2. Generalized Hyers-Ulam Stability of the Functional Equation 1.1:

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Volume 2010, Article ID 253040,22pages doi:10.1155/2010/253040

Research Article

Fuzzy Stability of an Additive-Quadratic-Quartic Functional Equation

Choonkil Park

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

Correspondence should be addressed to Choonkil Park,baak@hanyang.ac.kr Received 27 August 2009; Accepted 30 November 2009

Academic Editor: Yeol J. E. Cho

Copyrightq2010 Choonkil Park. 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 equation:fx2y fx−2y 2fxy 2f−x−y 2fx−y 2fy−x−4f−x−2fx f2y f−2y−4fy−4f−yin fuzzy Banach spaces.

1. Introduction and Preliminaries

Katsaras1defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view2–4. In particular, Bag and Samanta5, following Cheng and Mordeson6, gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Mich´alek type7. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces8.

We use the definition of fuzzy normed spaces given in5,9,10to investigate a fuzzy version of the generalized Hyers-Ulam stability forthe following functional equation

f x2y

f x−2y

2f xy

2f

−x−y 2f

x−y 2f

y−x

−4f−x−2fx f 2y

f

−2y

−4f y

−4f

−y 1.1

in the fuzzy normed vector space setting.

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Definition 1.1see5,9–11. LetXbe a real vector space. A functionN :X×R → 0,1is called a fuzzy norm onXif for allx, y∈Xand alls, t∈R,

N1Nx, t 0 fort≤0;

N2x0 if and only ifNx, t 1 for allt >0;

N3Ncx, t Nx, t/|c|ifc /0;

N4Nxy, st≥min{Nx, s, Ny, t};

N5Nx,·is a nondecreasing function ofRand lim_{t→ ∞}Nx, t 1;

N6forx /0,Nx,·is continuous onR.

The pairX, Nis called a fuzzy normed vector space.

The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in9,12.

Definition 1.2see5,9–11. LetX, Nbe a fuzzy normed vector space. A sequence{xn}in Xis said to be convergent or converge if there exists anx∈Xsuch that limn→ ∞Nxn−x, t 1 for allt > 0. In this case,xis called the limit of the sequence{xn}and we denote it byN- lim_n_{→ ∞}x_nx.

Definition 1.3see5,9,10. LetX, Nbe a fuzzy normed vector space. A sequence{xn}in X is called Cauchy if for eachε > 0 and eacht > 0 there exists ann₀ ∈ Nsuch that for all n≥n₀and allp >0, we haveNxnp−x_n, t>1−ε.

It is wellknown that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

We say that a mappingf : X → Y between fuzzy normed vector spacesX andY is continuous at a pointx₀ ∈ Xif for each sequence{xn}converging tox₀ inX, the sequence {fxn}converges tofx0. Iff :X → Yis continuous at eachx∈X, thenf:X → Yis said to be continuous onXsee8.

The stability problem of functional equations originated from a question of Ulam13 concerning the stability of group homomorphisms. Hyers14gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki 15for additive mappings and by Th. M. Rassias16for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias 16 has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability or as Hyers- Ulam-Rassias stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by G˘avruta17by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach.

The functional equation f

xy f

x−y

2fx 2f y

1.2

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof18for mappingsf :X → Y, whereX is a normed space andY is a Banach space. Cholewa19noticed that the theorem of Skof is

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still true if the relevant domainXis replaced by an Abelian group. Czerwik20proved 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 problemsee16,21–39.

In40, Lee et al. considered the following quartic functional equation:

f 2xy

f 2x−y

4f xy

4f x−y

24fx−6f y

. 1.3

It is easy to show that the functionfx x⁴ satisfies the functional equation1.3, which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping.

LetX be a set. A functiond:X×X → 0,∞is called a generalized metric onXifd 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 1.4see41,42. 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

d

Jⁿx, Jⁿ¹x

∞ 1.4

for all nonnegative integersnor there exists a positive integern₀such that 1dJⁿx, Jⁿ¹x<∞, for alln≥n0;

2the sequence{Jⁿx}converges to a fixed pointy^∗ofJ;

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

4dy, y^∗≤1/1−Ldy, Jyfor ally∈Y.

In 1996, G. Isac and Th. M. Rassias43were 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 authorssee12,44–48.

This paper is organized as follows. InSection 2, we prove the generalized Hyers-Ulam stability of the additive-quadratic-quartic functional equation1.1in fuzzy Banach spaces for an odd case. InSection 3, we prove the generalized Hyers-Ulam stability of the additive- quadratic-quartic functional equation1.1in fuzzy Banach spaces for an even case.

