7_854_xu.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 6, 2010, 1032-1047

ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE ON COMPLEX ANALYSIS: THEORY AND APPLICATIONS

DEDICATED TO PROFESSOR HARI M. SRIVASTAVA,

ON THE OCCASION OF HIS 70TH
BIRTHDAY

Generalized Ulam-Hyers Stability of a General Mixed

AQCQ-functional Equation in Multi-Banach Spaces: a Fixed

Point Approach

Tian Zhou Xu1,∗, John Michael Rassias 2, Wan Xin Xu 3

1 Department of Mathematics, School of Science, Beijing Institute of Technology, Beijing 100081,

Peoples Republic of China
2 Pedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrian

University of Athens, 4, Agamemnonos Str., Aghia Paraskevi, Athens 15342, Greece
3 School of Communication and Information Engineering, University of Electronic Science and Tech-

nology of China, Chengdu 611731, Peoples Republic of China

Abstract. Using the fixed point method, we investigate the generalized Hyers-Ulam stability of the

general mixed additive-quadratic-cubic-quartic functional equation

f (x + ny) + f (x − ny) = n2 f (x + y) + n2 f (x − y) + 2(1− n2) f (x)

+
n4 − n2

12
[ f (2y) + f (−2y)− 4 f (y)− 4 f (−y)]

for fixed integers n with n 6= 0,±1 in multi-Banach spaces.

2000 Mathematics Subject Classifications: 39B82, 39B52, 46B99

Key Words and Phrases: Fixed point alternative, Stability, Additive function, Quadratic function,

Cubic function, Quartic function, Multi-Banach space

∗Corresponding author.

Email addresses: xutianzhou�bit.edu.
n, tzxubit�gmail.
om (T. Xu), jrassias�primedu.uoa.gr,Ioannis.Rassias�primedu.uoa.gr, jrass�otenet.gr (J. Rassias), wxbit0930�gmail.
om,wxbit�sina.
om (W. Xu)

http://www.ejpam.com 1032 c© 2010 EJPAM All rights reserved.



T. Xu, J. Rassias, W. Xu / Eur. J. Pure Appl. Math, 3 (2010), 1032-1047 1033

1. Introduction

The concept of stability for a functional equation arises when one replaces a functional

equation by an inequality which acts as a perturbation of the equation. The first stability

problem concerning group homomorphisms was raised by Ulam [35] in 1940 and affirma-

tively solved by Hyers [17]. The result of Hyers was generalized by Rassias [28] for approxi-

mate linear mappings by allowing the Cauchy difference operator

C D f (x , y) = f (x+ y)−[ f (x)+ f (y)] to be controlled by ε(‖x‖p+‖y‖p). In 1994, a general-

ization of Rassias’ theorem was obtained by Găvruţa [11], who replaced ε(‖x‖p+ ‖y‖p) by a

general control function ϕ(x , y). In addition, J. M. Rassias et al.([29]-[32], [37]-[39]) gener-

alized the Hyers stability result by introducing two weaker conditions controlled by a product

of different powers of norms and a mixed product-sum of powers of norms, respectively. Re-

cently, several further interesting discussions, modifications, extensions, and generalizations

of the original problem of Ulam have been proposed (see, e.g., [2]-[3], [6], [8]-[16], [18],

[20]-[27], [33], [36]-[42] and the references therein).

The historical background and many important results for the Ulam-Hyers stability of var-

ious functional equations are surveyed in [4] (see also [19]. There are applications in actuar-

ial and financial mathematics, sociology and psychology, as well as in algebra and geometry

[1, 4, 19]. In addition, the motivation for studying these functional equations came from the

fact that recently polynomial equations have found applications in approximate checking, self-

testing, and self-correcting of computer programs that compute polynomials. The interested

reader should refer to [34] and [40] and references therein.

The functional equation

f (x + y) + f (x − y) = 2 f (x)+ 2 f (y) (1)

is said to be a quadratic functional equation because the quadratic function f (x) = x2 is a

solution of the functional equation (1). Every solution of the quadratic functional equation

is said to be a quadratic mapping. A quadratic functional equation was used to characterize

inner product spaces.

In 2001, J. M. Rassias [29] introduced the cubic functional equation

f (x + 2y)− 3 f (x + y) + 3 f (x)− f (x − y) = 6 f (y) (2)

and established the solution of the Ulam stability problem for these cubic mappings. It is easy

to show that the function f (x) = x3 satisfies the functional equation (2), which is called a

cubic functional equation and every solution of the cubic functional equation is said to be a

cubic mapping. The quartic functional equation

f (x + 2y) + f (x − 2y) = 4 f (x + y) + 4 f (x − y) + 6 f (x)+ 24 f (y) (3)

was introduced by J. M. Rassias [31]. It is easy to show that the function f (x) = x4 is

the solution of (3). Every solution of the quartic functional equation is said to be a quartic

mapping. C. Park [25] proved the generalized Hyers-Ulam stability of the following additive-

quadratic-cubic-quartic functional equation (briefly, AQCQ-functional equation)

f (x+2y)+ f (x−2y) = 4 f (x+ y)+4 f (x− y)−6 f (x)+ f (2y)+ f (−2y)−4 f (y)−4(−y) (4)



T. Xu, J. Rassias, W. Xu / Eur. J. Pure Appl. Math, 3 (2010), 1032-1047 1034

in non-Archimedean normed spaces.

