Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 817079, 16 pages
doi:10.1155/2011/817079
Research Article
A Fixed Point Approach to the Stability
of the Cauchy Additive and Quadratic Type
Functional Equation
Sun Sook Jin and Yang-Hi Lee
Department of Mathematics Education, Gongju National University of Education, Gongju 314-711,
Republic of Korea
Correspondence should be addressed to Yang-Hi Lee, yanghi2@hanmail.net
Received 2 June 2011; Accepted 18 July 2011
Academic Editor: Yuri Sotskov
Copyright q 2011 S. S. Jin and Y.-H. Lee. 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 investigate the stability of the functional equation 2fx y fx − y fy − x − 3fx −
f−x − 3fy − f−y 0 by using the fixed point theory in the sense of Cădariu and Radu.
1. Introduction
In 1940, Ulam 1 raised a question concerning the stability of homomorphisms as follow.
Given a group G1 , a metric group G2 with the metric d·, ·, and a positive number ε, does
there exist a δ > 0 such that if a mapping f : G1 → G2 satisfies the inequality
d f xy , fxf y < δ,
1.1
for all x, y ∈ G1 then there exists a homomorphism F : G1 → G2 with
d fx, Fx < ε,
1.2
for all x ∈ G1 ? When this problem has a solution, we say that the homomorphisms from G1 to
G2 are stable. In the next year, Hyers 2 gave a partial solution of Ulam’s problem for the case
of approximate additive mappings under the assumption that G1 and G2 are Banach spaces.
Hyers’ result was generalized by Aoki 3 for additive mappings and by Rassias 4 for
linear mappings by considering the stability problem with unbounded Cauchy’s differences.
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The paper of Rassias had much influence in the development of stability problems. The
terminology Hyers-Ulam-Rassias stability originated from this historical background. During
the last decades, the stability problems of functional equations have been extensively
investigated by a number of mathematicians, see 5–12.
Almost all subsequent proofs, in this very active area, have used Hyers’ method of 2.
Namely, the mapping F, which is the solution of a functional equation, is explicitly constructed, starting from the given mapping f, by the formulae Fx limn → ∞ 1/2n f2n x
or Fx limn → ∞ 2n fx/2n . We call it a direct method. In 2003, Cădariu and Radu 13
observed that the existence of the solution F for a functional equation and the estimation
of the difference with the given mapping f can be obtained from the fixed point theory
alternative. This method is called a fixed point method. In 2004, they applied this method
14 to prove stability theorems of the Cauchy functional equation:
f x y − fx − f y 0.
1.3
In 2003, they 15 obtained the stability of the quadratic functional equation:
f x y f x − y − 2fx − 2f y 0,
1.4
by using the fixed point method. Notice that if we consider the functions f1 , f2 : R → R
defined by f1 x ax and f2 x ax2 , where a is a real constant, then f1 satisfies 1.3,
and f2 holds 1.4, respectively. We call a solution of 1.3 an additive map, and a mapping
satisfying 1.4 is called a quadratic map. Now we consider the functional equation:
2f x y f x − y f y − x − 3fx − f−x − 3f y − f −y 0,
1.5
which is called the Cauchy additive and quadratic-type functional equation. The function f : R →
R defined by fx ax2 bx satisfies this functional equation, where a, b are real constants.
We call a solution of 1.5 a quadratic-additive mapping.
In this paper, we will prove the stability of the functional equation 1.5 by using the
fixed point theory. Precisely, we introduce a strictly contractive mapping with the Lipschitz
constant 0 < L < 1. Using the fixed point theory in the sense of Cădariu and Radu, together
with suitable conditions, we can show that the contractive mapping has the fixed point.
Actually the fixed point F becomes the precise solution of 1.5. In Section 2, we prove several
stability results of the functional equation 1.5 using the fixed point theory, see Theorems
2.3, 2.4, and 2.5. In Section 3, we use the results in the previous sections to get a stability
of the Cauchy functional equation 1.3 and that of the quadratic functional equation 1.4,
respectively.
2. Main Results
We recall the following result of the fixed point theory by Margolis and Diaz.
