Stability and superstability of generalized quadratic ternary derivations on non-Archimedean ternary Banach algebras: a fixed point approach

Using fixed point method, we prove the Hyers-Ulam stability and the superstability of generalized quadratic ternary derivations on non-Archimedean ternary Banach algebras. Indeed, we investigate the Hyers-Ulam stability and the superstability of the system of functional equations

in non-Archimedean ternary Banach algebras.

Mathematics Subject Classification 2010: Primary 39B52; 47H10; 39B72; 46L57; 17B40; 13N15; 17A40; 20N10.

The stability problem of functional equations had been first raised by Ulam [1]. This problem solved by Hyers [2] in the framework of Banach spaces. In 1978, Th.M. Rassias [3] provided a generalization of the Hyers' theorem by proving the existence of unique linear mappings near approximate additive mappings. Găvruta [4] obtained generalized result of the Th.M. Rassias' theorem which allows the Cauchy difference to be controlled by a general unbounded function.

Bourgin [5] proved the stability of ring homomorphisms between two unital Banach algebras and Badora [6] gave a generalization of the Bourgin result. The stability result concerning derivations between operator algebras was first obtained by Šemrl [7]. In [8], Badora proved the stability of functional equation

where f is a mapping on normed algebra A with the unit. Park et al. proved the stability of homomorphisms and derivations in Banach algebras, Banach ternary algebras, C*-algebras, Lie C*-algebras and C*-ternary algebras (see [9-14]).

Let be a ternary algebra. A mapping is called a quadratic ternary derivation if f is a quadratic mapping satisfies

for all .

A mapping is called a generalized quadratic ternary derivation if there exists a quadratic ternary derivation such that

for all .

Let denote a field and function (valuation absolute) | · | from into [0, ∞). A non-Archimedean valuation is a function | · | that satisfies the strong triangle inequality, namely,

for all . The associated field is referred to as a non-Archimedean field. Clearly, |1| = | - 1| = 1 and |n| ≤ 1 for all n ≥ 1. A trivial example of a non-Archimedean valuation is the function | · | taking everything except 0 into 1 and |0| = 0. We always assume in addition that | · | is non trivial, i.e., there exists such that |z| ≠ 0, 1.

Let X be a linear space over a field with a non-Archimedean nontrivial valuation | · |. A function ∥ · ∥ : X → [0, ∞) is said to be a non-Archimedean norm if it is a norm over with the strong triangle inequality (ultrametric), namely,

for all x, y ∈ X. Then (X, ∥ · ∥) is called a non-Archimedean space. In any such a space a sequence {x_n}_n∈ℕ is a Cauchy sequence if and only if {x_n+1-x_n}_n∈ℕ converges to zero. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.

A non-Archimedean ternary Banach algebra is a complete non-Archimedean space equipped with a ternary product (x,y,z) → [x,y,z] of into which is -linear in each variables and associative in the sense that

and satisfies the following:

(see [15-19]).

Arriola and Beyer [20] initiated the stability of functional equations in non-Archimedean spaces. In fact, they established stability of Cauchy functional equations over p-adic fields. After their results some papers (see, for instance, [21-27]) on the stability of other equations in such spaces have been published. Although different methods are known for establishing the stability of functional equations, almost all proofs depend on Hyers' method in [2]. In 2003, Radu [28] employed the alternative fixed point theorem, due to Diaz and Margolis [29], to prove the stability of Cauchy additive functional equation. Subsequently, this method was applied to investigate the Hyers-Ulam stability for Jensen functional equation [30], as well as for the Cauchy functional equation [31], by considering a general control function φ(x, y), with suitable properties. Using such an elegant idea, several authors applied the method to investigate the stability of some functional equations (see [12,32-34]).

Recently, Eshaghi Gordji and Khodaei [35] proved the Hyers-Ulam stability of the following quadratic functional equation

for nonzero fixed integers a, b. Recently, Eshaghi Gordji and Alizadeh [36,37] proved the Hyers-Ulam stability of homomorphisms and derivations on non-Archimedean Banach algebras.

In this paper, by using fixed point method, we establish the stability of generalized quadratic ternary derivations related to the quadratic functional equation (1.1) over non-Archimedean ternary Banach algebras.

In 1897, Hensel [38] discovered the p-adic numbers as a number theoretical analogue of power series in complex analysis. During the last three decades p-adic numbers have gained the interest of physicists for their research, in particular, in the problems coming from quantum physics, p-adic strings and superstrings [39,40]. A key property of p-adic numbers is that they do not satisfy the Archimedean axiom: For any x, y > 0, there exists n ∈ ℕ such that x < ny (see [41,42]).

Using the strong triangle inequality in the proof of the main result of [29], we get the following result:

Theorem 2.1. (Non-Archimedean Alternative Contraction Principle) Let (Ω,d) be a non-Archimedean generalized complete metric space and T :Ω → Ω a strictly contractive mapping (that is, d(T(x),T(y)) ≤ Ld(x, y) for all x, y ∈ T and a Lipschitz constant L < 1). Let x ∈ Ω. If either

(a) d(Tⁿ(x),Tⁿ⁺¹(x)) = ∞ for all n ≥ 0, or

(b) there exists some n₀ ≥ 0 such that d(Tⁿ(x),Tⁿ⁺¹(x)) < ∞ for all n ≥ n₀, then the sequence {Tⁿ(x)} is convergent to a unique fixed point x* of T in the set

and d(y, x*) ≤ d(y,T(y)) for all y in this set.