In [9, 33], the authors introduced a general mixed type functional equation

f (x + ny) + f (x − ny) = n2 f (x + y) + n2 f (x − y) + 2(1− n2) f (x)

+
n4 − n2

12
[ f (2y) + f (−2y)− 4 f (y)− 4 f (−y)] (5)

which is a generalized form of the additive-quadratic-cubic-quartic (4) and obtained its gen-

eral solution and generalized Hyers-Ulam stability for fixed integers n with n 6= 0,±1 in

Banach spaces.

The notion of multi-normed space was introduced by H. G. Dales and M. E. Polyakov [5].

This concept is somewhat similar to operator sequence space and has some connections with

operator spaces and Banach lattices. Motivations for the study of multi-normed spaces and

many examples were given in [5]. Also, the stability problems in multi-Banach spaces are

studied by Dales and Moslehian [6], Moslehian et al. ([21]-[23]) and Wang et al. [36].

In 1996, Isac and Rassias [18] were the first to provide applications of stability theory

of functional equations for the proof of new fixed point theorems with applications. The

stability problems of several various functional equations have been extensively investigated

by a number of authors using the fixed point method (see [2]-[3], [6]-[7], [20], [25]-[27],

[36], [38].

In this paper, we prove the generalized Hyers-Ulam stability of the general mixed AQCQ-

functional equation (5) in multi-Banach spaces using the fixed point method.

2. Preliminaries

We recall some preliminaries concerning multi-Banach space (see [5]-[6], [21]-[23]).

Let (E,‖ · ‖) be a complex linear space, and let k ∈ N. We denote by Ek the linear space

E ⊕ · · · ⊕ E consisting of k-tuples (x1, . . . , xk), where x1, . . . , xk ∈ E. The linear operations on

Ek are defined coordinate-wise. When we write (0, . . . , 0, x i, 0, . . . , 0) for an element in Ek,

we understand that x i appears in the ith coordinate. The zero elements of either E or Ek are

both denoted by 0 when there is no confusion. We denote by Nk the set {1,2, . . . , k} and by

Bk the group of permutations on Nk.

Definition 1. A multi-norm on {Ek, k ∈ N} is a sequence (‖ · ‖k) = (‖ · ‖k : k ∈ N) such that

‖ · ‖k is a norm on Ek for each k ∈ N, such that ‖x‖1 = ‖x‖ for each x ∈ E, and the following

axioms are satisfied for each k ∈ N with k ≥ 2:

(A1) ‖(xσ(1), . . . , xσ(k))‖k = ‖(x1, . . . , xk)‖k (σ ∈ Bk, x1, . . . , xk ∈ E);

(A2) ‖(α1 x1, . . . ,αk xk)‖k ≤ (maxi∈Nk
|αi|)‖(x1, . . . , xk)‖k(x i ∈ E,αi ∈ C, i = 1, . . . , k);

(A3) ‖(x1, . . . , xk−1, 0)‖k = ‖(x1, . . . , xk−1)‖k−1 (x1, . . . , xk−1 ∈ E);

(A4) ‖(x1, . . . , xk−1, xk−1)‖k = ‖(x1, . . . , xk−1)‖k−1 (x1, . . . , xk−1 ∈ E).



T. Xu, J. Rassias, W. Xu / Eur. J. Pure Appl. Math, 3 (2010), 1032-1047 1035

In this case, we say that ((Ek,‖ · ‖k) : k ∈ N) is a multi-normed space.

Suppose that ((Ek,‖ · ‖k) : k ∈ N) is a multi-normed space and take k ∈ N. It is easy to

show that

(a) ‖(x , . . . , x)‖k = ‖x‖ (x ∈ E);

(b) max
i∈Nk

‖x i‖ ≤ ‖(x1, . . . , xk)‖k ≤
k∑

i=1

‖x i‖ ≤ k max
i∈Nk

‖x i‖ (x1, . . . , xk ∈ E).

It follows from (b) that if (E,‖ · ‖) is a Banach space, then (Ek,‖ · ‖k) is a Banach space for

each k ∈ N; in this case ((Ek,‖ · ‖k) : k ∈ N) is said to be a multi-Banach space.

Now we state two important examples of multi-norms for arbitrary normed space E (see

[5]-[6], [21]-[23]).

Example 1. Let E be an arbitrary normed space. The sequence (‖ · ‖k : k ∈ N) on {Ek : k ∈ N}
defined by

‖(x1, . . . , xk)‖k :=max
i∈Nk

‖x i‖ (x1, . . . , xk ∈ E)

is a multi-norm called the minimum multi-norm. The terminology minimum is justified by prop-

erty (b).