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Theorem 2.1 see 16 or 17. Suppose that a complete generalized metric space X, d, which
means that the metric d may assume infinite values, and a strictly contractive mapping J : X → X
with the Lipschitz constant 0 < L < 1 are given. Then, for each given element x ∈ X, either
d J n x, J n1 x ∞,
∀n ∈ N ∪ {0}
2.1
or there exists a nonnegative integer k such that
1 dJ n x, J n1 x < ∞ for all n ≥ k,
2 the sequence {J n x} is convergent to a fixed point y∗ of J,
3 y∗ is the unique fixed point of J in Y : {y ∈ X, dJ k x, y < ∞},
4 dy, y∗ ≤ 1/1 − Ldy, Jy for all y ∈ Y .
Throughout this paper, let V be a real or complex linear space, and let Y be a Banach
space. For a given mapping f : V → Y , we use the following abbreviation:
Df x, y : 2f x y f x − y f y − x − 3fx − f−x − 3f y − f −y ,
2.2
for all x, y ∈ V . If f is a solution of the functional equation Df ≡ 0, see 1.5, we call it a
quadratic-additive mapping. We first prove the following lemma.
Lemma 2.2. If f : V → Y is a mapping such that Dfx, y 0 for all x, y ∈ V \ {0}, then f is a
quadratic-additive mapping.
Proof. By choosing x ∈ V \ {0}, we get
8f0 Df3x, 2x − Df4x, x − 2Df3x, x − Df2x, x Df−3x, −2x
− Df−4x, −x − 2Df−3x, −x − Df−2x, −x 4Dfx, −x 0.
2.3
Since f0 0, we easily obtain Dfx, y 0 for all x, y ∈ V .
Now we can prove some stability results of the functional equation 1.5.
Theorem 2.3. Let ϕ : V \ {0}2 → 0, ∞ be a given function. Suppose that the mapping f : V →
Y satisfies
Df x, y ≤ ϕ x, y ,
2.4
for all x, y ∈ V \ {0}. If there exist constants 0 < L, L < 1 such that ϕ has the property
1 ϕ x, y ≤ ϕ 2x, 2y ≤ 2Lϕ x, y ,
L
2.5
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for all x, y ∈ V \ {0}, then there exists a unique quadratic-additive mapping F : V → Y such that
3 ϕx, x ϕ−x, −x
fx − Fx ≤
,
161 − L
2.6
for all x ∈ V \ {0}. In particular, F is represented by
Fx lim
n→∞
f2n x f−2n x f2n x − f−2n x
,
2 · 4n
2n1
2.7
for all x ∈ V .
Proof. It follows from 2.5 that
ϕ 2n x, 2n y
0,
n→∞
2n
2.8
lim
for all x, y ∈ V \ {0}, and
8f0 lim Df 3x , 2x − Df 4x , x − 2Df 3x , x
n → ∞
2n 2n
2n 2n
2n 2n
2x x
3x 2x
4x x
,
,
−
,
−
Df
−
−
Df
−
2n 2n
2n 2n
2n 2n
x
x 3x x
2x x
− 2Df − n , − n − Df − n , − n 4Df n , − n 2
2
2
2
2
2 3x 2x
4x x
3x x
≤ lim ϕ n , n ϕ n , n 2ϕ n , n
n→∞
2 2
2 2
2 2
2x x
3x 2x
4x x
ϕ n , n ϕ − n ,− n ϕ − n ,− n
2 2
2
2
2
2
x
3x x
2x x
x
2ϕ − n , − n ϕ − n , − n 4ϕ n , − n
2
2
2
2
2
2
n
≤ lim L ϕ3x, 2x ϕ4x, x 2ϕ3x, x ϕ2x, x ϕ−3x, −2x
− Df
n→∞
ϕ−4x, −x 2ϕ−3x, −x ϕ−2x, −x 4ϕx, −x
2.9
0,
for all x ∈ V \ {0}. From this, we know that f0 0. Let S be the set of all mappings
g : V → Y with g0 0. We introduce a generalized metric on S by
d g, h inf K ∈ R | gx − hx ≤ K ϕx, x ϕ−x, −x ∀ x ∈ V \ {0} .
