2. Fixed Points and Functional Equations Associated with Inner Product Spaces

Volume 2010, Article ID 713675,14pages doi:10.1155/2010/713675

Research Article

Fixed Points, Inner Product Spaces, and Functional Equations

Choonkil Park

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

Correspondence should be addressed to Choonkil Park,[email protected] Received 1 February 2010; Revised 30 May 2010; Accepted 5 July 2010 Academic Editor: Marl`ene Frigon

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.

Rassias introduced the following equalityn

i,j1xi−xj²2nn

i1xi²,n

i1xi0, for a fixed integern≥3. LetV, Wbe real vector spaces. It is shown that, if a mappingf:V → Wsatisfies the following functional equationn

i,j1fxi−xj 2nn

i1fxifor allx1, . . . , xn∈Vwithn i1xi0, which is defined by the above equality, then the mappingf : V → W is realized as the sum of an additive mapping and a quadratic mapping. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

1. Introduction and Preliminaries

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 3for additive mappings and by Rassias 4for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias4has provided a lot of influence on the development of what we call the generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. A generalization of the Rassias theorem was obtained by G˘avrut¸a5by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.

A square norm on an inner product space satisfies the parallelogram equality

x y² x−y² 2x² 2y². 1.1

The functional equation

f x y

f x−y

2fx 2f y

1.2

is called a quadratic functional equation. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof6for mappingsf : X → Y, whereX is a normed space andY is a Banach space. Cholewa 7 noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik8proved the generalized Hyers-Ulam stability of the quadratic functional equation. The generalized Hyers-Ulam stability of the above quadratic functional equation and of two functional equations of quadratic type was obtained by C˘adariu and Radu9.

By a square norm on an inner product space, Rassias10introduced the following equality:

n i,j1

xi−xj²2n n

i1

xi², n

i1

xi0, 1.3

for a fixed integer n ≥ 3. By the above equality, we can define the following functional equation:

n i,j1

f x_i−x_j

2n n

i1

fxi 1.4

for allx₁, . . . , x_n∈V with_n

i1x_i0, whereV is a real vector space.

A square norm on an inner product space satisfies

3 i,j1

xi−xj²6 3

i1

xi² 1.5

for allx₁, x₂, x₃∈Rwithx₁ x₂ x₃ 0see10.

From the above equality we can define the following functional equation:

h x−y

h 2x y

h x 2y

3hx 3h y

3h x y

, 1.6

which is called a functional equation of quadratic type. In fact, hx ax² in Rsatisfies the functional equation of quadratic type. In particular, every solution of the functional equation of quadratic type is said to be a quadratic-type mapping. One can easily show that ifhsatisfies the quadratic functional equation thenhsatisfies the functional equation of quadratic type.

The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problemsee 11–24.

LetXbe a set. Then, a functiond:X×X → 0,∞is called a generalized metric onXif dsatisfies the following:

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 1.1see25,26. LetX, dbe a complete generalized metric space, and letJ :X → X be a strictly contractive mapping with the, Lipschitz constantL < 1. Then, for each given element x∈X, either

d

Jⁿx, J^{n 1}x

∞ 1.7

for all nonnegative integersnor there exists a positive integern₀such that 1dJⁿx, J^{n 1}x<∞,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, Isac and Rassias27were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using the fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authorssee17,28–31.

Throughout this paper, assume thatnis a fixed integer greater than 2. LetXbe a real normed vector space with norm · , and letY be a real Banach space with norm · .

In this paper, we investigate the functional equation 1.4. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation1.4in real Banach spaces.

2. Fixed Points and Functional Equations Associated with Inner Product Spaces

We investigate the functional equation1.4.

Lemma 2.1. LetV andWbe real vector spaces. If a mappingf :V → Wsatisfies n

i,j1

f xi−xj

2n n

i1

fxi 2.1

for allx1, . . . , xn ∈V with_n

i1xi 0, then the mappingf :V → Wis realized as the sum of an additive mapping and a quadratic-type mapping.