From now on, we assume that is a non-Archimedean ternary Banach algebra and ℓ ∈ {-1,1} is fixed. Also, we suppose that |4| < 1 and that 4 ≠ 0 in (i.e., the characteristic of is not 4). We denote [a, b, c] by [abc] in ternary Banach algebra .

Theorem 2.2. Let be two mappings with g(0) = f(0) = 0 for which there exists a function such that

for all and nonzero fixed integers a, b. Suppose that there exists L < 1 such that

for all . Then there exist a unique quadratic ternary derivation and a unique generalized quadratic ternary derivation (respected to d) such that

for all , where

for all .

Proof. By (2.2), one can show that

for all . Putting h = g in (2.1) and letting u = v = w = r = s = t = 0 in (2.1), we get

for all . Putting y = 0 in (2.5), we get

for all . Setting y = -y in (2.5), we get

for all . It follows from (2.5) and (2.7) that

for all . Putting y = by in (2.8), we get

for all . Setting x = 0 in (2.5), we get

for all . It follows from (2.9) and (2.10) that

for all . Replacing x and y by and in (2.5), respectively, we get

for all . Setting in (2.6), we get

for all . Putting in (2.11), we get

for all . It follows from (2.12), (2.13) and (2.14) that

for all . Let . For every g', h' ∈ Ω, define

It is easy to show that ρ is a complete generalized non-Archimedean metric on Ω (see [30,31,34]). We define J : Ω → Ω by for all and all g' ∈ Ω. One can show that

Hence J is a strictly contractive mapping on Ω with Lipschitz constant L. It follows from Theorem 2.1 that J has a unique fixed point in the set Λ = {g' ∈ Ω : ρ(g, g') < ∞}, where d is defined by

for all . It follows from (2.4) and (2.15) that

for all . This shows that D is quadratic.

If is another quadratic mapping which satisfies (2.3), then D' is a fixed point of J in Λ. The uniqueness of the fixed point of J in Λ implies that D = D'. Putting h = f, u = v = w = r = s = t = 0 in (2.4), we get

for all . By the same reasoning as above, we can show that the limit

exists for all . Moreover, we can show that d is a unique quadratic mapping on satisfying (2.3).

On the other hand, we have

for all . Therefore, d is a quadratic ternary derivation on . Also, we have

for all . It follows that D is a generalized quadratic ternary derivation (related to d) on .This completes the proof.

From now on, we use the following abbreviation for any mappings :

Remark. Let be the 2-adic number field. Let be a non-Archimedean Banach algebra on . Let ε be a nonnegative real number and let s be a real number such that s > 6 if ℓ = 1 and 0 < s < 2 if ℓ = -1. Suppose that the mappings satisfy g(0) = f(0) = 0 and

for all . Then there exist a unique quadratic ternary derivation and a unique generalized quadratic ternary derivation (respected to d) such that

max{||g(x) - D(x)||, ||f(x) - d(x)||}

for all , where i,j,k,m ≥ 1 are integers and gcd(k, 2) = gcd(m, 2) = 1.

Now, we have the following result on superstability of generalized quadratic ternary derivations on non-Archimedean ternary Banach algebras:

Corollary 2.3. Let p > 0 be a nonnegative real number such that and let j ∈ {3, 4, ..., 8} be fixed. Suppose that the mappings satisfy g(0) = f(0) = 0 and

for all , where a, b are positive fixed integers. Then f is a quadratic ternary derivation and g is a generalized quadratic ternary derivation related to f.

Proof. It follows from Theorem 2.2 by taking

for all and putting .

The authors declare that they have no competing interests.

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.

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-2012R1A1A2004299). Y.J. Cho was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2011-0021821).

1. Ulam, SM: Problems in Modern Mathematics, Science ed. Wiley, New York (1940)

2. 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

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

4. Găvruta, P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl. 184, 431–436 (1994). Publisher Full Text

5. Bourgin, DG: Approximately isometric and multiplicative transformations on continuous function rings. Duke Math J. 16, 385–397 (1949). Publisher Full Text

6. Badora, R: On approximate ring homomorphisms. J Math Anal Appl. 276, 589–597 (2002). Publisher Full Text

7. Šemrl, P: The functional equation of multiplicative derivation is superstable on standard operator algebras. Integral Equations Operator Theory. 18, 118–122 (1994). Publisher Full Text

8. Badora, R: On approximate derivations. Math Inequal Appl. 9, 167–173 (2006)

9. Cho, Y, Saadati, R, Vahidi, J: Approximation of homomorphisms and derivations on non-Archimedean Lie C*-algebras via fixed point method. Discrete Dynamics in Nature and Society. 2012, 9 Article ID 373904 (2012)

