Random Stability of an Additive Quadratic Quartic Functional Equation

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Volume 2010, Article ID 754210,18pages doi:10.1155/2010/754210

Research Article

Random Stability of an Additive-Quadratic-Quartic

Functional Equation

M. Mohamadi,

1

Y. J. Cho,

2

C. Park,

3

P. Vetro,

4

and R. Saadati

5

1_{Department of Mathematics, Islamic Azad University-Ayatollah Amoli Branch, Amol,} P.O. Box 678, Iran

2_{Department of Mathematics Education and the RINS, Gyeongsang National University,} Chinju 660-701, South Korea

3_{Department of Mathematics, Research Institute for Natural Sciences, Hanyang University,} Seoul 133-791, South Korea

4_{Dipartimento di Matematica ed Applicazioni, Universit`a degli Studi di Palermo, Via Archirafi,} 34 - 90123 Palermo, Italy

5_{Faculty of Sciences, Islamic Azad University-Ayatollah Amoli Branch, Amol, P.O. Box 678, Iran}

Correspondence should be addressed to C. Park,[email protected] R. Saadati,[email protected]

Received 7 December 2009; Accepted 8 February 2010

Academic Editor: Jong Kim

Copyrightq2010 M. Mohamadi 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.

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-quartic functional equationfx2y fx−2y 2fxy 2f−x−y

2fx−y 2fy−x−4f−x−2fx f2y f−2y−4fy−4f−yin complete random normed spaces.

1. Introduction

The stability problem of functional equations originated from a question of Ulam 1

concerning the stability of group homomorphisms. Hyers2gave a first aﬃrmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki

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The functional equation

fxyfx−y2fx 2fy 1.1

is called aquadratic functional equation. In particular, every solution of the quadratic functional equation is said to be aquadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof6for mappings f :X → Y, whereXis a normed space andY is a Banach space. Cholewa7noticed that the theorem of Skof is still true if the relevant domainX is replaced by an Abelian group. Czerwik8proved the generalized Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problemsee4,9–26.

In27, Lee et al. considered the following quartic functional equation

f2xyf2x−y4fxy4fx−y24fx−6fy. 1.2

It is easy to show that the functionfx x4_{satisfies the functional equation}_1.2_{, which is}

called aquartic functional equationand every solution of the quartic functional equation is said to be aquartic mapping.

LetXbe a set. A functiond :X×X → 0,∞is called ageneralized metriconX ifd

satisfies

1dx, y 0 if and only ifxy,

2dx, y dy, xfor allx, y∈X,

3dx, z≤dx, y dy, zfor allx, y, z∈X.

We recall a fundamental result in fixed point theory.

Theorem 1.128,29. LetX, dbe a complete generalized metric space and letJ :X → Xbe a

strictly contractive mapping with Lipschitz constantL <1. Then for each given elementx∈X, either

dJnx, Jn1x∞ 1.3

for all nonnegative integersnor there exists a positive integern0such that

1dJnx, Jn1x<∞, for alln≥n0,

2the sequence{Jn_x_}_{converges to a fixed point}_y∗_of_J_,

3y∗is the unique fixed point ofJin the setY {y∈X|dJn0x, y_<∞}_,

4dy, y∗≤1/1−Ldy, Jyfor ally∈Y.

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2. Preliminaries

In the sequel we adopt the usual terminology, notations and conventions of the theory of random normed spaces, as in37–41. Throughout this paper,Δis the space of all probability distribution functions that is, the space of all mappingsF:R∪ {−∞,∞} → 0,1, such that

F is left-continuous, non-decreasing on R,F0 0 andF∞ 1.D is a subset of Δ consising of all functionsF ∈Δ for whichl−F∞ 1, wherel−fxdenotes the left limit of the functionfat the pointx, that is,l−fx limt→x−ft. The spaceΔis partially ordered

by the usual point-wise ordering of functions, that is,F≤Gif and only ifFt≤Gtfor allt

inR. The maximal element forΔin this order is the distribution functionε0given by

ε0t ⎧ ⎨ ⎩

0, ift≤0,

1, ift >0.

2.1

Definition 2.140. A mappingT :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.

Typical examples of continuoust-norms areTPa, b ab,TMa, b mina, band

TLa, b maxab−1,0 theŁukasiewiczt-norm.

Recallsee42,43that ifT is at-norm and{xn}is a given sequence of numbers in

0,1,T_in₁xiis defined recurrently byT_i1₁xi x1andT_in₁xi TT_in−₁1xi, xnforn≥2.T_in∞xiis

defined asT_i∞₁x_ni.

