- Research Article
- Open Access
Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces
© Yeol Je Cho et al. 2010
Received: 29 July 2010
Accepted: 31 August 2010
Published: 7 September 2010
In 1940, the stability problem of functional equations originated from a question given by Ulam , that is, the stability of group homomorphisms.
In other words, under what condition, does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation.
In 1941, Hyers  gave a first affirmative answer to the question of Ulam for Banach spaces.
In 1978, Rassias  provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded. In 1991, Gajda  answered the question for the case , which was raised by Rassias. This new concept is known as the Hyers-Ulam-Rassias stability of functional equations (see [5–17]).
Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.3) was proved by Skof  for a mapping , where is a normed space and is a Banach space. Cholewa  noticed that the theorem of Skof is still true if the relevant domain is replaced by an abelian group. In , Czerwik proved the Hyers-Ulam-Rassias stability of (1.3) and Grabiec  generalized these results mentioned above.
where is a mapping from a real vector space into a real vector space 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 is thus called a cubic functional equation. Every solution of the cubic functional equation is said to be a cubic function. Also, Jun and Kim proved that a function between real vector spaces and is a solution of (1.5) if and only if there exits a unique function such that for all and is symmetric for each fixed one variable and is additive for fixed two variables.
Definition 1.1 (see ).
Definition 1.2 (see ).
Theorem 1.4 (see ).
where on random normed spaces. It is easy to see that the mapping is a solution of the functional equation (1.10). In Section 2, we investigate the general solution of functional equation (1.10) when is a mapping between vector spaces and, in Section 3, we establish the stability of the functional equation (1.10) in RN-spaces.
2. General Solutions
Before proceeding to the proof of Theorem 2.3 which is the main result in this section, we need the following two lemmas.
A function with satisfies (1.10) for all if and only if there exist functions and such that for all where the function is symmetric for each fixed one variable and is additive for fixed two variables and is symmetric biadditive.
It is clear that for all and it is easy to show that the functions and satisfy (1.10). Hence, by Lemmas 2.1 and 2.2, we know that the functions and are quadratic and cubic, respectively. Thus there exist a symmetric biadditive function such that for all and the function such that for all where the function is symmetric for each fixed one variable and is additive for fixed two variables. Therefore, we get for all
Conversely, let for all where the function is symmetric for each fixed one variable and is additive for fixed two variables and is biadditive. By a simple computation, one can show that the functions and satisfy the functional equation (1.10). Thus the function satisfies (1.10). This completes the proof.
3. Stability Problems
Now, we show that is a quadratic mapping. In fact, replacing with and , respectively, in (3.3) and then taking the limit as , we find that satisfies (1.10) for all Therefore, the mapping is quadratic.
Obviously, (3.38) follows from (3.40) and (3.42). This completes the proof.
This paper was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
- Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1964:xvii+150.MATHGoogle 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 ArticleMATHGoogle Scholar
- Rassias TM: 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
- Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991, 14(3):431–434. 10.1155/S016117129100056XMathSciNetView 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 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
- 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
- Gordji ME, Kaboli Gharetapeh S, Rashidi E, Karimi T, Aghaei M: Ternary Jordan derivations in -ternary algebras. Journal of Computational Analysis and Applications 2010, 12(2):463–470.MathSciNetMATHGoogle 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
- Găvruta P, Găvruta L: A new method for the generalized Hyers-Ulam-Rassias stability. International Journal of Nonlinear Analysis and Applications 2010, 1(2):11–18.MATHGoogle Scholar
- Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications. Volume 34. Birkhäuser Boston, Boston, Mass, USA; 1998:vi+313.MATHGoogle Scholar
- Isac G, Rassias TM: On the Hyers-Ulam stability of -additive mappings. Journal of Approximation Theory 1993, 72(2):131–137. 10.1006/jath.1993.1010MathSciNetView ArticleMATHGoogle Scholar
