Approximate Cauchy functional inequality in quasi-Banach spaces

Kim Hark-Mahn, Son, Son Eunyoung

Description

In this article, we prove the generalized Hyers-Ulam stability of the following Cauchy functional inequality: | | f ( x ) + f ( y ) + n f ( z ) | | ≤ n f x + y n + x in the class of mappings from n -divisible abelian groups to p -Banach spaces for any fixed positive integer n ≥ 2. In this article, we prove the generalized Hyers-Ulam stability of the following Cauchy functional inequality: | | f ( x ) + f ( y ) + n f ( z ) | | ≤ n f x + y n + x in the class of mappings from n -divisible abelian groups to p -Banach spaces for any fixed positive integer n ≥ 2.

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 5
• Langue: English

Extrait

Kim and Son Journal of Inequalities and Applications 2011, 2011:102
http://www.journalofinequalitiesandapplications.com/content/2011/1/102
RESEARCH Open Access
Approximate Cauchy functional inequality in
quasi-Banach spaces
*Hark-Mahn Kim and Eunyoung Son
* Correspondence: Abstract
[e-mail]
Department of Mathematics, In this article, we prove the generalized Hyers-Ulam stability of the following Cauchy
Chungnam National University, 79 functional inequality:
Daehangno, Yuseong-gu, Daejeon x +y305-764, Korea
||f(x)+f(y)+nf(z)|| ≤ nf +x
n
in the class of mappings from n-divisible abelian groups to p-Banach spaces for any
fixed positive integer n ≥ 2.
1 Introduction
The stability problem of functional equations originated from a question of Ulam [1]
concerning the stability of group homomorphisms.
We are given a group G and a metric group G with metric r (·,·). Given >0, does1 2
there exist a δ>0 such that if f : G ® G satisfies r(f(xy),f(x)f(y)) <δ for all x,y Î G ,1 2 1
then a homomorphism h : G ®G exists with r(f(x), h(x)) < for all x G ?1 2 1
In other words, we are looking for situations when the homomorphisms are stable, i.
e., if a mapping is almost a homomorphism, then there exists a true homomorphism
near it.
In 1941, Hyers [2] considered the case of approximately additive mappings between
Banach spaces and proved the following result. Suppose that E and E are Banach1 2
spaces and f : E ® E satisfies the following condition: there is a constant ≥ 0 such1 2
that
|f(x +y) −f(x) − (y)|| ≤ ε
nf(2 x)
for all x,y Î E . Then, the limit exists for all xÎ E , and it is a1 h(x) = lim 1n→∞ n2
unique additive mapping h:E ®E such that ||f(x)-h(x)|| ≤ .1 2
The method which was provided by Hyers, and which produces the additive mapping
h, was called a direct method. This method is the most important and most powerful
tool for studying the stability of various functional equations. Hyers’ theorem was
generalized by Aoki [3] and Bourgin [4] for additive mappings by considering an
unbounded Cauchy difference. In 1978, Rassias [5] also provided a generalization of
Hyers’ theorem for linear mappings which allows the Cauchy difference to be
p p
unbounded like this ||x|| +||y||.Let E and E be two Banach spaces and f : E ®1 2 1
E be a mapping such that f(tx) is continuous in t Î R for each fixed x.Assumethat2
© 2011 Kim and Son; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.Kim and Son Journal of Inequalities and Applications 2011, 2011:102 Page 2 of 11
http://www.journalofinequalitiesandapplications.com/content/2011/1/102
there exists > 0 and 0 ≤ p < 1 such that
p p||f(x +y) −f(x) −f(y)|| ≤ ε(||x|| + ||y|| ), ∀x,y ∈ E .1
Then, there exists a unique R-linear mapping T : E ® E such that1 2
2 p||f(x) −T(x)|| ≤ ||x||
p2 − 2
for all x Î E . A generalized result of Rassias’ theorem was obtained by Găvruta in1
[6] and Jung in [7]. In 1990, Rassias [8] during the 27th International Symposium on
Functional Equations asked the question whether such a theorem can also be proved
for p ≥ 1. In 1991, Gajda [9], following the same approach as in [5], gave an affirmative
solution to this question for p > 1. It was shown by Gajda [9], as well as by Rassias and
[001]emrl [10], that one cannot prove a Rassias’ type theorem when p = 1. The
counterexamples of Gajda [9], as well as of Rassias and [001]emrl [10], have stimulated
several mathematicians to invent new approximately additive or approximately linear
mappings. In particular, Rassias [11,12] proved a similar stability theorem in which he
p q
replaced the unbounded Cauchy difference by this factor ||x|| ||y|| for p,qÎ R with p
+ q ≠ 1.
Let G be an n-divisible abelian group n Î N (i.e., a ↦ na : G ® G is a surjection )
and X be a normed space with norm || · ||. Now, for a mapping f : G ®
X,weconsider the following generalized Cauchy-Jensen equation
x +y
f(x)+f(y)+nf(z)= nf +z , n ≥ 2
n
for all x,y, zÎ G, which has been introduced in [13].
Proposition 1.1. For a mapping f : G® X, the following statements are equivalent.
(a) f is additive,
x +y
(b) f(x)+f(y)+nf(z)= nf +z ,
n x +y
||f(x)+f(y)+nf(z)|| ≤ nf +z(c)
n
for all x, y, zÎ G.
As a special case for n = 2, the generalized Hyers-Ulam stability of functional
equation (b) and functional inequality (c) has been presented in [13]. We remark that there
are some interesting papers concerning the stability of functional inequalities and the
stability of functional equations in quasi-Banach spaces [14-18]. In this article, we are
going to improve the theorems given in [13] without using the oddness of approximate
additive functions concerning the functional inequality (c) for a more general case.
2 Generalized Hyers-Ulam stability of (c)
We recall some basic facts concerning quasi-Banach spaces and some preliminary
results. Let X be a real linear space. A quasi-norm is a real-valued function on X
satisfying the following:Kim and Son Journal of Inequalities and Applications 2011, 2011:102 Page 3 of 11
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(1) ||x|| ≥ 0 for all xÎ X and ||x|| = 0 if and only if x=0.
(2) ||lx|| = |l|||x|| for alllÎ R and all xÎ X.
(3) There is a constant M ≥ 1 such that ||x + y|| ≤ M(||x|| + ||y||) for all x,yÎ X.
The pair (X, || · ||) is called a quasi-normed space if || · || is a quasi-norm on X
[19,20]. The smallest possible M is called the modulus of concavity of || · ||. A
quasiBanach space is a complete quasi-normed space.
A quasi-norm || · || is called a p-norm (0 <p ≤ 1) if
p p p||x +y|| ≤||x|| + ||y||
for all x,yÎ X. In this case, a quasi-Banach space is called a p-Banach space.
p
Given a p-norm, the formula d(x,y):=||x - y|| gives us a translation invariant
metric on X. By the Aoki-Rolewicz theorem [20], each quasi-norm is equivalent to
some p-norm (see also [19]). Since it is much easier to work with p-norms, henceforth,
we restrict our attention mainly to p-norms. We observe that if x , x ,..., x are non-1 2 n
negative real numbers, then
pn n
px ≤ x ,i i
i=1 i=1
where 0 <p ≤ 1 [21].
From now on, let G be an n-divisible abelian group for some positive integer n ≥ 2,
and let Y be a p-Banach space with the modulus of concavity M.
Theorem 2.1. Suppose that a mapping f : G® Y with f(0) = 0 satisfies the functional
inequality
x +y
||f(x)+f(y)+nf(z)|| ≤nf +z + ϕ(x,y,z) (1)
n
3 +
for all x, y, zÎ G, and the perturbing function : G ®R satisfies
∞ pi i i ϕ(n x,n y,n z)
(x,y,z):= < ∞
ipn
i=0
for all x,y,zÎ G. Then, there exists a unique additive mapping h : G® Y, defined as
k kf(n x) −f(−n x)
, such thath(x) = lim
kk→∞ 2n
1
2M M (2)p||f(x) −h(x)|| ≤ [(nx,0,−x)+ (−nx,0,x)] + ϕ(x,−x,0)
2n 2
for all xÎ G.
Proof. Let y=-x, z = 0 in (1) and dividing both sides by 2, we have

