Nonlinear Random Stability of an ACQ Functional Equation

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Volume 2011, Article ID 194394,23pages doi:10.1155/2011/194394

Review Article

Nonlinear

L

-Random Stability of

an ACQ Functional Equation

Reza Saadati, M. M. Zohdi, and S. M. Vaezpour

Department of Mathematics, Science and Research Branch, Islamic Azad University, Ashrafi Esfahani Ave, Tehran 14778, Iran

Correspondence should be addressed to Reza Saadati,[email protected]

Received 9 December 2010; Accepted 6 February 2011

Academic Editor: Soo Hak Sung

Copyrightq2011 Reza 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.

We prove the generalized Hyers-Ulam stability of the following additive-cubic-quartic functional equation: 11fx2y 11fx−2y 44fxy 44fx−y 12f3y−48f2y 60fy−66fx

in complete latticetic random normed spaces.

1. Introduction

Random theory is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. It has also very useful applications in various fields, for example, population dynamics, chaos control, computer programming, nonlinear dynamical systems, nonlinear operators, statistical convergence, and so forth. The random topology proves to be a very useful tool to deal with such situations where the use of classical theories breaks down. The usual uncertainty principle of Werner Heisenberg leads to a generalized uncertainty principle, which has been motivated by string theory and noncommutative geometry. In strong quantum gravity regime space-time points are determined in a random manner. Thus impossibility of determining the position of particles gives the space-time a random structure. Because of this random structure, position space representation of quantum mechanics breaks down, and therefore a generalized normed space of quasiposition eigenfunction is required. Hence, one needs to discuss on a new family of random norms. There are many situations where the norm of a vector is not possible to be found and the concept of random norm seems to be more suitable in such cases, that is, we can deal with such situations by modeling the inexactness through the random norm1,2.

The stability problem of functional equations originated from a question of Ulam3

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answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki

5for additive mappings and by Th. M. Rassias6for linear mappings by considering an unbounded Cauchy diﬀerence. The paper of Th. M. Rassias6has provided a lot of influence in the development of what we callgeneralized Hyers-Ulam stabilityor asHyers-Ulam-Rassias stabilityof functional equations

11fx2y11fx−2y44fxy44fx−y12f3y

−48f2y60fy−66fx. 1.1

A generalization of the Th. M. Rassias theorem was obtained by G˘avrut¸a7by replacing the unbounded Cauchy diﬀerence by a general control function in the spirit of Th. M. Rassias approach.

The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problemsee6,8–24.

In25, Jun and Kim considered the following cubic functional equation:

f2xyf2x−y2fxy2fx−y12fx. 1.2

It is easy to show that the functionfx x3_{satisfies the functional equation}_1.2_{, which is} called acubic functional equation,and every solution of the cubic functional equation is said to be acubic mapping.

In26, Lee et al. considered the following quartic functional equation:

f2xyf2x−y4fxy4fx−y24fx−6fy. 1.3

It is easy to show that the functionfx x4satisfies the functional equation1.3, which is called aquartic functional equationand every solution of the quartic functional equation is said to be aquartic mapping.

The study of stability of functional equations is important problem in nonlinear sciences and application in solving integral equation via VIM 27–29 PDE and ODE30–

34. LetX be a set A functiond : X×X → 0,∞is called ageneralized metricon X if d 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 1.1see35,36. LetX, dbe a complete generalized metric space and letJ :X → X be a strictly contractive mapping with Lipschitz constantL <1. Then for each given elementx∈X, either

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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|dJn0_{x, y}_<∞}_;

4dy, y∗≤1/1−Ldy, Jyfor ally∈Y.

In 1996, Isac and Th. M. Rassias37were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authorssee38–43.

2. Preliminaries

The theory of random normed spaces RN-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 spaces has been recently studied by Alsina 44, Mirmostafaee and Moslehian45and Mirzavaziri and Moslehian40, Mihet¸ and Radu46, Mihet¸ et al.47,48, Baktash et al.49, and Saadati et al.50.