2.10
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It is easy to show that S, d is a generalized complete metric space. Now we consider the
mapping J : S → S, which is defined by
g2x − g−2x g2x g−2x
,
4
8
2.11
g2n x − g−2n x g2n x g−2n x
,
2 · 4n
2n1
2.12
Jgx :
for all x ∈ V . Notice
J n gx for all n ∈ N and x ∈ V . Let g, h ∈ S, and let K ∈ 0, ∞ be an arbitrary constant with
dg, h ≤ K. From the definition of d, we have
Jgx − Jhx 3 g2x − h2x 1 g−2x − h−2x
8
8
1 K ϕ2x, 2x ϕ−2x, −2x
2
≤ KL ϕx, x ϕ−x, −x ,
≤
2.13
for all x ∈ V \ {0}, which implies that
d Jg, Jh ≤ Ld g, h ,
2.14
for any g, h ∈ S. That is, J is a strictly contractive self-mapping of S with the Lipschitz
constant L. Moreover, by 2.4, we see that
fx − Jfx 1 −3Dfx, x Df−x, −x ≤ 3 ϕx, x ϕ−x, −x ,
16
16
2.15
for all x ∈ V \ {0}. It means that df, Jf ≤ 3/16 < ∞ by the definition of d. Therefore,
according to Theorem 2.1, the sequence {J n f} converges to the unique fixed point F : V → Y
of J in the set T {g ∈ S | df, g < ∞}, which is represented by 2.7 for all x ∈ V . By the
definition of F, together with 2.4 and 2.7, it follows that
Df 2n x, 2n y − Df −2n x, −2n y Df 2n x, 2n y Df −2n x, −2n y DF x, y lim n
n1
n → ∞
2·4
2
2n 1 n
ϕ 2 x, 2n y ϕ −2n x, −2n y
n → ∞ 2 · 4n
≤ lim
0,
2.16
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for all x, y ∈ V \ {0}. By Lemma 2.2, we have proved that
DF x, y 0,
2.17
for all x, y ∈ V .
Theorem 2.4. Let ϕ : V \ {0}2 → 0, ∞ be a given function. Suppose that the mapping f : V →
Y satisfies 2.5 for all x, y ∈ V \ {0}. If there exists a constant 0 < L < 1/2 such that ϕ has the
property
ϕ 2x, 2y ≤ 2Lϕ x, y
2.18
for all x, y ∈ V \ {0}, then there exists a unique quadratic-additive mapping F : V → Y satisfying
2.6 for all x ∈ V \ {0}. Moreover, if ϕx, y is continuous, then f itself is a quadratic-additive
mapping.
Proof. It follows from 2.18 that
lim ϕ 2n x, 2n y lim 2Ln ϕ x, y 0,
n→∞
n→∞
2.19
for all x, y ∈ V \ {0}, and
8f0 lim Df 3 · 2n x, 2n1 x − Df 2n2 x, 2n x − 2Df3 · 2n x, 2n x
n→∞
− Df 2n1 x, 2n x Df −3 · 2n x, −2n1 x
− Df −2n2 x, −2n x − 2Df−3 · 2n x, −2n x
− Df −2n1 x, −2n x 4Df2n x, −2n x
≤ lim ϕ 3 · 2n x, 2n1 x ϕ 2n2 x, 2n x 2ϕ3 · 2n x, 2n x
n→∞
ϕ 2n1 x, 2n x ϕ −3 · 2n x, −2n1 x ϕ −2n2 x, −2n x
2ϕ−3 · 2n x, −2n x ϕ −2n1 x, −2n x 4ϕ2n x, −2n x
≤ lim 2Ln ϕ3x, 2x ϕ4x, x 2ϕ3x, x ϕ2x, x ϕ−3x, −2x
n→∞
0,
ϕ−4x, −x 2ϕ−3x, −x ϕ−2x, −x 4ϕx, −x
2.20
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for all x ∈ V \ {0}. By the same method used in Theorem 2.3, we know that there exists a
unique quadratic-additive mapping F : V → Y satisfying 2.6 for all x ∈ V \ {0}. Since ϕ is
continuous, we get
