- Research Article
- Open Access
Fixed Points and Random Stability of a Generalized Apollonius Type Quadratic Functional Equation
© Min June Kim et al. 2011
- Received: 23 December 2010
- Accepted: 28 February 2011
- Published: 14 March 2011
Using the fixed-point method, we prove the generalized Hyers-Ulam stability of a generalized Apollonius type quadratic functional equation in random Banach spaces.
- Banach Space
- Functional Equation
- Normed Space
- Stability Problem
- Unique Fixed Point
The stability problem of functional equations was originated from a question of Ulam  concerning the stability of group homomorphisms. Hyers  gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki  for additive mappings and by Th. M. Rassias  for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias  has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa  by replacing the unbounded Cauchy difference by a general control function in the spirit of the Th. M. Rassias' approach.
On the other hand, in 1982–1998, J. M. Rassias generalized the Hyers' stability result by presenting a weaker condition controlled by a product of different powers of norms.
The functional equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The generalized Hyers-Ulam stability of the quadratic functional equation was proved by Skof  for mappings : , 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. Czerwik  proved the generalized Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [23–44]).
in Banach spaces.
Definition 2.1 (see ).
Definition 2.2 (see ).
Theorem 2.4 (see ).
Let be a generalized metric space. An operator : satisfies a Lipschitz condition with Lipschitz constant if there exists a constant such that for all . If the Lipschitz constant is less than 1, then the operator is called a strictly contractive operator. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We recall the following theorem by Diaz and Margolis.
In 1996, Isac and Th. M Rassias  were the first to provide applications of stability theory of functional equations for the proof of new fixed-point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [61–67]).
In this paper, we prove the generalized Hyers-Ulam stability of the generalized Apollonius type quadratic functional equation (1.3) in random Banach space by using the fixed point method.
where, as usual, . It is easy to show that is a generalized complete metric space (see [68, Lemma 2.1]).
for all and all . Letting in (3.13), we find that for all , which implies . By [45, Lemma 2.1], the mapping, : is quadratic.
The rest of the proof is similar to the proof of Theorem 3.1.
The fifth author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788) and by Hi Seoul Science Fellowship from Seoul Scholarship Foundation. This paper was dedictated to Professor Sehie Park.
- Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.Google 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
- 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
- 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
- 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 JM: On approximation of approximately linear mappings by linear mappings. Bulletin des Sciences Mathématiques 1984,108(4):445–446.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 the stability of the Euler-Lagrange functional equation. Chinese Journal of Mathematics 1992,20(2):185–190.MathSciNetMATHGoogle Scholar
- Rassias JM: On the stability of the non-linear Euler-Lagrange functional equation in real normed linear spaces. Journal of Mathematical and Physical Sciences 1994,28(5):231–235.MathSciNetMATHGoogle Scholar
- Rassias JM: On the stability of the general Euler-Lagrange functional equation. Demonstratio Mathematica 1996,29(4):755–766.MathSciNetMATHGoogle Scholar
- Rassias JM: Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings. Journal of Mathematical Analysis and Applications 1998,220(2):613–639. 10.1006/jmaa.1997.5856MathSciNetView ArticleMATHGoogle Scholar
- Ravi K, Arunkumar M, Rassias JM: Ulam stability for the orthogonally general Euler-Lagrange type functional equation. International Journal of Mathematics and Statistics 2008,3(A08):36–46.MathSciNetMATHGoogle Scholar
