Seoul 133-791, South Korea
Correspondence should be addressed to Choonkil Park,[email protected]
Received 20 July 2009; Accepted 3 November 2009
Recommended by Andrea Laforgia
We obtain the general solution and the stability result for the following functional equation in random normed spacesin the sense of Sherstnevunder arbitraryt-normsf2xy f2x−y
4fxy fx−y 2f2x−4fx−6fy.
Copyrightq2009 M. Eshaghi Gordji 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
The stability problem of functional equations originated from a question of Ulam1in 1940,
concerning the stability of group homomorphisms. LetG1,·be a group and let G2,∗, d
be a metric group with the metricd·,·.Given >0, does there exist aδ > 0 such that if a mappingh: G1 → G2 satisfies the inequalitydhx·y, hx∗hy < δfor allx, y ∈ G1,
then there exists a homomorphismH :G1 → G2withdhx, Hx< for allx∈ G1? In
other words, under what condition does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Hyers2 gave a first affirmative answer to the question of Ulam for Banach spaces. Letf:E → Ebe a mapping between Banach spaces such that
fxy−fx−fy≤δ 1.1
for allx, y∈Eand someδ >0.Then there exists a unique additive mappingT :E → Esuch that
for allx∈E.Moreover, ifftxis continuous int∈Rfor each fixedx∈E,thenT isR-linear. In 1978, Rassias3provided a generalization of Hyers’ theorem which allows the Cauchy difference to be unbounded. In 1991,Gajda 4 answered the question for the casep > 1, which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias stability of functional equationssee5–12. The functional equation
fxyfx−y2fx 2fy 1.3
is related to a symmetric biadditive mapping. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation1.3is said to be a quadratic mapping. It is well known that a mappingfbetween real vector spaces is quadratic if and only if there exits a unique symmetric biadditive mappingBsuch thatfx Bx, xfor allxsee5,13. The biadditive mappingBis given by
Bx, y 1
4
fxy−fx−y. 1.4
The Hyers-Ulam-Rassias stability problem for the quadratic functional equation1.3was proved by Skof for mappingsf:A → B, whereAis a normed space andBis a Banach space
see14. Cholewa15noticed that the theorem of Skof is still true if relevant domainAis replaced an abelian group. In16, Czerwik proved the Hyers-Ulam-Rassias stability of the functional equation1.3. Grabiec17has generalized the results mentioned above.
In18, Park and Bae considered the following quartic functional equation
fx2yfx−2y4fxyfx−y6fy−6fx. 1.5
In fact, they proved that a mappingf between two real vector spacesX andY is a solution of1.5if and only if there exists a unique symmetric multiadditive mappingM :X4 → Y
such thatfx Mx, x, x, xfor allx. It is easy to show that the functionfx x4satisfies
the functional equation1.5, which is called a quartic functional equationsee also19. In addition, Kim20has obtained the Hyers-Ulam-Rassias stability for a mixed type of quartic and quadratic functional equation.
The Hyers-Ulam-Rassias stability of different functional equations in random normed and fuzzy normed spaces has been recently studied in21–26. It should be noticed that in all these papers the triangle inequality is expressed by using the strongest triangular normTM.
The aim of this paper is to investigate the stability of the additive-quadratic functional equation in random normed spacesin the sense of Sherstnevunder arbitrary continuous
t-norms.
In this sequel, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in22,23,27–29. Throughout this paper,Δis the space of distribution functions, that is, the space of all mappingsF:R∪{−∞,∞} → 0,1such that
Fis left-continuous and nondecreasing onR, F0 0 andF∞ 1. Also,Dis a subset of
Δconsisting of all functionsF ∈ Δ for whichl−F∞ 1, wherel−fxdenotes the left
ordered by the usual point-wise ordering of functions, that is,F≤Gif and only ifFt≤Gt
for alltinR. The maximal element forΔin this order is the distribution functionε0given by
ε0t ⎧ ⎨ ⎩
0, ift≤0,
1, ift >0.
1.6
Definition 1.1see28. A mappingT :0,1×0,1 → 0,1is a continuous triangular norm
briefly, a continuoust-normifT satisfies 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.