Proof. Letgx: fx−f−x/2 andhx: fx f−x/2 for allx∈V. Then,gxis an odd mapping andhxis an even mapping satisfyingfx gx hxand2.1.

Lettingx₁ x, x₂ y, x₃ −x−y, andx₄ · · ·x_n 0 in2.1for the mappingg, we get

gx g y

−g x y

gx g y

g

−x−y

0 2.2

for allx, y∈V. So,gxis an additive mapping.

Lettingx₁ x, x₂ y, x₃ −x−y, andx₄ · · ·x_n 0 in2.1for the mappingh, we get

h x−y

h 2x y

h x 2y

3hx 3h y

3h x y

2.3

for allx, y∈V. So,hxis a quadratic-type mapping.

For a given mappingf:X → Y, we define

Dfx1, . . . , x_n: ⁿ

i,j1

f x_i−x_j

−2n n

i1

fxi 2.4

for allx1, . . . , xn∈X.

Letgx: fx−f−x/2 andhx: fx f−x/2 for allx∈X. Then,gxis an odd mapping andhxis an even mapping satisfyingfx gx hx. IfDfx1, . . . , x_n 0, thenDgx1, . . . , x_n 0 andDhx1, . . . , x_n 0.

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equationDfx1, . . . , x_n 0 in real Banach spaces.

Theorem 2.2. Let f : X → Y be a mapping with f0 0 for which there exists a function ϕ:Xⁿ → 0,∞such that there exists anL <1 such that

ϕx1, . . . , x_n≤ L

4ϕ2x1, . . . ,2x_n, 2.5

Dhx1, . . . , x_n ≤ϕx1, . . . , x_n 2.6 for allx1, . . . , xn∈Xwith_n

i1xi0. Then, there exists a unique quadratic-type mappingQ:X → Y satisfying

hx−Qx ≤ L 8−8Lϕ

⎛

⎜⎝x,−x,0, . . . , 0

n−2 times

⎞

⎟⎠ 2.7

for allx∈X.

Proof. Consider the set

S:

ψ:X −→Y

, 2.8

and introduce the generalized metric onS:

d ψ1, ψ2

inf

⎧⎨

⎩K∈R :ψ1x−ψ2x≤Kϕ

⎛

⎝x,−x,0, . . . , 0

n−2 times

⎞

⎠, ∀x∈X

⎫⎬

⎭. 2.9

By the same method given in17,28,32, one can easily show thatS, dis complete.

Now we consider the linear mappingJ:S → Ssuch that

Jψx:4ψx 2

2.10

for allx∈X.

It follows from the proof of Theorem 3.1 of25that d

Jψ₁, Jψ₂

≤Ld ψ₁, ψ₂

2.11 for allψ₁, ψ₂ ∈S.

Lettingx₁x, x₂−x, andx₃ · · ·x_n0 in2.6, we get

2h2x−8hx ≤ϕ

⎛

⎝x,−x,0, . . . , 0

n−2 times

⎞

⎠ 2.12

for allx∈X. It follows from2.12that hx−4hx 2

≤ 1 2ϕ

⎛

⎝x 2,−x

2,0, . . . , 0

n−2 times

⎞

⎠

≤ L 8ϕ

⎛

⎝x,−x,0, . . . , 0

n−2 times

⎞

⎠

2.13

for allx∈X. Hence,dh, Jh≤L/8.

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

1Qis a fixed point ofJ; that is,

Qx 2

1

4Qx 2.14

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

M

ψ ∈S:d ψ, h

<∞

. 2.15

This implies thatQis a unique mapping satisfying2.14such that there exists aK∈0,∞ satisfying

hx−Qx ≤Kϕ

⎛

⎝x,−x,0, . . . , 0

n−2 times

⎞

⎠ 2.16

for allx∈X.

