1. Introduction
The stability problem of functional equations originated from a question of Ulam [1] in 1940 concerning the stability of group homomorphisms.
Give a group
and a metric group
with the metric
. Given
, does there exist a
such that if
satisfies
for all
, then there is a homomorphism
with
for all
?
Hyers [2] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’s Theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias has provided a lot of influence in the development of what we call generalized Hyers-Ulam-Rassias stability of functional equations. Beginning around 1980, the stability problems of several functional equations and approximate homomorphisms have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [5] - [18] ).
The functional equation
is called the quadratic functional equation. Every solution of the quadratic functional equation is said to be a quadratic mapping. The Hyers-Ulam stability for quadratic functional equation was first proved by Skof [5] for mappings acting between a normed space and a Banach space. P. W. Cholewa [6] showed that Skof’s Theorem is also valid if the normed space is replaced with an abelian group.
Now we recall some basic facts concerning
-Banach spaces. We fixed real numbers
with
and p with
. Let
or
. Let X be linear space over
. A quasi-β-norm
is a real-valued function on X satisfying the following conditions:
(i)
;
if and only if
;
(ii)
;
(iii) There is a constant
such that
.
The pair
is called a quasi-β-normed space if
is a quasi-β-norm on X. The smallest possible K is called the module of concavity of
. A quasi-β-Banach space is a complete quasi-β-normed space.
A quasi-β-norm
is called a
-norm if
for all
. In this case, a quasi-
-Banach space is called a
-Banach space. For more details and related stability results on
-Banach spaces, we refer to [19] [20] . Recently, L. Gǎvruta and P. Gǎvruta [21] studied the approximation of functions in Banach space. In this paper, we will consider this problem in
-Banach spaces and extend previous result for quadratic functional equations.
2. Main Results
Given
and
. Throughout this paper we always assume that X is a linear space, Y is a
-Banach space and
is a mapping.
Definition 2.1. Let
be a mapping. We say f is Φ-approximable by a quadratic map if there exists a quadratic mapping
such that
(1)
for all
. In this case, we say that Q is the quadratic Φ-approximation of f.
The following result is our main result in this paper.
Theorem 2.2. Let
and suppose
. Then f is Φ-approximable by a quadratic map if and only if the following two condition hold:
(i)
,
;
(ii) There exists
such that
In this case, the quadratic Φ-approximation of f is unique and is given by
for all
.
Proof. We first assume that f is Φ-approximable by a quadratic map. Then for
, we have
and
It follows that
for all
. Hence
for all
. By letting
, we obtain condition (i) since
. Since Q is quadratic, we have
for all
. We take
in the first position, then for all
, we have
and the condition (ii) holds.
Conversely we suppose that (i) and (ii) hold. It follows from condition (ii) that for all
, we have
(2)
Then
is a Cauchy sequence. Indeed, by using
replace x, we get
and by multipling
, for all
, we have
Hence, for all
,
as
. Since Y is a
-Banach space, the limit
exists. Let
in relation (2), we get condition (1).
Now we show that Q satisfies the required conditions. From the hypothesis, for all
,
Hence for all
,
Therefore
and Q is a quadratic map. Now we show the uniqueness of Q. We suppose that Q satisfies
for all
and there exists a
satisfying
Since Q and
are quadratic mappings, we have
for all
. Hence for all
,
Since
, for all
, we have
Hence for all
,
. This completes the proof. ,
Corollary 2.3. Let
be a mapping satisfying
and
for all
where
. Suppose
a function with
and satisfying
(3)
for all
. Then there exists a unique quadratic function
such that
which is defined
for all
.
Proof. Replace x and y by
in (3), we have
Dividing by
, we have
(4)
Replacing x by
in (4), we get
(5)
Then we have
for all
. We claim that
(6)
holds for all
and
. When
, this is obviously by (4). Suppose (6) holds when
, i.e. for all
,
Then for
, we have
for all
. By induction, (6) is true for all
and
. Replacing
by
in (3) and multiplying both side by
, we have
Since
we have
for all
. Hence for all
,
It follows from Theorem 2.2 (with
there) that there exists a unique quadratic function Q such that
for all
. ,
Theorem 2.4. Let
. Suppose
. Then f is Φ-approximable by a quadratic map if and only if the following two condition
(i)
;
(ii) There exists a
such that
hold for all
. In this case, the quadratic Φ-approximation of f is unique and is given by
Proof. The proof is similar to that of Theorem 2.2 and we omit it. ,
Corollary 2.5. Let
be a mapping such that
for all
. Let
. Suppose
all
. Let
a function with
and satisfying
for all
. Then there exists a unique quadratic function
such that
for all
.
Proof. The proof is similar to that of Corollary 2.3 and we omit it. ,
Funding
This article is partially supported by NSFC (11871303 and 11671133) and NSF of Shandong Province (ZR2019MA039).