Typical examples of continuoust-norms areTPa, b ab,TMa, b mina, band
TLa, b maxab−1,0 the Lukasiewiczt-norm. Recallsee30,31that ifT is a t
-norm and{xn}is a given sequence of numbers in0,1, thenTn
i1xiis defined recurrently by
T1
i1xi x1andTin1xi TTin−11xi, xnforn≥2.Ti∞nxiis defined asTi∞1xni−1.It is known31
that for the Lukasiewiczt-norm, the following implication holds:
lim
n→ ∞TL ∞
i1xni−11⇐⇒
∞
n1
1−xn<∞. 1.7
Definition 1.2see29. Arandom normed spacebriefly, RN-spaceis a tripleX, μ, T, where
Xis a vector space,T is a continuoust-norm, andμis a mapping fromXintoD such that the following conditions hold:
RN1μxt ε0tfor allt >0 if and only ifx0; RN2μαxt μxt/|α|for allx∈X,α /0;
RN3μxyts≥Tμxt, μysfor allx, y∈Xandt, s≥0.
Every normed spaceX, · defines a random normed spaceX, μ, TM, where
μxt
t
tx 1.8
for allt >0,andTMis the minimumt-norm. This space is called the induced random normed
space.
Definition 1.3. LetX, μ, Tbe an RN-space.
1A sequence{xn}inXis said to beconvergenttoxinXif, for every >0 andλ >0, there exists a positive integerNsuch thatμxn−x>1−λwhenevern≥N.
3An RN-spaceX, μ, Tis said to becompleteif and only if every Cauchy sequence in
Xis convergent to a point inX.
Theorem 1.4see28. IfX, μ, Tis an RN-space and{xn}is a sequence such thatxn → x, then
limn→ ∞μxnt μxtalmost everywhere.
Recently, Gordji et al. establish the stability of cubic, quadratic and additive-quadratic functional equations in RN-spacessee32,33.
In this paper, we deal with the following functional equation:
f2xyf2x−y4fxyfx−y2f2x−4fx−6fy 1.9
on RN-spaces. It is easy to see that the functionfx ax4bx2is a solution of1.9.
InSection 2, we investigate the general solution of the functional equation1.9when
f is a mapping between vector spaces and in Section 3, we establish the stability of the functional equation1.9in RN-spaces.
2. General Solution
We need the following lemma for solution of 1.9. Throughout this section, X and Y are vector spaces.
Lemma 2.1. If a mappingf:X → Y satisfies1.9for allx, y∈X,thenfis quadratic-quartic.
Proof. We show that the mappingsg :X → Ydefined bygx:f2x−16fxandh:X → Y defined byhx:f2x−4fxare quadratic and quartic, respectively.
Lettingxy0 in1.9, we havef0 0. Puttingx0 in1.9, we getf−y fy. Thus the mappingfis even. Replacingyby 2yin1.9, we get
f2x2yf2x−2y4fx2yfx−2y2f2x−4fx−6f2y 2.1
for allx, y∈X. Interchangingxwithyin1.9, we obtain
f2yxf2y−x4fyxfy−x2f2y−4fy−6fx 2.2
for allx, y∈X. Sincefis even, by2.2, one gets
fx2yfx−2y4fxyfx−y2f2y−4fy−6fx 2.3
for allx, y∈X.It follows from2.1and2.3that
f2xy−16fxyf2x−y−16fx−y
for allx, y∈X. This means that
gxygx−y2gx 2gy 2.5
for allx, y∈X. Therefore, the mappingg:X → Yis quadratic. To prove thath:X → Y is quartic, we have to show that
hx2yhx−2y4hxyhx−y6hy−6hx 2.6
for allx, y∈X.Sincefis even, the mappinghis even. Now if we interchangexwithyin the last equation, we get
h2xyh2x−y4hxyhx−y6hx−6hy 2.7
for allx, y∈X. Thus, it is enough to prove thathsatisfies2.7. Replacingxandyby 2xand 2yin1.9, respectively, we obtain
f22xyf22x−y4f2xyf2x−y2f4x−4f2x−6f2y
2.8
for allx, y∈X. Sinceg2x 4gxfor allx∈X,
f4x 20f2x−64fx 2.9
for allx∈X. By2.8and2.9, we get
f22xyf22x−y4f2xyf2x−y32f2x−4fx−6f2y
2.10
for allx, y∈X. By multiplying both sides of1.9by 4, we get
4f2xyf2x−y16fxyfx−y8f2x−4fx−24fy 2.11
for allx, y∈X. If we subtract the last equation from2.10, we obtain
h2xyh2x−yf22xy−4f2xyf22x−y−4f2x−y
4f2xy−4fxy4f2x−y−4fx−y
24f2x−4fx−6f2y−4fy
4hxyhx−y6hx−6hy
2.12
for allx, y∈X.