a2 b2
lim ϕ a1 · 2n a2 x, b1 · 2n b2 y ≤ lim 2Ln ϕ a1 n x, b1 n y
n→∞
n→∞
2
2
0 · ϕ a1 x, b1 y 0,
2.21
for all x, y ∈ V \{0} and for any fixed integers a1 , a2 , b1 , b2 with a1 , b1 /
0. Therefore, we obtain
2fx − Fx ≤ lim Df2n 1x, −2n x − DF2n 1x, −2n x
n→∞
n1
2 1 x F − f − 2n1 1 x F−f
3 f − F 2n 1x f − F −2n 1x
3 f − F −2n x f − F 2n x
2.22
3
lim ψ 2n1 1 x
81 − L n → ∞
2ψ2n 1x 2ψ2n x
≤ lim ϕ 2n 1x, −2n x n→∞
0,
for all x ∈ V \ {0}, where ψx is defined by ψx ϕx, x ϕ−x, −x. Since f0 0 F0,
we have shown that f ≡ F. This completes the proof of this theorem.
We continue our investigation with the next result.
Theorem 2.5. Let ϕ : V \ {0}2 → 0, ∞. Suppose that f : V → Y satisfies the inequality
Dfx, y ≤ ϕx, y for all x, y ∈ V \ {0}. If there exists 0 < L < 1 such that the mapping ϕ has the
property
Lϕ 2x, 2y ≥ 4ϕ x, y
2.23
for all x, y ∈ V \ {0}, then there exists a unique quadratic-additive mapping F : V → Y such that
fx − Fx ≤
L
ϕx, x ϕ−x, −x ,
81 − L
2.24
for all x ∈ V \ {0}. In particular, F is represented by
Fx lim
n→∞
for all x ∈ V .
n−1
2
x
x 4n x x f n f − n
f n −f − n
,
2
2
2
2
2
2.25
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Proof. By the similar method used to prove f0 0 in the proof of Theorem 2.3, we can easily
show that f0 0. Let the set S, d be as in the proof of Theorem 2.3. Now we consider the
mapping J : S → S defined by
Jgx : g
x
x
x
x −g −
2 g
g −
,
2
2
2
2
2.26
for all g ∈ S and x ∈ V . Notice that
x
x 4n x x J n gx 2n−1 g n − g − n
g n g − n
2
2
2
2
2
2.27
and J 0 gx gx, for all x ∈ V . Let g, h ∈ S, and let K ∈ 0, ∞ be an arbitrary constant with
dg, h ≤ K. From the definition of d, we have
x
x x x Jgx − Jhx 3
−h
−h − g
g −
2
2
2
2
x x
x x ≤ 4K ϕ ,
ϕ − ,−
2 2
2 2
≤ LK ϕx, x ϕ−x, −x ,
2.28
for all x ∈ V \ {0}. So
d Jg, Jh ≤ Ld g, h ,
2.29
for any g, h ∈ S. That is, J is a strictly contractive self-mapping of S with the Lipschitz
constant L. Also we see that
x x fx − Jfx 1 ,
Df
2
2 2
x x 1 x x
≤
ϕ ,
ϕ − ,−
2
2 2
2 2
≤
2.30
L
ϕx, x ϕ−x, −x ,
8
for all x ∈ V \ {0}, which implies that df, Jf ≤ L/8 < ∞. Therefore, according to
Theorem 2.1, the sequence {J n f} converges to the unique fixed point F of J in the set
T : {g ∈ S | df, g < ∞}, which is represented by 2.25. Since
d f, F ≤
L
1
d f, Jf ≤
,
1−L
81 − L
2.31
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the inequality 2.24 holds. From the definition of Fx, 2.4, and 2.23, we have
DF x, y lim 2n−1 Df x , y − Df − x , − y
n
n
n
n
n→∞
2 2
2
2
≤ lim
n→∞
x
y 4n x y Df n , n Df − n , − n 2
2 2
2
2
x
x y
y ϕ n, n ϕ − n,− n
2 2
2
2
n
2 4
2
n
2.32
0,
for all x, y ∈ V \ {0}. By Lemma 2.2, F is quadratic additive.