It is known43that for theŁukasiewiczt-norm the following implication holds:

lim

n→ ∞TL ∞

i1xni1⇐⇒

∞

n1

1−xn<∞. 2.2

Definition 2.241. ARandom Normed spacebriefly, RN-spaceis a tripleX, μ, T, whereX

is 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 allx∈X,α /0;

RN3μxyts≥Tμxt, μysfor allx, y∈Xandt, s≥0.

Definition 2.3. LetX, μ, Tbe a RN-space.

1A sequence{xn}inXis said to beconvergenttoxinXif, for every >0 andλ >0,

there exists positive integerNsuch thatμxn−x>1−λwhenevern≥N.

2A sequence {xn}inX is called Cauchyif, for every > 0 andλ > 0, there exists

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

Xis convergent to a point inX. A complete RN-space is said to be random Banach space.

Theorem 2.4 40. If X, μ, Tis a RN-space and {xn}is a sequence such thatxn → x, then

limn→ ∞μxnt μxtalmost everywhere.

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 physics. The generalized Hyers-Ulam stability of diﬀerent functional equations in random normed spaces, RN-spaces and fuzzy normed RN-spaces has been recently studied in, Alsina 44, Mirmostafaee, Mirzavaziri and Moslehian33,45–47, Mihet¸ and Radu38,39,48,49, Mihet, Saadati and Vaezpour50,51, Baktash et al.52and Saadati et al.53.

3. Generalized Hyers-Ulam Stability of the Functional Equation

f

x

2

y

f

x

−

2

y

2

f

x

y

2

f

−

x

−

y

2

f

x

−

y

2

f

y

−

x

−

4

f

−

x

−

2

f

x

f

2

y

f

−

2

y

−

4

f

y

−

4

f

−

y

: An Odd Case

One can easily show that an odd mapping f : X → Y satisfiesfx2y fx−2y 2fxy2f−x−y2fx−y2fy−x−4f−x−2fxf2yf−2y−4fy−4f−y if and only if the odd mapping mappingf :X → Y is an additive mapping, that is,

fx2yfx−2y2fx. 3.1

One can easily show that an even mappingf :X → Ysatisfiesfx2y fx−2y 2fxy2f−x−y2fx−y2fy−x−4f−x−2fxf2yf−2y−4fy−4f−y if and only if the even mappingf:X → Y is a quadratic-quartic mapping, that is,

fx2yfx−2y4fxy4fx−y−6fx 2f2y−8fy. 3.2

It was shown in54, Lemma 2.1thatgx:f2x−4fxandhx :f2x−16fxare quartic and quadratic, respectively, and thatfx 1/12gx−1/12hx.

For a given mappingf:X → Y, we define

Dfx, y:fx2yfx−2y−2fxy−2f−x−y−2fx−y−2fy−x 4f−x 2fx−f2y−f−2y4fy4f−y 3.3

for allx, y∈X.

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Theorem 3.1. LetXbe a linear space,Y, μ, TMbe a complete RN-space andΦbe a mapping from

X2_to_D_{Φx, y}_{is denoted by}_Φ

x,ysuch that, for some0< α <1/3,

Φ3x,3yt≤Φx,yαt

x, y∈X, t >0 3.4

Letf:X → Y be an odd mapping satisfying

μ_Dfx,yt≥Φx,yt 3.5

for allx, y∈Xand allt >0. Then

Ax: lim

n→ ∞3 n_f

x

3n

3.6

exists for eachx∈Xand defines a unique additive mappingA:X → Ysuch that

μfx−Axt≥Φx,x

1−3α α t

3.7

for allx∈Xand allt >0.

Proof. Lettingxyin3.5, we get

μf3x−3fxt≥Φx,xt 3.8

for allx∈Xand allt >0. Consider the set

S:g:X−→Y, 3.9

and introduce the generalized metric onS:

dg, hinfu∈R:μ_gx₋_hxut≥Φx,xt,∀x∈X,∀t >0

, 3.10

where, as usual, inf∅ ∞. It is easy to show thatS, dis complete.See the proof of Lemma 2.1 in38.

Now we consider the linear mappingJ:S → Ssuch that

Jgx:3g

x

3

3.11

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Letg, h∈Sbe given such thatdg, h< ε. Then

μ_gx₋_hxεt≥Φx,xt 3.12

for allx∈Xand allt >0. Hence

μJgx−Jhx3αεt μ3gx/3−3hx/33αεt

μgx/3−hx/3αεt

≥Φx/3,x/3αt

≥Φx,xt

3.13

for allx∈Xand allt >0. Sodg, h< εimplies thatdJg, Jh≤3αε. This means that

dJg, Jh≤3αdg, h 3.14

for allg, h∈S.