LetL L,≥Lbe a complete lattice, that is, a partially ordered set in which every

nonempty subset admits supremum and infimum, and 0L infL, 1L supL. The space of latticetic random distribution functions, denoted byΔ_L, is defined as the set of all mappings

F : Ê ∪ {−∞,∞} → L such thatF is left continuous and nondecreasing on Ê, F0

0L, F∞ 1L.

D_L ⊆ Δ_Lis defined asD_L {F ∈Δ_L :l−F∞ 1L}, wherel−fxdenotes the left limit of the functionf at the pointx. The spaceΔ_Lis partially ordered by the usual point-wise ordering of functions, that is,F≥Gif and only ifFt≥LGtfor alltinÊ. The maximal

element forΔ_Lin this order is the distribution function given by

ε0t

⎧ ⎨ ⎩

0L, if t≤0,

1L, if t >0.

2.1

Definition 2.1see51. Atriangular normt-normonLis a mappingT:L2 → Lsatisfying the following conditions:

a ∀x∈L Tx,1L x boundary condition;

b ∀x, y∈L2 Tx, y Ty, x commutativity;

c ∀x, y, z∈L3 Tx,Ty, z TTx, y, z associativity;

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Let{xn}be a sequence inLwhich converges tox∈Lequipped order topology. The

t-normTis said to be acontinuoust-normif

lim

n→ ∞T

xn, y

Tx, y, 2.2

for ally∈L.

At-normTcan be extendedby associativityin a unique way to ann-array operation taking forx1, . . . , xn∈Lnthe valueTx1, . . . , xndefined by

T0

i1xi1, Tni1xiT

Tn−1 i1xi, xn

Tx1, . . . , xn. 2.3

Tcan also be extended to a countable operation taking for any sequencexnn∈NinL

the value

T∞

i1xi lim n→ ∞T

n

i1xi. 2.4

The limit on the right side of2.4exists since the sequenceTn_i₁xin∈is nonincreasing

and bounded from below.

Note that we putTT wheneverL 0,1. IfTis at-norm thenx_Tnis defined for all x∈0,1andn∈N∪ {0}by 1, ifn0 andTx_Tn−1, x, ifn≥1. At-normT is said to beof Hadˇzi´c-typewe denote byT ∈ Hif the familyx_Tn_n_∈_Nis equicontinuous atx1cf.52.

Definition 2.2see 51. A continuoust-norm T on L 0,12 is said to be continuous t-representableif there exist a continuoust-norm∗and a continuoust-conormon0,1such that, for allx x1, x2,y y1, y2∈L,

Tx, yx1∗y1, x2y2. 2.5

For example,

Ta, b a1b1,min{a2b2,1},

Ma, b min{a1, b1},max{a2, b2}

2.6

for alla a1, a2,b b1, b2∈0,12are continuoust-representable. Define the mappingT∧fromL2toLby

T∧x, y

⎧ ⎨ ⎩

x, ify≥Lx,

y, ifx≥Ly.

2.7

Recallsee52,53that if{xn}is a given sequence inL,T∧ni1xi is defined recurrently by

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A negation onLis any decreasing mappingN :L → LsatisfyingN0L 1Land

N1L 0L. IfNNx x, for allx ∈ L, thenNis called an involutive negation. In the following,Lis endowed with afixednegationN.

Definition 2.3. Alatticetic random normed spaceis a tripleX, μ,T∧, whereX is a vector space andμis a mapping fromXintoD_Lsuch that the following conditions hold:

LRN1μxt ε0tfor allt >0 if and only ifx0;

LRN2μαxt μxt/|α|for allxinX,α /0 andt≥0;

LRN3μxyts≥LT∧μxt, μysfor allx, y∈Xandt, s≥0.

We note that fromLPN2it follows thatμ−xt μxt x∈X, t≥0.

Example 2.4. LetL 0,1×0,1and operation≤Lbe defined by

L{a1, a2:a1, a2∈0,1×0,1, a1a2≤1},

a1, a2≤Lb1, b2⇐⇒a1≤b1, a2≥b2, ∀a a1, a2, b b1, b2∈L.