Example 2. Let E be an arbitrary normed space and let {(‖ · ‖α
k

: k ∈ N) : α ∈ A} be the

(non-empty) family of all multi-norms on {Ek : k ∈ N}. For k ∈ N, consider

‖|(x1, . . . , xk)‖|k := sup
α∈A

‖(x1, . . . , xk)‖
α
k (x1, . . . , xk ∈ E).

Then (‖| · ‖|k : k ∈ N) is a multi-norm on {Ek : k ∈ N}, called the maximum multi-norm.

Definition 2. Let ((Ek,‖ · ‖k) : k ∈ N) be a multi-normed space. A sequence {xn} in E is a

multi-null sequence if, for each ǫ > 0, there exists n0 ∈ N such that

sup
k∈N
‖(xn, . . . , xn+k−1)‖k < ǫ(n≥ n0).

Let x ∈ E. We say that the sequence {xn} is multi-convergent to x in E if {xn− x} is a multi-null

sequence. In this case, x is called the limit of the sequence {xn} and we denote it by lim
n→∞

xn = x.

For explicitly later use, We recall a fundamental result in fixed point theory.

Let X be a set. A function d : X × X → [0,∞] is called a generalized metric on X if d

satisfies: (1) d(x , y) = 0 if and only if x = y; (2) d(x , y) = d(y, x) for all x , y ∈ X ; (3)

d(x , y)≤ d(x , z) + d(y, z) for all x , y, z ∈ X .

Theorem 1 (The fixed point alternative theorem, see [2, 7, 20, 25, 38]). Let (Ω, d) be a

complete generalized metric space and J : Ω→ Ω be a strictly contractive mapping with Lipschitz

constant 0≤ L < 1, that is

d(J x , J y)≤ Ld(x , y) for all x ∈ X .



T. Xu, J. Rassias, W. Xu / Eur. J. Pure Appl. Math, 3 (2010), 1032-1047 1036

Then, for each given x ∈ Ω, either

d(J mx , J m+1x) =∞ for all m ≥ 0,

or

d(J mx , J m+1 x)<∞ for all m≥ m0,

for some nonnegative integer m0. Actually, if the second alternative holds, then the sequence

{J m x} converges to a fixed point y∗ of J and

(i) y∗ is the unique fixed point of J in the set ∆= {y ∈ Ω : d(J m0 x , y)<∞};

(ii) d(y, y∗)≤ 1

1−L
d(y, J y) for all y ∈∆.

3. Generalized Hyers-Ulam stability of the functional equation

In this section, we investigate the stability of the mixed type functional equation (5) in

multi-Banach spaces. For convenience, we use the following abbreviation for a given mapping

f : E→ F :

D f (x , y) := f (x + ny) + f (x − ny)− n2 f (x + y)− n2 f (x − y)− 2(1− n2) f (x)

−
n4 − n2

12
[ f (2y) + f (−2y)− 4 f (y)− 4 f (−y)]

for all x , y ∈ X .

Theorem 2. Let E be a linear space and let ((F k,‖ · ‖k) : k ∈ N) be a multi-Banach space.

Suppose that ǫ ≥ 0 and f : E→ F is an odd mapping satisfying

sup
k∈N
‖(D f (x1, y1), . . . , D f (xk, yk))‖k ≤ ǫ (6)

for all x1, . . . , xk, y1, . . . , yk ∈ E. Then there exists a unique additive mapping A : E → F such

that

sup
k∈N
‖( f (2x1)− 8 f (x1)− A(x1), . . . , f (2xk)− 8 f (xk)− A(xk))‖k ≤

9n2+ 4

n4 − n2
ǫ (7)

for all x1, . . . , xk ∈ E.

Proof. Let x1, . . . , xk, y1, . . . , yk ∈ E. Using the oddness of f and (6), we have

sup
k∈N
‖( f (x1 + ny1) + f (x1− ny1)− n2 f (x1+ y1)− n2 f (x1− y1)

−2(1− n2) f (x1), . . . , f (xk + nyk) + f (xk − nyk)− n2 f (xk + yk)

−n2 f (xk − yk)− 2(1− n2) f (xk))‖k ≤ ǫ.

(8)

Replacing yi by x i(i ∈ Nk) in (8), we get

sup
k∈N
‖( f ((1+ n)x1) + f ((1− n)x1)− n2 f (2x1)− 2(1− n2) f (x1), . . . ,

f ((1+ n)xk) + f ((1− n)xk)− n2 f (2xk)− 2(1− n2) f (xk))‖k ≤ ǫ.
(9)



T. Xu, J. Rassias, W. Xu / Eur. J. Pure Appl. Math, 3 (2010), 1032-1047 1037

Replacing x i by 2x i(i ∈ Nk) in (9), we get

sup
k∈N
‖( f (2(1+ n)x1) + f (2(1− n)x1)− n2 f (4x1)− 2(1− n2) f (2x1), . . . ,

f (2(1+ n)xk) + f (2(1− n)xk)− n2 f (4xk)− 2(1− n2) f (2xk))‖k ≤ ǫ.
(10)