- Cao H-X, Lv J-R, Rassias JM: Superstability for generalized module left derivations and generalized module derivations on a Banach module. I. Journal of Inequalities and Applications 2009, 2009:-10.Google Scholar
- Cao H-X, Lv J-R, Rassias JM: Superstability for generalized module left derivations and generalized module derivations on a Banach module. II. Journal of Inequalities in Pure and Applied Mathematics 2009,10(3, article 85):1–8.MathSciNetMATHGoogle Scholar
- Gordji ME, Zolfaghari S, Rassias JM, Savadkouhi MB: Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces. Abstract and Applied Analysis 2009, 2009:-14.Google Scholar
- Ravi K, Rassias JM, Arunkumar M, Kodandan R: Stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation. Journal of Inequalities in Pure and Applied Mathematics 2009,10(4, article 114):1–29.MathSciNetMATHGoogle Scholar
- Gordji ME, Khodaei H: On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations. Abstract and Applied Analysis 2009, 2009:-11.Google Scholar
- Savadkouhi MB, Gordji ME, Rassias JM, Ghobadipour N: Approximate ternary Jordan derivations on Banach ternary algebras. Journal of Mathematical Physics 2009,50(4):-9.Google Scholar
- Skof F: Proprietà locali e approssimazione di operatori. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113–129. 10.1007/BF02924890MathSciNetView ArticleGoogle 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
- 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
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.View ArticleMATHGoogle 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
- 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 ArticleMATHGoogle Scholar
- Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar
- Najati A: Hyers-Ulam stability of an -Apollonius type quadratic mapping. Bulletin of the Belgian Mathematical Society. Simon Stevin 2007,14(4):755–774.MathSciNetMATHGoogle Scholar
- Najati A: Fuzzy stability of a generalized quadratic functional equation. Communications of the Korean Mathematical Society 2010,25(3):405–417. 10.4134/CKMS.2010.25.3.405MathSciNetView ArticleMATHGoogle Scholar
- Najati A, Moradlou F: Hyers-Ulam-Rassias stability of the Apollonius type quadratic mapping in non-Archimedean spaces. Tamsui Oxford Journal of Mathematical Sciences 2008,24(4):367–380.MathSciNetMATHGoogle Scholar
- Najati A, Park C: The Pexiderized Apollonius-Jensen type additive mapping and isomorphisms between -algebras. Journal of Difference Equations and Applications 2008,14(5):459–479. 10.1080/10236190701466546MathSciNetView ArticleMATHGoogle Scholar
- Oh S-Q, Park C-G: Linear functional equations in a Hilbert module. Taiwanese Journal of Mathematics 2003,7(3):441–448.MathSciNetMATHGoogle Scholar
- Park C-G: On the stability of the linear mapping in Banach modules. Journal of Mathematical Analysis and Applications 2002,275(2):711–720. 10.1016/S0022-247X(02)00386-4MathSciNetView ArticleMATHGoogle Scholar
- Park C-G: Modified Trif's functional equations in Banach modules over a -algebra and approximate algebra homomorphisms. Journal of Mathematical Analysis and Applications 2003,278(1):93–108. 10.1016/S0022-247X(02)00573-5MathSciNetView ArticleMATHGoogle Scholar
- Park C-G: Lie -homomorphisms between Lie -algebras and Lie -derivations on Lie -algebras. Journal of Mathematical Analysis and Applications 2004,293(2):419–434. 10.1016/j.jmaa.2003.10.051MathSciNetView ArticleMATHGoogle Scholar
- Park C-G: Homomorphisms between Lie -algebras and Cauchy-Rassias stability of Lie -algebra derivations. Journal of Lie Theory 2005,15(2):393–414.MathSciNetMATHGoogle Scholar
- Park C-G: Approximate homomorphisms on -triples. Journal of Mathematical Analysis and Applications 2005,306(1):375–381. 10.1016/j.jmaa.2004.12.043MathSciNetView ArticleMATHGoogle Scholar
- Park C-G: Homomorphisms between Poisson -algebras. Bulletin of the Brazilian Mathematical Society 2005,36(1):79–97. 10.1007/s00574-005-0029-zMathSciNetView ArticleMATHGoogle Scholar
- Park C-G: Isomorphisms between unital -algebras. Journal of Mathematical Analysis and Applications 2005,307(2):753–762. 10.1016/j.jmaa.2005.01.059MathSciNetView ArticleMATHGoogle Scholar
- Park CG, Hou JC, Oh SQ: Homomorphisms between -algebras and Lie -algebras. Acta Mathematica Sinica 2005,21(6):1391–1398. 10.1007/s10114-005-0629-yMathSciNetView ArticleMATHGoogle Scholar