2.8

ThenL,≤Lis a complete latticesee51. In this complete lattice, we denote its units by 0L

0,1and 1L 1,0. LetX, · be a normed space. LetTa, b min{a1, b1},max{a2, b2} for alla a1, a2,b b1, b2∈0,1×0,1andμbe a mapping defined by

μxt t

tx,

x

tx

, ∀t∈Ê

_. _2.9

ThenX, μ,Tis a latticetic random normed space.

IfX, μ,T∧is a latticetic random normed space, then

V{Vε, λ:ε >L0L, λ∈L\ {0L,1L}}, Vε, λ {x∈X:Fxε>LNλ} 2.10

is a complete system of neighborhoods of null vector for a linear topology onXgenerated by the normF.

Definition 2.5. LetX, μ,T∧be a latticetic random normed space.

1A sequence {xn} in X is said to be convergentto x in X if, for every t > 0 and

ε∈L\ {0L}, there exists a positive integerNsuch thatμxn−xt>LNεwhenever

n≥N.

2A sequence{xn}inX is calledCauchy sequenceif, for everyt >0 andε∈L\ {0L}, there exists a positive integerNsuch thatμxn−xmt>LNεwhenevern≥m≥N.

3A latticetic random normed spacesX, μ,T∧is said to becompleteif and only if every Cauchy sequence inXis convergent to a point inX.

Theorem 2.6. IfX, μ,T∧ is a latticetic random normed space and {xn} is a sequence such that

xn → x, thenlimn→ ∞μxnt μxt.

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Lemma 2.7. LetX, μ,T∧be a latticetic random normed space andx∈X. If

μxt C, ∀t >0, 2.11

thenC1Landx0.

Proof. Letμxt Cfor allt > 0. Since Ranμ ⊆ D_L, we haveC 1L, and byLRN1we conclude thatx0.

3. Generalized Hyers-Ulam Stability of the Functional Equation

1.1 :

An Odd Case

One can easily show that an even mappingf :X → Y satisfies1.1if and only if the even mappingf:X → Y is a quartic mapping, that is,

f2xyf2x−y4fxy4fx−y24fx−6fy, 3.1

and that an odd mappingf :X → Ysatisfies1.1if and only if the odd mappingf:X → Y is an additive-cubic mapping, that is,

fx2yfx−2y4fxy4fx−y−6fx. 3.2

It was shown in Lemma 2.2 of55thatgx:f2x−2fxandhx:f2x−8fxare cubic and additive, respectively, and thatfx 1/6gx−1/6hx.

For a given mappingf:X → Y, we define

Dfx, y:11fx2y11fx−2y−44fxy−44fx−y

−12f3y48f2y−60fy66fx 3.3

for allx, y∈X.

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

Theorem 3.1. LetXbe a linear space,Y, μ,T∧a complete LRN -space andΦa mapping fromX2 toD_LΦx, yis denoted byΦx,ysuch that, for some0< α <1/8,

Φ2x,2yt≤LΦx,yαt

x, y∈X, t >0. 3.4

Letf:X → Y be an odd mapping satisfying

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for allx, y∈Xand allt >0. Then

Cx: lim

n→ ∞8

n _f x

2n−1

−2fx 2n

3.6

exists for eachx∈Xand defines a cubic mappingC:X → Y such that

μf2x−2fx−Cxt≥ T∧ Φ0,x

33−264α

17α t

,Φ2x,x

33−264α

17α t

3.7

for allx∈Xand allt >0.

Proof. Lettingx0 in3.5, we get

μ12f3y−48f2y60fyt≥LΦ0,yt 3.8

for ally∈Xand allt >0.