Replacing x i and yi by 2x i and x i(i ∈ Nk) in (8), respectively, we get

sup
k∈N
‖( f ((2+ n)x1) + f ((2− n)x1)− n2 f (3x1)− n2 f (x1)− 2(1− k2) f (2x1), . . . ,

f ((2+ n)xk) + f ((2− n)xk)− n2 f (3xk)− n2 f (xk)− 2(1− k2) f (2xk))‖k ≤ ǫ.
(11)

Replacing yi by 2x i(i ∈ Nk) in (8), we get

sup
k∈N
‖( f ((1+ 2n)x1) + f ((1− 2n)x1)− n2 f (3x1) + n2 f (x1)− 2(1− n2) f (x1), . . . ,

f ((1+ 2n)xk) + f ((1− 2n)xk)− n2 f (3xk) + n2 f (xk)− 2(1− n2) f (xk))‖k ≤ ǫ.
(12)

Replacing yi by 3x i(i ∈ Nk) in (8), we get

sup
k∈N
‖( f ((1+ 3n)x1) + f ((1− 3n)x1)− n2 f (4x1) + n2 f (2x1)− 2(1− n2) f (x1), . . . ,

f ((1+ 3n)xk) + f ((1− 3n)xk)− n2 f (4xk) + n2 f (2xk)− 2(1− n2) f (xk))‖k ≤ ǫ.
(13)

Replacing x i and yi by (1+ n)x i and x i(i ∈ Nk) in (8), respectively, we have

sup
k∈N
‖( f ((1+ 2n)x1) + f (x1)− n2 f ((2+ n)x1)− n2 f (nx1)

−2(1− n2) f ((1+ n)x1), . . . , f ((1+ 2n)xk) + f (xk)

−n2 f ((2+ n)xk)− n2 f (nxk)− 2(1− n2) f ((1+ n)xk))‖k ≤ ǫ.

(14)

Again replacing x i and yi by (1− n)x i and x i(i ∈ Nk) in (8), respectively, we have

sup
k∈N
‖( f (x1) + f ((1− 2n)x1)− n2 f ((2− n)x1) + n2 f (nx1)

−2(1− n2) f ((1− n)x1), . . . , f (xk) + f ((1− 2n)xk)

−n2 f ((2− n)xk) + n2 f (nxk)− 2(1− n2) f ((1− n)xk))‖k ≤ ǫ.

(15)

By (14) and (15), we have

sup
k∈N
‖( f ((1+ 2n)x1) + f ((1− 2n)x1) + 2 f (x1)− n2 f ((2+ n)x1)

−2(1− n2) f ((1+ n)x1)− n2 f ((2− n)x1)− 2(1− n2) f ((1− n)x1),

. . . , f ((1+ 2n)xk) + f ((1− 2n)xk)+ 2 f (xk)− n2 f ((2+ n)xk)

−n2 f ((2− n)xk)− 2(1− n2) f ((1+ n)xk)− 2(1− n2) f ((1− n)xk))‖k ≤ 2ǫ.

(16)

Replacing x i and yi by (1+ 2n)x i and x i(i ∈ Nk) in (8), respectively, we have

sup
k∈N
‖( f ((1+ 3n)x1) + f ((1+ n)x1)− n2 f (2(1+ n)x1)− n2 f (2nx1)

−2(1− n2) f ((1+ 2n)x1), . . . , f ((1+ 3n)xk) + f ((1+ n)xk)

−n2 f (2(1+ n)xk)− n2 f (2nxk)− 2(1− n2) f ((1+ 2n)xk))‖k ≤ ǫ.

(17)



T. Xu, J. Rassias, W. Xu / Eur. J. Pure Appl. Math, 3 (2010), 1032-1047 1038

Again replacing x i and yi by (1− 2n)x i and x i(i ∈ Nk) in (8), respectively, we obtain

sup
k∈N
‖( f ((1− 3n)x1) + f ((1− n)x1)− n2 f (2(1− n)x1) + n2 f (2nx1)

−2(1− n2) f ((1− 2n)x1), . . . , f ((1− 3n)xk) + f ((1− n)xk)

−n2 f (2(1− n)xk) + n2 f (2nxk)− 2(1− n2) f ((1− 2n)xk))‖k ≤ ǫ.

(18)

By (17) and (18), we have

sup
k∈N
‖( f ((1+ 3n)x1) + f ((1− 3n)x1) + f ((1+ n)x1) + f ((1− n)x1)

−n2 f (2(1+ n)x1)− n2 f (2(1− n)x1)− 2(1− n2) f ((1+ 2n)x1)

−2(1− n2) f ((1− 2n)x1), . . . , f ((1+ 3n)xk) + f ((1− 3n)xk)

+ f ((1+ n)xk) + f ((1− n)xk)− n2 f (2(1+ n)xk)− n2 f (2(1− n)xk)

−2(1− n2) f ((1+ 2n)xk)− 2(1− n2) f ((1− 2n)xk))‖k ≤ 2ǫ.