- Park C-G, Park W-G: On the Jensen's equation in Banach modules. Taiwanese Journal of Mathematics 2002,6(4):523–531.MathSciNetMATHGoogle Scholar
- Rassias ThM: The problem of S. M. Ulam for approximately multiplicative mappings. Journal of Mathematical Analysis and Applications 2000,246(2):352–378. 10.1006/jmaa.2000.6788MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM: 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 ThM, Šemrl P: On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proceedings of the American Mathematical Society 1992,114(4):989–993. 10.1090/S0002-9939-1992-1059634-1MathSciNetView ArticleMATHGoogle Scholar
- Park C-G, Rassias ThM: Hyers-Ulam stability of a generalized Apollonius type quadratic mapping. Journal of Mathematical Analysis and Applications 2006,322(1):371–381. 10.1016/j.jmaa.2005.09.027MathSciNetView ArticleMATHGoogle Scholar
- Hadžić O, Pap E: Fixed Point Theory in PM Spaces. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001.MATHGoogle Scholar
- Šerstnev AN: On the notion of a random normed space. Doklady Akademii Nauk SSSR 1963, 149: 280–283.MathSciNetGoogle Scholar
- Schweizer B, Sklar A: Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics. North-Holland, New York, NY, USA; 1983:xvi+275.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
- Mirmostafaee AK, Mirzavaziri M, Moslehian MS: Fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems 2008,159(6):730–738. 10.1016/j.fss.2007.07.011MathSciNetView ArticleMATHGoogle Scholar
- Miheţ D: The fixed point method for fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems 2009,160(11):1663–1667. 10.1016/j.fss.2008.06.014MathSciNetView ArticleMATHGoogle 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. Mathematica Slovaca. In pressGoogle Scholar
- Mirmostafaee AK: A fixed point approach to almost quartic mappings in quasi fuzzy normed spaces. Fuzzy Sets and Systems 2009,160(11):1653–1662. 10.1016/j.fss.2009.01.011MathSciNetView ArticleMATHGoogle Scholar
- Mirmostafaee AK, Moslehian MS: Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets and Systems 2008,159(6):720–729. 10.1016/j.fss.2007.09.016MathSciNetView ArticleMATHGoogle Scholar
- Mirmostafaee AK, Moslehian MS: Fuzzy approximately cubic mappings. Information Sciences 2008,178(19):3791–3798. 10.1016/j.ins.2008.05.032MathSciNetView ArticleMATHGoogle Scholar
- Mirmostafaee AK, Moslehian MS: Stability of additive mappings in non-Archimedean fuzzy normed spaces. Fuzzy Sets and Systems 2009,160(11):1643–1652. 10.1016/j.fss.2008.10.011MathSciNetView ArticleMATHGoogle Scholar
- Cădariu L, Radu V: Fixed points and the stability of Jensen's functional equation. Journal of Inequalities in Pure and Applied Mathematics 2003.,4(1, article 4):Google Scholar
- 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/S0002-9904-1968-11933-0MathSciNetView ArticleMATHGoogle Scholar
- 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/S0161171296000324MathSciNetView ArticleMATHGoogle Scholar
- Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. In Iteration Theory (ECIT '02), Grazer Math. Ber.. Volume 346. Karl-Franzens-Univ. Graz, Graz, Austria; 2004:43–52.Google Scholar
- Cădariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory and Applications 2008, 2008:-15.Google Scholar
- Lee JR, Kim J-H, Park C: A fixed point approach to the stability of an additive-quadratic-cubic-quartic functional equation. Fixed Point Theory and Applications 2010, 2010:-16.Google Scholar
- 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/s00574-006-0016-zMathSciNetView ArticleMATHGoogle Scholar
- Park C: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory and Applications 2007, 2007:-15.Google Scholar
- Park C: Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach. Fixed Point Theory and Applications 2008, 2008:-9.Google Scholar
- Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003,4(1):91–96.MathSciNetMATHGoogle Scholar
- Miheţ D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces. Journal of Mathematical Analysis and Applications 2008,343(1):567–572. 10.1016/j.jmaa.2008.01.100MathSciNetView ArticleMATHGoogle Scholar
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