It follows from3.8that

μfx−3fx/3αt≥Φx,xt, 3.15

for allx∈Xand allt >0. So

df, Jf≤α < 1

3. 3.16

ByTheorem 1.1, there exists a mappingA:X → Y satisfying the following:

1Ais a fixed point ofJ, that is,

A

x

3

1

3Ax 3.17

for allx∈X. The mappingAis a unique fixed point ofJin the set

Mh∈S:dh, g<∞. 3.18

This implies thatAis a unique mapping satisfying3.17such that there exists au∈0,∞

satisfying

μfx−Axut≥Φx,xt, 3.19

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2dJn_{f, A} _→ _{0 as}_n _{→ ∞. This implies the equality}

lim

n→ ∞3 n_f

x

3n

Ax 3.20

for allx∈X. Sincef:X → Y is odd,A:X → Yis an odd mapping;

3df, A≤1/1−3αdf, Jfwithf∈M, which implies the inequality

df, A≤ α

1−3α, 3.21

from which it follows

μfx−Ax

α

1−3αt

≥Φx,xt. 3.22

This implies that the inequality3.7holds. Now, we have,

μ₃n_Df_x/₃n_,y/₃nt μ_Df_x/₃n_,y/₃n

t

3n

≥Φx/3n_,y/₃n

t

3n

3.23

for allx, y∈X, allt >0 and alln∈N. So, we obtain by3.4

μ₃n_Df_x/₃n_,y/₃nt≥Φx,y

t 3αn

3.24

for allx, y∈X, allt >0 and alln∈N.

Since limn→ ∞Φx,yt/3αn 1 for allx, y ∈ X and all t > 0, byTheorem 2.4, we

deduce that

μ_DA_x,yt 1 3.25

for allx, y∈Xand allt >0. Thus the mappingA:X → Yis additive, as desired.

Corollary 3.2. Letθ≥0and letpbe a real number withp >3. LetXbe a normed vector space with norm · . Letf :X → Ybe an odd mapping satisfying

μDfx,yt≥ t

tθxpyp 3.26

for allx, y∈Xand allt >0. Then

Ax: lim

n→ ∞3 n_f

x

3n

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exists for eachx∈Xand defines an additive mappingA:X → Ysuch that

μfx−Axt≥ 3 p₋₃_t

3p₋₃_t₂_.₃p_θ_xp 3.28

Proof. The proof follows fromTheorem 3.1by taking

Φx,yt:

t

tθxpyp 3.29

for allx, y∈X. Then we can chooseα3−p_{and we get the desired result.}

Similarly, we can obtain the following. We will omit the proof.

X2_to_D₍_{Φx, y}_{is denoted by}_Φ

x,y)such that, for some0< α <3,

Φx/3,y/3t≤Φx,yαt

x, y∈X, t >0. 3.30

Letf:X → Y be an odd mapping satisfying3.5. Then

Ax: lim

n→ ∞

1 3nf3

n_x _3.31

μfx−Axt≥Φx,x3−αt 3.32

Corollary 3.4. Letθ≥0and letpbe a real number with0< p <1. LetXbe a normed vector space with norm · . Letf :X → Y be an odd mapping satisfying3.26. Then

Ax: lim

n→ ∞3

−n_f₃n_x

3.33

μ_fx₋_Axt≥ 3−3

p_t

3−3p_t₂_θ_xp 3.34

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μDfx,yt≥ t

tθxpyp 3.35

for allx, y∈X. Then we can chooseα3pand we get the desired result.

4. Generalized Hyers-Ulam Stability of the Functional Equation

f

x

2

y

f

x

−

2

y

2

f

x

y

2

f

−

x

−

y

2

f

x

−

y

2

f

y

−

x

−

4

f

−

x

−

2

f

x

f

2

y

f

−

2

y

−

4

f

y

−

4

f

−

y

: An Even Case

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equationDfx, y 0 in random Banach spaces: an even case.