(19)

By (9), (11), (12) and (16), we get

sup
k∈N
‖( f (3x1)− 4 f (2x1) + 5 f (x1), . . . , f (3xk)− 4 f (2xk) + 5 f (xk))‖k ≤

3n2 + 1

n4 − n2
ǫ. (20)

By (9), (10), (12), (13) and (19), we get

sup
k∈N
‖( f (4x1)− 2 f (3x1)− 2 f (2x1) + 6 f (x1), . . . , f (4xk)− 2 f (3xk)

−2 f (2xk) + 6 f (xk))‖k ≤
3n2+2

n4−n2 ǫ.
(21)

By (20) and (21), we get

sup
k∈N
‖( f (4x1)− 10 f (2x1)+ 16 f (x1), . . . , f (4xk)− 10 f (2xk)+ 16 f (xk))‖k ≤

9n2+ 4

n4 − n2
ǫ. (22)

Consider the set Ω := {g | g : E→ F, g(0) = 0} and introduce the generalized metric on Ω,

d(g,h) = inf{α > 0| sup
k∈N
‖(g(x1)− h(x1), . . . , g(xk)− h(xk))‖k ≤ α,∀x1, . . . , xk ∈ E, k ∈ N}.

It is easy to show that (Ω, d) is a generalized complete metric space [see 20, Lemma 2.1].

Define J : Ω → Ω by J g(x) = g(2x)/2 for all x ∈ E. Let g,h ∈ Ω be given such that

d(g,h)< β , by the definition,

sup
k∈N
‖(g(x1)− h(x1), . . . , g(xk)− h(xk))‖k ≤ β for all x1, . . . , xk ∈ E, k ∈ N.

Hence
sup
k∈N
‖(J g(x1)− Jh(x1), . . . , J g(xk)− Jh(xk))‖k

≤ 1

2
sup
k∈N
‖(g(2x1)− h(2x1), . . . , g(2xk)− h(2xk))‖k ≤

β

2



T. Xu, J. Rassias, W. Xu / Eur. J. Pure Appl. Math, 3 (2010), 1032-1047 1039

for all x1, . . . , xk ∈ E, k ∈ N. By definition, d(J g, Jh) ≤ β/2. Therefore, d(J g, Jh) ≤ 1

2
d(g,h)

for all g,h ∈ Ω. This means that J is a strictly contractive self-mapping of Ω with Lipschitz

constant 1/2.

Now, let f̃ : E → F be the mapping defined by f̃ (x) := f (2x)− 8 f (x) for each x ∈ E. By

(22), we get

sup
k∈N
‖( f̃ (2x1)− 2 f̃ (x1), . . . , f̃ (2xk)− 2 f̃ (xk))‖k ≤

9n2+ 4

n4 − n2
ǫ. (23)

Multiplying (23) by 1/2, we obtain

sup
k∈N
‖(J f̃ (x1)− f̃ (x1), . . . , J f̃ (xk)− f̃ (xk))‖k ≤

9n2+ 4

2(n4− n2)
ǫ. (24)

Then d(J f̃ , f̃ ) ≤ (9n2 + 4)/(2(n4− n2))ǫ and therefore, by Theorem 1, J has a unique fixed

point A : E→ F in the set ∆= {h ∈ Ω : d( f̃ ,h) <∞}. This implies that A(2x) = 2A(x) and

A(x) = lim
m→∞

J m f̃ (x) = lim
m→∞

1

2m
f̃ (2mx) (25)

for all x ∈ E. Since f̃ : E→ F is odd, A : E→ F is an odd mapping. Moreover,

d( f̃ ,A)≤
1

1− L
d( f̃ , J f̃ )≤

9n2+ 4

n4− n2
ǫ.

This implies that the inequality (7) holds. Also we have

‖DA(x , y)‖= lim
m→∞

1

2m
‖D f (2m+1 x , 2m+1 y)− 8D f (2mx , 2m y)‖ ≤ lim

m→∞

9ǫ

2m
= 0,

and A satisfies (5). By Theorem 2.2 of [33], the function x → A(2x)−8A(x) is additive. Hence

A(2x) = 2A(x) implies that A is an additive mapping.

If T is another additive mapping satisfying (7). Then T is a fixed point of J in∆. However,

by Theorem 1, J has only one fixed point in ∆, hence A= T . This completes the proof.

Theorem 3. Let E be a linear space and let ((F k,‖ · ‖k) : k ∈ N) be a multi-Banach space.

Suppose that ǫ ≥ 0 and f : E→ F is an odd mapping satisfying

sup
k∈N
‖(D f (x1, y1), . . . , D f (xk, yk))‖k ≤ ǫ

for all x1, . . . , xk, y1, . . . , yk ∈ E. Then there exists a unique cubic mapping C : E→ F such that

sup
k∈N
‖( f (2x1)− 2 f (x1)− C(x1), . . . , f (2xk)− 2 f (xk)− C(xk))‖k ≤

9n2+ 4

7(n4− n2)
ǫ

for all x1, . . . , xk ∈ E.



