- Research Article
- Open Access
Stability of Approximate Quadratic Mappings
© Hark-Mahn Kim et al. 2010
- Received: 23 October 2009
- Accepted: 1 February 2010
- Published: 14 February 2010
We investigate the general solution of the quadratic functional equation , in the class of all functions between quasi- -normed spaces, and then we prove the generalized Hyers-Ulam stability of the equation by using direct method and fixed point method.
- Functional Equation
- Normed Space
- Fixed Point Theorem
- Stability Theorem
- Quadratic Functional Equation
In 1940, Ulam  gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.
Let be a group and let be a metric group with metric . Given , does there exist a such that if satisfies for all , then a homomorphism exists with for all ?
In 1941, the first result concerning the stability of functional equations was presented by Hyers . And then Aoki  and Bourgin  have investigated the stability theorems of functional equations with unbounded Cauchy differences. In 1978, Th. M. Rassias  provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded. It was shown by Gajda  as well as by Th. M. Rassias and Šemrl  that one cannot prove the Rassias' type theorem when . Găvruta  obtained generalized result of Th. M. Rassias' Theorem which allow the Cauchy difference to be controlled by a general unbounded function. J. M. Rassias [9, 10] established a similar stability theorem linear and nonlinear mappings with the unbounded Cauchy difference.
Let and be real vector spaces. A function is called a quadratic function if and only if is a solution function of the quadratic functional equation:
It is well known that a function between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function such that for all , where the mapping is given by . See [11, 12] for the details. The Hyers-Ulam stability of the quadratic functional (1.1) was first proved by Skof  for functions , where is a normed space and is a Banach space. Cholewa  demonstrated that Skof's theorem is also valid if is replaced by an abelian group. Czerwik  proved the Hyers-Ulam stability of quadratic functional (1.1) by the similar way to Th. M. Rassias control function . According to the theorem of Borelli and Forti , we obtain the following generalization of stability theorem for the quadratic functional (1.1): let be an abelian group and a Banach space; let be a mapping with satisfying the inequality
for all . Assume that one of the following conditions
holds for all , then there exists a unique quadratic function such that
In this paper, we consider a new quadratic functional equation
for all vectors in quasi- -normed spaces. First, we note that a function is a solution of the functional (1.5) in the class of all functions between vector spaces if and only if the function is quadratic. Further, we investigate the generalized Hyers-Ulam stability of (1.5) by using direct method and fixed point method. As a result of the paper, we have a much better possible estimation of approximate quadratic mappings by quadratic mappings than that of Czerwik  and Skof .
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  (see also ), each quasinorm is equivalent to some -norm. In , 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  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.
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 . Therefore letting , one has for all , completing the proof of uniqueness.
In this case, since and so by assumption.
Replacing by in (2.6), we obtain
for all .
Thereafter, applying the same argument as in the proof of Theorem 2.1, we obtain the desired result.
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. . In 1996, Isac and Th. M. Rassias  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  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  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.
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
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
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.
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.
In the following corollary, we have a stability result of (1.5) in the sense of Th. M. Rassias.
for all , where if .
If , then we get by putting in (3.1). Letting for all and then applying Theorem 2.1 we obtain easily the desired results.
for all .
In the following corollary, we have a stability result of (1.5) in the sense of Hyers.
for all .
In the next corollary, we get a stability result of (1.5) in the sense of J. M. Rassias.
for all and all if , where if .
Letting and applying Theorems 2.1 and 2.2, we get the results.
This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (no. 2009-0070940).
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