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:

(A_{1}) for all ;

(A_{2}) there exists a nonnegative integer such that

(A_{2.1}) for all ;

(A_{2.2}) the sequence converges to a fixed point of ;

(A_{2.3}) is the unique fixed point of in the set

(A_{2.4}) 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.