T. Xu, J. Rassias, W. Xu / Eur. J. Pure Appl. Math, 3 (2010), 1032-1047 1040

Proof. The proof is similar to that of Theorem 2.

Theorem 4. Let E be a linear space and let ((F k,‖ · ‖k) : k ∈ N) be a multi-Banach space.

Suppose that ǫ ≥ 0 and f : E→ F is an even mapping with f (0) = 0, satisfying condition

sup
k∈N
‖(D f (x1, y1), . . . , D f (xk, yk))‖k ≤ ǫ (26)

for all x1, . . . , xk, y1, . . . , yk ∈ E. Then there exists a unique quadratic mapping B : E → F such

that

sup
k∈N
‖( f (2x1)− 16 f (x1)− B(x1), . . . , f (2xk)− 16 f (xk)− B(xk))‖k ≤

8n2+ 2

n4 − n2
ǫ (27)

for all x1, . . . , xk ∈ E.

Proof. Let x1, . . . , xk, y1, . . . , yk ∈ E. Using the evenness of f and from (26), we have

sup
k∈N
‖( f (x1+ ny1) + f (x1− ny1)− n2 f (x1+ y1)− n2 f (x1− y1)− 2(1− n2) f (x1)

− n4−n2

12
[2 f (2y1)− 8 f (y1)], . . . , f (xk+ nyk) + f (xk − nyk)− n2 f (xk + yk)

−n2 f (xk− yk)− 2(1− n2) f (xk)−
n4−n2

12
[2 f (2yk)− 8 f (yk)])‖k ≤ ǫ.

(28)

Interchanging x i and yi(i ∈ Nk) in (28), we get

sup
k∈N
‖( f (nx1+ y1) + f (nx1− y1)− n2 f (x1+ y1)− n2 f (x1− y1)− 2(1− n2) f (y1)

− n4−n2

12
[2 f (2x1)− 8 f (x1)], . . . , f (nxk + yk) + f (nxk− yk)− n2 f (xk+ yk)

−n2 f (xk− yk)− 2(1− n2) f (yk)−
n4−n2

12
[2 f (2xk)− 8 f (xk)])‖k ≤ ǫ.

(29)

Letting yi = 0(i ∈ Nk) in (29), we get

sup
k∈N
‖(2 f (nx1)− 2n2 f (x1)−

n4−n2

12
[2 f (2x1)− 8 f (x1)], . . . ,

2 f (nxk)− 2n2 f (xk)−
n4−n2

12
[2 f (2xk)− 8 f (xk)])‖k ≤ ǫ.

(30)

Putting yi = x i(i ∈ Nk) in (29), we have

sup
k∈N
‖( f ((n+ 1)x1) + f ((n− 1)x1)− n2 f (2x1)− 2(1− n2) f (x1)

− n4−n2

12
[2 f (2x1)− 8 f (x1)], . . . , f ((n+ 1)xk) + f ((n− 1)xk)

−n2 f (2xk)− 2(1− n2) f (xk)−
n4−n2

12
[2 f (2xk)− 8 f (xk)])‖k ≤ ǫ.

(31)

Replacing x i by 2x i(i ∈ Nk) in (30), we get

sup
k∈N
‖(2 f (2nx1)− 2n2 f (2x1)−

n4−n2

12
[2 f (4x1)− 8 f (2x1)], . . . ,

2 f (2nxk)− 2n2 f (2xk)−
n4−n2

12
[2 f (4xk)− 8 f (2xk)])‖k ≤ ǫ.

(32)



T. Xu, J. Rassias, W. Xu / Eur. J. Pure Appl. Math, 3 (2010), 1032-1047 1041

Letting yi = nx i(i ∈ Nk) in (29), we get

sup
k∈N
‖( f (2nx1)− n2 f ((1+ n)x1)− n2 f ((1− n)x1)− 2(1− n2) f (nx1)

− n4−n2

12
[2 f (2x1)− 8 f (x1)], . . . , f (2nxk)− n2 f ((1+ n)xk)

−n2 f ((1− n)xk)− 2(1− n2) f (nxk)−
n4−n2

12
[2 f (2xk)− 8 f (xk)])‖k ≤ ǫ.

(33)

By (30)-(33), we obtain

sup
k∈N
‖( f (4x1)−20 f (2x1)+64 f (x1), . . . , f (4x1)−20 f (2x1)+64 f (x1))‖k ≤

24n2+ 6

n4− n2
ǫ. (34)

Consider the set Ω := {g | g : E→ F, g(0) = 0} and introduce the generalized metric on Ω,

d(g,h) = inf{α > 0| sup
k∈N
‖(g(x1)− h(x1), . . . , g(xk)− h(xk))‖k ≤ α,∀x1, . . . , xk ∈ E, k ∈ N}.

It is easy to show that (Ω, d) is a generalized complete metric space [see 20, Lemma 2.1].

Define J : Ω → Ω by J g(x) = g(2x)/4 for all x ∈ E. Let g,h ∈ Ω be given such that

d(g,h)< β , by the definition,

sup
k∈N
‖(g(x1)− h(x1), . . . , g(xk)− h(xk))‖k ≤ β for all x1, . . . , xk ∈ E, k ∈ N.

