- Research Article
- Open Access
Quadratic-Quartic Functional Equations in RN-Spaces
© M. Eshaghi Gordji et al. 2009
- Received: 20 July 2009
- Accepted: 3 November 2009
- Published: 1 December 2009
We obtain the general solution and the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary -norms .
- Banach Space
- Vector Space
- General Solution
- Abelian Group
- Functional Equation
The stability problem of functional equations originated from a question of Ulam  in 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 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. Hyers  gave a first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that
for all and some Then there exists a unique additive mapping such that
for all Moreover, if is continuous in for each fixed then is -linear. In 1978, 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]). The functional equation
is related to a symmetric biadditive mapping. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation (1.3) is said to be a quadratic mapping. It is well known that a mapping between real vector spaces is quadratic if and only if there exits a unique symmetric biadditive mapping such that for all (see [5, 13]). The biadditive mapping is given by
The Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.3) was proved by Skof for mappings , where is a normed space and is a Banach space (see ). Cholewa  noticed that the theorem of Skof is still true if relevant domain is replaced an abelian group. In , Czerwik proved the Hyers-Ulam-Rassias stability of the functional equation (1.3). Grabiec  has generalized the results mentioned above.
In , Park and Bae considered the following quartic functional equation
In fact, they proved that a mapping between two real vector spaces and is a solution of (1.5) if and only if there exists a unique symmetric multiadditive mapping such that for all . It is easy to show that the function satisfies the functional equation (1.5), which is called a quartic functional equation (see also ). In addition, Kim  has obtained the Hyers-Ulam-Rassias stability for a mixed type of quartic and quadratic functional equation.
The Hyers-Ulam-Rassias stability of different functional equations in random normed and fuzzy normed spaces has been recently studied in [21–26]. It should be noticed that in all these papers the triangle inequality is expressed by using the strongest triangular norm .
The aim of this paper is to investigate the stability of the additive-quadratic functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary continuous -norms.
In this sequel, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [22, 23, 27–29]. Throughout this paper, is the space of distribution functions, that is, the space of all mappings such that is left-continuous and nondecreasing on and . Also, 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 point-wise 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 .
Typical examples of continuous -norms are , and (the Lukasiewicz -norm). Recall (see [30, 31]) that if is a -norm and is a given sequence of numbers in , then is defined recurrently by and for . is defined as It is known  that for the Lukasiewicz -norm, the following implication holds:
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:
(RN1) for all if and only if ;
(RN2) for all , ;
(RN3) for all and
for all and is the minimum -norm. This space is called the induced random normed space.
Let be an RN-space.
A sequence in is said to be convergent to in if, for every and , there exists a positive integer such that whenever .
A sequence in is called a Cauchy sequence if, for every and , there exists a positive integer such that whenever .
An 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.
In this paper, we deal with the following functional equation:
on RN-spaces. It is easy to see that the function is a solution of (1.9).
In Section 2, we investigate the general solution of the functional equation (1.9) when is a mapping between vector spaces and in Section 3, we establish the stability of the functional equation (1.9) in RN-spaces.
We need the following lemma for solution of (1.9). Throughout this section, and are vector spaces.
If a mapping satisfies (1.9) for all then is quadratic-quartic.
We show that the mappings defined by and defined by are quadratic and quartic, respectively.
Letting in (1.9), we have . Putting in (1.9), we get . Thus the mapping is even. Replacing by in (1.9), we get
for all . Therefore, the mapping is quadratic.
To prove that is quartic, we have to show that
for all .
Therefore, the mapping is quartic. This completes the proof of the lemma.
for all .
for all The proof of the converse is obvious.
Throughout this section, assume that is a real linear space and is a complete RN-space.
for all and all
Since the right-hand side of the inequality (3.17) tends to as and tend to infinity, the sequence is a Cauchy sequence. Thus we may define for all .
Now we show that is a quadratic mapping. Replacing with and in (3.1), respectively, we get
Taking the limit as , we find that satisfies (1.9) for all . By Lemma 2.1, the mapping is quadratic.
Letting the limit as in (3.16), we get (3.3) by (3.10).
Finally, to prove the uniqueness of the quadratic mapping subject to (3.3), let us assume that there exists another quadratic mapping which satisfies (3.3). Since for all and all from (3.3), it follows that
for all and all . Letting in (3.19), we conclude that , as desired.
for all and all
Since the right-hand side of (3.36) tends to as and tend to infinity, the sequence is a Cauchy sequence. Thus we may define for all .
Now we show that is a quartic mapping. Replacing with and in (3.20), respectively, we get
Taking the limit as , we find that satisfies (1.9) for all . By Lemma 2.1 we get that the mapping is quartic.
Letting the limit as in (3.35), we get (3.22) by (3.29).
Finally, to prove the uniqueness of the quartic mapping subject to let us assume that there exists a quartic mapping which satisfies (3.22). Since and for all and all from (3.22), it follows that
for all and all . Letting in (3.38), we get that , as desired.
for all and all
for all and all . Hence we obtain (3.41) by letting and for all The uniqueness property of and is trivial.
C. Park 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).
- Ulam SM: Problems in Modern Mathematics. Science edition, 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 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, 34. Birkhäuser, Basel, Switzerland; 1998:vi+313.View ArticleMATHGoogle 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
- Kannappan Pl: Quadratic functional equation and inner product spaces. Results in Mathematics 1995,27(3–4):368–372.MathSciNetView ArticleMATHGoogle Scholar
- Skof F: Proprieta' locali e approssimazione di operatori. Milan Journal of Mathematics 1983,53(1):113–129.MathSciNetGoogle Scholar
- Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984,27(1–2):76–86.MathSciNetView ArticleMATHGoogle Scholar
- Czerwik S: 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
- Park W, Bae J: 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
- Kim H: On the stability problem for a mixed type of quartic and quadratic functional equation. Journal of Mathematical Analysis and Applications 2006,324(1):358–372. 10.1016/j.jmaa.2005.11.053MathSciNetView 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, 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
- 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
- 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
- Chang SS, 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
- Gordji ME, Rassias JM, Savadkouhi MB: Stability of a mixed type additive and quadratic functional equation in random normed spaces. preprint preprintGoogle Scholar
- Gordji ME, Rassias JM, Savadkouhi MB: Approximation of the quadratic and cubic functional equation in RN-spaces. European Journal of Pure and Applied Mathematics 2009,2(4):494–507.MathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.