Replacingxby 2yin3.5, we get

μ11f4y−56f3y114f2y−104fyt≥LΦ2y,yt 3.9

for ally∈Xand allt >0. By3.8and3.9,

μf4y−10f2y16fy 14₃₃t₁₁1 t

≥LT∧ μ14/3312f3y−48f2y60fy 14₃₃t

, μ1/1111f4y−56f3y114f2y−104fy ₁₁1 t

≥LT∧

Φ0,yt,Φ2y,yt

3.10

for ally∈Xand allt >0. Lettingy:x/2 andgx:f2x−2fxfor allx∈X, we get

μgx−8gx/2 17₃₃t

≥LT∧Φ0,x/2t,Φx,x/2t 3.11

for allx∈Xand allt >0. Consider the set

S:g:X−→Y, 3.12

and introduce the generalized metric onS:

dg, hinfu∈Ê _:_μ

gx−hxut≥LT∧Φ0,xt,Φ2x,xt, ∀x∈X, ∀t >0

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where, as usual, inf∅ ∞. It is easy to show thatS, dis complete.See the proof of Lemma 2.1 of46.

Now we consider the linear mappingJ:S → Ssuch that

Jgx:8gx 2

3.14

for allx∈X.

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

μgx−hxεt≥LT∧Φ0,xt,Φ2x,xt 3.15

for allx∈Xand allt >0. Hence

μJgx−Jhx8αεt μ8gx/2−8hx/28αεt

μgx/2−hx/2αεt

≥LT∧Φ0,x/2αt,Φx,x/2αt

≥LT∧Φ0,xt,Φ2x,xt

3.16

for allx∈Xand allt >0. Sodg, h εimplies that

dJg, Jh≤8αε. 3.17

This means that

dJg, Jh≤8αdg, h 3.18

for allg, h∈S.

It follows from3.11that

μgx−8gx/2 17 33αt

≥LT∧Φ0,xt,Φ2x,xt 3.19

for allx∈Xand allt >0. So

dg, Jg≤ 17

33α. 3.20

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

1Cis a fixed point ofJ, that is,

Cx 2

1

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for allx∈X. Sinceg:X → Y is odd,C:X → Y is an odd mapping. The mapping Cis a unique fixed point ofJin the set

Mg∈S:df, g<∞. 3.22

This implies thatCis a unique mapping satisfying3.21such that there exists a u∈0,∞satisfying

μgx−Cxut≥LT∧Φ0,xt,Φ2x,xt 3.23

for allx∈Xand allt >0;

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

lim

n→ ∞8

n_gx

2n

Cx 3.24

for allx∈X;

3dg, C≤1/1−8αdg, Jg, which implies the inequality

dg, C≤ 17α

33−264α. 3.25

This implies that inequality3.7holds.

FromDgx, y Df2x,2y−2Dfx, y, by3.5, we deduce that

μDf2x,2yt≥LΦ2x,2yt, μ−2Dfx,yt μDfx,y

t 2

≥LΦx,y

t 2

, 3.26

and so, byLRN3and3.4, we obtain

μDgx,y3t≥LT∧μDf2x,2yt, μ−2Dfx,y2t≥LT∧Φ2x,2yt,Φx,yt

≥LΦ2x,2yt.

3.27

It follows that

μ8n_Dg_x/₂n_,y/₂n3t μ_Dg_x/₂n_,y/₂n 3 t

8n

≥LΦx/2n−1_,y/₂n−1

t 8n

≥L · · · ≥LΦx,y

1 8

t

8αn−1

3.28

for allx, y∈X, allt >0 and alln∈Æ. Since 0<8α <1,

lim

n→ ∞Φx,y t

8αn

1L 3.29

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μDCx,yt 1L 3.30

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

Corollary 3.2. Letθ≥0and letpbe a real number withp >3. LetXbe a normed vector space with norm · and letX, μ,T∧be an LRN -space in whichL 0,1andT∧min. Letf:X → Y be an odd mapping satisfying

