- Research Article
- Open Access
Stability of Mixed Type Cubic and Quartic Functional Equations in Random Normed Spaces
© M. Eshaghi Gordji and M. B. Savadkouhi 2009
- Received: 22 June 2009
- Accepted: 5 August 2009
- Published: 31 August 2009
We obtain the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary -norms .
- Banach Space
- Functional Equation
- Additive Mapping
- Triangle Inequality
- Mixed Type
for all Moreover if is continuous in for each fixed then is linear. In Rassias  provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded. In Gajda  answered the question for the case , which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias stability of functional equations (see [5–12]).
and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.3) The function satisfies the functional equation (1.3) which is thus called a cubic functional equation. Every solution of the cubic functional equation is said to be a cubic function. Jun and Kim proved that a function between real vector spaces X and Y is a solution of (1.3) 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.
In fact they proved that a function between real vector spaces and is a solution of (1.4) if and only if there exists a unique symmetric multiadditive function such that for all (see also [15–18]). It is easy to show that the function satisfies the functional equation (1.4) which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic function.
In the sequel we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [19–21]. Throughout this paper, is the space of distribution functions that is, the space of all mappings , such that is leftcontinuous and nondecreasing on and is a subset of consisting of all functions for which , where denotes the left limit of the function at the point , that is, . The space is partially ordered by the usual pointwise ordering of functions, that is, if and only if for all in . The maximal element for in this order is the distribution function given by
Definition 1.1 (see ).
A mapping is a continuous triangular norm (briefly, a continuous -norm) if satisfies the following conditions:
(a) is commutative and associative;
(b) is continuous;
(c) for all ;
(d) whenever and for all .
Definition 1.2 (see ).
A random normed space (briefly, RN-space) is a triple , where is a vector space, is a continuous -norm, and is a mapping from into such that, the following conditions hold:
for all if and only if ;
for all , ;
for all and
for all and is the minimum -norm. This space is called the induced random normed space.
Let be a RN-space.
(1)A sequence in is said to be convergent to in if, for every and , there exists positive integer such that whenever .
(2)A sequence in is called Cauchy sequence if, for every and , there exists positive integer such that whenever .
(3)A RN-space is said to be complete if and only if every Cauchy sequence in is convergent to a point in .
Theorem 1.4 (see ).
If is an RN-space and is a sequence such that , then almost everywhere.
on random normed spaces. It is easy to see that the function is a solution of the functional equation (1.8) In the present paper we establish the stability of the functional equation (1.8) in random normed spaces.
for all and all
for all and all
Taking the limit as , we find that satisfies (1.8) for all Therefore the mapping is cubic.
To prove (2.3) take the limit as in (2.9) Finally, to prove the uniqueness of the cubic function subject to (2.3) let us assume that there exists a cubic function which satisfies (2.3) Since and for all and from (2.3) it follows that
for all and all . By letting in above inequality, we find that .
for all and all
Taking the limit as , we find that satisfies (1.8) for all Hence, the mapping is quartic.
To prove (2.14) take the limit as in (2.20) Finally, to prove the uniqueness property of subject to (2.14) let us assume that there exists a quartic function which satisfies (2.14) Since and for all and from (2.14) it follows that
for all and all . Taking the limit as , we find that .
for all and all
Obviously, (2.25) follows from (2.28) and (2.31).
The second 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.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 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
- 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
- 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, 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.MATHGoogle 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
- 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: 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
- 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
- Park W-G, Bae J-H: On a bi-quadratic functional equation and its stability. Nonlinear Analysis: Theory, Methods & Applications 2005,62(4):643–654. 10.1016/j.na.2005.03.075MathSciNetView ArticleMATHGoogle Scholar
- Chung JK, Sahoo PK: On the general solution of a quartic functional equation. Bulletin of the Korean Mathematical Society 2003,40(4):565–576.MathSciNetView ArticleMATHGoogle Scholar
- Lee SH, Im SM, Hwang IS: Quartic functional equations. Journal of Mathematical Analysis and Applications 2005,307(2):387–394. 10.1016/j.jmaa.2004.12.062MathSciNetView ArticleMATHGoogle Scholar
- Najati A: On the stability of a quartic functional equation. Journal of Mathematical Analysis and Applications 2008,340(1):569–574. 10.1016/j.jmaa.2007.08.048MathSciNetView ArticleMATHGoogle Scholar
- Park C-G: On the stability of the orthogonally quartic functional equation. Bulletin of the Iranian Mathematical Society 2005,31(1):63–70.MathSciNetMATHGoogle Scholar
- Chang S, Cho YJ, Kang SM: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science, Huntington, NY, USA; 2001:x+338.MATHGoogle 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
- 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 Publishers, 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
- 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
- Miheţ D: The probabilistic stability for a functional equation in a single variable. Acta Mathematica Hungarica 2009,123(3):249–256. 10.1007/s10474-008-8101-yMathSciNetView 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 2009.Google Scholar
- Saadati R, Vaezpour SM, Cho YJ: 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
- Eshaghi Gordji M, Ebadian A, Zolfaghari S: Stability of a functional equation deriving from cubic and quartic functions. Abstract and Applied Analysis 2008, 2008:-17.Google Scholar
- Eshaghi Gordji M, 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(11):5629–5643. 10.1016/j.na.2009.04.052MathSciNetView ArticleMATHGoogle Scholar
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