Remark 2.6. If ϕ satisfies the equality ϕx, y ϕ−x, −y for all x, y ∈ V \ {0} in Theorems
2.3, 2.4, and 2.5, then the inequalities 2.6 and 2.24 can be replaced by
fx − Fx ≤ ϕx, x ,
41 − L
fx − Fx ≤ Lϕx, x ,
81 − L
2.33
for all x ∈ V \ {0}, respectively.
3. Applications
For a given mapping f : V → Y , we use the following abbreviations:
Af x, y : f x y − fx − f y ,
Qf x, y : f x y f x − y − 2fx − 2f y ,
3.1
for all x, y ∈ V . Using Theorems 2.3, 2.4, and 2.5 we will show the stability results of the
additive functional equation Af ≡ 0 and the quadratic functional equation Qf ≡ 0 in the
following corollaries.
Corollary 3.1. Let fi : V → Y, i 1, 2, 3, be mappings for which there exist functions φi : V \
{0}2 → 0, ∞, i 1, 2, 3, such that
Afi x, y ≤ φi x, y ,
3.2
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for all x, y ∈ V \ {0}. If there exists 0 < L < 1 such that
1 φ1 x, y ≤ φ1 2x, 2y ≤ 2Lφ1 x, y ,
L
φ2 2x, 2y ≤ Lφ2 x, y ,
4φ3 x, y ≤ Lφ3 2x, 2y
3.3
3.4
3.5
for all x, y ∈ V \ {0}, then there exist unique additive mappings Fi : V → Y, i 1, 2, 3, such that
3 φ1 x, x φ1 x, −x φ1 −x, x φ1 −x, −x
f1 x − F1 x ≤
,
81 − L
3 φ2 x, x φ2 x, −x φ2 −x, x φ2 −x, −x
f2 x − F2 x ≤
,
81 − L/2
L φ3 x, x φ3 x, −x φ3 −x, x φ3 −x, −x
f3 x − F3 x ≤
,
41 − L
3.6
3.7
3.8
for all x ∈ V \ {0}. In particular, the mappings Fi , i 1, 2, 3, are represented by
f1 2n x
,
n→∞
2n
F1 x lim
f2 2n x
,
n→∞
2n
x
F3 x lim 2n f3 n ,
n→∞
2
F2 x lim
3.9
3.10
3.11
for all x ∈ V . Moreover, if φ2 x, y is continuous, then f2 is itself an additive mapping.
Proof. Notice that
Dfi x, y 2Afi x, y Afi x, −y Afi y, −x ,
3.12
for all x, y ∈ V and i 1, 2, 3. Put
ϕi x, y : 2φi x, y φi x, −y φi y, −x ,
3.13
for all x, y ∈ V and i 1, 2, 3, then ϕ1 satisfies 2.5, ϕ2 satisfies 2.18, and ϕ3 satisfies
2.23. Therefore, Dfi x, y ≤ ϕi x, y, for all x, y ∈ V \ {0} and i 1, 2, 3. According to
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Theorem 2.3, there exists a unique mapping F1 : V → Y satisfying 3.6, which is represented
by 2.7. Observe that, by 3.2 and 3.3,
n
n
f1 2n x f1 −2n x lim f1 2 x f1 −2 x − f1 0 lim n1
n1
n→∞
n→∞
2
2
lim
1 Af1 2n x, −2n x
n → ∞ 2n1
≤ lim
1
n → ∞ 2n1
φ1 2n x, −2n x
3.14
Ln
φ1 x, −x
n→∞ 2
≤ lim
0,
as well as
n n
f1 2n x f1 −2n x ≤ lim 2 L φ1 x, −x 0,
lim n → ∞ 2 · 4n
n → ∞
2 · 4n
3.15
for all x ∈ V \ {0}. From this and 2.7, we get 3.9. Moreover, we have
