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A Fixed Point Approach to the Stability of a Quadratic Functional Equation in Algebras
Advances in Difference Equations volume 2009, Article number: 256165 (2009)
Abstract
We use a fixed point method to investigate the stability problem of the quadratic functional equation in algebras.
1. Introduction and Preliminaries
In 1940, the following question concerning the stability of group homomorphisms was proposed by Ulam [1]: Under what conditions does there exist a group homomorphism near an approximately group homomorphism? In 1941, Hyers [2] considered the case of approximately additive functions , where and are Banach spaces and satisfies Hyers inequality
for all . Aoki [3] and Th. M. Rassias [4] provided a generalization of the Hyers' theorem for additive mappings and for linear mappings, respectively, by allowing the Cauchy difference to be unbounded (see also [5]).
Theorem 1.1 (Th. M. Rassias).
Let be a mapping from a normed vector space into a Banach space subject to the inequality
for all , where and are constants with and . Then the limit
exists for all and is the unique additive mapping which satisfies
for all . If then inequality (1.2) holds for and (1.4) for . Also, if for each the mapping is continuous in , then is linear.
The result of the Th. M. Rassias theorem has been generalized by Gvruţa [6] who permitted the Cauchy difference to be bounded by a general control function. During the last three decades a number of papers and research monographs have been published on various generalizations and applications of the generalized HyersUlam stability to a number of functional equations and mappings (see [7–20]). We also refer the readers to the books [21–25]. A quadratic functional equation is a functional equation of the following form:
In particular, every solution of the quadratic equation (1.5) is said to be a quadratic mapping. It is well known that a mapping between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive mapping such that for all (see [16, 21, 26, 27]. The biadditive mapping is given by
The HyersUlam stability problem for the quadratic functional equation (1.5) was studied by Skof [28] for mappings where is a normed space and is a Banach space. Cholewa [8] noticed that the theorem of Skof is still true if we replace by an Abelian group. Czerwik [9] proved the generalized HyersUlam stability of the quadratic functional equation (1.5). Grabiec [11] has generalized these results mentioned above. Jun and Lee [14] proved the generalized HyersUlam stability of a Pexiderized quadratic functional equation.
Let be a set. A function is called a generalized metric on if satisfies

(i)
if and only if ;

(ii)
for all ;

(iii)
for all
We recall the following theorem by Margolis and Diaz.
Theorem 1.2 (see [29]).
Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either
for all nonnegative integers or there exists a nonnegative integer such that

(1)
for all ;

(2)
the sequence converges to a fixed point of ;

(3)
is the unique fixed point of in the set ;

