We investigate the Hyers-Ulam stability of the quadratic functional equation on restricted domains. Applying these results, we study of an asymptotic behavior of these quadratic mappings.
1. Introduction
The question concerning the stability of group homomorphisms was posed by Ulam [1]. Hyers [2] solved the case of approximately additive mappings on Banach spaces. Aoki [3] provided a generalization of the Hyers' theorem for additive mappings. In [4], Rassias generalized the result of Hyers for linear mappings by allowing the Cauchy
difference to be unbounded (see also [5]). The result of Rassias has been generalized by G
vruţa [6] who permitted the norm of the Cauchy difference 
to be bounded by a general control function under some conditions. This stability
concept is also applied to the case of various functional equations by a number of
authors. For more results on the stability of functional equations, see [7–32]. We also refer the readers to the books [33–37].
It is easy to see that the function 
defined by 
with 
an arbitrary constant is a solution of the functional equation
(11)So, it is natural that each equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to
be a quadratic function. It is well known that a function 
between real vector spaces 
and 
is quadratic if and only if there exists a unique symmetric biadditive function 
such that 
for all 
(see [21, 33, 35]).
A stability theorem for the quadratic functional equation (1.1) was proved by Skof
[38] for functions 
, where 
is a normed space and 
is a Banach space. Cholewa [11] noticed that the result of Skof holds (with the same proof) if 
is replaced by an abelian group 
. In [12], Czerwik generalized the result of Skof by allowing growth of the form 
for the norm of 
, where 
and 
. In 1998, Jung [39] investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted
domains (see also [40–42]). Rassias [43] investigated the Hyers-Ulam stability of mixed type mappings on restricted domains.
In [44], the authors considered the asymptoticity of Hyers-Ulam stability close to the asymptotic
derivability.
2. Stability of (1.1) on Restricted Domains
In this section, we investigate the Hyers-Ulam stability of the functional equation (1.1) on a restricted domain. As an application, we use the result to the study of an asymptotic behavior of that equation.
Theorem 2.1.
Given a real normed vector space 
and a real Banach space 
, let 
and 
with 
be fixed. If a mapping 
satisfies the inequality
(21)for all 
such that 
, where 
, then there exists a unique quadratic mapping 
such that
(22)for all 
with 
and 
. Moreover, if 
is measurable or if 
is continuous in 
for each fixed 
, then 
for all 
and 
.
Proof.
Letting 
in (2.1), we get
(23)for all 
with 
. If we put 
with 
and 
in (2.1), we obtain
(24)It follows from (2.3) and (2.4) that
(25)for all 
with 
. Replacing 
by 
in (2.5), we infer the inequality
(26)for all 
with 
and all integers 
. Therefore,
(27)for all 
with 
and all integers 
. It follows from (2.7) that the sequence 
converges for all 
with 
. Let us denote 
for all 
with 
. It is clear that
(28)for all 
with 
. Letting 
and 
in (2.7), we get
(29)for all 
with 
.
Now, suppose that 
such that 
, then by (2.1) and the definition of 
, we obtain
(210)We have to extend the mapping 
to the whole space 
. Given any 
with 
, let 
denote the largest integer such that 
. Consider the mapping 
defined by 
and
(211)Let 
with 
and let 
. We have two cases.
Case 1.
If 
, we have from (2.8) that
(212)Case 2.
If 
, then 
is the largest integer satisfying 
, and we have
(213)Therefore, 
for all 
with 
. From the definition of 
and (2.8), it follows that 
for all 
. Now, suppose that 
with 
and choose a positive integer 
such that 
. By the definition of 
and its property, we have
(214)So by the definition of 
, we have
(215)for all 
with 
. Since 
, (2.15) holds true for 
. Let 
with 
. It follows from (2.1) and (2.15) that
(216)Letting 
in (2.16), we get 
for all 
with 
. Since 
, the same is true for 
. So, 
is even and this implies that (2.16) is true for all 
. Therefore, 
is quadratic. By the definition 
when 
, thus (2.2) follows from (2.9). To prove the uniqueness of 
, let 
be another quadratic mapping satisfying (2.2) for all 
. Let 
with 
and choose a positive integer 
such that 
, then
(217)for all 
. Since 
and 
are quadratic, we get
(218)for all 
. Therefore, 
. Since 
, we have 
for all 
. The proof of our last assertion follows from the proof of Theorem 1 in [12].