Af1 2n x, 2n y φ1 2n x, 2n y
≤ Ln φ1 x, y ,
≤
n
n
2
2
3.16
for all x, y ∈ V \ {0}. Taking the limit as n → ∞ in the above inequality and using F1 0 0,
we get
AF1 x, y 0,
3.17
for all x, y ∈ V . According to Theorem 2.4, there exists a unique mapping F2 : V → Y
satisfying 3.7, which is represented by 2.7. By using the similar method to prove 3.9, we
can show that F2 is represented by 3.10. In particular, if φ2 x, y is continuous, then ϕ2 is
continuous on V \ {0}2 , and we can say that f2 is an additive map by Theorem 2.4. On the
other hand, according to Theorem 2.5, there exists a unique mapping F3 : V → Y satisfying
3.8 which is represented by 2.25. Observe that, by 3.2 and 3.5,
x
x
x −x lim 22n−1 f
f
lim 22n−1 Af3 n , − n 3
3
n
n
n→∞
n→∞
2
2
2
2
x
x
≤ lim 22n−1 φ3 n , − n
n→∞
2
2
Ln
φ3 x, −x 0,
n→∞ 2
≤ lim
3.18
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as well as
lim 2
n→∞
x
Ln
−x φ3 x, −x 0,
f3 2n f3 2n ≤ nlim
→ ∞ 2n1
n−1 3.19
for all x ∈ V \ {0}. From these and 2.25, we get 3.11. Moreover, we have
x y x y Ln n
2 Af3 n , n ≤ 2n φ3 n , n ≤ n φ3 x, y ,
2 2
2 2
2
3.20
for all x, y ∈ V \ {0}. Taking the limit as n → ∞ in the above inequality and using F3 0 0,
we get
AF3 x, y 0,
3.21
for all x, y ∈ V .
Corollary 3.2. Let fi : V → Y, i 1, 2, 3, be mappings for which there exist functions φi : V \
{0}2 → 0, ∞, i 1, 2, 3, such that
Qfi x, y ≤ φi x, y ,
3.22
for all x, y ∈ V \ {0}. If there exists 0 < L < 1 such that the mapping φ1 satisfies 3.3, φ2 satisfies
3.4, and φ3 satisfies 3.5 for all x, y ∈ V \ {0}, then there exist unique quadratic mappings Fi :
V → Y, i 1, 2, 3, such that
3 φ1 x, x φ1 x, −x φ1 −x, x φ1 −x, −x
f1 x − F1 x ≤
,
161 − L
3 φ2 x, x φ2 x, −x φ2 −x, x φ2 −x, −x
f2 x − F2 x ≤
,
161 − L/2
L φ3 x, x φ3 x, −x φ3 −x, x φ3 −x, −x
f3 x − F3 x ≤
,
81 − L
3.23
3.24
3.25
for all x ∈ V \ {0}. In particular, the mappings Fi , i 1, 2, 3, are represented by
f1 2n x
,
n→∞
4n
F1 x lim
f2 2n x
,
n→∞
4n
x
F3 x lim 4n f3 n ,
n→∞
2
F2 x lim
for all x ∈ V . Moreover, if φ2 x, y is continuous, then f2 itself is a quadratic mapping.
3.26
3.27
3.28
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13
Proof. Notice that
1
Dfi x, y Qfi x, y Qfi x, −y Qfi y, −x ,
2
3.29
for all x, y ∈ V and i 1, 2, 3. Put ϕi x, y : φi x, y 1/2φi x, −y φi y, −x, for
all x, y ∈ V and i 1, 2, 3, then ϕ1 satisfies 2.5, ϕ2 satisfies 2.18, and ϕ3 satisfies 2.23.