Theorem 4.1. LetX be a linear space, let Y, μ, TMbe a complete RN-space and Φbe a mapping

fromX2_to_D₍_{Φx, y}_{is denoted by}_Φ_{x,y) such that, for some}₀_{< α <}₁_/₁₆_,

Φx,yαt≥Φ2x,2yt

x, y∈X, t >0. 4.1

Letf:X → Y be an even mapping satisfyingf0 0and3.5. Then

Qx: lim

n→ ∞16 n

f

x

2n−1

−4fx

2n

4.2

exists for eachx∈Xand defines a quartic mappingQ:X → Y such that

μf2x−4fx−Qxt≥TM

Φx,x

1−16α

5α t

,Φ2x,x

1−16α

5α t

4.3

Proof. Lettingxyin3.5, we get

μ_f3y−6f2y15fyt≥Φy,yt 4.4

for ally∈Xand allt >0.

Replacingxby 2yin3.5, we get

μ_f₄_y₋₄_f₃_y₄_f₂_y₄_f_yt≥Φ2y,yt 4.5

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By4.4and4.5,

μf4x−20f2x64fx5t

≥TM

μ4f3x−6f2x15fx4t, μf4x−4f3x4f2x4fxt

≥TMΦx,xt,Φ2x,xt

4.6

for allx∈Xand allt >0. Lettinggx:f2x−4fxfor allx∈X, we get

μ_gx₋16gx/25t≥TMΦx/2,x/2t,Φx,x/2t 4.7

for allx∈Xand allt >0.

LetS, dbe the generalized metric space defined in the proof ofTheorem 3.1. Now we consider the linear mappingJ:S → Ssuch that

Jhx:16hx

2

4.8

for all x ∈ X. It is easy to see that J is a strictly contractive self-mapping on S with the Lipschitz constant 16α.

It follows from4.7that

μgx−16gx/25αt≥TMΦx,xt,Φ2x,xt 4.9

dg, Jg≤5α≤ 5

16 <∞. 4.10

ByTheorem 1.1, there exists a mappingQ:X → Ysatisfying the following:

1Qis a fixed point ofJ, that is,

Qx

2

1

16Qx 4.11

for allx∈X. Sinceg :X → Y is even withg0 0,Q :X → Y is an even mapping with

Q0 0. The mappingQis a unique fixed point ofJin the set

Mh∈S:dh, g<∞. 4.12

This implies thatQis a unique mapping satisfying4.11such that there exists au∈0,∞

satisfying

μgx−Qxut≥TMΦx,xt,Φ2x,xt 4.13

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2dJn_{g, Q} _→ _{0 as}_n _{→ ∞. This implies the equality}

lim

n→ ∞16 n_gx

2n

Qx 4.14

for allx∈X;

3dh, Q≤1/1−16αdh, Jhfor everyh∈M, which implies the inequality

dg, Q≤ 5α

1−16α. 4.15

This implies that the inequality4.3holds.

Proceeding as in the proof ofTheorem 3.1, we obtain that the mappingQ : X → Y

satisfiesfx2y fx−2y 2fxy 2f−x−y 2fx−y 2fy−x−4f−x− 2fx f2y f−2y−4fy−4f−y.

Now, we have

Q2x−16Qxlim

n→ ∞

16n_g

x

2n−1

−16n1_gx

2n

16 lim

n→ ∞

16n−1_g

x

2n−1

−16n_gx

2n

0

4.16

for everyx∈X. Since the mappingx → Q2x−4Qxis quarticsee54, Lemma 2.1, we get that the mappingQ:X → Yis quartic.

Corollary 4.2. Letθ≥0and letpbe a real number withp >4. LetXbe a normed vector space with norm · . Letf :X → Ybe an even mapping satisfyingf0 0and3.26. Then

Qx: lim

n→ ∞16 n

f

x

2n−1

−4fx

2n

4.17

μ_f2x−4fx−Qxt≥ 2 p₋₁₆_t

2p₋₁₆_t₅₁₂p_θ_xp 4.18

μDfx,yt≥ t

tθxpyp 4.19

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x,y) such that, for some0< α <16,

Φx,yαt≥Φx/2,y/2t

x, y∈X, t >0. 4.20

Qx: lim

n→ ∞

1 16n

f2n1_x₋₄_f₂n_x _4.21

μ_f2x−4fx−Qxt≥TM

Φx,x

16−α

5 t

,Φ2x,x

16−α

5 t

4.22

Corollary 4.4. Letθ≥0and letpbe a real number with0< p <4. LetXbe a normed vector space with norm · . Letf :X → Y be an even mapping satisfyingf0 0and3.26. Then

Qx: lim

n→ ∞

1 16n

f2n1x−4f2nx 4.23

μf2x−4fx−Qxt≥ 16−2 p_t

16−2p_t₅₁₂p_θ_xp 4.24

μDfx,yt≥ t

tθxpyp 4.25

for allx, y∈X. Then we can chooseα2pand we get the desired result.

