On the stability of a mixed type quadratic-additive functional equation
© Lee et al.; licensee Springer 2013
Received: 28 December 2012
Accepted: 19 June 2013
Published: 5 July 2013
In this paper we establish the general solutions of the following mixed type quadratic-additive functional equation:
in the class of functions between real vector spaces. Moreover, we prove the generalized Hyers-Ulam-Rassias stability of this equation in Banach spaces.
The stability problems of functional equations go back to 1940 when Ulam  proposed the following question: This question was solved affirmatively by Hyers  under the assumption that is a Banach space. He proved that if f is a mapping between Banach spaces satisfying for some fixed , then there exists a unique additive mapping A such that . In 1978, Rassias  generalized Hyers’ result to the unbounded Cauchy difference. Since then, the stability problems of various functional equations have been extensively studied and generalized by a number of authors (see [4–9]).
for all ?
and investigated the generalized Hyers-Ulam-Rassias stability of this equation in quasi-Banach spaces.
2 General solutions of (1.2)
Throughout this section, X and Y will be real vector spaces. In order to solve the general solutions of (1.2), we need the following two lemmas.
Lemma 2.1 If an even mapping satisfies (1.2) for all , then f is quadratic.
for all . □
Lemma 2.2 If an odd mapping satisfies (1.2) for all , then f is additive.
for all . □
Now we are ready to establish the general solutions of (1.2).
for all .
for all . By Lemmas 2.1 and 2.2 we have the result.
(Sufficiency) This is obvious. □
3 Stability of (1.2)
for all . We note that the condition ( ) implies ( ). Similarly, the condition (ℬ) implies ( ). One of the conditions ( ), (ℬ) will be needed to derive a quadratic mapping, and one of the conditions ( ), ( ) will be required to derive an additive mapping in the following theorem.
for all .
for all , where .
for all and . Taking the limit as , we conclude that for all .
for all . The rest of the proof is similar to that of Case 1.
for all .
for all . Using a similar argument to that of Case 1, we can easily see the uniqueness of A.
for all . Similarly, we can show the uniqueness of A. □
From the theorem above, we have the following corollary immediately.
for all ( if ), where if .
for all ( if ). □
for all .
- Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1964.Google Scholar
- Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222-224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar
- Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297-300. 10.1090/S0002-9939-1978-0507327-1View ArticleGoogle Scholar
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge; 2002.View ArticleGoogle Scholar
- Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431-436. 10.1006/jmaa.1994.1211MathSciNetView ArticleGoogle Scholar
- Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Boston; 1998.View ArticleGoogle Scholar
- Jung S-M Springer Optimization and Its Applications. In Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York; 2011.View ArticleGoogle Scholar
- Kannappan P: Functional Equations and Inequalities with Applications. Springer, New York; 2009.View ArticleGoogle Scholar
- Sahoo PK, Kannappan P: Introduction to Functional Equations. CRC Press, Boca Raton; 2011.Google Scholar
- Kannappan P: Quadratic functional equation and inner product spaces. Results Math. 1995, 27: 368-372. 10.1007/BF03322841MathSciNetView ArticleGoogle Scholar
- Jung S-M: On the Hyers-Ulam stability of the functional equations that have the quadratic property. J. Math. Anal. Appl. 1998, 222: 126-137. 10.1006/jmaa.1998.5916MathSciNetView ArticleGoogle Scholar
- Jun K-W, Kim H-M: On the stability of an n -dimensional quadratic and additive functional equation. Math. Inequal. Appl. 2006, 9: 153-165.MathSciNetGoogle Scholar
- Najati A, Moghimi MB: Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces. J. Math. Anal. Appl. 2008, 337: 399-415. 10.1016/j.jmaa.2007.03.104MathSciNetView ArticleGoogle Scholar
- Gordji ME, Ebadian A, Zolfaghari S: Stability of a mixed type cubic and quartic functional equation. Abstr. Appl. Anal. 2008., 2008: Article ID 801904Google Scholar
- Gordji ME, Bavand-Savadkouhi M: Stability of cubic and quartic functional equations in non-Archimedean spaces. Acta Appl. Math. 2010, 110: 1321-1329. 10.1007/s10440-009-9512-7MathSciNetView ArticleGoogle Scholar
- Gordji ME, Kaboli Gharetapeh S, Rassias JM, Zolfaghari S: Solution and stability of a mixed type additive, quadratic and cubic functional equation. Adv. Differ. Equ. 2009., 2009: Article ID 826130Google Scholar
- Gordji ME, Savadkouhi MB: Stability of mixed type cubic and quartic functional equations in random normed spaces. J. Inequal. Appl. 2009., 2009: Article ID 527462Google Scholar
- Gordji ME, Abbaszadeh S, Park C: On the stability of generalized mixed type quadratic and quartic functional equation in quasi-Banach spaces. J. Inequal. Appl. 2009., 2009: Article ID 153084Google Scholar
- Gordji ME, Savadkouhi MB, Park C: Quadratic-quartic functional equations in RN-spaces. J. Inequal. Appl. 2009., 2009: Article ID 868423Google Scholar
- Gordji ME: Stability of a functional equation deriving from quartic and additive functions. Bull. Korean Math. Soc. 2010, 47: 491-502. 10.4134/BKMS.2010.47.3.491MathSciNetView ArticleGoogle Scholar
- Gordji ME, Kaboli Gharetapeh S, Moslehian MS, Zolfaghari S: Stability of a mixed type additive, quadratic, cubic and quartic functional equation. Springer Optim. Appl. 35. In Nonlinear Analysis and Variational Problems. Springer, New York; 2010:65-80.View ArticleGoogle Scholar
- Lee Y-S: On the stability of a mixed type functional equation in generalized functions. Adv. Differ. Equ. 2012., 2012: Article ID 16Google Scholar
- Najati A, Eskandani GZ: A fixed point method to the generalized stability of a mixed additive and quadratic functional equation in Banach modules. J. Differ. Equ. Appl. 2010, 16: 773-788.MathSciNetView ArticleGoogle Scholar
- Nakmahachalasint P: On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations. Int. J. Math. Math. Sci. 2007., 2007: Article ID 63239Google Scholar
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