f(x)+f(−x) ϕ(x,−x,0) ≤ (3) 2 2
for all xÎ G. Replacing x by nx and letting y = 0 and z=-x in (1), we get
||f(nx) +nf(−x)|| ≤ ϕ(nx,0,−x) (4)Kim and Son Journal of Inequalities and Applications 2011, 2011:102 Page 4 of 11
http://www.journalofinequalitiesandapplications.com/content/2011/1/102
for all xÎ G. Replacing x by -x in (4), one has
||f(−nx) +nf(x)|| ≤ ϕ(−nx,0,x) (5)
f(x) −f(−x)
for all xÎ G. Put . Combining (4) and (5) yieldsg(x)=
2
M
||ng(x) −g(nx)|| ≤ (ϕ(nx,0,−x)+ ϕ(−nx,0,x))
2
that is,

1 M
g(x) − g(nx) ≤ (ϕ(nx,0,−x)+ ϕ(−nx,0,x)) (6) n 2n
for all xÎ G. It follows from (6) that
pl m g(nx g(n x) − l mn n
m−1 p 1 1k k+1 ≤ g(n x) − g(n x) k k+1n n
k=l
(7) m−1 p 1 1k k+1 = g(n x) − g(n x) kpn n
k=1
m−1 p M k+1 k p k+1 k p≤ [ϕ(n x,0,−n x) + ϕ(−n x,0,n x) ]
(k+1)p p2 nk=1
for all nonnegative integers m and l withm>l ≥0and x Î G. Since the right-hand
mg(n x
side of (7) tends to zero as l® ∞, we obtain the sequence is Cauchy for all x
mn

mg(n x
Î G. Because of the fact that Y is complete, it follows that the sequence
conmn
verges in Y. Therefore, we can define a function h : G® Y by
m m mg(n x) f(n x) −f(−n x)
h(x) = lim = lim , x ∈ G.
m mm→∞ m→∞n 2n
Moreover, letting l = 0 and taking m® ∞ in (7), we get
1
f(x) −f(−x) M (8)p −h(x) ≤||g(x) −