μDfx,yt≥ t

tθxpyp 3.31

for allx, y∈Xand allt >0. Then

Cx: lim

n→ ∞8

n _f x

2n−1

−2fx 2n

3.32

μf2x−2fx−Cxt≥

332p₋₈_t

332p₋₈_t₁₇₁₂p_θ_xp 3.33

Proof. The proof follows fromTheorem 3.1by taking

Φx,yt:

t

tθxpyp 3.34

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

Theorem 3.3. LetXbe a linear space,Y, μ,T∧a complete LRN -space andΦa mapping fromX2 toD_LΦx, yis denoted byΦx,ysuch that, for some0< α <8,

Φx,yαt≥LΦx/2,y/2t

x, y∈X, t >0. 3.35

Letf:X → Y be an odd mapping satisfying1.1. Then

Cx: lim

n→ ∞ 1 8n

f2n1x−2f2nx 3.36

μf2x−2fx−Cxt≥ T∧ Φ0,x

264−33α

17 t

,Φ2x,x

264−33α

17 t

3.37

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

Jgx: 1

8g2x 3.38

for allx∈X.

μJgx−Jhx α₈εt

μ1/8g2x−1/8h2x α₈εt

μg2x−h2xαεt

≥LT∧Φ0,2xαt,Φ4x,2xαt

3.40

dJg, Jh≤ α

8ε. 3.41

This means that

dJg, Jh≤ α 8d

g, h 3.42

for allg, h∈S.

μgx−1/8g2x ₂₆₄17 t

for allx∈Xand allt >0. Sodg, Jg≤17/264.

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

1Cis a fixed point ofJ, that is,

C2x 8Cx 3.44

for allx∈X. Sinceg:X → Yis odd,C:X → Y is an odd mapping. The mapping Cis a unique fixed point ofJin the set

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This implies thatCis a unique mapping satisfying3.44such that there exists a u∈0,∞satisfying

μgx−Cxut≥LT∧Φ0,xt,Φ2x,xt 3.46

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

lim

n→ ∞ 1 8ng2

n_x _C_x _3.47

for allx∈X;

3dg, C≤1/1−α/8dg, Jg, which implies the inequality

dg, C≤ 17

264−33α. 3.48

The rest of the proof is similar to the proof ofTheorem 3.1.

Corollary 3.4. Letθ ≥ 0, and letp be a real number with0 < p < 3. LetX be a normed vector space with norm · , and letX, μ,T∧be an LRN-space in whichL 0,1and T∧ min. Let f:X → Ybe an odd mapping satisfying3.31. Then

Cx: lim

n→ ∞ 1 8n

f2n1x−2f2nx 3.49

μf2x−2fx−Cxt≥

338−2p_t

338−2p_t₁₇₁₂p_θ_xp 3.50

Φx,yt: t

tθxpyp 3.51

for allx, y∈X. Then we can chooseα2p_{, and we get the desired result.}

Theorem 3.5. LetXbe a linear space,X, μ,T∧an LRN-space and letΦbe a mapping fromX2to D_L Φx, yis denoted byΦx,ysuch that, for some0< α <1/2,

Φx,yαt≥LΦ2x,2yt

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Ax: lim

n→ ∞2

n _f x

2n−1

−8fx 2n

3.53

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

μf2x−8fx−Axt≥LT∧ Φ0,x

33−66α 17α t

,Φ2x,x

33−66α 17α t

3.54

Proof. LetS, dbe the generalized metric space defined in the proof ofTheorem 3.1. Lettingy:x/2 andhx:f2x−8fxfor allx∈Xin3.10, we get

μhx−2hx/2 17₃₃t

≥LT∧Φ0,x/2t,Φx,x/2t 3.55

for allx∈Xand allt >0.

Jhx:2hx 2

3.56

for allx∈X.

μJgx−Jhx2αεt μ2gx/2−2hx/22αεt

≥LT∧Φ0,x/2αt,Φx,x/2αt

3.58

for allx∈Xand allt >0. Sodg, h εimplies thatdJg, Jh≤2αε. This means that

dJg, Jh≤2αdg, h 3.59

for allg, h∈S.