Throughout this paper, assume that X is a vector space and thatY, N is a fuzzy Banach space.

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2. Generalized Hyers-Ulam Stability of the Functional Equation 1.1:

An Odd Case

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

f x2y

f x−2y

2fx. 2.1

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

f x2y

f x−2y

4f xy

4f x−y

−6fx 2f 2y

−8f y

. 2.2

It was shown in49, 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

Df x, y

:f x2y

f x−2y

−2f xy

−2f

−x−y

−2f x−y

−2f y−x 4f−x 2fx−f

2y

−f

−2y 4f

y 4f

−y

2.3

for allx, y∈X.

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

Theorem 2.1. Letϕ:X² → 0,∞be a function such that there exists anL <1 with

ϕ x, y

≤ L 3ϕ

3x,3y 2.4

for allx, y∈X. Letf:X → Ybe an odd mapping satisfying

N Df

x, y , t

≥ t

tϕ

x, y 2.5

for allx, y∈Xand allt >0. Then

Ax:N- lim

n→ ∞3ⁿf x

3ⁿ

2.6

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

N

fx−Ax, t

≥ 3−3Lt

3−3LtLϕx, x 2.7

for allx∈Xand allt >0.

Proof. Lettingxyin2.5, we get

N

f3x−3fx, t

≥ t

tϕx, x 2.8

for allx∈Xand allt >0.

Consider the set

S:

g:X−→Y 2.9

and introduce the generalized metric onS

d g, h

inf

μ∈R :N

gx−hx, μt

≥ t

tϕx, x, ∀x∈X, ∀t >0

, 2.10

where, as usual, infφ ∞. It is easy to show thatS, dis complete.see the proof of Lemma 2.1 of50.

Now we consider the linear mappingJ:S → Ssuch that

Jgx:3g x

3

2.11

for allx∈X.

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

N

gx−hx, εt

≥ t

tϕx, x 2.12

for allx∈Xand allt >0. Hence

N

Jgx−Jhx, Lεt N

3g

x 3

−3h x

3

, Lεt

N

g x

3

−h x

3

,L 3εt

≥ Lt/3

Lt/3ϕx/3, x/3 ≥ Lt/3 Lt/3 L/3ϕx, x t

tϕx, x

2.13

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

Jg, Jh

≤Ld g, h

2.14

for allg, h∈S.

It follows from2.8that

N

fx−3f x

3

,L 3t

≥ t

tϕx, x 2.15

for allx∈Xand allt >0. Sodf, Jf≤L/3.

ByTheorem 1.4, there exists a mappingA:X → Y satisfying the following.

1Ais a fixed point ofJ, that is,

A x

3

1

3Ax 2.16

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

M

g∈S:d f, g

<∞ . 2.17

This implies thatAis a unique mapping satisfying2.16such that there exists aμ∈0,∞ satisfying

N

fx−Ax, μt

≥ t

tϕx, x 2.18

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

N- lim

n→ ∞3ⁿf x

3ⁿ

Ax 2.19

for allx∈X;

3df, A≤1/1−Ldf, Jf, which implies the inequality

d f, A

≤ L

3−3L. 2.20

This implies that inequality2.7holds.

By2.5,

N

3ⁿDf x

3ⁿ, y 3ⁿ

,3ⁿt

≥ t

tϕ

x/3ⁿ, y/3ⁿ 2.21

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for allx, y∈X, allt >0,and alln∈N. So

N

3ⁿDf x

3ⁿ, y 3ⁿ

, t

≥ t/3ⁿ t/3ⁿ Lⁿ/3ⁿϕ

x, y 2.22

for allx, y∈X, allt >0,and alln∈N. Since limn→ ∞t/3ⁿ/t/3ⁿ Lⁿ/3ⁿϕx, y 1 for allx, y∈Xand allt >0,

N DA

x, y , t

1 2.23

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

Corollary 2.2. Letθ≥0 and letpbe a real number withp >1. LetXbe a normed vector space with norm · . Letf :X → Ybe an odd mapping satisfying

N Df

x, y , t

≥ t

tθ

x ^py^p 2.24

for allx, y∈Xand allt >0. Then

Ax:N− lim

n→ ∞3ⁿf x

3ⁿ

2.25

N

fx−Ax, t

≥ 3^p−3t

3^p−3t2θ x ^p 2.26

Proof. The proof follows fromTheorem 2.1by taking ϕ

x, y :θ

x ^py^p

2.27

for allx, y∈X. Then we can chooseL3^1−pand we get the desired result.