x,y)such that, for some0< α <1/4,

Φx,yαt≥Φ2x,2yt

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Tx: lim

n→ ∞4 n

f

x

2n−1

−16fx

2n

4.27

exists for eachx∈Xand defines a quadratic mappingT:X → Ysuch that

μf2x−16fx−Txt≥TM

Φx,x

1−4α

5α t

,Φ2x,x

1−4α

5α t

4.28

Proof. LetS, dbe the generalized metric space defined in the proof ofTheorem 4.1. Lettinggx:f2x−16fxfor allx∈Xin4.6, we get

μgx−4gx/25t≥TMΦx/2,x/2t,Φx,x/2t 4.29

for allx∈Xand allt >0.

It is easy to see that the linear mappingJ:S → Ssuch that

Jhx:4hx

2

, 4.30

for allx∈X, is a strictly contractive self-mapping with the Lipschitz constant 4α. It follows from4.29that

μ_gx₋4gx/25αt≥TMΦx,xt,Φ2x,xt 4.31

dg, Jg≤5α <∞. 4.32

ByTheorem 1.1, there exists a mappingT :X → Y satisfying the following.

1Tis a fixed point ofJ, that is,

Tx

2

1

4Tx 4.33

for allx∈X. Sinceg :X → Y is even withg0 0,T :X → Y is an even mapping with

T0 0. The mappingTis a unique fixed point ofJin the set

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This implies thatT is a unique mapping satisfying4.33such that there exists au ∈0,∞

satisfying

μgx−Txut≥TMΦx,xt,Φ2x,xt 4.35

for allx∈Xand allt >0;

2dJn_{g, T} _→ _{0 as}_n _{→ ∞. This implies the equality}

lim

n→ ∞4 n_gx

2n

Tx 4.36

for allx∈X;

3dh, T≤ 1

1−4αdh, Jhfor eachh∈M, which implies the inequality

dg, T≤ 5α

1−4α. 4.37

This implies that the inequality4.28holds.

Proceeding as in the proof ofTheorem 4.1, we obtain that the mappingT : X → Y

satisfiesfx2y fx−2y 2fxy 2f−x−y 2fx−y 2fy−x−4f−x− 2fx f2y f−2y−4fy−4f−y.

Now, we have

T2x−4Tx lim

n→ ∞

4ng

x

2n−1

−4n1gx

2n

4 lim

n→ ∞

4n−1g

x

2n−1

−4ngx

2n

0

4.38

for everyx∈X. Since the mappingx → T2x−16Txis quadraticsee54, Lemma 2.1, we get that the mappingT :X → Y is quadratic.

Corollary 4.6. Letθ≥0and letpbe a real number withp >2. LetXbe a normed vector space with norm · . Letf :X → Ybe an even mapping satisfyingf0 0and3.26. Then

Tx: lim

n→ ∞4 n

f

x

2n−1

−16fx

2n

4.39

μf2x−16fx−Txt≥ 2 p₋₄_t

2p₋₄_t₅₁₂p_θ_xp 4.40

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Φx,yt: t

tθxpyp 4.41

for allx, y∈X. Then we can chooseα2−p_{and we get the desired result.}

x,y) such that, for some0< α <4,

Φx,yαt≥Φx/2,y/2t

x, y∈X, t >0. 4.42

Tx: lim

n→ ∞

1 4n

f2n1x−16f2nx 4.43

μ_f2x−16fx−Txt≥TM

Φx,x

4−α

5 t

,Φ2x,x

4−α

5 t

4.44

Corollary 4.8. Letθ≥0and letpbe a real number with0< p <2. LetXbe a normed vector space with norm · . Letf :X → Y be an even mapping satisfyingf0 0and3.26. Then

Tx: lim

n→ ∞

1 4n

f2n1x−16f2nx 4.45

μf2x−16fx−Txt≥

4−2p_t

4−2p_t₅₁₂p_θ_xp 4.46

Φx,yt:

t

tθxpyp 4.47

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Acknowledgments

The authors would like to thank referees for giving useful suggestions for the improvement of this paper. The first author is supported by Islamic Azad University-Ayatollah Amoli Branch, Amol, Iran. The second author was supported by the Korea Research Foundation Grant funded by the Korean GovernmentKRF-2008-313-C00050and the third author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and TechnologyNRF-2009-0070788. The fourth author is supported by Universit`a degli Studi di Palermo, R.S. ex 60%.

References

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