Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 326951, 19 pages
doi:10.1155/2011/326951
Research Article
Solutions and the Generalized
Hyers-Ulam-Rassias Stability of a Generalized
Quadratic-Additive Functional Equation
M. Janfada and R. Shourvazi
Department of Mathematics, Sabzevar Tarbiat Moallem University, P.O. Box 397, Sabzevar, Iran
Correspondence should be addressed to M. Janfada, mjanfada@gmail.com
Received 23 January 2011; Revised 5 April 2011; Accepted 26 April 2011
Academic Editor: D. Anderson
Copyright q 2011 M. Janfada and R. Shourvazi. 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 study general solutions and generalized Hyers-Ulam-Rassias stability of the following n
dimensional functional equation f ki1 xi k − 2 ki1 fxi ki1 kj1,j>i fxi xj , k ≥ 3,
on non-Archimedean normed spaces.
1. Introduction
In 1960, Ulam 1 proposed the following question: under what conditions does there exist
an additive mapping near an approximately additive mapping? Hyers 2 showed that if X
and Y are Banach spaces, > 0 and f : X → Y are a mapping such that
f x y − fx − f y < 1.1
for all x, y ∈ X, then there exists a unique additive mapping T : X → Y such that fx −
T x < , for all x ∈ X.
Let f : X → Y be a mapping such that ftx is continuous in t ∈ R , for each fixed
x ∈ X. Suppose for some > 0 and p ∈ 0, 1,
f x y − fx − f y < xp yp ,
1.2
2
Abstract and Applied Analysis
for all x, y ∈ X. Rassias 3 showed that there exists a unique R-linear mapping T : X → Y
such that
fx − T x <
2
xp ,
2 − 2p
1.3
for all x ∈ X. Găvruţa 4 generalized the results of Rassias.
The functional equation fx y fx − y 2fx 2fy is called the quadratic
functional equation. In particular every solution of the quadratic functional equation is said
to be a quadratic mapping see 5, 6. It is well known that a mapping f between real vector
spaces is quadratic if and only if there exists a unique symmetric biadditive mapping B such
that fx Bx, x, for all x see 7–9.
A generalized Hyers-Ulam stability problem for the quadratic functional equation was
proved by Skof 10 for mappings f : X → Y , where X is a normed space and Y is a Banach
space. Cholewa 11 noticed that the theorem of Skof is still true if the relevant domain X is
replaced by an Abelian group. In 12, Czerwik proved the generalized Hyers-Ulam stability
of the quadratic functional equation. Borelli and Forti 13 generalized the stability result as
follows cf. 14, 15: let G be an Abelian group and X a Banach space. Assume that a mapping
f : G → X satisfies the functional inequality
f x y f x − y − 2fx − 2f y ≤ ϕ x, y ,
1.4
for all x, y ∈ G, and ϕ : G × G → 0, ∞ is a function such that
φ x, y :
∞ ϕ 2i x, 2i y
i0
4i1
< ∞,
1.5
for all x, y ∈ G. Then there exists a unique quadratic mapping Q : G → X with the property
fx − Qx ≤ φx, x, for all x ∈ G.
Stability of the quadratic functional was also studied by many other authors in various
cases see, e.g., 16–25.
Let X and Y be some given vector spaces, and let f : X → Y be a given mapping. For
any k ≥ 3, define
Dfx1 , . . . , xk : f
k
i1
xi
k − 2
k
k
k fxi −
f xi xj ,
i1
i1 j1,j>i
1.6
where xi ∈ X, i 1, . . . , k. One can see that the quadratic function f : R → R defined by
fx x2 and any additive mapping satisfy not only the following functional equation:
f x y z fx f y fz f x y f y z fz x
1.7
but also
Dfx1 , . . . , xk 0,
1.8
for all xi ∈ R . So it is natural that these functional equations are called quadratic additive.
Abstract and Applied Analysis
3
The functional equation 1.7 was first solved by Kannappan. In fact he proved that a
mapping f on a real vector space is a solution of 1.7 if and only if there exists a symmetric
biadditive mapping B and an additive mapping A such that fx Bx, x Ax, for
any x see 9. The stability problem for 1.7 is also studied in 26. Moreover 1.7 was
pexiderized and solved by Kannappan 9.
In 27, solutions and the generalized Hyers-Ulam-Rassias stability of the functional
equation 1.8 have been studied for k 3.
The generalized Hyers-Ulam-Rassias stability problem for the functional equation
1.8 was first considered by Bae and Park 28. Also solutions and the generalized UlamGăvruţa-Rassias stability of this functional equation were studied by Nakmahachalasint 29.