ϕ x, y

≤3Lϕ x

3,y 3

2.28

for allx, y∈X. Letf:X → Ybe an odd mapping satisfying2.5. Then

Ax:N− lim

n→ ∞

1

3ⁿf3ⁿx 2.29

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exists for eachx∈Xand defines an additive mappingC:X → Y such that

N

fx−Ax, t

≥ 3−3Lt

3−3Ltϕx, x 2.30

Proof. LetS, dbe the generalized metric space defined in the proof ofTheorem 2.1.

Consider the linear mappingJ:S → Ssuch that

Jfx: 1

3f3x 2.31

for allx∈X.

N

gx−hx, εt

≥ t

tϕx, x 2.32

N

Jgx−Jhx, Lεt N

1

3g3x−1

3h3x, Lεt

N

g3x−h3x,3Lεt

≥ 3Lt

3Ltϕ3x,3x ≥ 3Lt 3Lt3Lϕx, x t

tϕx, x

2.33

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

d Jg, Jh

≤Ld g, h

2.34

for allg, h∈S.

N

fx−1

3f3x,1 3t

≥ t

tϕx, x 2.35

for allx∈Xand allt >0. Sodf, Jf≤1/3.

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ByTheorem 1.4, there exists a mappingA:X → Y satisfying the following.

1Ais a fixed point ofJ, that is,

A3x 3Ax 2.36

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

M

g∈S:d f, g

<∞ . 2.37

This implies thatAis a unique mapping satisfying2.36such that there exists aμ∈0,∞ satisfying

N

fx−Ax, μt

≥ t

tϕx, x 2.38

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

N- lim

n→ ∞

1

3ⁿf3ⁿx Ax 2.39

for allx∈X;

3df, A≤1/1−Ldf, Jf, which implies the inequality

d f, A

≤ 1

3−3L. 2.40

This implies that the inequality2.30holds.

The rest of the proof is similar to the proof ofTheorem 2.1.

Corollary 2.4. Letθ≥0 and letpbe a real number with 0< p <1. LetXbe a normed vector space with norm · . Letf :X → Y be an odd mapping satisfying2.24. Then

Ax:N- lim

n→ ∞

1

3ⁿf3ⁿx 2.41

N

fx−Ax, t

≥ 3−3^pt

3−3^pt2θ x ^p 2.42

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Proof. The proof follows fromTheorem 2.3by taking

ϕ x, y

:θ

x ^py^p

2.43

for allx, y∈X. Then we can chooseL3^p−1and we get the desired result.

3. 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 fuzzy Banach spaces: an even case.

ϕ x, y

≤ L 16ϕ

2x,2y 3.1

for allx, y∈X. Letf:X → Ybe an even mapping satisfyingf0 0 and2.5. Then

Qx:N- lim

n→ ∞16ⁿ

f x

2ⁿ⁻¹

−4fx 2ⁿ

3.2

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

N

f2x−4fx−Qx, t

≥ 16−16Lt

16−16Lt5L

ϕx, x ϕ2x, x 3.3

Proof. Lettingxyin2.5, we get

N f

3y

−6f 2y

15f y

, t

≥ t

tϕ

y, y 3.4

for ally∈Xand allt >0.

Replacingxby 2yin2.5, we get

N f

4y

−4f 3y

4f 2y

4f y

, t

≥ t

tϕ

2y, y 3.5

for ally∈Xand allt >0.

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By3.4and3.5,

N

f4x−20f2x 64fx,4tt

≥min N

4

f3x−6f2x 15fx ,4t

, N

f4x−4f3x 4f2x 4fx, t

≥ t

tϕx, x ϕ2x, x

3.6

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

N

gx−16gx 2

,5t

≥ t

tϕx/2, x/2 ϕx, x/2 3.7

Consider the set

S:

g:X−→Y 3.8

and introduce the generalized metric onS

d g, h

inf

μ∈R:N

gx−hx, μt

≥ t

tϕx, x ϕ2x, x, ∀x∈X, ∀t >0

, 3.9

where, as usual, infφ ∞. It is easy to show thatS, dis complete.see the proof of Lemma 2.1 of50.

Jgx:16gx 2

3.10

for allx∈X.