We now introduce one of the fundamental results of fixed point theory by Margolis and Diaz.
Theorem 2.2 (see [22]).
Let 
be a complete generalized metric space and let 
be a strictly contractive mapping with Lipschitz constant 
. If there exists a nonnegative integer 
such that 
for some 
, then the following are true:
(1)the sequence 
converges to a fixed point 
of 
,
(2)
is the unique fixed point of 
in
(219)(3)
for all 
.
By using the idea of Cădariu and Radu [45], we applied a fixed point method to the investigation of the generalized Hyers-Ulam stability of the functional equation (1.1) on a restricted domain.
Theorem 2.3.
Given a real normed vector space 
and a real Banach space 
, let 
be fixed and let 
be a mapping which satisfies the inequality (2.1) for all 
, where 
is a function such that
(220)for all 
, where 
is a constant number, then there exists a unique quadratic mapping 
such that
(221)for all 
with 
, where
(222)and 
for all 
. Moreover, if 
is measurable or if 
is continuous in 
for each fixed 
, then 
for all 
and 
.
Proof.
It follows from (2.20) that
(223)for all 
. Let 
. Letting 
for 
in (2.1), we get the following inequalities:
(224)
(225)
(226)
(227)It follows from (2.24) and (2.25) that
(228)By (2.26) and (2.27), we have
(229)It follows from (2.25) and (2.29) that
(230)Using (2.28) and (2.30), we have
(231)By (2.24), we get
(232)Hence, we obtain from (2.31) and (2.32) that
(233)So, it follows from (2.28) and (2.33) that
(234)for all 
. Let 
. We introduce a generalized metric on 
as follows:
(235)We assert that 
is a generalized complete metric space. Let 
be a Cauchy sequence in 
and 
be given, then there exists an integer 
such that 
for all 
. This implies that 
for all 
and all 
. Therefore, 
is a Cauchy sequence in 
for all 
. Since 
is a Banach space, 
converges for all 
. Thus, we can define a function 
by
(236)Since
(237)for all 
and all 
, we get 
for all 
. That is, the Cauchy sequence 
converges to 
in 
. Hence, 
is complete. We now consider the mapping 
defined by
(238)Let 
and let 
be an arbitrary constant with 
. From the definition of 
, we have
(239)for all 
. By the assumption (2.20) and the last inequality, we have
(240)for all 
. So 
. That is, 
is a strictly contractive on 
. It follows from (2.34) that 
. Therefore, according to Theorem 2.2, there exists a function 
such that the sequence 
converges to 
and 
. Indeed,
(241)and 
, for all 
. Also, 
is the unique fixed point of 
in the set 
and
(242)By (2.1), (2.23) and using the definition of 
, we get
(243)for all 
. We will define a mapping 
such that 
. Similar to the proof of Theorem 2.1 for a given 
with 
, let 
denote the largest integer such that 
. Consider the mapping 
defined by 
and
(244)Let 
with 
and let 
. We have two cases.
Case 1.
. Since 
for all 
, we have
(245)Case 2.
If 
, then 
is the largest integer satisfying 
, and we have
(246)Therefore, 
for all 
with 
. Using 
for all 
and the definition of 
, we get that 
for all 
. Now, suppose that 
with 
and choose a positive integer 
such that 
. By the definition of 
and its property, we have
(247)So by the definition of 
, we have
(248)for all 
with 
. Since 
, (2.48) holds true for 
. Let 
with 
, 
, 
. It follows from (2.1), (2.23), and (2.48) that
(249)Since 
and 
for all 
, we conclude that (2.49) is true for all 
. Let 
with 
. Putting 
in (2.49), we get 
. Therefore, by letting 
in (2.49), we get 
for all 
with 
. Since 
, the same is true for 
. So, 
is even and this implies that (2.49) is true for all 
. Therefore, 
is quadratic. To prove the uniqueness of 
, let 
be another quadratic mapping satisfying (2.21), for all 
. Let 
with 
and choose a positive integer 
such that 
, then
(250)for all 
. Since 
and 
are quadratic, we get
(251)for all 
. Therefore, (2.23) implies that 
. Since 
, we have 
for all 
. Our last assertion is trivial in view of Theorem 2.1.