2One hasdJ^mh, Q → 0 asm → ∞. This implies the equality

dlim→ ∞4^dh x

2^d

Qx 2.17

for allx∈X. Sincehis an even mapping,Q:X → Yis an even mapping.

Moreover,one has3dh, Q≤1/1−Ldh, Jh, which implies the inequality

dh, Q≤ L

8−8L. 2.18

This implies that inequality2.7holds.

It follows from2.5,2.6, and2.17that

DQx1, . . . , x_n lim

d→ ∞4^d Dh

x₁ 2^d, . . . ,x_n

2^d

≤ lim

d→ ∞4^dϕ x1

2^d, . . . ,xn

2^d

0

2.19

for allx1, . . . , xn ∈ X with _n

i1xi 0. So, DQx1, . . . , xn 0 for allx1, . . . , xn ∈ X with _n

i1x_i0. ByLemma 2.1, the mappingQ:X → Y is a quadratic-type mapping.

Therefore, there exists a unique quadratic-type mappingQ :X → Y satisfying2.7.

Corollary 2.3. Letp >2 andθ≥0 be real numbers, and letf:X → Y be a mapping such that

Dhx1, . . . , x_n ≤θ n j1

x_j^p 2.20 for allx₁, . . . , x_n∈Xwith_n

i1x_i0. Then, there exists a unique quadratic-type mappingQ:X → Y satisfying

hx−Qx ≤ θ

2^p−4x^p 2.21

for allx∈X.

Proof. The proof follows fromTheorem 2.2by taking

ϕx1, . . . , x_n:θ n

j1

x_j^p 2.22

for allx1, . . . , xn∈X. Then, we can chooseL2^2−p, and we get the desired result.

Remark 2.4. Letf :X → Y be a mapping for which there exists a functionϕ:Xⁿ → 0,∞ satisfying2.6andf0 0 such that there exists anL <1 such that

ϕx1, . . . , x_n≤4Lϕx₁

2 , . . . ,x_n 2

2.23

for allx1, . . . , xn ∈Xwith_n

i1xi 0. By a similar method to the proof ofTheorem 2.2, one can show that there exists a unique quadratic-type mappingQ:X → Ysatisfying

hx−Qx ≤ 1 8−8Lϕ

⎛

⎝x,−x,0, . . . , 0

n−2 times

⎞

⎠ 2.24

for allx∈X.

For the casep <2, one can obtain a similar result toCorollary 2.3: letp < 2 andθbe positive real numbers, and letf:X → Y be a mapping satisfying2.20. Then, there exists a unique quadratic-type mappingQ:X → Y satisfying

hx−Qx ≤ θ

4−2^px^p 2.25

for allx∈X.

Theorem 2.5. Letf:X → Ybe a mapping for which there exists a functionϕ:Xⁿ → 0,∞such that there exists anL <1 such that

ϕx1, . . . , x_n≤ L

2ϕ2x1, . . . ,2x_n, 2.26

Dgx1, . . . , xn≤ϕx1, . . . , xn 2.27 for allx₁, . . . , x_n ∈ Xwith_n

i1x_i 0. Then, there exists a unique additive mappingA: X → Y satisfying

gx−Ax≤ L 4n−4nLϕ

⎛

⎜⎝x, x,−2x,0, . . . , 0

n−3 times

⎞

⎟⎠ 2.28

for allx∈X.

Proof. Consider the set

S:

ψ:X −→Y

, 2.29

and introduce the generalized metric onS:

d ψ1, ψ2

inf

⎧⎨

⎩K∈R :ψ1x−ψ2x≤Kϕ

⎛

⎝x, x,−2x,0, . . . , 0

n−3 times

⎞

⎠, ∀x∈X

⎫⎬

⎭. 2.30

By the same method given in17,28,32, one can easily show thatS, dis complete.

