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,baak@hanyang.ac.kr
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 equalityni,j1xi−xj22nni1xi2,ni1xi0, for a fixed
integern≥3. LetV, Wbe real vector spaces. It is shown that, if a mappingf:V → Wsatisfies the following functional equationni,j1fxi−xj 2nni1fxifor allx1, . . . , xn∈Vwithni1xi0,
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 stabilityorHyers-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
The functional equation
fx y fx−y2fx 2fy 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−xj2 2n
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
fxi−xj
2n
n
i1
fxi 1.4
for allx1, . . . , xn∈V with
n
i1xi0, whereV is a real vector space. A square norm on an inner product space satisfies
3
i,j1
xi−xj2 6
3
i1
xi2 1.5
for allx1, x2, x3∈Rwithx1 x2 x3 0see10.
From the above equality we can define the following functional equation:
hx−y h2x y hx 2y3hx 3hy 3hx 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 aquadratic-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
LetXbe a set. Then, a functiond:X×X → 0,∞is called ageneralized metriconXif
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
dJnx, Jn 1x∞ 1.7
for all nonnegative integersnor there exists a positive integern0such that
1dJnx, Jn 1x<∞,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
fxi−xj
2n
n
i1
fxi 2.1
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 · · ·xn 0 in2.1for the mappingg, we get
gx gy−gx ygx gy g−x−y0 2.2
for allx, y∈V. So,gxis an additive mapping.
Lettingx1 x, x2 y, x3 −x−y, andx4 · · ·xn 0 in2.1for the mappingh, we get
hx−y h2x y hx 2y3hx 3hy 3hx y 2.3
for allx, y∈V. So,hxis a quadratic-type mapping.
For a given mappingf:X → Y, we define
Dfx1, . . . , xn: n
i,j1
fxi−xj
−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, . . . , xn 0, thenDgx1, . . . , xn 0 andDhx1, . . . , xn 0.
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equationDfx1, . . . , xn 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 <1such that
ϕx1, . . . , xn≤
L
4ϕ2x1, . . . ,2xn, 2.5
Dhx1, . . . , xn ≤ϕx1, . . . , xn 2.6
for allx1, . . . , xn∈Xwithin1xi0. Then, there exists a unique quadratic-type mappingQ:X →
Y satisfying
hx−Qx ≤ L
8−8Lϕ
⎛ ⎜
⎝x,−x,0 , . . . ,0 n−2times
⎞ ⎟
⎠ 2.7
for allx∈X.
Proof. Consider the set
and introduce thegeneralized metriconS:
dψ1, 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
dJψ1, Jψ2
≤Ldψ1, ψ2
2.11
for allψ1, ψ2 ∈S.
Lettingx1x, x2−x, andx3 · · ·xn0 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
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 hasdJmh, Q → 0 asm → ∞. This implies the equality
lim d→ ∞4
dh
x
2d
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, . . . , xn lim d→ ∞4
dDh
x1 2d, . . . ,
xn 2d
≤ lim d→ ∞4
dϕ
x1 2d, . . . ,
xn 2d
0
2.19
for allx1, . . . , xn ∈ X with ni1xi 0. So, DQx1, . . . , xn 0 for allx1, . . . , xn ∈ X with
n
i1xi0. 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 >2andθ≥0be real numbers, and letf:X → Y be a mapping such that
Dhx1, . . . , xn ≤θ n
j1
xjp 2.20
for allx1, . . . , xn∈Xwith
n
i1xi0. Then, there exists a unique quadratic-type mappingQ:X →
Y satisfying
hx−Qx ≤ θ
2p−4x
p 2.21
Proof. The proof follows fromTheorem 2.2by taking
ϕx1, . . . , xn:θ n
j1
xjp 2.22
for allx1, . . . , xn∈X. Then, we can chooseL22−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, . . . , xn≤4Lϕ
x
1 2 , . . . ,
xn 2
2.23
for allx1, . . . , xn ∈Xwithni1xi 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
4−2px
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 <1such that
ϕx1, . . . , xn≤
L
2ϕ2x1, . . . ,2xn, 2.26
Dgx1, . . . , xn≤ϕx1, . . . , xn 2.27
for allx1, . . . , xn ∈ Xwith
n
i1xi 0. Then, there exists a unique additive mappingA: X → Y satisfying
gx−Ax≤ L
4n−4nLϕ
⎛ ⎜
⎝x, x,−2x,0 , . . . ,0 n−3times
⎞ ⎟
⎠ 2.28
Proof. Consider the set
S:ψ:X −→Y, 2.29
and introduce thegeneralized metriconS:
dψ1, 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
dJψ1, Jψ2
≤Ldψ1, 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
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
Mψ∈S:dψ, 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 hasdJmg, A → 0 asm → ∞. This implies the equality
lim d→ ∞2
dg
x
2d
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
dg, A≤ L
4n−4nL. 2.39
This implies that inequality2.28holds.