Moreover,
Dfi x, y ≤ ϕi x, y ,
3.30
for all x, y ∈ V \ {0} and i 1, 2, 3. According to Theorem 2.3, there exists a unique mapping
F1 : V → Y satisfying 3.23 which is represented by 2.7. Observe that
f1 2n x − f1 −2n x n−1
n−1
n−1
n−1
lim 1 2
−2
x,
−2
x
−
Qf
x,
2
x
lim Qf
1
1
n → ∞ 2n1
n → ∞
2n1
1 ≤ lim n1 φ1 2n−1 x, −2n−1 x φ1 −2n−1 x, 2n−1 x
n→∞2
x x Ln x x φ1 , −
φ1 − ,
≤ lim
n→∞ 2
2 2
2 2
3.31
0,
as well as
n f1 2n x − f1 −2n x ≤ lim L φ1 x , − x φ1 − x , x
lim 0,
n→∞
n → ∞ 2n1
2 · 4n
2 2
2 2
3.32
for all x ∈ V \ {0}. From this and 2.7, we get 3.26 for all x ∈ V . Moreover, we have
Qf1 2n x, 2n y φ1 2n x, 2n y Ln ≤ n φ1 x, y ,
≤
n
n
4
4
2
3.33
for all x, y ∈ V \ {0}. Taking the limit as n → ∞ in the above inequality, we get
QF1 x, y 0
3.34
for all x, y ∈ V \ {0}. Using F1 0 0, we have
QF1 x, 0 0,
y y
y y
,−
F1 − ,
0,
QF1 0, y −F1
2 2
2 2
for all x, y ∈ V \ {0}. Therefore, QF1 x, y 0 for all x, y ∈ V .
3.35
14
Journal of Applied Mathematics
Next, by Theorem 2.4, there exists a unique mapping F2 : V → Y satisfying 3.24,
which is represented by 2.7. By using the similar method to prove 3.26, we can show
that F2 is represented by 3.27. In particular, φ2 x, y is continuous, then ϕ2 is continuous
on V \ {0}2 , and we can say that f2 is a quadratic map by Theorem 2.4. On the other hand,
according to Theorem 2.5, there exists a unique mapping F3 : V → Y satisfying 3.25 which
is represented by 2.25. Observe that
x
x x
x x
x
n
4 f3 n − f3 − n 4 Qf3 n1 , − n1 − Qf3 − n1 , n1 2
2
2
2
2
2
x
x
x
x
≤ 4n φ3 n1 , − n1 φ3 − n1 , n1
2
2
2
2
x x
x x φ3 − ,
,
≤ Ln φ3 , −
2 2
2 2
n
3.36
for all x ∈ V \ {0}. It leads us to get
x
x lim 4n f3 n − f3 − n
0,
n→∞
2
2
x
x lim 2n f3 n − f3 − n
0,
n→∞
2
2
3.37
for all x ∈ V \ {0}. From these and 2.25, we obtain 3.28. Moreover, we have
x y x y
n
4 Qf3 n , n ≤ 4n φ3 n , n ≤ Ln φ3 x, y ,
2 2
2 2
3.38
for all x, y ∈ V \ {0}. Taking the limit as n → ∞ in the above inequality and using F3 0 0,
we get
QF3 x, y 0,
3.39
for all x, y ∈ V .
Now, we obtain Hyers-Ulam-Rassias stability results in the framework of normed
spaces using Theorems 2.3 and 2.4.
Corollary 3.3. Let X be a normed space, and let Y be a Banach space. Suppose that the mapping
f : X → Y satisfies the inequality
Df x, y ≤ θ xp yp
3.40
for all x, y ∈ X \ {0}, where θ ≥ 0 and p ∈ −∞, 0 ∪ 0, 1 ∪ 2, ∞. Then there exists a unique
quadratic-additive mapping F : X → Y such that
fx − Fx ≤
⎧
⎪
⎪
⎨
θ
xp
2p − 4
⎪
⎪
⎩ θ xp
2 − 2p
if p > 2,
if 0 < p < 1
for all x ∈ X \ {0}. Moreover if p < 0, then f is itself a quadratic-additive mapping.