Now we consider the linear mappingJ:S → Ssuch that

Jψx:2ψx 2

2.31

for allx∈X.

It follows from the proof of Theorem 3.1 of25that

d

Jψ₁, Jψ₂

≤Ld ψ₁, ψ₂

2.32

for allψ₁, ψ₂ ∈S.

Lettingx1x2xandx3 · · ·xn0 in2.27, we get

4ngx−2ng2x≤ϕ

⎛

⎝x, x,−2x,0, . . . , 0

n−3 times

⎞

⎠ 2.33

for allx∈X. It follows from2.33that

gx−2gx 2

≤ 1 2nϕ

⎛

⎝x 2,x

2,−x,0, . . . , 0

n−3 times

⎞

⎠

≤ L 4nϕ

⎛

⎝x, x,−2x,0, . . . , 0

n−3 times

⎞

⎠

2.34

for allx∈X. Hence,dg, Jg≤L/4n.

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

1Ais a fixed point ofJ; that is,

Ax 2

1

2Ax 2.35

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

M

ψ∈S:d ψ, g

<∞

. 2.36

This implies thatAis a unique mapping satisfying2.35such that there exists aK∈0,∞ satisfying

gx−Ax≤Kϕ

⎛

⎝x, x,−2x,0, . . . , 0

n−3 times

⎞

⎠ 2.37

for allx∈X.

2One hasdJ^mg, A → 0 asm → ∞. This implies the equality

dlim→ ∞2^dg x

2^d

Ax 2.38

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

3Moreoverdg, A≤1/1−Ldg, Jg, which implies the inequality

d g, A

≤ L

4n−4nL. 2.39

This implies that inequality2.28holds.

It follows from2.26,2.27, and2.38that

DAx1, . . . , x_n lim

d→ ∞n^d Dg

x1

2^d, . . . ,xn

2^d

≤ lim

d→ ∞2^dϕ x1

2^d, . . . ,xn

2^d

0

2.40

for allx₁, . . . , x_n ∈ X with _n

i1x_i 0. So, DAx1, . . . , x_n 0 for allx₁, . . . , x_n ∈ X with _n

i1x_i0. ByLemma 2.1, the mappingA:X → Y is an additive mapping.

Therefore, there exists a unique additive mappingA : X → Y satisfying2.28, as desired.

Corollary 2.6. Letp > 1 andθ ≥ 0 be real numbers, and letf : X → Y be a mapping satisfying 2.20. Then, there exists a unique additive mappingA:X → Y satisfying

gx−Ax≤ 2^p 2θ

2n2^p−2x^p 2.41

for allx∈X.

Proof. The proof follows fromTheorem 2.5by taking

ϕx1, . . . , x_n:θ n

j1

x_j^p 2.42

for allx₁, . . . , x_n∈X. Then, we can chooseL2^1−p, and we get the desired result.

Remark 2.7. Letf :X → Y be a mapping for which there exists a functionϕ:Xⁿ → 0,∞ satisfying2.27such that there exists anL <1 such that

ϕx1, . . . , x_n≤2Lϕx₁

2 , . . . ,x_n 2

2.43

for allx₁, . . . , x_n ∈ X. By a similar method to the proof of Theorem 2.5, one can show that there exists a unique additive mappingA:X → Y satisfying

gx−Ax≤ 1 4n−4nLϕ

⎛

⎝x, x,−2x,0, . . . , 0

n−3 times

⎞

⎠ 2.44

for allx∈X.

For the casep <1, one can obtain a similar result toCorollary 2.6: letp < 1 andθbe positive real numbers, and letf:X → Y be a mapping satisfying2.20. Then, there exists a unique additive mappingA:X → Y satisfying

gx−Ax≤ 2 2^pθ

2n2−2^px^p 2.45

for allx∈X.