It follows from2.26,2.27, and2.38that
DAx1, . . . , xn lim d→ ∞n
dDg
x1 2d, . . . ,
xn 2d
≤ lim d→ ∞2
dϕ
x1 2d, . . . ,
xn 2d
0
2.40
for allx1, . . . , xn ∈ X with
n
i1xi 0. So, DAx1, . . . , xn 0 for allx1, . . . , xn ∈ X with
n
i1xi0. ByLemma 2.1, the mappingA:X → Y is an additive mapping.
Corollary 2.6. Letp > 1andθ ≥ 0be real numbers, and letf : X → Y be a mapping satisfying
2.20. Then, there exists a unique additive mappingA:X → Y satisfying
gx−Ax≤ 2p 2θ 2n2p−2x
p 2.41
for allx∈X.
Proof. The proof follows fromTheorem 2.5by taking
ϕx1, . . . , xn:θ n
j1
xjp 2.42
for allx1, . . . , xn∈X. Then, we can chooseL21−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, . . . , xn≤2Lϕ
x
1 2 , . . . ,
xn 2
2.43
for allx1, . . . , xn ∈ 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 2pθ
2n2−2px
p 2.45
for allx∈X.
Since
Dfx1, . . . , xn≤ϕx1, . . . , xn,
Dhx1, . . . , xn ≤ 1
2ϕx1, . . . , xn 1
2ϕ−x1, . . . ,−xn,
Dgx1, . . . , xn≤ 1
2ϕx1, . . . , xn 1
2ϕ−x1, . . . ,−xn.
Note that
L
4ϕ2x1, . . . ,2xn≤
L
2ϕ2x1, . . . ,2xn. 2.47
Combining Theorems2.2and2.5, we obtain the following result.
Theorem 2.8. Letf :X → Y be a mapping satisfyingf0 0for which there exists a function
ϕ:Xn → 0,∞satisfying2.5and
Dfx1, . . . , xn≤ϕx1, . . . , xn 2.48
for allx1, . . . , xn ∈ X withni1xi 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−2times
⎞ ⎟ ⎠ ϕ
⎛ ⎜
⎝−x, x,0 , . . . ,0 n−2times
⎞ ⎟ ⎠
⎞ ⎟ ⎠
L
8n−8nL
⎛ ⎜ ⎝ϕ
⎛ ⎜
⎝x, x,−2x,0 , . . . ,0 n−3times
⎞ ⎟ ⎠ ϕ
⎛ ⎜
⎝−x,−x,2x,0 , . . . ,0 n−3times
⎞ ⎟ ⎠
⎞ ⎟ ⎠
2.49
for allx∈X.
Corollary 2.9. Letp >2andθbe positive real numbers, and letf :X → Ybe a mapping such that
Dfx1, . . . , xn≤θn i1
xip 2.50
for allx1, . . . , xn ∈ X withni1xi 0. Then, there exist an additive mappingA : X → Y and a quadratic-type mappingQ:X → Y such that
fx−Ax−Qx≤
1 2p−4
2p 2 2n2p−2
θxp 2.51
Proof. Defineϕx1, . . . , xn θin1xip, and apply Theorem 2.8to get the desired result.
Note that
2Lϕx1
2 , . . . ,
xn 2
≤4Lϕx1
2 , . . . ,
xn 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≤2Lϕ
x1
2 , . . . ,
xn 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 4−2p
2 2p 2n2−2p
θxp 2.55
for allx∈X.
Acknowledgment
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