Indeed for its solutions the following theorem is proved.
Theorem 1.1 see 29, Theorem 2.1. Let n > 2 be a positive integer, and let X and Y be vector
spaces. A mapping f : X → Y satisfies the functional equation 1.8 if and only if the even part of
f, defined by fe x 1/2fx f−x for all x ∈ X, satisfies the classical quadratic functional
equation and the odd part of f, defined by fo x 1/2fx − f−x for all x ∈ X, satisfies the
Cauchy functional equation fx y fx fy.
In Section 2 of this paper, we will prove that the functional equation 1.8 is equivalent
to the functional equation 1.7. In Section 3, first we prove the generalized Hyers-UlamRassias stability of the functional equation 1.8 in non-Archimedean normed spaces, and
then as a consequence of this result, we prove the generalized Hyers-Ulam-Rassias stability
of 1.8 in non-Archimedean normed spaces. Finally using the methods of Theorem 3.1.
in 29, directly the generalized Hyers-Ulam-Rassias stability of 1.8 will be proved in
non-Archimedean normed spaces. The stability problem in non-Archimedean case has been
studied by many authors, for example, see 30–34.
First we need some preliminaries in non-Archimedean normed space.
Let K be a field. A non-Archimedean absolute value on K is a function | · | : K → R
such that for any a, b ∈ K we have
i |a| ≥ 0 and equality holds if and only if a 0,
ii |ab| |a||b|,
iii |a b| ≤ max{|a|, |b|}.
Condition iii is called the strong triangle inequality. By ii, we have |1| | − 1| 1. Thus, by
induction, it follows from iii that |n| ≤ 1, for each integer n. We always assume in addition
that | · | is nontrivial, that is,
iv there is an a0 ∈ K such that |a0 | / 0, 1.
Let X be a linear space over a scalar field K with a non-Archimedean nontrivial
valuation | · |. A function · : X → R is a non-Archimedean norm valuation if it is a
norm over K with the strong triangle inequality ultrametric; namely,
x y ≤ max x, y ,
x, y ∈ X .
Then X, · is called a non-Archimedean normed space.
1.9
4
Abstract and Applied Analysis
By a complete non-Archimedean normed space we mean one in which every Cauchy
sequence is convergent.
Thanks to the inequality
xn − xm ≤ max xj1 − xj : m ≤ j ≤ n − 1 n > m
1.10
a sequence {xn } is Cauchy if and only if {xn1 − xn } converges to zero in a non-Archimedean
space.
2. Solution of the Functional Equation 1.8
Throughout this section, X and Y will be some vector spaces. The following theorem proves
that the functional equation 1.8 is equivalent to the functional equation 1.7, and so every
solution of the functional equation 1.8 is a summation of a quadratic and an additive
mappings.
Theorem 2.1. Let X and Y be common domain and range of the f’s in the functional equations 1.7
and 1.8. Then the functional equation 1.8 is equivalent to 1.7.
Proof. We can easily see that 1.8 implies 1.7. Now, suppose a mapping f : X → Y satisfies
1.7, for all x, y ∈ X. Using mathematical induction, we are going to show that, for any k ≥ 3,
k
k
k
k xi k − 2 fxi f xi xj .