- Khodaei H, Rassias ThM: Approximately generalized additive functions in several variables. International Journal of Nonlinear Analysis and Applications 2010, 1: 22–41.MATHGoogle Scholar
- Park C, Najati A: Generalized additive functional inequalities in Banach algebras. International Journal of Nonlinear Analysis and Applications 2010, 1: 54–62.MATHGoogle Scholar
- Park C, Gordji ME: Comment on approximate ternary Jordan derivations on Banach ternary algebras [ Bavand Savadkouhi et al. J. Math. Phys. 50, 042303 (2009)]. Journal of Mathematical Physics 2010, 51:-7.Google Scholar
- Rassias TM: On the stability of functional equations and a problem of Ulam. Acta Applicandae Mathematicae 2000, 62(1):23–130. 10.1023/A:1006499223572MathSciNetView ArticleMATHGoogle Scholar
- Rassias TM: 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
- Kannappan P: Quadratic functional equation and inner product spaces. Results in Mathematics. Resultate der Mathematik 1995, 27(3–4):368–372.MathSciNetView ArticleMATHGoogle Scholar
- Skof F: Local properties and approximation of operators. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113–129. 10.1007/BF02924890MathSciNetView ArticleMATHGoogle Scholar
- Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984, 27(1–2):76–86.MathSciNetView ArticleMATHGoogle Scholar
- Czerwik St: On the stability of the quadratic mapping in normed spaces. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1992, 62: 59–64. 10.1007/BF02941618MathSciNetView 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
- 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
- Chang S-S, Cho YJ, Kang SM: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science Publishers, Huntington, NY, USA; 2001:x+338.MATHGoogle 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 2009, 71(11):5629–5643. 10.1016/j.na.2009.04.052MathSciNetView ArticleMATHGoogle Scholar
- Schweizer B, Sklar A: Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics. North-Holland Publishing, New York, NY, USA; 1983:xvi+275.Google Scholar
- Sherstnev AN: On the notion of a random normed space. Doklady Akademii Nauk SSSR 1963, 149: 280–283.MathSciNetMATHGoogle Scholar
- Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and Its Applications. Volume 536. Kluwer Academic, Dordrecht, The Netherlands; 2001:x+273.Google Scholar
- Hadžić O, Pap E, Budinčević M: Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces. Kybernetika 2002, 38(3):363–382.MathSciNetMATHGoogle Scholar
- Baktash E, Cho YJ, Jalili M, Saadati R, Vaezpour SM: On the stability of cubic mappings and quadratic mappings in random normed spaces. Journal of Inequalities and Applications 2008, 2008:-11.Google Scholar
- Gordji ME, Ghaemi MB, Majani H: Generalized Hyers-Ulam-Rassias theorem in Menger probabilistic normed spaces. Discrete Dynamics in Nature and Society 2010, 2010:-11.Google Scholar
- Khodaei H, Kamyar M: Fuzzy approximately additive mappings. International Journal of Nonlinear Analysis and Applications 2010, 1: 44–53.MATHGoogle Scholar
- Mohamadi M, Cho YJ, Park C, Vetro F, Saadati R: Random stability on an additive-quadratic-quartic functional equation. Journal of Inequalities and Applications 2010, 2010:-18.Google Scholar
- Miheţ D, Saadati R, Vaezpour SM: The stability of the quartic functional equation in random normed spaces. Acta Applicandae Mathematicae 2010, 110(2):797–803. 10.1007/s10440-009-9476-7MathSciNetView ArticleMATHGoogle Scholar
- Miheţ D, Saadati R, Vaezpour SM: The stability of an additive functional equation in Menger probabilistic -normed spaces. Mathematics, Slovak. In pressGoogle Scholar
- Park C: Fuzzy stability of an additive-quadratic-quartic functional equation. Journal of Inequalities and Applications 2010, 2010:-22.Google Scholar
- Park C: Fuzzy stability of additive functional inequalities with the fixed point alternative. Journal of Inequalities and Applications 2009, 2009:-17.Google Scholar
- Park C: A fixed point approach to the fuzzy stability of an additive-quadratic-cubic functional equation. Fixed Point Theory and Applications 2009, 2009:-24.Google Scholar
- Saadati R, Vaezpour SM, Cho YJ: Erratum: a note to paper "On the stability of cubic mappings and quartic mappings in random normed spaces". Journal of Inequalities and Applications 2009, 2009:-6.Google Scholar
- Shakeri S, Saadati R, Park C: Stability of the quadratic functional equation in non-Archimedean −fuzzy normed spaces. International Journal of Nonlinear Analysis and Applications 2010, 1: 72–83.MATHGoogle Scholar
- Zhang S-S, Rassias JM, Saadati R: Stability of a cubic functional equation in intuitionistic random normed spaces. Applied Mathematics and Mechanics. English Edition 2010, 31(1):21–26. 10.1007/s10483-010-0103-6MathSciNetView ArticleMATHGoogle Scholar
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