Abstract
Using the fixed point method, we prove the HyersUlam stability of an additivequadraticcubicquartic functional equation in matrix normed spaces.
MSC: 47L25, 47H10, 39B82, 46L07, 39B52.
Keywords:
operator space; fixed point; HyersUlam stability; additivequadraticcubicquartic functional equation1 Introduction and preliminaries
The abstract characterization given for linear spaces of bounded Hilbert space operators in terms of matricially normed spaces[1] implies that quotients, mapping spaces and various tensor products of operator spaces may again be regarded as operator spaces. Owing in part to this result, the theory of operator spaces is having an increasingly significant effect on operator algebra theory (see [2]).
The proof given in [1] appealed to the theory of ordered operator spaces [3]. Effros and Ruan [4] showed that one can give a purely metric proof of this important theorem by using the technique of Pisier [5] and Haagerup [6] (as modified in [7]).
The stability problem of functional equations originated from a question of Ulam [8] concerning the stability of group homomorphisms.
The functional equation
is called the Cauchy additive functional equation. In particular, every solution of the Cauchy additive functional equation is said to be an additive mapping. Hyers [9] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [10] for additive mappings and by TM Rassias [11] for linear mappings by considering an unbounded Cauchy difference. A generalization of the TM Rassias theorem was obtained by Găvruta [12] by replacing the unbounded Cauchy difference by a general control function in the spirit of TM Rassias’ approach.
In 1990, TM Rassias [13] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for . In 1991, Gajda [14], following the same approach as in TM Rassias [11], gave an affirmative solution to this question for . It was shown by Gajda [14], as well as by TM Rassias and Šemrl [15], that one cannot prove a TM Rassias’ type theorem when (cf. the books of Czerwik [16], Hyers et al.[17]).
In 1982, JM Rassias [18] followed the innovative approach of the TM Rassias’ theorem [11] in which he replaced the factor by for with .
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. A HyersUlam stability problem for the quadratic functional equation was proved by Skof [19] for mappings , where X is a normed space and Y is a Banach space. Cholewa [20] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [21] proved the HyersUlam stability of the quadratic functional equation.
In [22], Jun and Kim considered the following cubic functional equation:
It is easy to show that the function satisfies functional equation (1.1), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping.
In [23], Lee et al. considered the following quartic functional equation:
It is easy to show that the function satisfies functional equation (1.2), which is called a quartic functional equation, and every solution of the quartic functional equation is said to be a quartic mapping. 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 [2433]).
We will use the following notations:
is that component is 1 and the other components are zero;
is that component is x and the other components are zero;
Note that is a matrix normed space if and only if is a normed space for each positive integer n and holds for , and , and that is a matrix Banach space if and only if X is a Banach space and is a matrix normed space.
Let E, F be vector spaces. For a given mapping and a given positive integer n, define by
Let X be a set. A function is called a generalized metric on X if d satisfies
We recall a fundamental result in fixed point theory.
Letbe a complete generalized metric space and letbe a strictly contractive mapping with Lipschitz constant. Then, for each given element, either
for all nonnegative integersnor there exists a positive integersuch that
(2) the sequenceconverges to a fixed pointofJ;
(3) is the unique fixed point ofJin the set;
In 1996, Isac and Rassias [36] 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 [3743]).
In this paper, we prove the HyersUlam stability of the following additivequadraticcubicquartic functional equation:
in matrix normed spaces by using the fixed point method.
One can easily show that an odd mapping satisfies (1.3) if and only if the odd mapping is an additivecubic mapping, i.e.,
It was shown in [[44], Lemma 2.2] that and are cubic and additive, respectively, and that .
One can easily show that an even mapping satisfies (1.3) if and only if the even mapping is a quadraticquartic mapping, i.e.,
It was shown in [[45], Lemma 2.1] that and are quartic and quadratic, respectively, and that .
Throughout this paper, let be a matrix normed space and be a matrix Banach space.
2 HyersUlam stability of AQCQfunctional equation (1.3) in matrix normed spaces: odd mapping case
In this section, we prove the HyersUlam stability of AQCQfunctional equation (1.3) in matrix normed spaces for an odd mapping case.
Lemma 2.1Letbe a matrix normed space. Then:
Proof (1) Since and , . Since , . So .
(3) By
we get the result. □
Theorem 2.2Letbe a function such that there exists anwith
for all. Letbe an odd mapping satisfying
for all. Then there exists a unique additive mappingsuch that
Proof Let and except for in (2.2).
By (2.4) and (2.5),
for all . Replacing by and letting in (2.6), we get
Consider the set
and introduce the generalized metric on S:
where, as usual, . It is easy to show that is complete (see [46,47]).
Now we consider the linear mapping such that
for all . So implies that . This means that
By Theorem 1.1, there exists a mapping satisfying the following:
(1) A is a fixed point of J, i.e.,
for all . The mapping A is a unique fixed point of J in the set
This implies that A is a unique mapping satisfying (2.8) such that there exists a satisfying
(2) as . This implies the equality
(3) , which implies the inequality
So
It follows from (2.1) and (2.2) that
for all . Hence for all a, b. So is additive.
By Lemma 2.1 and (2.9),
for all . Thus is a unique additive mapping satisfying (2.3), as desired. □
Corollary 2.3Letr, θbe positive real numbers with. Letbe an odd mapping such that
for all. Then there exists a unique additive mappingsuch that
Proof The proof follows from Theorem 2.2 by taking for all . Then we can choose and we get the desired result. □
Theorem 2.4Letbe a function such that there exists anwith
for all. Letbe an odd mapping satisfying (2.2). Then there exists a unique additive mappingsuch that
Proof Let be the generalized metric space defined in the proof of Theorem 2.2.
Now we consider the linear mapping such that
It follows from (2.7) that
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 2.5Letr, θbe positive real numbers with. Letbe an odd mapping satisfying (2.10). Then there exists a unique additive mappingsuch that
Proof The proof follows from Theorem 2.4 by taking for all . Then we can choose and we get the desired result. □
Theorem 2.6Letbe a function such that there exists anwith
for all. Letbe an odd mapping satisfying (2.2). Then there exists a unique cubic mappingsuch that
Proof Let be the generalized metric space defined in the proof of Theorem 2.2.
Replacing by and letting in (2.6), we get
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 2.7Letr, θbe positive real numbers with. Letbe an odd mapping satisfying (2.10). Then there exists a unique cubic mappingsuch that
Proof The proof follows from Theorem 2.6 by taking for all . Then we can choose and we get the desired result. □
Theorem 2.8Letbe a function such that there exists anwith
for all. Letbe an odd mapping satisfying (2.2). Then there exists a unique cubic mappingsuch that
Proof Let be the generalized metric space defined in the proof of Theorem 2.2.
It follows from (2.11) that
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 2.9Letr, θbe positive real numbers with. Letbe an odd mapping satisfying (2.10). Then there exists a unique cubic mappingsuch that
Proof The proof follows from Theorem 2.8 by taking for all . Then we can choose and we get the desired result. □
3 HyersUlam stability of AQCQfunctional equation (1.3) in matrix normed spaces: even mapping case
In this section, we prove the HyersUlam stability of AQCQfunctional equation (1.3) in matrix normed spaces for an even mapping case.
Theorem 3.1Letbe a function such that there exists anwith
for all. Letbe an even mapping satisfyingand (2.2). Then there exists a unique quadratic mappingsuch that
Proof Let be the generalized metric space defined in the proof of Theorem 2.2.
By (3.1) and (3.2),
for all . Replacing by and letting in (3.3), we get
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 3.2Letr, θbe positive real numbers with. Letbe an even mapping satisfying (2.10). Then there exists a unique quadratic mappingsuch that
Proof The proof follows from Theorem 3.1 by taking for all . Then we can choose and we get the desired result. □
Theorem 3.3Letbe a function such that there exists anwith
for all. Letbe an even mapping satisfyingand (2.2). Then there exists a unique quadratic mappingsuch that
Proof Let be the generalized metric space defined in the proof of Theorem 2.2.
It follows from (3.4) that
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 3.4Letr, θbe positive real numbers with. Letbe an even mapping satisfying (2.10). Then there exists a unique quadratic mappingsuch that
Proof The proof follows from Theorem 3.3 by taking for all . Then we can choose and we get the desired result. □
Theorem 3.5Letbe a function such that there exists anwith
for all. Letbe an even mapping satisfyingand (2.2). Then there exists a unique quartic mappingsuch that
Proof Let be the generalized metric space defined in the proof of Theorem 2.2.
Replacing by and letting in (3.3), we get
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 3.6Letr, θbe positive real numbers with. Letbe an even mapping satisfying (2.10). Then there exists a unique quartic mappingsuch that
Proof The proof follows from Theorem 3.5 by taking for all . Then we can choose and we get the desired result. □
Theorem 3.7Letbe a function such that there exists anwith
for all. Letbe an even mapping satisfyingand (2.2). Then there exists a unique quartic mappingsuch that
Proof Let be the generalized metric space defined in the proof of Theorem 2.2.
It follows from (3.5) that
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 3.8Letr, θbe positive real numbers with. Letbe an even mapping satisfying (2.10). Then there exists a unique quartic mappingsuch that
Proof The proof follows from Theorem 3.7 by taking for all . Then we can choose and we get the desired result. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Acknowledgements
CP was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF2012R1A1A2004299), and DYS was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF20100021792).
References