f
i1
i1
2.1
i1 j1,j>i
k−1
Let M f k−1
i1 xi k − 3
i1 fxi , and suppose 2.1 holds for k − 1, we prove that 2.1
is valid for k. Let x1 , x2 , . . . , xk ∈ X be given and k odd. By the assumption of induction, we
have
k
k f xi xj
i1 j1,j>i
k−1
k
k−1
f
xi k − 3 fxi fxi xk i1
M
i1
i1
k−1/2
fx2i−1 xk fx2i xk fx2i−1 x2i − fx2i−1 x2i i1
M
k−1/2
k−1/2
fx2i−1 x2i fx2i−1 x2i xk fx2i−1 fx2i fxk −
i1
i1
M fx1 x2 xk fx3 x4 xk fx1 x2 x3 x4 − fx1 x2 x3 x4 k−1/2
i3
−
k−1/2
fx2i−1 x2i i1
fx2i−1 x2i xk k−1
i1
fxi k−1
fxk 2
Abstract and Applied Analysis
4
M fxk f
xi xk fx1 x2 fx3 x4 fx5 x6 xk 5
i1
k−1/2
6
6
4
xi − f
xi − f
xi fx2i−1 x2i xk f
i1
k−1
i1
i1
i4
k−1/2
k−1
fxk −
fx2i−1 x2i 2
i1
i1
4
6
6
M fxk f
xi xk fx5 x6 xk f
xi − f
xi
fxi i1
i1
i1
k−1/2
k−1/2
4
k−1
k−1
−f
fxk −
xi fx2i−1 x2i xk fxi fx2i−1 x2i 2
i3
i1
i4
i1
6
4
6
M fxk f
xi xk f
xi fx5 x6 fxk − f
xi
i1
i1
i1
k−1/2
k−1/2
4
k−1
k−1
−f
fxk −
xi fx2i−1 x2i xk fxi fx2i−1 x2i 2
i3
i1
i4
i1
6
8
8
M 2fxk f
xi xk fx7 x8 xk f
xi − f
xi
i1
i1
i1
k−1/2
k−1/2
6
k−1
k−1
−f
fxk −
xi fx2i−1 x2i xk fxi fx2i−1 x2i 2
i5
i1
i1
i4
8
6
8
6
M 3fxk f
xi xk f
xi fx7 x8 − f
xi − f
xi
i1
k−1/2
i1
k−1
i1
i1
k−1/2
k−1
fxk −
fx2i−1 x2i 2
i5
i1
i4
k−1/2
8
8
xi xk − f
xi fx2i−1 x2i xk M 3fxk f
fx2i−1 x2i xk fxi i1
k−1
i1
i5
k−1/2
k−1
fxk −
fx2i−1 x2i 2
i5
i1
k−1
k−1
k−1
k−3
k−1
fxk f
fxk ··· M xi xk − f
xi fxi 2
2
i1
i1
i1
k−1
k
k−1
k − 3 fxi k − 2fxk f
xi fxi fxi i1
k
k
f
xi k − 2 fxi .
i1
i1
i1
i1
2.2
The proof of the case that k is even is very similar and is omitted. Thus 1.7 and 1.8 are
equivalent.
6
Abstract and Applied Analysis
3. Generalized Hyers-Ulam-Rassias Stability of
the Functional Equation 1.8
In this section, we assume that X and Y are a linear space and a complete non-Archimedean
normed space, respectively, over a scalar field K with a non-Archimedean nontrivial
valuation | · |.
Lemma 3.1. Assume that a mapping f : X → Y satisfies the following inequality:
f x y z fx f y fz − f x y − f y z − fz x ≤ ϕ x, y, z ,
3.1
where ϕ : X × X × X → 0, ∞ is an arbitrary mapping and x, y, z ∈ X. Then,
n
n
fx − 2 1 f2n x 2 − 1 f−2n x
2n1
2n1
2
2
i1
2 − 1 i
i
i
≤ max max
ϕ −2 x, −2 x, 2 x , |2|ϕ0, 0, 0 : 0 ≤ i < n ,
|2|2i3
i1
2 1 i
i
i
,
max
ϕ 2 x, 2 x, −2 x , |2|ϕ0, 0, 0 : 0 ≤ i < n
|2|2i3
3.2
for all x ∈ X and n ∈ N.
Proof. If we replace x, y and z in 3.1 by 0, we get f0 ≤ ϕ0, 0, 0. Putting x y −z in
3.1, it yields
3fx f−x − f2x ≤ max ϕx, x, −x, |2|ϕ0, 0, 0 .
3.3
By substituting −x for x in 3.3, we get
3f−x fx − f−2x ≤ max ϕ−x, −x, x, |2|ϕ0, 0, 0 .
3.4
We use induction on n to prove our claim. By 3.3 and 3.4, we have
fx − 3 f2x 1 f−2x
8
8
1 3 ≤ max 3fx f−x − f2x, −3f−x − fx f−2x
8
8
3 1 ≤ max ϕx, x, −x, |2|ϕ0, 0, 0 , ϕ−x, −x, x, |2|ϕ0, 0, 0 .
8
8
3.5
Abstract and Applied Analysis
7
This proves the validity of the inequality 3.2 for n 1. Now, assume that inequality 3.2
holds true for some n ∈ N. By using 3.3, 3.4, and the following relation
fx −
2n1 1 n1 2n1 − 1 n1 f 2 x 2n3 f −2 x
22n3
2
fx −
2n 1
2n − 1
f2n x 2n1 f−2n x
2n1
2
2
2n1 1 2n3 3f2n x f−2n x − f 2n1 x
2
−
3.6
2n1 − 1 n
n
n1
3f−2
x
f2
x
−
f
−2
x
,
22n3
we get inequality 3.2 for n 1. This completes the proof.