N

gx−hx, εt

≥ t

tϕx, x ϕ2x, x 3.11

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N

Jgx−Jhx, Lεt N

16gx 2

−16hx 2

, Lεt N

gx

2

−hx 2

, L 16εt

≥ Lt/16

Lt/16ϕx/2, x/2ϕx, x/2

≥ Lt/16

Lt/16L/16

ϕx, xϕ2x, x t

tϕx, x ϕ2x, x

3.12

d Jg, Jh

≤Ld g, h

3.13

for allg, h∈S.

N

gx−16gx 2

,5L 16t

≥ t

tϕx, x ϕ2x, x 3.14

for allx∈Xand allt >0. Sodg, Jg≤5L/16.

ByTheorem 1.4, there exists a mappingQ:X → Y satisfying the following.

1Qis a fixed point ofJ, that is,

Qx 2

1

16Qx 3.15

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

M

g∈S:d f, g

<∞ . 3.16

This implies thatQis a unique mapping satisfying3.15such that there exists aμ∈0,∞ satisfying

N

gx−Qx, μt

≥ t

tϕx, x ϕ2x, x 3.17

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2dJⁿg, Q → 0 asn → ∞. This implies the equality

N- lim

n→ ∞16ⁿgx 2ⁿ

Qx 3.18

for allx∈X.

3dg, Q≤1/1−Ldg, Jg, which implies the inequality

d g, Q

≤ 5L

16−16L. 3.19

The rest of the proof is similar to that of the proof ofTheorem 2.1.

Corollary 3.2. Letθ≥0 and letpbe a real number withp >4. LetXbe a normed vector space with norm · . Letf :X → Ybe an even mapping satisfyingf0 0 and2.24. Then

Qx:N- lim

n→ ∞16ⁿ

f x

2ⁿ⁻¹

−4fx 2ⁿ

3.20

N

≥ 2^p−16t

2^p−16t532^pθ x ^p 3.21

ϕ x, y

:θ

x ^py^p

3.22

ϕ x, y

≤16Lϕx 2,y

2

3.23

Qx:N- lim

n→ ∞

1 16ⁿ

f

2ⁿ¹x

−4f2ⁿx

3.24

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N

≥ 16−16Lt

16−16Lt5ϕx, x 5ϕ2x, x 3.25

Jgx: 1

16g2x 3.26

for allx∈X.

N

gx−hx, εt

≥ t

tϕx, x ϕ2x, x 3.27

N

1

16g2x− 1

16h2x, Lεt

N

g2x−h2x,16Lεt

≥ 16Lt

16Ltϕ2x,2x ϕ4x,2x ≥ 16Lt

16Lt16L

ϕx, x ϕ2x, x t

tϕx, x ϕ2x, x

3.28

d Jg, Jh

≤Ld g, h

3.29

for allg, h∈S.

N

gx− 1

16g2x, 5 16t

≥ t

tϕx, x ϕ2x, x 3.30

for allx∈Xand allt >0. Sodg, Jg≤5/16.

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ByTheorem 1.4, there exists a mappingQ:X → Y satisfying the following.

1Qis a fixed point ofJ, that is,

Q2x 16Qx 3.31

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

M

g∈S:d f, g

<∞ . 3.32

This implies thatQis a unique mapping satisfying3.31such that there exists aμ∈0,∞ satisfying

N

gx−Qx, μt

≥ t

tϕx, x ϕ2x, x 3.33

2dJⁿg, Q → 0 asn → ∞. This implies the equality

N- lim

n→ ∞

1

16ⁿg2ⁿx Qx 3.34

for allx∈X;

3dg, Q≤1/1−Ldg, Jg, which implies the inequality

d g, Q

≤ 5

16−16L. 3.35

Corollary 3.4. Letθ≥0 and letpbe a real number with 0< p <4. LetXbe a normed vector space with norm · . Letf :X → Y be an even mapping satisfyingf0 0 and2.24. Then

Qx:N- lim

n→ ∞

1 16ⁿ

f

2ⁿ¹x

−4f2ⁿx

3.36

N

≥ 16−2^pt

16−2^pt532^pθ x ^p 3.37

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ϕ x, y

:θ

x ^py^p

3.38

ϕ x, y

≤ L 4ϕ

2x,2y 3.39

Tx:N- lim

n→ ∞4ⁿ

f x

2ⁿ⁻¹

−16fx 2ⁿ

3.40

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

N

f2x−16fx−Tx, t

≥ 4−4Lt

4−4Lt5L

ϕx, x ϕ2x, x 3.41

Lettinghx:f2x−16fxfor allx∈Xin3.6, we get

N

hx−4hx 2

,5t

≥ t

tϕx/2, x/2 ϕx, x/2 3.42

Jhx:4hx 2

3.43

for allx∈X.