Ruan, ZJ: Subspaces of algebras. J. Funct. Anal.. 76, 217–230 (1988). Publisher Full Text

Effros, E, Ruan, ZJ: On approximation properties for operator spaces. Int. J. Math.. 1, 163–187 (1990). Publisher Full Text

Choi, MD, Effros, E: Injectivity and operator spaces. J. Funct. Anal.. 24, 156–209 (1977). Publisher Full Text

Effros, E, Ruan, ZJ: On the abstract characterization of operator spaces. Proc. Am. Math. Soc.. 119, 579–584 (1993). Publisher Full Text

Pisier, G: Grothendieck’s theorem for noncommutative algebras with an appendix on Grothendieck’s constants. J. Funct. Anal.. 29, 397–415 (1978). PubMed Abstract  Publisher Full Text

Haagerup, U: Decomp. of completely bounded maps (unpublished manuscript)

Effros, E: On Multilinear Completely Bounded Module Maps, pp. 479–501. Am. Math. Soc., Providence (1987)

Ulam, SM: A Collection of the Mathematical Problems, Interscience, New York (1960)

Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA. 27, 222–224 (1941). PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Aoki, T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn.. 2, 64–66 (1950). Publisher Full Text

Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc.. 72, 297–300 (1978). Publisher Full Text

Gǎvruta, P: A generalization of the HyersUlamRassias stability of approximately additive mappings. J. Math. Anal. Appl.. 184, 431–436 (1994). Publisher Full Text

Rassias, TM: Problem 16; 2. Report of the 27th international symp. on functional equations. Aequ. Math.. 39, 292–293 (1990) 309

Gajda, Z: On stability of additive mappings. Int. J. Math. Math. Sci.. 14, 431–434 (1991). Publisher Full Text

Rassias, TM, Šemrl, P: On the behaviour of mappings which do not satisfy HyersUlam stability. Proc. Am. Math. Soc.. 114, 989–993 (1992). Publisher Full Text

Czerwik, S: Functional Equations and Inequalities in Several Variables, World Scientific, Singapore (2002)

Hyers, DH, Isac, G, Rassias, TM: Stability of Functional Equations in Several Variables, Birkhäuser, Basel (1998)

Rassias, JM: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal.. 46, 126–130 (1982). Publisher Full Text

Skof, F: Proprietà locali e approssimazione di operatori. Rend. Semin. Mat. Fis. Milano. 53, 113–129 (1983). Publisher Full Text