In the following two theorems, the generalized Hyers-Ulam-Rassias stability of 1.7
is proved under the approximately even and approximately odd conditions, respectively.
Theorem 3.2. Assume that a mapping f : X → Y satisfies the following inequality:
f x y z fx f y fz − f x y − f y z − fz x ≤ ϕ x, y, z ,
fx − f−x ≤ ψx,
3.7
where ϕ : X × X × X → 0, ∞ and ψ : X → 0, ∞ are mappings such that
lim
ϕ 2n x, 2n y, 2n z
|2|2n
n→∞
0,
i1
2 − 1 i
i
i
ϕ x, y, z lim max
ϕ 2 x, 2 y, 2 z , |2|ϕ0, 0, 0 : 0 ≤ i < n < ∞,
n→∞
|2|2i3
3.8
i1
2 − 1 i
i
i
lim lim max
ϕ 2 x, 2 y, 2 z , |2|ϕ0, 0, 0 : k ≤ i < n k 0,
k→∞n→∞
|2|2i3
lim
n→∞
ψ2n x
|2|2n
0,
3.9
for all x, y, z ∈ X. Then there exists a unique quadratic mapping Q : X → Y which satisfies 1.7
and the inequality
fx − Qx ≤ max ϕ−x,
−x, x, Φx,
x, −x ,
3.10
8
Abstract and Applied Analysis
where
i1
2 1 i
i
i
x, y, z lim max
ϕ 2 x, 2 y, 2 z , |2|ϕ0, 0, 0 : 0 ≤ i < n ,
Φ
n→∞
|2|2i3
3.11
for all x, y, z ∈ X. If moreover f is measurable or ftx is continuous in t for each fixed x ∈ X then
Qtx t2 Qx, for all x ∈ X and t ∈ R .
Proof. It follows from 3.2, the second condition in 3.7 and Lemma 3.1, that
n
fx − f2 x 22n n
2n 1
2n − 1
2 −1
2n − 1
n
n
n
n
fx − 2n1 f2 x 2n1 f−2 x −
f−2 x − 2n1 f2 x 2n1
2
2
2
2
i1
2 − 1 i
i
i
≤ max max
:0≤i<n ,
ϕ
−2
x,
−2
x,
2
x
,
0,
0
|2|ϕ0,
|2|2i3
i1
2 1 |2n − 1|
i
i
i
n
max
ϕ 2 x, 2 x, −2 x , |2|ϕ0, 0, 0 : 0 ≤ i < n , 2n1 ψ2 x .
|2|2i3
|2|
3.12
By 3.12, for n ≥ m, we have
f2n x f2m x 22n − 22m n−m m 1 1
m 2m f
2
·
2
x
−
f2
x
22n−m
|2|
i1
2 − 1 1
im
im
im
≤ 2m max max
ϕ −2 x, −2 x, 2 x , |2|ϕ0, 0, 0 : 0 ≤ i < n − m ,
|2|
|2|2i3
i1
2 1 im
im
im
max
ϕ 2 x, 2 x, −2 x , |2|ϕ0, 0, 0 : 0 ≤ i < n − m ,
|2|2i3
|2n−m − 1| n−m m ψ 2
·2 x .
22n−m1 3.13
Replacing n, m with n 1, n, respectively, in 3.13, we can easily get
f 2n1 x f2n x ϕ−2n x, −2n x, 2n x |2|ϕ0, 0, 0
1
,
,
2n1 −
≤ max
2
22n |2|3
|2|2n
|2|2n
1 ψ 2n1 x
|3| ϕ2n x, 2n x, −2n x |2|ϕ0, 0, 0
,
.
,
|2| |2|2n1
|2|3
|2|2n
|2|2n
3.14
Abstract and Applied Analysis
9
The sequence {f2n x/22n } is a Cauchy sequence, since the right-hand side of the inequality
3.14 tends to 0 as n tends to ∞. Therefore, we may apply a direct method for definition of
Q. Define
f2n x
,
n → ∞ 22n
3.15
Qx lim
for all x ∈ X. From the first condition in 3.7, it follows that
f 2n x y z f2n x f 2n y f2n z f 2n x y
−
22n
22n
22n
22n
22n
3.16
n
n
n
f 2n y z
f2n z x ϕ 2 x, 2 y, 2 z
−
−
,
≤
22n
22n
|2|2n
for all x, y, z ∈ X and for all n ∈ N. Therefore, by letting n → ∞ in the last inequality, from
3.8, it is clear that Q is a solution of 1.7. Analogously, by the second condition in 3.7, Q is
even. By putting z −y in 1.7 and taking account of Q0 0, we see Q as an even solution
of 1.7 and so is quadratic. According to |2|n ≤ 1 for all n ∈ N, we get
|2n − 1|
|2|2n1
|2n |
n
ψ2 x ≤ max
|2|2n1
n
ψ2 x,
1
|2|2n1
n
ψ2 x
1
|2|2n1
ψ2n x.
