Now, we consider some basic concepts concerning quasi-
-normed spaces and some preliminary results. We fix a real number
with
and let
denote either
or
. Let
be a linear space over
. A quasi-
-norm
is a real-valued function on
satisfying the following.
(1)
for all
and
if and only if
.
(2)
for all
and all
.
(3) There is a constant
such that
for all
.
The pair
is called a
space if
is a quasi-
-norm on
. The smallest possible
is called the modulus of concavity of
. A
space is a complete quasi-
-normed space. A quasi-
-norm
is called a
-norm
if
for all
. In this case, a quasi-
-Banach space is called a
-Banach space. We can refer to [24, 25] for the concept of quasinormed spaces and
-Banach spaces. Given a
-norm, the formula
gives us a translation invariant metric on
. By the Aoki-Rolewicz theorem [25] (see also [24]), each quasinorm is equivalent to some
-norm. In [26], Tabor has investigated a version of the D. H. Hyers, Th. M. Rassias, and Z. Gajda theorem (see [5, 6]) in quasibanach spaces. Recently, J. M. Rassias and Kim [27] have obtained stability results of general additive equations in quasi-
-normed spaces.
From now on, let
be a quasi-
-normed space with norm
and let
be a
-Banach space with norm
unless we give any specific reference. Now, we are ready to investigate the generalized Hyers-Ulam stability problem for the functional (1.5) using direct method.
Theorem 2.1.
Assume that a function
satisfies
for all
and that
satisfies the following control conditions
for all
. Then there exists a unique quadratic function
satisfying
for all
, where
. The function
is defined as
for all 
Proof.
Putting
in (2.2), we get
. Replacing
by
in (2.2), we obtain
for all
. Dividing (2.6) by
, we get
for all
where
. Now letting
and dividing
in (2.7), we have
for all
. Therefore we prove from the inequality (2.8) that for any integers
with 
Since the right-hand side of (2.9) tends to zero as
, the sequence
is Cauchy for all
and thus converges by the completeness of
. Define
by
Letting
in (2.2), respectively, and dividing both sides by
and after then taking the limit in the resulting inequality, we have
and so the function
is quadratic.
Taking the limit in (2.9) with
as
, we obtain that
which yields the estimation (2.4).
To prove the uniqueness of the quadratic function
subject to (2.4), let us assume that there exists a quadratic function
which satisfies (1.5) and the inequality (2.4). Obviously, we obtain that
for all
. Hence it follows from (2.4) that
for all
. Therefore letting
, one has
for all
, completing the proof of uniqueness.
Theorem 2.2.
Assume that a function
satisfies
for all
and that
satisfies conditions
for all
. Then there exists a unique quadratic function
satisfying
for all
. The function
is given by
for all 
Proof.
In this case,
since
and so
by assumption.
Replacing
by
in (2.6), we obtain
for
. Therefore we prove from inequality (2.19) that for any integers
with 
for all
. Since the right-hand side of (2.20) tends to zero as
, the sequence
is Cauchy for all
and thus converges by the completeness of
. Define
by
for all
.
Thereafter, applying the same argument as in the proof of Theorem 2.1, we obtain the desired result.
We now introduce a fundamental result of fixed point theory. We refer to [28] for the proof of it, and the reader is referred to papers [29–31].
Theorem 2.3.
Let
be a generalized complete metric space (i.e.,
may assume infinite values). Assume that
is a strictly contractive operator with the Lipschitz constant
Then for a given element
one of the following assertions is true:
(A1)
for all
;
(A2) there exists a nonnegative integer
such that
(A2.1)
for all
;
(A2.2) the sequence
converges to a fixed point
of
;
(A2.3)
is the unique fixed point of
in the set 
(A2.4)
for all 
For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. [32]. In 1996, Isac and Th. M. Rassias [33] applied the stability theory of functional equations to prove fixed point theorems and study some new applications in nonlinear analysis. Cădariu and Radu [29, 31] and Radu [34] applied the fixed point theorem of alternative to the investigation of Cauchy and Jensen functional equations. Recently, Jung et al. [35–40] and Jung and Rassias [41] have obtained the generalized Hyers-Ulam stability of functional equations via the fixed point method.
Now we are ready to investigate the generalized Hyers-Ulam stability problem for the functional (1.5) using the fixed point method.
Theorem 2.4.
Let
be a function with
for which there exists a function
such that there exists a constant
satisfying the inequalities
for all
. Then there exists a unique quadratic function
defined by
such that
for all 
Proof.
Let us define
to be the set of all functions
and introduce a generalized metric
on
as follows:
Then it is easy to show that
is complete (see [37, Proof of Theorem
]). Now we define an operator
by
for all
First, we assert that
is strictly contractive with constant
on
. Given
, let
be an arbitrary constant with
that is,
Then it follows from (2.23) that
for all
that is,
for any
with
Thus we see that
for any
and so
is strictly contractive with constant
on
.
Next, if we put
in (2.22) and we divide both sides by
, then we get
for all
which implies 
Thus applying Theorem 2.3 to the complete generalized metric space
with contractive constant
, we see from
of Theorem 2.3 that there exists a function
which is a fixed point of
, that is,
such that
as
By mathematical induction we know that
for all 
Since
as
by
of Theorem 2.3, there exists a sequence
such that
as
and
for every
Hence, it follows from the definition of
that
for all
This implies
for all 
In turn, it follows from (2.22) and (2.23) that
for all
, which implies that
is a solution of (1.5) and so the mapping
is quadratic.
By
of Theorem 2.3, we obtain
which yields the inequality (2.24).
To prove the uniqueness of
, assume now that
is another quadratic mapping satisfying the inequality (2.24). Then
is a fixed point of
with
in view of the inequality (2.24). This implies that
and so
by
of Theorem 2.3. The proof is complete.
By a similar way, one can prove the following theorem using the fixed point method.
Theorem 2.5.
Let
be a function with
for which there exists a function
such that there exists a constant
satisfying the inequalities
for all
. Then there exists a unique quadratic function
defined by
such that
for all 
Proof.
We use the same notations for
and
as in the proof of Theorem 2.4. Thus
is a complete generalized metric space. Let us define an operator
by
for all
Then it follows from (2.35) that
for all
that is,
Thus we see that
for any
and so
is strictly contractive with constant
on
.
Next, if we put
in (2.34) and we divide both sides by
, then we get by virtue of (2.35)
for all
which implies
Thereafter, applying the same argument as in the proof of Theorem 2.4, we obtain the desired results.