Theorem 2.2. A mappingf : X → Y satisfies1.9for allx, y ∈ X if and only if there exist a unique symmetric multiadditive mappingM:X4 → Y and a unique symmetric bi-additive mapping
B:X×X → Y such that
fx Mx, x, x, x Bx, x 2.13
for allx∈X.
Proof. Letfsatisfy1.9and assume thatg, h:X → Yare mappings defined by
gx:f2x−16fx, hx:f2x−4fx 2.14
for allx∈X.ByLemma 2.1, we obtain that the mappingsgandhare quadratic and quartic, respectively, and
fx 1
12hx− 1
12gx 2.15
for allx∈X.
Therefore, there exist a unique symmetric multiadditive mappingM:X4 → Y and a
unique symmetric bi-additive mappingB:X×X → Y such that1/12hx Mx, x, x, x
and−1/12gx Bx, xfor allx∈X5,18. So
fx Mx, x, x, x Bx, x 2.16
for allx∈X.The proof of the converse is obvious.
3. Stability
Throughout this section, assume thatXis a real linear space andY, μ, Tis a complete RN-space.
Theorem 3.1. Letf : X → Y be a mapping withf0 0for which there isρ : X×X → D
(ρx, yis denoted byρx,y) with the property:
μf2xyf2x−y−4fxy−4fx−y−2f2x8fx6fyt≥ρx,yt 3.1
for allx, y∈Xand allt >0.If
lim
n→ ∞T ∞
i1
Tρ2ni−1x,2ni−1x
22ni1t, T
ρ2ni−1x,2·2ni−1x
22nit
4
, ρ0,2ni−1x
22nit
3
1,
lim
n→ ∞ρ2nx,2ny
22nt1
for allx, y∈Xand allt >0, then there exists a unique quadratic mappingQ1:X → Y such that
μf2x−16fx−Q1xt≥T
∞
i1
T
ρ2i−1x,2i−1x
2i1t, T
ρ2i−1x,2·2i−1x
2it
4
, ρ0,2i−1x
2it
3
3.3
for allx∈Xand allt >0.
Proof. Puttingyxin3.1, we obtain
μf3x−6f2x15fxt≥ρx,xt 3.4
for allx∈Xand allt >0.Lettingy2xin3.1, we get
μf4x−4f3x4f2x8fx−4f−xt≥ρx,2xt 3.5
for allx∈Xand allt >0.Puttingx0 in3.1, we obtain
μ3fy−3f−yt≥ρ0,yt 3.6
for ally∈Xand allt >0.Replacingybyxin3.6, we see that
μ3fx−3f−xt≥ρ0,xt 3.7
for allx∈Xand allt >0.It follows from3.5and3.7that
μf4x−4f3x4f2x4fxt≥T
ρx,2x
t
2
, ρ0,x
2t
3
3.8
for allx∈Xand allt >0.If we add3.4to3.8, then we have
μf4x−20f2x64fxt≥T
ρx,x2t, T
ρx,2x
t
4
, ρ0,x
t
3
. 3.9
Let
ψx,xt T
ρx,x2t, T
ρx,2x
t
4
, ρ0,x
t
3
3.10
for allx∈Xand allt >0. Then we get
for allx∈Xand allt >0.Letg :X → Y be a mapping defined bygx:f2x−16fx. Then we conclude that
μg2x−4gxt≥ψx,xt 3.12
for allx∈Xand allt >0. Thus we have
μg2x/22−gxt≥ψx,x
22t 3.13
for allx∈Xand allt >0.Hence
μg2k1x/22k1−g2kx/22kt≥ψ2kx,2kx
22k1t 3.14
for allx∈X, allt >0 and allk∈N.This means that
μg2k1x/22k1−g2kx/22k
t
2k1
≥ψ2kx,2kx
2k1t 3.15
for allx∈X,allt >0 and allk∈N.By the triangle inequality, from 1>1/21/22· · ·1/2n,
it follows that
μg2nx/22n−gxt≥Tkn1
μg2kx/22k−g2k−1x/22k−1
t
2k
≥Tin1ψ2i−1x,2i−1x
2it 3.16
for allx∈ Xand allt > 0.In order to prove the convergence of the sequence{g2nx/22n},
we replacexwith 2mxin3.16to obtain that
μg2nmx/22nm−g2mx/22mt≥Tin1
ψ2im−1x,2im−1x
2i2mt. 3.17
Since the right-hand side of the inequality 3.17 tends to 1 as m and n tend to infinity, the sequence {g2nx/22n} is a Cauchy sequence. Thus we may define Q
1x
limn→ ∞g2nx/22nfor allx∈X.
