- Research Article
- Open Access
Solution and Stability of a Mixed Type Additive, Quadratic, and Cubic Functional Equation
© M. Eshaghi Gordji et al. 2009
- Received: 24 January 2009
- Accepted: 26 June 2009
- Published: 17 August 2009
We obtain the general solution and the generalized Hyers-Ulam-Rassias stability of the mixed type additive, quadratic, and cubic functional equation .
- Banach Space
- General Solution
- Functional Equation
- Quadratic Function
- Unique Function
The stability problem of functional equations originated from a question of Ulam  in 1940, concerning the stability of group homomorphisms. Let be a group, and let be a metric group with the metric Given , does there exist a , such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all ? In other words, under what condition does there exist a homomorphism near an approximate homomorphism?
for all Moreover if is continuous in for each fixed then is linear (see also ). In 1950, Aoki  generalized Hyers' theorem for approximately additive mappings. In 1978, Th. M. Rassias  provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded. This new concept is known as Hyers-Ulam-Rassias stability of functional equations (see [2–24]).
is related to symmetric biadditive function. In the real case it has among its solutions. Thus, it has been called quadratic functional equation, and each of its solutions is said to be a quadratic function. Hyers-Ulam-Rassias stability for the quadratic functional equation (1.3) was proved by Skof for functions , where is normed space and Banach space (see [25–28]).
and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.5).
The function satisfies the functional equation (1.5), which explains why it is called cubic functional equation.
Jun and Kim proved that a function between real vector spaces and is a solution of (1.5) if and only if there exists a unique function such that for all and is symmetric for each fixed one variable and is additive for fixed two variables (see also [31–33]).
It is easy to see that the function is a solution of the functional equation (1.6). In the present paper we investigate the general solution and the generalized Hyers-Ulam-Rassias stability of the functional equation (1.6).
In this section we establish the general solution of functional equation (1.6).
Let , be vector spaces, and let be a function. Then satisfies (1.6) if and only if there exists a unique additive function , a unique symmetric and biadditive function and a unique symmetric and 3-additive function such that for all .
This completes the proof of theorem.
The following corollary is an alternative result of Theorem 2.1.
If is even function, then is quadratic.
If is odd function, then is cubic-additive.
for an upper bound
for all And analogously, as in the case , we can show that the function defined by is unique quadratic function satisfying (1.6) and (3.7).
for all By using (3.26), we may define a mapping as for all Similar to Theorem 3.1, we can show that is the unique cubic function satisfying (1.6) and (3.25).
for all On the other hand and are cubic, then Therefore by (3.35) we obtain that for all Again by (3.35) we have for all
The proof is similar to the proof of Theorem 3.4.
Now we establish the generalized Hyers-Ulam-Rassias stability of functional equation (1.6) as follows.
for all .
Let for all Then and for all Hence in view of Theorem 3.1 there exists a unique quadratic function satisfying (3.7). Let for all Then and for all From Theorem 3.4, it follows that there exist a unique cubic function and a unique additive function satisfying (3.29). Now it is obvious that (3.40) holds true for all and the proof of theorem is complete.
It follows from Theorem 3.6 by taking for all .
for all .
By Theorem 3.8, we are going to investigate the following stability problem for functional equation (1.6).
for all .
By Corollary 3.9, we solve the following Hyers-Ulam stability problem for functional equation (1.6).
for all .
The authors would like to express their sincere thanks to referees for their invaluable comments. The first author would like to thank the Semnan University for its financial support. Also, the fourth author would like to thank the office of gifted students at Semnan University for its financial support.
- Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1940.Google Scholar
- Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.View ArticleGoogle Scholar
- Bourgin DG: Classes of transformations and bordering transformations. Bulletin of the American Mathematical Society 1951, 57: 223-237. 10.1090/S0002-9904-1951-09511-7MathSciNetView ArticleMATHGoogle Scholar
- 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/00210064MathSciNetView ArticleMATHGoogle Scholar
- 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/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar
- 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.View ArticleGoogle Scholar
- Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984,27(1-2):76-86.MathSciNetView ArticleMATHGoogle Scholar
- Forti G-L: Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations. Journal of Mathematical Analysis and Applications 2004,295(1):127-133. 10.1016/j.jmaa.2004.03.011MathSciNetView ArticleMATHGoogle Scholar
- Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991,14(3):431-434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar
- Găvruţa P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994,184(3):431-436. 10.1006/jmaa.1994.1211MathSciNetView ArticleMATHGoogle Scholar
- 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.27.4.222MathSciNetView ArticleGoogle Scholar
- Isac G, Rassias ThM:On the Hyers-Ulam stability of -additive mappings. Journal of Approximation Theory 1993,72(2):131-137. 10.1006/jath.1993.1010MathSciNetView ArticleMATHGoogle Scholar
- Jun K-W, Kim H-M: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. Journal of Mathematical Analysis and Applications 2002,274(2):267-278.MathSciNetView ArticleGoogle Scholar
- Maligranda L: A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions—a question of priority. Aequationes Mathematicae 2008,75(3):289-296. 10.1007/s00010-007-2892-8MathSciNetView ArticleMATHGoogle Scholar
- Rassias JM: On a new approximation of approximately linear mappings by linear mappings. Discussiones Mathematicae 1985, 7: 193-196.MathSciNetMATHGoogle Scholar
- Rassias JM: On approximation of approximately linear mappings by linear mappings. Bulletin des Sciences Mathématiques. Deuxième Série 1984,108(4):445-446.MATHGoogle Scholar
- Rassias JM: Complete solution of the multi-dimensional problem of Ulam. Discussiones Mathematicae 1994, 14: 101-107.MathSciNetMATHGoogle Scholar
- Rassias JM: On the stability of a multi-dimensional Cauchy type functional equation. In Geometry, Analysis and Mechanics. World Scientific, River Edge, NJ, USA; 1994:365-376.Google Scholar
- Rassias JM: Solution of a stability problem of Ulam. In Functional Analysis, Approximation Theory and Numerical Analysis. World Scientific, River Edge, NJ, USA; 1994:241-249.View ArticleGoogle Scholar
- Rassias JM: Solution of a stability problem of Ulam. Discussiones Mathematicae 1992, 12: 95-103.MathSciNetMATHGoogle Scholar
- Rassias JM: Solution of a problem of Ulam. Journal of Approximation Theory 1989,57(3):268-273. 10.1016/0021-9045(89)90041-5MathSciNetView ArticleMATHGoogle Scholar
- Rassias JM: On approximation of approximately linear mappings by linear mappings. Journal of Functional Analysis 1982,46(1):126-130. 10.1016/0022-1236(82)90048-9MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM (Ed): Functional Equations and Inequalities, Mathematics and Its Applications. Volume 518. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:xii+336.Google Scholar
- 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.2000.7046MathSciNetView ArticleMATHGoogle Scholar
- Grabiec A: The generalized Hyers-Ulam stability of a class of functional equations. Publicationes Mathematicae Debrecen 1996,48(3-4):217-235.MathSciNetMATHGoogle Scholar
- Rassias JM, Rassias MJ: Refined Ulam stability for Euler-Lagrange type mappings in Hilbert spaces. International Journal of Applied Mathematics & Statistics 2007, 7: 126-132.MathSciNetGoogle Scholar
- Rassias JM: On the stability of the Euler-Lagrange functional equation. Chinese Journal of Mathematics 1992,20(2):185-190.MathSciNetMATHGoogle Scholar
- Skof F: Proprieta' locali e approssimazione di operatori. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113-129. 10.1007/BF02924890MathSciNetView ArticleMATHGoogle Scholar
- Rassias JM: Solution of the Ulam stability problem for cubic mappings. Glasnik Matematički. Serija III 2001,36(56)(1):63-72.Google Scholar
- Rassias JM: Solution of the Ulam problem for cubic mappings. Analele Universităţii din Timişoara. Seria Matematică-Informatică 2000,38(1):121-132.MATHGoogle Scholar
- Gordji ME: Stability of a functional equation deriving from quartic and additive functions. to appear in Bulletin of the Korean Mathematical SocietyGoogle Scholar
- Gordji ME, Ebadian A, Zolfaghari S: Stability of a functional equation deriving from cubic and quartic functions. Abstract and Applied Analysis 2008, 2008:-17.Google Scholar
- Gordji ME, Khodaei H: Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2009, 71: 5629-5643. 10.1016/j.na.2009.04.052View ArticleMATHGoogle Scholar
- Brzdęk J, Pietrzyk A: A note on stability of the general linear equation. Aequationes Mathematicae 2008,75(3):267-270. 10.1007/s00010-007-2894-6MathSciNetView ArticleMATHGoogle Scholar
- Pietrzyk A: Stability of the Euler-Lagrange-Rassias functional equation. Demonstratio Mathematica 2006,39(3):523-530.MathSciNetMATHGoogle Scholar
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