Cholewa, PW: Remarks on the stability of functional equations. Aequ. Math.. 27, 76–86 (1984). Publisher Full Text

Czerwik, S: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb.. 62, 59–64 (1992). Publisher Full Text

Jun, K, Kim, H: The generalized HyersUlamRassias stability of a cubic functional equation. J. Math. Anal. Appl.. 274, 867–878 (2002). Publisher Full Text

Lee, S, Im, S, Hwang, I: Quartic functional equations. J. Math. Anal. Appl.. 307, 387–394 (2005). Publisher Full Text

Aczel, J, Dhombres, J: Functional Equations in Several Variables, Cambridge University Press, Cambridge (1989)

Amyari, M, Park, C, Moslehian, MS: Nearly ternary derivations. Taiwan. J. Math.. 11, 1417–1424 (2007)

Chou, CY, Tzeng, JH: On approximate isomorphisms between Banach ∗algebras or algebras. Taiwan. J. Math.. 10, 219–231 (2006)

Eshaghi Gordji, M, Savadkouhi, MB: Stability of a mixed type cubicquartic functional equation in nonArchimedean spaces. Appl. Math. Lett.. 23, 1198–1202 (2010). Publisher Full Text

Isac, G, Rassias, TM: On the HyersUlam stability of ψadditive mappings. J. Approx. Theory. 72, 131–137 (1993). Publisher Full Text

Jun, K, Lee, Y: A generalization of the HyersUlamRassias stability of the Pexiderized quadratic equations. J. Math. Anal. Appl.. 297, 70–86 (2004). Publisher Full Text

Jung, S: HyersUlamRassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor (2001)

Park, C: Homomorphisms between Poisson algebras. Bull. Braz. Math. Soc.. 36, 79–97 (2005). Publisher Full Text

Park, C, Ghaleh, SG, Ghasemi, K: nJordan ∗homomorphisms in algebras. Taiwan. J. Math.. 16, 1803–1814 (2012)

Rassias, JM: Solution of a problem of Ulam. J. Approx. Theory. 57, 268–273 (1989). Publisher Full Text

Cădariu, L, Radu, V: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math.. 4(1), (2003) Article ID 4

Diaz, J, Margolis, B: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc.. 74, 305–309 (1968). Publisher Full Text

Isac, G, Rassias, TM: Stability of ψadditive mappings: applications to nonlinear analysis. Int. J. Math. Math. Sci.. 19, 219–228 (1996). Publisher Full Text

Cădariu, L, Radu, V: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber.. 346, 43–52 (2004)

Cădariu, L, Radu, V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl.. 2008, (2008) Article ID 749392

Jung, Y, Chang, I: The stability of a cubic type functional equation with the fixed point alternative. J. Math. Anal. Appl.. 306, 752–760 (2005). Publisher Full Text

Mirzavaziri, M, Moslehian, MS: A fixed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc.. 37, 361–376 (2006). Publisher Full Text

Park, C: Fixed points and HyersUlamRassias stability of CauchyJensen functional equations in Banach algebras. Fixed Point Theory Appl.. 2007, (2007) Article ID 50175

Park, C: Generalized HyersUlam stability of functional equations: a fixed point approach. Taiwan. J. Math.. 14, 1591–1608 (2010)

Radu, V: The fixed point alternative and the stability of functional equations. Fixed Point Theory. 4, 91–96 (2003)

Eshaghi Gordji, M, KaboliGharetapeh, S, Park, C, Zolfaghari, S: Stability of an additivecubicquartic functional equation. Adv. Differ. Equ.. 2009, (2009) Article ID 395693

Eshaghi Gordji, M, Abbaszadeh, S, Park, C: On the stability of a generalized quadratic and quartic type functional equation in quasiBanach spaces. J. Inequal. Appl.. 2009, (2009) Article ID 153084

Miheţ, D, Radu, V: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl.. 343, 567–572 (2008). Publisher Full Text

Park, C: Generalized HyersUlamRassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl.. 2008, (2008) Article ID 493751