Hence
sup
k∈N
‖(J g(x1)− Jh(x1), . . . , J g(xk)− Jh(xk))‖k

≤ 1

4
sup
k∈N
‖(g(2x1)− h(2x1), . . . , g(2xk)− h(2xk))‖k ≤

β

4

for all x1, . . . , xk ∈ E, k ∈ N. By definition, d(J g, Jh) ≤ β/4. Therefore, d(J g, Jh) ≤ 1

4
d(g,h)

for all g,h ∈ Ω. This means that J is a strictly contractive self-mapping of Ω with Lipschitz

constant 1/4.

Now, let f̃ : E→ F be the mapping defined by f̃ (x) := f (2x)−16 f (x) for each x ∈ E. By

(34), we get

sup
k∈N
‖( f̃ (2x1)− 4 f̃ (x1), . . . , f̃ (2xk)− 4 f̃ (xk))‖k ≤

24n2+ 6

n4− n2
ǫ. (35)

Multiplying (35) by 1/4, we obtain

sup
k∈N
‖(J f̃ (x1)− f̃ (x1), . . . , J f̃ (xk)− f̃ (xk))‖k ≤

24n2+ 6

4(n4− n2)
ǫ. (36)

Then d(J f̃ , f̃ )≤ ǫ(24n2+ 6)/(4(n4− n2)) and therefore, by Theorem 1, J has a unique fixed

point B : E→ F in the set ∆= {h ∈ Ω : d( f̃ ,h) <∞}. This implies that B(2x) = 4B(x) and

B(x) = lim
m→∞

J m f̃ (x) = lim
m→∞

1

4m
f̃ (2mx) (37)



T. Xu, J. Rassias, W. Xu / Eur. J. Pure Appl. Math, 3 (2010), 1032-1047 1042

for all x ∈ E. Since f̃ : E→ F is even, B : E→ F is an even mapping. Moreover,

d( f̃ ,A)≤
1

1− L
d( f̃ , J f̃ )≤

8n2+ 2

n4− n2
ǫ.

This implies that the inequality (27) holds. Also we have

‖DB(x , y)‖ = lim
m→∞

1

4m ‖D f (2m+1 x , 2m+1 y)− 16D f (2mx , 2m y)‖

≤ lim
m→∞

17ǫ

4m = 0,

and B satisfies (5). By Theorem 2.1 of [33], the mapping x → B(2x)− 16B(x) is quadratic.

Hence B(2x) = 4B(x) implies that B is a quadratic mapping.

The rest of the proof is similar to that of Theorem 2.

Theorem 5. Let E be a linear space and let ((F k,‖ · ‖k) : k ∈ N) be a multi-Banach space.

Suppose that ǫ ≥ 0 and f : E→ F is an even mapping with f (0) = 0, satisfying condition

sup
k∈N
‖(D f (x1, y1), . . . , D f (xk, yk))‖k ≤ ǫ

for all x1, . . . , xk, y1, . . . , yk ∈ E. Then there exists a unique quartic mapping Q : E → F such

that

sup
k∈N
‖( f (2x1)− 4 f (x1)−Q(x1), . . . , f (2xk)− 4 f (xk)−Q(xk))‖k ≤

8n2+ 2

5(n4− n2)
ǫ

for all x1, . . . , xk ∈ E.

Proof. The proof is similar to that of Theorem 4.

Theorem 6. Let E be a linear space and let ((F k,‖ · ‖k) : k ∈ N) be a multi-Banach space.

Suppose that ǫ ≥ 0 and f : E→ F is an odd mapping satisfying

sup
k∈N
‖(D f (x1, y1), . . . , D f (xk, yk))‖k ≤ ǫ (38)

for all x1, . . . , xk, y1, . . . , yk ∈ E. Then there exist a unique additive mapping A : E → F and a

unique cubic mapping C : E→ F such that

sup
k∈N
‖( f (x1)− A(x1)− C(x1), . . . , f (xk)− A(xk)− C(xk))‖k ≤

4(9n2+ 4)

21(n4− n2)
ǫ (39)

for all x1, . . . , xk ∈ E.

Proof. By Theorems 2 and 3, there exist a unique additive mapping A0 : E → F and a

unique cubic mapping C0 : E→ F such that

sup
k∈N
‖( f (2x1)− 8 f (x1)− A0(x1), . . . , f (2xk)− 8 f (xk)− A0(xk))‖k ≤

9n2+ 4

n4 − n2
ǫ (40)



T. Xu, J. Rassias, W. Xu / Eur. J. Pure Appl. Math, 3 (2010), 1032-1047 1043

and

sup
k∈N
‖( f (2x1)− 2 f (x1)− C0(x1), . . . , f (2xk)− 2 f (xk)− C0(xk))‖k ≤

9n2+ 4

7(n4− n2)
ǫ (41)

for all x1, . . . , xk ∈ E. Now from (40) and (41), one can see that

sup
k∈N
‖(6 f (x1) + A0(x1)− C0(x1), . . . , 6 f (xk) + A0(xk)− C0(xk))‖k ≤

8(9n2+ 4)

7(n4− n2)
ǫ

for all x1, . . . , xk ∈ E. Thus we obtain (39) by defining A(x) = −A0(x)/6 and C(x) = C0(x)/6.