Corollary 2.4.
Given a real normed vector space 
and a real Banach space 
, let 
and 
with 
be fixed. Suppose that a mapping 
satisfies the inequality (2.1) for all 
, then there exists a unique quadratic mapping 
such that
(252)for all 
with 
and 
. Moreover, if 
is measurable or if 
is continuous in 
for each fixed 
, then 
for all 
and 
.
Remark 2.5.
We may replace the condition (2.20) by
(253)for all 
and 
. Using the direct method, there exists a unique quadratic mapping 
such that
(254)for all 
with 
. For the case 
, where 
and 
, we have
(255)Using ideas from the papers [39, 43], we prove the generalized Hyers-Ulam stability of (1.1) on restricted domains. We first prove the following lemma.
Lemma 2.6.
Given a real normed vector space 
and a real Banach space 
, let 
and 
be fixed. If a mapping 
satisfies the inequality
(256)for all 
with 
, then
(257)for all 
, where
(258)Proof.
Assume that 
. If 
, then we choose a 
with 
. Otherwise, let
(259)It is clear that 
and
(260)Also
(261)Therefore,
(262)So, we get
(263)So, 
satisfies (2.57) for all 
.
Theorem 2.7.
Given a real normed vector space 
and a real Banach space 
, let 
and 
with 
be given. Assume that a mapping 
satisfies the inequality (2.56) for all 
with 
, then there exists a unique quadratic mapping 
such that 
and
(264)for all 
.
Proof.
By Lemma 2.6, 
satisfies (2.57) for all 
. Letting 
in (2.57), we get
(265)for all 
, where
(266)We can use the argument given in the proof of Theorem 2.1 to arrive the inequality
(267)for all 
and all integers 
. It follows from (2.67) that the sequence 
converges for all 
. So, we can define the mapping 
by 
for all 
. Letting 
and 
in (2.67), we get (2.64).
For the case 
and 
in Theorem 2.7, it is obvious that our inequality (2.64) is sharper than the corresponding
inequalities of Jung [39] and Rassias [43].
Skof [38] has proved an asymptotic property of the additive mappings, and Jung [39] has proved an asymptotic property of the quadratic mappings (see also [41]). Using the method in [39], the proof of the following corollary follows from Theorem 2.7 by letting 
and 
.
Corollary 2.8 (see [39]).
Given a real normed vector space 
and a real Banach space 
, a mapping 
satisfies (1.1) if and only if the asymptotic condition
(268)holds true.
3. 
-Asymptotically Quadratic Mappings
We apply our results to the study of 
-asymptotical derivatives. Let 
be a real normed vector space and let 
be a real Banach space 
. Let 
be arbitrary.
Definition 3.1.
A mapping 
is called 
-asymptotically close to a mapping 
if and only if 
.
Definition 3.2.
A mapping 
is called 
-asymptotically derivable if the mapping 
is 
-asymptotically close to a quadratic mapping 
. In this case, we say that 
is a 
-asymptotical derivative of 
.
Definition 3.3.
A mapping 
is called 
-asymptotically quadratic if and only if, for every 
, there exists 
such that
(31)for all 
with 
.
Definition 3.4.
A mapping 
is called quadratic outside a ball if there exists 
such that 
for all 
with 
.
We have the following result.
Theorem 3.5.
If 
is quadratic outside a ball and 
is 
-asymptotically close to 
, then 
is 
-asymptotically quadratic.
The following result follows from Corollary 2.4.
Corollary 3.6.
If 
is quadratic outside a ball and 
is 
-asymptotically close to 
, then 
has a 
-asymptotical derivative.
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