Since

Dfx1, . . . , x_n≤ϕx1, . . . , x_n, Dhx1, . . . , xn ≤ 1

2ϕx1, . . . , xn 1

2ϕ−x1, . . . ,−xn, Dgx1, . . . , x_n≤ 1

2ϕx1, . . . , x_n 1

2ϕ−x1, . . . ,−xn.

2.46

Note that

L

4ϕ2x1, . . . ,2x_n≤ L

2ϕ2x1, . . . ,2x_n. 2.47

Combining Theorems2.2and2.5, we obtain the following result.

Theorem 2.8. Letf :X → Y be a mapping satisfyingf0 0 for which there exists a function ϕ:Xⁿ → 0,∞satisfying2.5and

Dfx1, . . . , x_n≤ϕx1, . . . , x_n 2.48 for allx₁, . . . , x_n ∈ X with_n

i1x_i 0. Then, there exist an additive mappingA : X → Y and a quadratic type mappingQ:X → Ysuch that

fx−Ax−Qx≤ L 16−16L

⎛

⎜⎝ϕ

⎛

⎜⎝x,−x,0, . . . , 0

n−2 times

⎞

⎟⎠ ϕ

⎛

⎜⎝−x, x,0, . . . , 0

n−2 times

⎞

⎟⎠

⎞

⎟⎠

L 8n−8nL

⎛

⎜⎝ϕ

⎛

⎜⎝x, x,−2x,0, . . . , 0

n−3 times

⎞

⎟⎠ ϕ

⎛

⎜⎝−x,−x,2x,0, . . . , 0

n−3 times

⎞

⎟⎠

⎞

⎟⎠ 2.49

for allx∈X.

Corollary 2.9. Letp >2 andθbe positive real numbers, and letf :X → Ybe a mapping such that

Dfx1, . . . , xn≤θ n

i1

xi^p 2.50

for allx1, . . . , xn ∈ X with_n

i1xi 0. Then, there exist an additive mappingA : X → Y and a quadratic-type mappingQ:X → Y such that

fx−Ax−Qx≤ 1

2^p−4

2^p 2 2n2^p−2

θx^p 2.51

for allx∈X.

Proof. Defineϕx1, . . . , xn θ_n

i1xi^p, and apply Theorem 2.8to get the desired result.

Note that

2Lϕx₁

2 , . . . ,x_n 2

≤4Lϕx₁ 2 , . . . ,x_n

2

. 2.52

Combining Remarks2.4and2.7, we obtain the following result.

Remark 2.10. Letf :X → Y be a mapping for which there exists a functionϕ:Xⁿ → 0,∞ satisfying2.48andf0 0 such that there exists anL <1 such that

ϕx1, . . . , xn≤2Lϕx₁

2 , . . . ,x_n 2

2.53

for allx1, . . . , xn ∈ X. By a similar method to the proof of Theorem 2.8, one can show that there exist an additive mappingA:X → Y and a quadratic-type mappingQ:X → Y such that

fx−Ax−Qx≤ 1 16−16L

⎛

⎝ϕ

⎛

⎝x,−x,0, . . . , 0

n−2 times

⎞

⎠ ϕ

⎛

⎝−x, x,0, . . . , 0

n−2 times

⎞

⎠

⎞

⎠

1 8n−8nL

⎛

⎝ϕ

⎛

⎝x, x,−2x,0, . . . , 0

n−3 times

⎞

⎠ ϕ

⎛

⎝−x,−x,2x,0, . . . , 0

n−3 times

⎞

⎠

⎞

⎠ 2.54

for allx∈X.

For the casep <1, one can obtain a similar result toCorollary 2.9: letp < 1 andθbe positive real numbers, and letf :X → Y be a mapping satisfying2.50. Then, there exist an additive mappingA:X → Yand a quadratic-type mappingQ:X → Y satisfying

fx−Ax−Qx≤ 1

4−2^p

2 2^p 2n2−2^p

θx^p 2.55

for allx∈X.

Acknowledgment

This paper 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.

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