3.41
Journal of Applied Mathematics
15
Proof. This corollary follows from Theorems 2.3, 2.4, and 2.5, and Remark 2.6, by putting
p ϕ x, y : θ xp y ,
3.42
for all x, y ∈ X \ {0} with L 2p−1 < 1 if p < 1, L 22−p < 1 if p > 2, and L 2−p < 1 if
p > 0.
Corollary 3.4. Let X be a normal space let and Y be a Banach space. Suppose that the mapping
f : X → Y satisfies the inequality
Df x, y ≤ θxp yq ,
3.43
for all x, y ∈ X \ {0}, where θ ≥ 0 and p q ∈ −∞, 0 ∪ 0, 1 ∪ 2, ∞. Then there exists a unique
quadratic-additive mapping F : X → Y such that
⎧
pq
⎪
⎪ θx
⎨ 22pq − 4
fx − Fx ≤
pq
⎪
⎪ θx
⎩
22 − 2pq if p q > 2,
if 0 < p q < 1,
3.44
for all x ∈ X \ {0}. Moreover, if p q < 0, then f is itself a quadratic-additive mapping.
Proof. This corollary follows from Theorems 2.3, 2.4, 2.5 and Remark 2.6, by putting
q
ϕ x, y : θxp y ,
3.45
for all x, y ∈ X \ {0} with L 2pq−1 < 1 if p q < 1, L 22−p−q < 1 if p q > 2, and L 2−p−q < 1
if p q > 0.
References
1 S. M. Ulam, A collection of Mathematical Problems, Interscience Publishers, New York, NY, USA, 1968.
2 D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of
Sciences of the United States of America, vol. 27, pp. 222–224, 1941.
3 T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical
Society of Japan, vol. 2, pp. 64–66, 1950.
4 T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American
Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
5 P. Găvruţa, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive
mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994.
6 K.-W. Jun and Y.-H. Lee, “A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized
quadratic equations. II,” Kyungpook Mathematical Journal, vol. 47, no. 1, pp. 91–103, 2007.
7 G. H. Kim, “On the stability of functional equations with square-symmetric operation,” Mathematical
Inequalities & Applications, vol. 4, no. 2, pp. 257–266, 2001.
8 H.-M. Kim, “On the stability problem for a mixed type of quartic and quadratic functional equation,”
Journal of Mathematical Analysis and Applications, vol. 324, no. 1, pp. 358–372, 2006.
9 Y.-H. Lee, “On the stability of the monomial functional equation,” Bulletin of the Korean Mathematical
Society, vol. 45, no. 2, pp. 397–403, 2008.
16
Journal of Applied Mathematics
10 Y.-H. Lee and K.-W. Jun, “A generalization of the Hyers-Ulam-Rassias stability of Jensen’s equation,”
Journal of Mathematical Analysis and Applications, vol. 238, no. 1, pp. 305–315, 1999.
11 Y.-H. Lee and K.-W. Jun, “A generalization of the Hyers-Ulam-Rassias stability of the Pexider
equation,” Journal of Mathematical Analysis and Applications, vol. 246, no. 2, pp. 627–638, 2000.
12 Y.-H. Lee and K.-W. Jun, “On the stability of approximately additive mappings,” Proceedings of the
American Mathematical Society, vol. 128, no. 5, pp. 1361–1369, 2000.
13 L. Cădariu and V. Radu, “Fixed points and the stability of Jensen’s functional equation,” Journal of
Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 4, 2003.
14 L. Cădariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,”
in Iteration Theory, vol. 346 of Grazer Mathematische Berichte, pp. 43–52, Karl-Franzens-Universitaet
Graz, Graz, Austria, 2004.
15 L. Cădariu and V. Radu, “Fixed points and the stability of quadratic functional equations,” Analele
Universităţii din Timişoara, vol. 41, no. 1, pp. 25–48, 2003.
16 B. Margolis and J. B. Diaz, “A fixed point theorem of the alternative, for contractions on a generalized
complete metric space,” Bulletin of the American Mathematical Society, vol. 74, pp. 305–309, 1968.
17 I. A. Rus, “Principles and applications of fixed point theory,” Editura Dacia, Cluj-Napoca, 1979
Romanian.
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