Now we show that Q1 is a quadratic mapping. Replacingx, y with 2nx and 2ny in 3.1, respectively, we get
μg2n2xyg2n2x−y−4g2nxy−4g2nx−y−2g2n1x8g2nx6g2ny/4nt
≥ρ2nx,2ny
22nt.
3.18
Taking the limit asn → ∞, we find thatQ1satisfies1.9for allx, y∈X. ByLemma 2.1, the
mappingQ1:X → Y is quadratic.
Finally, to prove the uniqueness of the quadratic mappingQ1 subject to3.3, let us
assume that there exists another quadratic mappingQ1which satisfies3.3. SinceQ12nx
22nQ
1x, Q12nx 22nQ1xfor allx∈Xand alln∈N,from3.3, it follows that
μQ1x−Q1x2t
μQ12nx−Q12nx
22n1t
≥TμQ12nx−g2nx
22nt, μg2nx−Q
12nx
22nt
≥T
Ti∞1
T
ρ2ni−1x,2ni−1x
22ni1t, T
ρ2ni−1x,2·2ni−1x
22nit
4
, ρ0,2ni−1x
22nit
3
,
Ti∞1
T
ρ2ni−1x,2ni−1x
22ni1t, T
ρ2ni−1x,2ni−1x
22nit
4
, ρ0,2ni−1x
22nit
3
3.19
for allx∈Xand allt >0. Lettingn → ∞in3.19, we conclude thatQ1 Q1, as desired.
Theorem 3.2. Letf : X → Y be a mapping withf0 0for which there isρ : X×X → D
(ρx, yis denoted byρx,y) with the property:
μf2xyf2x−y−4fxy−4fx−y−2f2x8fx6fyt≥ρx,yt 3.20
for allx, y∈Xand allt >0.If
lim
n→ ∞T ∞
i1
T
ρ2ni−1x,2ni−1x
24n3i1t,
T
ρ2ni−1x,2·2ni−1x
24n3it
4
, ρ0,2ni−1x
24n3it
3
1,
lim
n→ ∞ρ2nx,2ny
24nt1
3.21
for allx, y∈Xand allt >0, then there exists a unique quartic mappingQ2 :X → Ysuch that
μf2x−4fx−Q2xt≥T
∞
i1
T
ρ2i−1x,2i−1x
23i1t, T
ρ2i−1x,2·2i−1x
23it
4
, ρ0,2i−1x
23it
3
3.22
for allx∈Xand allt >0.
Proof. Puttingyxin3.20, we obtain
for allx∈Xand allt >0. Lettingy2xin3.20, we get
μf4x−4f3x4f2x8fx−4f−xt≥ρx,2xt 3.24
for allx∈Xand allt >0. Puttingx0 in3.20, we obtain
μ3fy−3f−yt≥ρ0,yt 3.25
for ally∈Xand allt >0. Replacingybyxin3.25, we get
μ3fx−3f−xt≥ρ0,xt 3.26
for allx∈Xand allt >0. It follows from3.5and3.26that
μf4x−4f3x4f2x4fxt≥T
ρx,2x
t
2
, ρ0,x
2t
3
3.27
for allx∈Xand allt >0. If we add3.23to3.27, then we have
μf4x−20f2x64fxt≥T
ρx,x2t, T
ρx,2x
t
4
, ρ0,x
t
3
. 3.28
Let
ψx,xt T
ρx,x2t, T
ρx,2x
t
4
, ρ0,x
t
3
3.29
for allx∈Xand allt >0. Then we get
μf4x−20f2x64fxt≥ψx,xt 3.30
for allx ∈X and allt >0.Leth:X → Y be a mapping defined byhx : f2x−4fx. Then we conclude that
μh2x−16hxt≥ψx,xt 3.31
for allx∈Xand allt >0. Thus we have
μh2x/24−hxt≥ψx,x
for allx∈Xand allt >0.Hence
μh2k1x/24k1−h2kx/24kt≥ψ2kx,2kx
24k1t 3.33
for allx∈X, allt >0 and allk∈N.This means that
μh2k1x/24k1−h2kx/24k
t
2k1
≥ψ2kx,2kx
23k1t 3.34
for allx∈X,allt >0 and allk∈N.By the triangle inequality, from 1>1/21/22· · ·1/2n,
it follows that
μh2nx/24n−hxt≥Tkn1
μh2kx/24k−h2k−1x/24k−1
t
2k
≥Tin1ψ2i−1x,2i−1x
23it
3.35
for allx∈ Xand allt >0.In order to prove the convergence of the sequence{h2nx/24n}, we replacexwith 2mxin3.35to obtain that
μh2nmx/24nm−h2mx/24mt≥Tin1
ψ2im−1x,2im−1x
23i4mt. 3.36
Since the right-hand side of 3.36 tends to 1 as m and n tend to infinity, the sequence
{h2nx/24n}is a Cauchy sequence. Thus we may defineQ
2x limn→ ∞h2nx/24nfor
allx∈X.