3.17
Then from 3.12 and definition of Q one may see that inequality 3.10 holds true.
Now, let T : X → Y be another quadratic mapping which satisfies 1.7 and inequality
3.10. Obviously, we have
Q2n x 4n Qx,
T 2n x 4n T x,
3.18
for all x ∈ X and n ∈ N. Hence, it follows from 3.10 that
Qx − T x |2|−2n Q2n x − T 2n x
≤ |2|−2n max Q2n x − f2n x, T 2n x − f2n x
≤ max
3.19
n
n x, 2n x, −2n x
ϕ−2
x, −2n x, 2n x Φ2
,
,
2n
|2|
|2|2n
for all x ∈ X and n ∈ N. Applying 3.8 and letting n → ∞ in the preceding inequality, we
immediately conclude the uniqueness of Q. The proof of the last assertion in the theorem
goes through in the same way as that of Theorem 1 12.
10
Abstract and Applied Analysis
Theorem 3.3. Assume that a mapping f : X → Y satisfies the system of inequalities
f x y z fx f y fz −f x y − f y z − fz x ≤ ϕ x, y, z ,
fx f−x ≤ ψx,
3.20
for all x, y, z ∈ X, where ϕ : X × X × X → 0, ∞ and ψ : X → 0, ∞ are mappings with
conditions of Theorem 3.2. Then with the notations of Theorem 3.2, there exists a unique additive
mapping F : X → Y satisfying the inequality
fx − Fx ≤ max ϕ−x,
−x, x, Φx,
x, −x
3.21
for all x ∈ X.
Proof. From 3.2, the second condition in 3.20, and Lemma 3.1, we get
n
fx − f2 x n
2
n
2n 1
2n − 1
2n − 1
2 −1
n
n
n
n
fx
−
f2
x
f−2
x
−
f−2
x
f2
x
2n1
2n1
2n1
2n1
2
2
2
2
i1
2 − 1 i
i
i
≤ max max
ϕ −2 x, −2 x, 2 x , |2|ϕ0, 0, 0 : 0 ≤ i < n ,
|2|2i3
i1
2 1 |2n − 1|
i
i
i
n
max
ϕ 2 x, 2 x, −2 x , |2|ϕ0, 0, 0 : 0 ≤ i < n , 2n1 ψ2 x .
|2|2i3
|2|
3.22
For n ≥ m, we have
f2n x f2m x −
2n
2m n−m m 1 1
m m n−m f 2
· 2 x − f2 x
|2| 2
i1
2 − 1 1
im
im
im
≤ m max max
ϕ −2 x, −2 x, 2 x , |2|ϕ0, 0, 0 : 0 ≤ i < n − m ,
|2|
|2|2i3
i1
2 1 im
im
im
max
ϕ 2 x, 2 x, −2 x , |2|ϕ0, 0, 0 : 0 ≤ i < n − m ,
|2|2i3
|2n−m − 1| n−m m ψ 2
·2 x .
22n−m1 3.23
Abstract and Applied Analysis
11
Therefore,
f 2n1 x f2n x 1 ϕ−2n x, −2n x, 2n x |2|ϕ0, 0, 0
−
,
,
≤ max
2n1
2n |2|n
|2|n
|2|3
|3|
|2|3
ϕ2n x, 2n x, −2n x |2|ϕ0, 0, 0
1 ψ 2n1 x
.
,
, 2
|2|n
|2|n
|2|
|2|n1
3.24
Conditions of ϕ and ψ and |4| ≤ |2| imply that sequence {f2n x/2n } is a Cauchy sequence.
Now, for any x ∈ X, define
f2n x
.
n→∞
2n
3.25
Fx lim
Similarly, as in the proof of Theorem 3.2, due to 3.20, we may see that the mapping F
satisfies 1.7 and is an odd function. By putting z −y in 1.7, considering the oddness
of F, and letting u x y, v x − y, we get
2F
u v
2
Fu Fv.