μhx−2hx/2 17₃₃αt

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

1Ais a fixed point ofJ, that is,

Ax 2

1

2Ax 3.61

for allx∈X. Sinceh:X → Y is odd,A:X → Yis an odd mapping. The mapping Ais a unique fixed point ofJin the set

Mg∈S:df, g<∞. 3.62

This implies thatAis a unique mapping satisfying 3.61such that there exists a u∈0,∞satisfying

μhx−Axut≥LT∧Φ0,xt,Φ2x,xt 3.63

2dJnh, A → 0 asn → ∞. This implies the equality

lim

n→ ∞2

n_hx

2n

Ax 3.64

for allx∈X;

3dh, A≤1/1−2αdh, Jh, which implies the inequality

dh, A≤ 17α

33−66α. 3.65

The rest of the proof is similar to the proof ofTheorem 3.1.

Corollary 3.6. Letθ≥0, and letpbe a real number withp >1. LetXbe a normed vector space with norm · , and letX, μ,T_∧be an LRN-space in whichL 0,1andT_∧min. Letf:X → Y be an odd mapping satisfying3.31. Then

Ax: lim

n→ ∞2

n _f x

2n−1

−8fx 2n

3.66

μf2x−8fx−Axt≥ 332

p₋₂_t

332p₋₂_t₁₇₁₂p_θ_xp 3.67

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Φx,yt: t

tθxpyp 3.68

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

Theorem 3.7. LetXbe a linear space,X, μ,T∧an LRN-space and letΦbe a mapping fromX2to D_L Φx, yis denoted byΦx,ysuch that, for some0< α <2,

x, y∈X, t >0. 3.69

Ax: lim

n→ ∞

1 2n

f2n1x−8f2nx 3.70

μf2x−8fx−Axt≥LT∧ Φ0,x 66−33α

17 t

,Φ2x,x 66−33α

17 t

3.71

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

Jhx: 1

2h2x 3.72

for allx∈X.

μJgx−JhxLεt μ1/2g2x−1/2h2x

_α

2εt

μg2x−h2xαεt

≥LT∧Φ0,2xαt,Φ4x,2xαt

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dJg, Jh≤ α

2ε. 3.75

This means that

dJg, Jh≤ α 2d

g, h 3.76

for allg, h∈S.

μhx−1/2h2x 17₆₆t

for allx∈Xand allt >0. Sodh, Jh≤17/66.

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

1Ais a fixed point ofJ, that is,

A2x 2Ax 3.78

for allx∈X. Sinceh:X → Y is odd,A:X → Yis an odd mapping. The mapping Ais a unique fixed point ofJin the set

Mg∈S:df, g<∞. 3.79

This implies thatAis a unique mapping satisfying 3.78such that there exists a u∈0,∞satisfying

μhx−Axut≥LT∧Φ0,xt,Φ2x,xt 3.80

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

lim

n→ ∞ 1 2nh2

n_x _A_x _3.81

for allx∈X;

3dh, A≤1/1−α/2dh, Jh, which implies the inequality

dh, A≤ 17

66−33α. 3.82

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Corollary 3.8. Letθ ≥ 0, and letp be a real number with0 < p < 1. LetX be a normed vector space with norm · , and letX, μ,T∧be an LRN-space in whichL 0,1and T∧ min. Let f:X → Ybe an odd mapping satisfying3.31. Then

Ax: lim

n→ ∞ 1 2n

f2n1x−8f2nx 3.83

μf2x−8fx−Axt≥

332−2p_t

332−2p_t₁₇₁₂p_θ_xp 3.84

Φx,yt: t

tθxpyp 3.85

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

4. Generalized Hyers-Ulam Stability of the Functional Equation

1.1 :

An Even Case

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

Theorem 4.1. LetXbe a linear space,X, μ,T∧an LRN-space and letΦbe a mapping fromX2to D_L Φx, yis denoted byΦx,ysuch that, for some0< α <1/16,