Now we show thatQ2is a quartic mapping. Replacingx, ywith 2nxand 2nyin3.20,
respectively, we get
μh2n2xyh2n2x−y−4h2nxy−4h2nx−y−2h2n1x8h2nx6h2ny/16nt
≥ρ2nx,2ny
24nt.
3.37
Taking the limit asn → ∞, we find thatQ2satisfies1.9for allx, y∈X. ByLemma 2.1we
get that the mappingQ2:X → Y is quartic.
Finally, to prove the uniqueness of the quartic mapping Q2 subject to 3.24, let
us assume that there exists a quartic mapping Q2 which satisfies 3.22. Since Q22nx
24nQ
2xandQ22nx 24nQ2xfor allx∈Xand alln∈N,from3.22, it follows that
μQ2x−Q2x2t
μQ22nx−Q22nx
24n1t
≥TμQ22nx−h2nx
24nt, μh2nx−Q
22nx
24nt,
≥T
Ti∞1
T
ρ2ni−1x,2ni−1x
24n3i1t, T
ρ2ni−1x,2·2ni−1x
24n3it
4
, ρ0,2ni−1x
24n3it
3
,
Ti∞1
T
ρ2ni−1x,2ni−1x
24n3i1tT
ρ2ni−1x,2·2ni−1x
24n3it
4
, ρ0,2ni−1x
24n3it
3
3.38
for allx∈Xand allt >0. Lettingn → ∞in3.38, we get thatQ2Q2, as desired.
Theorem 3.3. Letf : X → Y be a mapping withf0 0for which there isρ : X×X → D
(ρx, yis denoted byρx,y) with the property:
μf2xyf2x−y−4fxy−4fx−y−2f2x8fx6fyt≥ρx,yt 3.39
for allx, y∈Xand allt >0.If
lim
n→ ∞T ∞
i1
T
ρ2ni−1x,2ni−1x
24n3i1t,
T
ρ2ni−1x,2·2ni−1x
24n3it
4
, ρ0,2ni−1x
24n3it
3
1,
lim
n→ ∞ρ2nx,2ny
22nt1
3.40
for allx, y∈Xand allt >0, then there exist a unique quadratic mappingQ1:X → Yand a unique
quartic mappingQ2:X → Y such that
μfx−Q1x−Q2xt
≥T
Ti∞1
T
ρ2i−1x,2i−1x
2it
12
, T
ρ2i−1x,2·2i−1x
2it
4·24
, ρ0,2i−1x
2it
3·24
,
Ti∞1
T
ρ2i−1x,2i−1x
23it
24
, T
ρ2i−1x,2·2i−1x
23it
4·24
, ρ0,2i−1x
23it
3·24
for allx∈Xand allt >0.
Proof. By Theorems3.1and3.2, there exist a quadratic mappingQ1 : X → Y and a quartic mappingQ2:X → Y such that
μf2x−16fx−Q1xt≥Ti∞1
T
ρ2i−1x,2i−1x
2i1t, T
ρ2i−1x,2·2i−1x
2it
4
, ρ0,2i−1x
2it
3
,
μf2x−4fx−Q2xt≥Ti∞1
T
ρ2i−1x,2i−1x
23i1t, T
ρ2i−1x,2·2i−1x
23it
4
, ρ0,2i−1x
23it
3
3.42
for allx∈Xand allt >0. It follows from the last inequalities that
μfx1/12Q1x−1/12Q2xt
≥T
μf2x−16fx−Q1x
t
24
, μf2x−4fx−Q2x
t 24 ≥T
Ti∞1
T
ρ2i−1x,2i−1x
2it
12
, T
ρ2i−1x,2·2i−1x
2it
4·24
, ρ0,2i−1x
2it
3·24
,
Ti∞1
T
ρ2i−1x,2i−1x
23it
24
, T
ρ2i−1x,2·2i−1x
23it
4·24
, ρ0,2i−1x
23it
3·24
3.43
for allx ∈ X and allt > 0. Hence we obtain3.41by lettingQ1x −1/12Q1xand
Q2x 1/12Q2xfor allx∈X.The uniqueness property ofQ1andQ2is trivial.
Acknowledgment
C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology NRF-2009-0070788.
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