10. Park, C: On an approximate automorphism on a C*-algebra. Proc Amer Math Soc. 132, 1739–1745 (2004). Publisher Full Text

11. Park, C, Hou, J, Oh, S: Homomorphisms between Lie JC*-algebras and Lie C*-algebra. Acta Math Sin. 21, 1391–1398 (2005). Publisher Full Text

12. Park, C, Rassias, JM: Stability of the Jensen type functional equation in C*-algebras: a fixed point approach. Abstr Appl Anal. 2009, 17 Article ID 360432 (2009)

13. Park, C, Rassias, TM: Homomorphisms in C*-ternary algebras and JB*-triples. J Math Anal Appl. 337, 13–20 (2008). Publisher Full Text

14. Park, C, Rassias, TM: Homomorphisms and derivations in proper JCQ*-triples. J Math Anal Appl. 337, 1404–1414 (2008). Publisher Full Text

15. Haag, R, Kastler, D: An algebraic approach to quantum field theory. J Math Phys. 5, 848–861 (1964). Publisher Full Text

16. Bagarello, F, Morchio, G: Dynamics of mean-field spin models from basic results in abstract differential equations. J Stat Phys. 66, 849–866 (1992). Publisher Full Text

17. Murphy, GJ: Non-Archimedean Banach Algebras. (Thesis) University of Cambridge (1977)

18. Narici, L: Non-Archimedian Banach spaces and algebras. Arch Math (Basel). 19, 428–435 (1968). Publisher Full Text

19. Shilkret, N: Non-Archimedian Banach Algebras. Thesis (Ph.D.), Polytechnic Univ., Pro-Quest LLC (1968)

20. Arriola, LM, Beyer, WA: Stability of the Cauchy functional equation over p-adic fields. Real Analysis Exchange. 31, 125–132 (2005)

21. Cho, Y, Park, C, Saadati, R: Functional inequalities in non-Archimedean Banach spaces. Appl Math Lett. 10, 1238–1242 (2010)

22. Cho, Y, Saadati, R: Lattictic non-Archimedean random stability of ACQ-functional equation. Adv Difference Equ. 2011, 12 Article No. 31 (2011)

    Article No. 31

    BioMed Central Full Text

23. Eshaghi Gordji, M, Khodaei, H, Khodabakhsh, R: General quartic-cubic-quadratic functional equation in non-Archimedean normed spaces. UPB Sci Bull (Series A). 72, 69–84 (2010)

24. Eshaghi Gordji, M, Savadkouhi, MB: Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces. Appl Math Lett. 23, 1198–1202 (2010). Publisher Full Text

25. Najati, A, Moradlou, F: Hyers-Ulam-Rassias stability of the Aopollonius type quadratic mapping in non-Archimedean spaces. Tamsui Oxford J Math Sci. 24, 367–380 (2008)

26. Saadati, R, Cho, Y, Vahidi, J: The stability of the quartic functional equation in various spaces. Comput Math Appl. 60, 1994–2002 (2010). Publisher Full Text

27. Saadati, R, Park, C: Non-Archimedean -fuzzy normed spaces and stability of functional equations. Comput Math Appl. 60, 2488–2496 (2010). Publisher Full Text

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

29. Diaz, JB, Margolis, B: A fixed point theorem of the alternative for contractions on the generalized complete metric space. Bull Amer Math Soc. 126, 305–309 (1968)

30. Cădariu, L, Radu, V: Fixed points and the stability of Jensen's functional equation. J Inequal Pure Appl Math. 4, 7 no. 1, Article ID 4 (2003)

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

32. Cădariu, L, Radu, V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008, 15 Article ID 749392 (2008)

33. Jung, S, Kim, T: A fixed point approach to the stability of the cubic functional equation. Bol Soc Mat Mexicana. 12, 51–57 (2006)

34. Mirmostafaee, AK: Hyers-Ulam stability of cubic mappings in non-Archimedean normed spaces. Kyungpook Math J. 50, 315–327 (2010). Publisher Full Text

35. Eshaghi Gordji, M, Khodaei, H: On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations. Abstr Appl Anal. 2009, 11 Article ID 923476 (2009)

36. Eshaghi Gordji, M: Nearly ring homomorphisms and nearly ring derivations on non-Archimedean Banach algebras. Abstr Appl Anal. 2010, 12 Article ID 393247 (2010)

37. Eshaghi Gordji, M, Alizadeh, Z: Stability and superstability of ring homomorphisms on non-Archimedean Banach algebra. Abstr Appl Anal. 2011, 10 Article ID 123656 (2011)

38. Hensel, K: Uber eine neue Begrundung der Theorie der algebraischen Zahlen. Jahresber Deutsch Math Verein. 6, 83–88 (1897)

39. Khrennikov, A: p-adic Valued Distributions in Mathematical Physics. Kluwer Academic Publishers, Dordrecht (1994)

40. Khrennikov, A: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic Publishers, Dordrecht (1997)

41. Gouvêa, FQ: p-adic Numbers. Springer-Verlag, Berlin (1997)

42. Vladimirov, VS, Volovich, IV, Zelenov, EI: p-Adic Analysis and Mathematical Physics. World Scientific (1994)