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N

gx−hx, εt

≥ t

tϕx, x ϕ2x, x 3.44

for allx∈Xand allt >0. Hence N

4gx 2

−4hx 2

, Lεt N

gx

2

−hx 2

,L 4εt

≥ Lt/4

Lt/4ϕx/2, x/2 ϕx/2, x

≥ Lt/4

Lt/4 L/4

ϕx, x ϕ2x, x t

tϕx, x ϕ2x, x

3.45

Jg, Jh

≤Ld g, h

3.46

for allg, h∈S.

N

hx−4hx 2

,5L 4 t

≥ t

tϕx, x ϕ2x, x 3.47

for allx∈Xand allt >0. Sodh, Jh≤5L/4.

ByTheorem 1.4, there exists a mappingT :X → Y satisfying the following.

1Tis a fixed point ofJ, that is,

Tx 2

1

4Tx 3.48

for allx∈X. Sinceh:X → Y is even,T :X → Y is an even mapping. The mappingTis a unique fixed point ofJin the set

M

g∈S:d f, g

<∞ . 3.49

This implies thatT is a unique mapping satisfying3.48such that there exists aμ ∈0,∞ satisfying

N

hx−Tx, μt

≥ t

tϕx, x ϕ2x, x 3.50

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

N- lim

n→ ∞4ⁿhx 2ⁿ

Tx 3.51

for allx∈X.

3dh, T≤1/1−Ldh, Jh, which implies the inequality

dh, T≤ 5L

4−4L. 3.52

Corollary 3.6. Letθ≥0 and letpbe a real number withp >2. LetXbe a normed vector space with norm · . Letf :X → Ybe an even mapping satisfyingf0 0 and2.24. Then

Tx:N- lim

n→ ∞4ⁿ

f x

2ⁿ⁻¹

−16fx 2ⁿ

3.53

N

f2x−16fx−Tx, t

≥ 2^p−4t

2^p−4t532^pθ x ^p 3.54

ϕ x, y

:θ

x ^py^p

3.55

ϕ x, y

≤4Lϕx 2,y

2

3.56

Tx:N- lim

n→ ∞

1 4ⁿ

f

2ⁿ¹x

−16f2ⁿx

3.57

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N

f2x−16fx−Tx, t

≥ 4−4Lt

4−4Lt5ϕx, x 5ϕ2x, x 3.58

Jhx: 1

4h2x 3.59

for allx∈X.

N

gx−hx, εt

≥ t

tϕx, x ϕ2x, x 3.60

N

Jgx−Jhx, Lεt N

1

4g2x−1

4h2x, Lεt

N

g2x−h2x,4Lεt

≥ 4Lt

4Ltϕ2x,2x ϕ4x,2x ≥ 4Lt

4Lt4L

ϕx, x ϕ2x, x t

tϕx, x ϕ2x, x

3.61

Jg, Jh

≤Ld g, h

3.62

for allg, h∈S.

N

hx−1

4h2x,5 4t

≥ t

tϕx, x ϕ2x, x 3.63

for allx∈Xand allt >0. Sodh, Jh≤5/4.

ByTheorem 1.4, there exists a mappingT :X → Y satisfying the following.

1Tis a fixed point ofJ, that is,

T2x 4Tx 3.64

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for allx∈X. Sinceh:X → Y is even,T :X → Y is an even mapping. The mappingTis a unique fixed point ofJin the set

M

g∈S:d f, g

<∞ . 3.65

This implies thatT is a unique mapping satisfying3.64such that there exists aμ ∈0,∞ satisfying

N

hx−Tx, μt

≥ t

tϕx, x ϕ2x, x 3.66

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

N- lim

n→ ∞

1

4ⁿh2ⁿx Tx 3.67

for allx∈X.

3dh, T≤1/1−Ldh, Jh, which implies the inequality

dh, T≤ 5

4−4L. 3.68

Corollary 3.8. Letθ≥0 and letpbe a real number with 0< p <2. LetXbe a normed vector space with norm · . Letf :X → Y be an even mapping satisfyingf0 0 and2.24. Then

Tx:N- lim

n→ ∞

1 4ⁿ

f

2ⁿ¹x

−16f2ⁿx

3.69

N

f2x−16fx−Tx, t

≥ 4−2^pt

4−2^pt532^pθ x ^p 3.70

Proof. The proof follows fromTheorem 3.7by taking ϕ

x, y :θ

x ^py^p

3.71

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Acknowledgment

This work was supported by the Hanyang University in 2009.

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