3.26
According to 35, the mapping F is additive, since F0 0. The validity of inequality 3.21
follows directly from 3.22 and the definition of F. Now, let G : X → Y be another additive
mapping which satisfies 3.21. It then follows from 3.21 that
Fx − Gx |2|−n F2n x − G2n x
≤ |2|−n max F2n x − f2n x, G2n x − f2n x
≤ max
3.27
n x, 2n x, −2n x
ϕ−2
x, −2 x, 2 x Φ2
,
,
n
|2|
|2|n
n
n
n
for all x ∈ X and n ∈ N. This implies the uniqueness of F.
Remark 3.4. The approximately even condition guarantees the quadratic property of Q,
whereas the approximately odd condition guarantees the additive behavior of F.
Corollary 3.5. Let k ∈ N and k ≥ 3. Assume that a mapping f : X k → Y satisfies the following
inequalities:
k
k
k
k f
xi k − 2 fxi −
f xi xj ≤ φx1 , . . . , xk ,
i1
i1
i1 j1,j>i
3.28
fx − f−x ≤ ψx,
3.29
12
Abstract and Applied Analysis
where φ : X k → 0, ∞ and ψ : X → 0, ∞ are mappings such that
lim
φ 2n x, 2n y, 2n z, 0, . . . , 0
|2|2n
n→∞
0,
i1
2 − 1 i
i
i
φ 2 x, 2 y, 2 z, 0, . . . , 0 , |2|φ0, . . . , 0 : 0 ≤ i < n < ∞,
ϕ x, y, z lim max
n→∞
|2|2i3
i1−k
2
− 1 i
i
i
lim lim max
φ 2 x, 2 y, 2 z, 0, . . . , 0 , |2|φ0, . . . , 0 : k ≤ i < n k 0,
k→∞n→∞
|2|2i3
lim
n→∞
ψ2n x
|2|2n
0,
3.30
for all x, y, z ∈ X. Then there exists a unique quadratic mapping Q : X → Y which satisfies 1.8
and the following inequality:
fx − Qx ≤ max ϕ−x,
−x, x, Φx,
x, −x ,
3.31
where
|2k − 4k − 3|
ϕ x, y, z max φ x, y, z, 0, . . . , 0 ,
φ0, . . . , 0 ,
|k − 2k − 1|
i1
2 − 1 i
i
i
ϕ 2 x, 2 y, 2 z , |2|ϕ0, 0, 0 , 0 ≤ i < n ,
ϕ x, y, z lim max
n→∞
|2|2i3
i1
2 1 i
i
i
Φ x, y, z lim max
φ 2 x, 2 y, 2 z, , |2|φ0, 0, 0 , 0 ≤ i < n .
n→∞
|2|2i3
3.32
Proof. By letting xi 0, i 1, . . . , k, in 3.28 we get
f0 ≤
|2|
φ0, . . . , 0,
|k − 1k − 2|
3.33
and also from 3.28 and 3.33 we have
fx1 x2 x3 fx1 fx2 fx3 − fx1 x2 − fx2 x3 |2k − 4k − 3|
≤ max φx1 , x2 , x3 , 0, . . . , 0,
φ0, . . . , 0 .
|k − 2k − 1|
3.34
Now by considering ϕx, y, z max{φx, y, z, 0, . . . , 0, |2k − 4k − 3|/|k − 2k − 1|φ0,
. . . , 0}, we may see that ϕ satisfies 3.8, and so using Theorem 3.2, we get 3.31.
Abstract and Applied Analysis
13
Remark 3.6. In the previous corollary, if one replaces 3.29 with
fx f−x ≤ ψx,
3.35
where ψ : X → 0, ∞ is a mapping satisfying
lim
ψ2n x
n→∞
|2|2n
0,
3.36
then using Theorem 3.3 and a similar argument with Corollary 3.5, we may find a unique
additive mapping F : X → Y which satisfies 1.8 and the inequality
fx − Fx ≤ max ϕ−x,
−x, x, Φx,
x, −x .
3.37
In the following theorem, using 29 Theorem 3.1, the generalized Hyers-Ulam-Rassias
stability of 1.8 is proved directly in non-Archimedean normed spaces. Note that the
conditions on φ in this theorem are different from the previous conditions.