(4)
for all .
Throughout this paper will be a algebra. We denote by the unique positive element such that for each positive element . Also, we denote by and the set of real, complex, and rational numbers, respectively. In this paper, we use a fixed point method (see [7, 15, 17]) to investigate the stability problem of the quadratic functional equation
in algebras. A systematic study of fixed point theorems in nonlinear analysis is due to Hyers et al. [30] and Isac and Rassias [13].
2. Solutions of (1.8)
Theorem 2.1.
Let be a linear space. If a mapping satisfies and the functional equation (1.8), then is quadratic.
Proof.
Letting and in (1.8) respectively, we get
for all It follows from (1.8) and (2.1) that
for all Letting in (2.2), we get
for all Thus (2.2) implies that
for all Hence is quadratic.
Remark 2.2.
A quadratic mapping does not satisfy (1.8) in general. Let be the mapping defined by for all It is clear that is quadratic and that does not satisfy (1.8).
Corollary 2.3.
Let be a linear space. If a mapping satisfies the functional equation (1.8), then there exists a symmetric biadditive mapping such that for all
3. Generalized HyersUlam Stability of (1.8) in Algebras
In this section, we use a fixed point method (see [7, 15, 17]) to investigate the stability problem of the functional equation (1.8) in algebras.
For convenience, we use the following abbreviation for a given mapping
for all where is a linear space.
Theorem 3.1.
Let be a linear space and let be a mapping with for which there exists a function such that
for all . If there exists a constant such that
for all , then there exists a unique quadratic mapping such that
for all where
Moreover, if is continuous in for each fixed , then is quadratic, that is, for all and all
Proof.
Replacing and by and in (3.2), respectively, we get
for all Replacing and by and in (3.2), respectively, we get
for all It follows from (3.6) and (3.7) that
for all Letting in (3.8), we get
for all By (3.3) we have for all Let be the set of all mappings with . We can define a generalized metric on as follows:
is a generalized complete metric space [7].
Let be the mapping defined by
Let and let be an arbitrary constant with . From the definition of , we have
for all . Hence
for all . So
for any . It follows from (3.9) that . According to Theorem 1.2, the sequence converges to a fixed point of , that is,
and for all . Also,
and is the unique fixed point of in the set . Thus the inequality (3.4) holds true for all . It follows from the definition of , (3.2), and (3.3) that
for all By Theorem 2.1, the function is quadratic.
Moreover, if is continuous in for each fixed , then by the same reasoning as in the proof of [4] is quadratic.
Corollary 3.2.
Let and be nonnegative real numbers and let be a mapping with such that
for all Then there exists a unique quadratic mapping such that
for all . Moreover, if is continuous in for each fixed , then is quadratic.
The following theorem is an alternative result of Theorem 3.1 and we will omit the proof.
Theorem 3.3.
Let be a mapping with for which there exists a function satisfying (3.2) for all If there exists a constant such that
for all , then there exists a unique quadratic mapping such that
for all , where is defined as in Theorem 3.1. Moreover, if is continuous in for each fixed , then is quadratic.
Corollary 3.4.
Let and be nonnegative real numbers and let be a mapping with such that
for all . Then there exists a unique quadratic mapping such that
for all . Moreover, if is continuous in for each fixed , then is quadratic.
For the case we use the Gajda's example [31] to give the following counterexample (see also [9]).
Example 3.5.
Let be defined by
Consider the function by the formula
It is clear that is continuous and bounded by on . We prove that
for all To see this, if or then
Now suppose that Then there exists a positive integer such that
Thus
Hence
for all It follows from the definition of and (3.28) that
Thus satisfies (3.26). Let be a quadratic function such that
for all where is a positive constant. Then there exists a constant such that for all . So we have
for all Let with If , then for all So
which contradicts (3.33).
References
 1.
Ulam SM: A Collection of Mathematical Problems. Interscience Publishers, New York, NY, USA; 1960:xiii+150.
 2.
Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222
 3.
Aoki T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan 1950, 2: 64–66. 10.2969/jmsj/00210064
 4.
Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297–300. 10.1090/S00029939197805073271
 5.
Bourgin DG: Classes of transformations and bordering transformations. Bulletin of the American Mathematical Society 1951, 57: 223–237. 10.1090/S000299041951095117
 6.
Găvruţa P: A generalization of the HyersUlamRassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994,184(3):431–436. 10.1006/jmaa.1994.1211
 7.
Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. In Iteration Theory, Grazer Mathematische Berichte. Volume 346. KarlFranzensUniversitaet Graz, Graz, Austria; 2004:43–52.
 8.
Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984,27(1–2):76–86.
 9.
Czerwik S: On the stability of the quadratic mapping in normed spaces. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1992, 62: 59–64. 10.1007/BF02941618
 10.
Faĭziev VA, Rassias ThM, Sahoo PK: The space of additive mappings on semigroups. Transactions of the American Mathematical Society 2002,354(11):4455–4472. 10.1090/S0002994702030362
 11.
Grabiec A: The generalized HyersUlam stability of a class of functional equations. Publicationes Mathematicae Debrecen 1996,48(3–4):217–235.
 12.
Hyers DH, Rassias ThM: Approximate homomorphisms. Aequationes Mathematicae 1992,44(2–3):125–153. 10.1007/BF01830975
 13.
Isac G, Rassias ThM: Stability of additive mappings: applications to nonlinear analysis. International Journal of Mathematics and Mathematical Sciences 1996,19(2):219–228. 10.1155/S0161171296000324
 14.
Jun KW, Lee YH: On the HyersUlamRassias stability of a Pexiderized quadratic inequality. Mathematical Inequalities & Applications 2001,4(1):93–118.
 15.
Jung SM, Kim TS: A fixed point approach to the stability of the cubic functional equation. Boletín de la Sociedad Matemática Mexicana 2006,12(1):51–57.
 16.
Kannappan Pl: Quadratic functional equation and inner product spaces. Results in Mathematics 1995,27(3–4):368–372.
 17.
Mirzavaziri M, Moslehian MS: A fixed point approach to stability of a quadratic equation. Bulletin of the Brazilian Mathematical Society 2006,37(3):361–376. 10.1007/s005740060016z
 18.
Park CG: On the stability of the linear mapping in Banach modules. Journal of Mathematical Analysis and Applications 2002,275(2):711–720. 10.1016/S0022247X(02)003864
 19.
Rassias ThM: On a modified HyersUlam sequence. Journal of Mathematical Analysis and Applications 1991,158(1):106–113. 10.1016/0022247X(91)90270A
 20.
Rassias ThM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications 2000,251(1):264–284. 10.1006/jmaa.2000.7046
 21.
Aczél J, Dhombres J: Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications. Volume 31. Cambridge University Press, Cambridge, UK; 1989:xiv+462.
 22.
Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.
 23.
Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications. Volume 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.
 24.
Jung SM: HyersUlamRassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.
 25.
Rassias ThM: Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:x+224.
 26.
Amir D: Characterizations of Inner Product Spaces, Operator Theory: Advances and Applications. Volume 20. Birkhäuser, Basel, Switzerland; 1986:vi+200.
 27.
Jordan P, von Neumann J: On inner products in linear, metric spaces. Annals of Mathematics 1935,36(3):719–723. 10.2307/1968653
 28.
Skof F: Local properties and approximation of operators. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113–129. 10.1007/BF02924890
 29.
Diaz JB, Margolis B: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bulletin of the American Mathematical Society 1968, 74: 305–309. 10.1090/S000299041968119330
 30.
Hyers DH, Isac G, Rassias ThM: Topics in Nonlinear Analysis & Applications. World Scientific, River Edge, NJ, USA; 1997:xiv+699.
 31.
Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991,14(3):431–434. 10.1155/S016117129100056X
Acknowledgment
The third author was supported by Korea Research Foundation Grant funded by the Korean Government (KRF2008313C00041).
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Moghimi, M.B., Najati, A. & Park, C. A Fixed Point Approach to the Stability of a Quadratic Functional Equation in Algebras. Adv Differ Equ 2009, 256165 (2009). https://doi.org/10.1155/2009/256165
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Keywords
 Banach Space
 Functional Equation
 Positive Element
 Group Homomorphism
 Real Vector Space