Φx,yαt≥LΦ2x,2yt

x, y∈X, t >0. 4.1

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

Qx: lim

n→ ∞16

n_fx

2n

4.2

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

μfx−Qxt≥LT∧ Φ0,x 22−352α

13α t

,Φx,x 22−352α

13α t

4.3

Proof. Lettingx0 in3.5, we get

μ12f3y−70f2y148fyt≥LΦ0,yt 4.4

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Lettingxyin3.5, we get

μf3y−4f2y−17fyt≥LΦy,yt 4.5

for ally∈Xand allt >0. By4.4and4.5,

μf2y−16fy

1 22t

12 22t

≥LT∧ μ1/2212f3y−70f2y148fy ₂₂1t

, μ12/22f3y−4f2y−17fy 12₂₂t

≥LT∧Φ0,yt,Φy,yt

4.6

for ally∈Xand allt >0. Consider the set

S:g:X−→Y, 4.7

and introduce the generalized metric onS

dg, hinfu∈Ê :N

gx−hx, ut≥LT∧Φ0,xt,Φx,xt, ∀x∈X, ∀t >0

, 4.8

where, as usual, inf∅ ∞. It is easy to show thatS, dis complete.See the proof of Lemma 2.1 of46.

Jgx:16gx 2

4.9

for allx∈X.

μgx−hxεt≥LT∧Φ0,xt,Φx,xt 4.10

μJgx−Jhx16αεt μ16gx/2−16hx/216αεt

≥LT∧Φ0,x/2αt,Φx/2,x/2αt

≥LT∧Φ0,xt,Φx,xt

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dJg, Jh≤16αε. 4.12

This means that

dJg, Jh≤16αdg, h 4.13

for allg, h∈S.

μfx−16fx/2 13₂₂αt

≥LT∧Φ0,xt,Φx,xt 4.14

for allx∈Xand allt >0. Sodf, Jf≤13α/22.

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

1Qis a fixed point ofJ, that is,

Qx 2

1

16Qx 4.15

for allx ∈ X. Sincef : X → Y is even, Q : X → Y is an even mapping. The mappingQis a unique fixed point ofJin the set

Mg∈S:df, g<∞. 4.16

This implies thatQis a unique mapping satisfying 4.15such that there exists a u∈0,∞satisfying

μfx−Qxut≥LT∧Φ0,xt,Φx,xt 4.17

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

lim

n→ ∞16 n_fx

2n

Qx 4.18

for allx∈X;

3df, Q≤1/1−16αdf, Jf, which implies the inequality

df, Q≤ 13α

22−352α. 4.19

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Corollary 4.2. Letθ≥0, and letpbe a real number withp >4. LetXbe a normed vector space with norm · , and letX, μ,T∧be an LRN-space in whichL 0,1andT∧min. Letf:X → Y be an even mapping satisfyingf0 0and3.31. Then

Qx: lim

n→ ∞16

n_fx

2n

4.20

μfx−Qxt≥ 112

p₋₁₆_t

112p₋₁₆_t₁₃_θ_xp 4.21

Φx,yt: t

tθxpyp 4.22

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

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

Theorem 4.3. LetXbe a linear space,X, μ,T∧an LRN -space and letΦbe a mapping fromX2to D_L Φx, yis denoted byΦx,ysuch that, for some0< α <16,

x, y∈X, t >0. 4.23

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

Qx: lim

n→ ∞ 1 16nf2

n_x _4.24

μfx−Qxt≥LT∧ Φ0,x 352−22α

13 t

,Φx,x 352−22α

13 t

4.25

Corollary 4.4. Letθ ≥ 0, and letp be a real number with0 < p < 4. LetX be a normed vector space with norm · , and letX, μ,T∧be an LRN-space in whichL 0,1and T∧ min. Let f:X → Ybe an even mapping satisfyingf0 0and3.31. Then

Qx: lim

n→ ∞ 1 16nf2

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μfx−Qxt≥

1116−2p_t

1116−2p_t₁₃_θ_xp 4.27

Φx,yt:

t

tθxpyp 4.28

for allx, y∈X. Then we can chooseα2p_{, and we get the desired result.}

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