Theorem 3.7. Let k > 2 be a positive integer, X a non-Archimedean vector space, and Y a
non-Archimedean Banach space. Let φ : X k → 0, ∞ be an even function. Define ϕx φx, x, −x, 0, . . . , 0 for all x ∈ X. If
φ2m x1 , 2m x2 , . . . , 2m xk 0,
m→∞
|4|m
i ϕ 2x
ϕx
lim max
: 0 ≤ i < m < ∞,
m→∞
|4|i
i ϕ 2x
: k ≤ i < m k 0,
lim lim max
k→∞m→∞
|4|i
3.38
x2
xk , . . . , m 0,
m
m→∞
2
2
x
ϕx
lim max |2|i ϕ i : 1 ≤ i < m 1 < ∞,
m→∞
2
x
lim lim max |2|i ϕ i : 1 k ≤ i < m k 1 0,
k→∞m→∞
2
3.39
lim
or
lim |2|m φ
x
1
,
2m
for all x1 , x2 , . . . , xk ∈ X, and a mapping f : X → Y satisfies f0 0 and
Dfx1 , . . . , xk ≤ φx1 , . . . , xk ,
3.40
14
Abstract and Applied Analysis
for all x1 , x2 , . . . , xk ∈ X, then there exists a unique function T : X → Y that satisfies functional
equation 1.8, and if condition 3.38 holds,
fe x − Te x ≤ 1 ϕx,
|2|3
3.41
fo x − To x ≤ 1 ϕ1 x,
|2|2
3.42
where ϕ1 x limm → ∞ max{ϕ2i x/|2|i : 0 ≤ i < m} or, if condition 3.39 holds,
fe x − Te x ≤ 1 ϕx,
|2|3
3.43
fo x − To x ≤ 1 ϕ2 x,
|2|2
3.44
where ϕ2 x limm → ∞ max{|2|i ϕx/2i : 1 ≤ i < m 1}. The function T is given by
T x ⎧
⎪
⎨ lim 4−m fe 2m x 2−m fo 2m x if conditions 3.38 hold,
m→∞
⎪
⎩ lim 4m fe 2−m x 2m fo 2−m x if conditions 3.39 hold,
3.45
m→∞
for all x ∈ X.
Proof. We will prove the theorem for a function φ satisfying condition 3.38 and accordingly
inequalities 3.41 and 3.42. A proof for conditions 3.39 and inequalities 3.43 and 3.44
can be reproduced in a similar manner. Setting x1 , x2 , . . . , xk x, x, −x, 0, 0, . . . , 0 in 3.40
and simplifying, we have
3fx f−x − f2x ≤ ϕx.
3.46
3f−x fx − f−2x ≤ ϕ−x ϕx.
3.47
Replacing x by −x, we get
Then,
4fe x − fe 2x 1 3fx f−x − f2x 3f−x fx − f−2x |2|
1
3fx f−x − f2x, 1 3f−x fx − f−2x
≤ max
|2|
|2|
≤
ϕx
,
|2|
Abstract and Applied Analysis
15
2fo x − fo 2x 1 3fx f−x − f2x − 3f−x fx − f−2x |2|
≤ max
≤
1
3fx f−x − f2x, 1 3f−x fx − f−2x
|2|
|2|
ϕx
.
|2|
3.48
Rewrite the inequality on fe as fe x − fe 2x/4 ≤ 1/|2|3 ϕx for all x ∈ X. Then by
replacing x by 2m x and dividing both sides by 4m we have
f 2m x fe 2m1 x 1 ϕ2m x
e
≤
−
.
4m
4m1 |2|3 |4|m
3.49
i m
ϕ 2x
x
f
2
1
e
fe x −
≤
max
:0≤i<m ,
4m |2|3
|4|i
3.50
Thus
for every positive integer m. If we rewrite the inequality for fo as fo x − fo 2x/2 ≤
1/|2|2 ϕx and repeat the same steps as in the case of fe , we will have
f 2m x fo 2m1 x 1 ϕ2m x
o
≤
−
,
2m
2m1 |2|2 |2|m
3.51
i m
ϕ 2x
x
f
2
1
o
≤
fo x −
max
:0≤i<m ,
2m |2|2
|2|i
3.52
for every positive integer m. By definition of φ, relations 3.49 and 3.38, the sequence
{fe 2m x/4m } is a Cauchy sequence in a Banach space. Let Te x limm → ∞ fe 2m x/4m for all x ∈ X, thus
i ϕ 2x
fe x − Te x ≤ 1 lim max
:0≤i<m .
|2|3 m → ∞
|4|i
3.53
By similar way we have
i ϕ 2x
fo x − To x ≤ 1 lim max
:0≤i<m .