The uniqueness of A and C is easy to show.

Theorem 7. Let E be a linear space and let ((F k,‖ · ‖k) : k ∈ N) be a multi-Banach space.

Suppose that ǫ ≥ 0 and f : E→ F is an even mapping with f (0) = 0, satisfying condition

sup
k∈N
‖(D f (x1, y1), . . . , D f (xk, yk))‖k ≤ ǫ (42)

for all x1, . . . , xk, y1, . . . , yk ∈ E. Then there exist a unique quadratic mapping B : E → F and a

unique quartic mapping Q : E→ F such that

sup
k∈N
‖( f (x1)− B(x1)−Q(x1), . . . , f (xk)− B(xk)−Q(xk))‖k ≤

4n2+ 1

5(n4− n2)
ǫ (43)

for all x1, . . . , xk ∈ E.

Proof. By Theorems 4 and 5, there exist a unique quadratic mapping B0 : E → F and a

unique quartic mapping Q0 : E→ F such that

sup
k∈N
‖( f (2x1)− 16 f (x1)− B0(x1), . . . , f (2xk)− 16 f (xk)− B0(xk))‖k ≤

8n2+ 2

n4 − n2
ǫ (44)

and

sup
k∈N
‖( f (2x1)− 4 f (x1)−Q0(x1), . . . , f (2xk)− 4 f (xk)−Q0(xk))‖k ≤

8n2+ 2

5(n4− n2)
ǫ (45)

for all x1, . . . , xk ∈ E. Now from (44) and (45), one can see that

sup
k∈N
‖(12 f (x1) + B0(x1)−Q0(x1), . . . , 12 f (xk) + B0(xk)−Q0(xk))‖k ≤

6(8n2+ 2)

5(n4− n2)
ǫ

for all x1, . . . , xk ∈ E. Thus we obtain (43) by defining B(x) = −B0(x)/12 and

Q(x) = Q0(x)/12. The uniqueness of B and Q is easy to show.



T. Xu, J. Rassias, W. Xu / Eur. J. Pure Appl. Math, 3 (2010), 1032-1047 1044

Theorem 8. Let E be a linear space and let ((F k,‖ · ‖k) : k ∈ N) be a multi-Banach space.

Suppose that ǫ ≥ 0 and f : E→ F is a mapping with f (0) = 0, satisfying condition

sup
k∈N
‖(D f (x1, y1), . . . , D f (xk, yk))‖k ≤ ǫ (46)

for all x1, . . . , xk, y1, . . . , yk ∈ E. Then there exist a unique additive mapping A : E→ F, a unique

quadratic mapping B : E→ F, a unique cubic mapping C : E→ F, and a unique quartic mapping

Q : E→ F such that

sup
k∈N
‖( f (x1)− A(x1)− B(x1)− C(x1)−Q(x1), . . . ,

f (xk)− A(xk)− B(xk)− C(xk)−Q(xk))‖k ≤
164n2+101

105(n4−n2)
ǫ

(47)

for all x1, . . . , xk ∈ E.

Proof. Let fo(x) =
1

2
[ f (x)− f (−x)] for all x ∈ E. Then fo(0) = 0, fo(x) = − fo(−x) .

Hence

sup
k∈N
‖(D fo(x1, y1), . . . , D fo(xk, yk))‖k ≤ ǫ

for all x1, . . . , xk, y1, . . . , yk ∈ E. By Theorem 6, there exist a unique additive mapping

A : E→ F and a unique cubic mapping C : E→ F such that

sup
k∈N
‖( f (x1)− A(x1)− C(x1), . . . , f (xk)− A(xk)− C(xk))‖k ≤

4(9n2+ 4)

21(n4− n2)
ǫ (48)

for all x1, . . . , xk ∈ E. Let fe(x) =
1

2
[ f (x)+ f (−x)] for all x ∈ E. Then

fe(0) = 0, fe(x) = fe(−x) and

sup
k∈N
‖(D fe(x1, y1), . . . , D fe(xk, yk))‖k ≤ ǫ

for all x1, . . . , xk, y1, . . . , yk ∈ E. By Theorem 7, there exist a unique quadratic mapping

B : E→ F and a unique quartic mapping Q : E→ F such that

sup
k∈N
‖( f (x1)− B(x1)−Q(x1), . . . , f (xk)− B(xk)−Q(xk))‖k ≤

4n2+ 1

5(n4− n2)
ǫ (49)

for all x1, . . . , xk ∈ E. By (48) and (49), we get (47). This completes the proof.

ACKNOWLEDGEMENTS The first author was supported by the National Natural Science

Foundation of China (Grant No. 10671013, 60972089).



REFERENCES 1045

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