|2|2 m → ∞
|2|i
3.54
16
Abstract and Applied Analysis
Define T x Te x To x, for all x ∈ X. In order to show that T satisfies 1.8, it is enough
to show that Te and To satisfy 1.8. For convenience, define Dfe and Dfo as the even part
and the odd part of Df in 1.8, respectively. For Te , consider
1 1 m
m
Df2m x1 , . . . , 2m xk Df−2m x1 , . . . , −2m xk m Dfe 2 x1 , . . . , 2 xk m
|4|
|4| |2|
≤
1
φ2m x1 , . . . , 2m xk .
|4|m |2|
3.55
As m tend to infinity, the left-hand side approaches DTe x1 , . . . , xk and by conditions 3.38,
the right-hand side approaches 0. Thus, DTe x1 , . . . , xk 0. A similar argument shows that
DTo x1 , . . . , xk 0 since
lim
1
m → ∞ |2|m
φ2m x1 , . . . , 2m xk 0.
3.56
Hence, T Te To satisfies 1.8 as desired. Now using Theorem 1.1, one can see that Te and To
are quadratic and additive, respectively. Also letting m → ∞ in 3.50 one may obtain 3.43.
To prove the uniqueness of T , suppose that there exists another function S : X → Y
that satisfies 1.8 and inequalities 3.41 and 3.42 with S instead of T . Then,
Sx − T x ≤ max{Se x − Te x, So x − To x}.
3.57
It is straightforward to show that every solution of the quadratic functional equation fx y fx − y 2fx 2fy has the quadratic property fnx n2 fx and every solution
of the linear functional equation fx y fx fy has the linear property fnx nfx
for every positive integer n and for every x in the domain. We thus obtain
1 1 m
m m
m Se 2 x − fe 2 x , m So 2 x − fo 2 x
Sx − T x ≤ lim max
m→∞
|4|m
|2|
1
1
m
m
ϕ2
x, 2 m ϕ1 2 x ,
≤ lim max
m→∞
|2|3 |4|m
|2| |2|
3.58
for all x ∈ X. This completes the proof.
Corollary 3.8. If |2| < 1, a mapping f : X → Y satisfies f0 0 and the inequality
Dfx1 , x2 , . . . , xk ≤ ,
3.59
for some > 0 and for all x1 , x2 , . . . , xk ∈ X, then there exists a unique mapping T : X → Y that
satisfies functional equation 1.8 and for all x ∈ X,
fe x − Te x ≤ ,
|2|
fo x − To x ≤ .
|2|
3.60
Abstract and Applied Analysis
17
Proof. Let φx1 , x2 , . . . , xk . Then condition 3.39 in Theorem 3.7 holds. Hence, it follows
from Theorem 3.7 that there exists a unique mapping T : X → Y such that
fe x − Te x ≤ 1 lim max |4|i ϕ x : 1 ≤ i < m 1 1 |4|,
2i
|2|3 m → ∞
|2|3
fo x − To x ≤ 1 lim max |2|i ϕ x : 1 ≤ i < m 1 1 |2|.
2i
|2|3 m → ∞
|2|2
3.61
Corollary 3.9. Let p be a positive real number with 0 < p < 1 or p > 2 and |2| < 1. If a mapping
f : X → Y satisfies the inequality,
n
Dfx1 , x2 , . . . , xk ≤ xi p ,
3.62
i1
for some > 0 and for all x1 , x2 , . . . , xk ∈ X, then there exists a unique mapping T : X → Y that
satisfies functional equation 1.8 and for all x ∈ X,
fe x − Te x ≤ 3 xp ,
|2|3
fe x − Te x ≤
3
|2|p1
p
x ,
fo x − To x ≤ 3 xp ,
|2|2
fo x − To x ≤
3
|2|p1
p
x ,
for p > 2,
3.63
for 0 < p < 1.
Proof. Substituting x1 x2 · · · xk 0 into 3.62, we get
n
f0 k − 2kf0 f0.
2
3.64
It follows that 1 kk − 2 > k2 , since k > 2. Hence, f0 0. Let φx1 , x2 , . . . , xn n
i1 xi p . If 0 < p < 1, then condition 3.39 in Theorem 3.7 holds, and it follows that
p p p x x −x 3
fe x − Te x ≤ 1 |4| p1 xp ,
3
2
2
2
|2|
|2|
p p p x x −x 3
fo x − To x ≤ 1 |2| p1 xp .
2 2
2
2
|2|
|2|
3.65
If p > 2, we apply Theorem 3.7 with condition 3.38 to get a similar result.
Acknowledgment
The authors would like to appreciate the referees for their contributions and their very useful
suggestions.
18
Abstract and Applied Analysis
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