- Research
- Open Access
- Published:
Fixed points and quadratic equations connected with homomorphisms and derivations on non-Archimedean algebras
Advances in Difference Equations volume 2012, Article number: 128 (2012)
Abstract
We apply the fixed point method to prove the stability of the systems of functional equations

on non-Archimedean Banach algebras. Moreover, we give some applications of our results in non-Archimedean Banach algebras over p-adic numbers.
MSC:34K36, 46S40, 47S40, 39B82, 39B52, 26E50.
1 Introduction and preliminaries
The story of the stability of functional equations dates back to 1925 when a stability result appeared in the celebrated book by Gy. Pólya and G. Szegő [1]. In 1940, Ulam [2] posed the famous Ulam stability problem which was partially solved by Hyers [3] in the framework of Banach spaces. Later, Aoki [4] considered the stability problem with unbounded Cauchy differences. In 1978, Th. M. Rassias [5] provided a generalization of the Hyers’ theorem by proving the existence of unique linear mappings near approximate additive mappings. Gǎvruta [6] obtained a generalized result of Th. M. Rassias’ theorem which allows the Cauchy difference to be controlled by a general unbounded function. On the other hand, J. M. Rassias [7–10] considered the Cauchy difference controlled by a product of different powers of norm. However, there was a singular case; for this singularity, a counterexample was given by Gǎvruta [11].
Bourgin [12] proved the stability of ring homomorphisms in two unital Banach algebras. Badora [13] gave a generalization of the Bourgin’s result. The stability result concerning derivations on operator algebras was first obtained by Šemrl [14]. In [15], Badora proved the stability of functional equation , where f is a mapping on normed algebra A with unit.
Let , be two algebras. A mapping is called a quadratic homomorphism if f is a quadratic mapping satisfying for all . For instance, let be commutative. Then the mapping , defined by (), is a quadratic homomorphism.
A mapping is called a quadratic derivation if f is a quadratic mapping satisfying for all . For instance, consider the algebra of matrices
Then it is easy to see that the mapping , defined by , is a quadratic derivation. We note that quadratic derivations and ring derivations are different.
Arriola and Beyer in [16] initiated the stability of functional equations in non-Archimedean spaces. In fact they established the stability of Cauchy functional equations over p-adic fields. After their results, some papers (see for instance [17–24]) 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 the Hyers’s method in [3]. In 2003, Radu [25] employed the alternative fixed point theorem, due to Diaz and Margolis [26], to prove the stability of the Cauchy additive functional equation. Subsequently, this method was applied to investigate the Hyers-Ulam stability for the Jensen functional equation [27], as well as for the Cauchy functional equation [28], by considering a general control function with suitable properties. Using such an elegant idea, several authors applied the method to investigate the stability of some functional equations, see [29–33].
Recently, Eshaghi and Khodaei [34] considered the following quadratic functional equation:
and proved the Hyers-Ulam stability of the above functional equation in classical Banach spaces. Recently, Eshaghi [35] proved the Hyers-Ulam stability of homomorphisms and derivations on non-Archimedean Banach algebras.
In the present paper, we adopt the idea of CÇŽdariu and Radu [28] to establish the stability of quadratic homomorphisms and quadratic derivations related to the quadratic functional equation (1.1) over non-Archimedean Banach algebras. Some applications of our results in non-Archimedean Banach algebras over p-adic numbers will be exhibited.
Now, we recall some notations and basic definitions used later on in the paper.
In 1897, Hensel [36] 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 problems coming from quantum physics, p-adic strings and superstrings [37, 38]. A key property of p-adic numbers is that they do not satisfy the Archimedean axiom: For , there exists such that (see [39, 40]).
Example 1.1 Let p be a prime number. For any nonzero rational number such that m and n are coprime to the prime number p, define the p-adic absolute value . Then ∥ is a non-Archimedean norm on . The completion of with respect to ∥ is denoted by and is called the p-adic number field. Note that if , then in for each integer n.
Let denote a field and function (valuation absolute) from into . 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, and for all . A trivial example of a non-Archimedean valuation is the function taking everything except 0 into 1 and . We always assume in addition that is nontrivial, i.e., there is a such that .
Let X be a linear space over a field with a non-Archimedean nontrivial valuation . A function is said to be a non-Archimedean norm if it is a norm over with the strong triangle inequality (ultrametric); namely, for all . Then is called a non-Archimedean space. In any such a space a sequence is Cauchy if and only if converges to zero. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent. A non-Archimedean Banach algebra is a complete non-Archimedean algebra which satisfies for all . For more details the reader is referred to [41, 42].
Let X be a nonempty set and satisfying: if and only if , and (strong triangle inequality), for all . Then is called a non-Archimedean generalized metric space. is called complete if every d-Cauchy sequence in X is d-convergent.
Using the strong triangle inequality in the proof of the main result of [26], we get to the following result:
Theorem 1.2 (Non-Archimedean Alternative Contraction Principle)
If is a non-Archimedean generalized complete metric space and a strictly contractive mapping (that is , for all and a Lipschitz constant ). Let , then either
-
(i)
for all , or
-
(ii)
there exists some such that for all ; the sequence is convergent to a fixed point of T; is the unique fixed point of T in the set
and for all y in this set.
2 Non-Archimedean approximately quadratic homomorphisms and quadratic derivations
Hereafter, unless otherwise stated, we will assume that and are two non-Archimedean Banach algebras. Also, let ; and we assume that in (i.e., the characteristic of is not 4).
Theorem 2.1 Let be fixed and let be a mapping with for which there exists a function such that
for all and nonzero fixed integers a, b. If there exists an such that
for all . Then there exists a unique quadratic homomorphism such that
for all , where
Proof Letting in (2.1), we get
for all . Setting in (2.4), we get
for all . It follows from (2.4) and (2.5) that
for all . Putting in (2.4), we get
for all . Setting in (2.4), we get
for all . Putting in (2.6), we get
for all . It follows from (2.8) and (2.9) that
for all . Replacing x and y by and in (2.4), respectively, we get
for all . Setting in (2.7), we get
for all . Putting in (2.10), we get

for all . It follows from (2.11), (2.12) and (2.13) that
for all . For every , define
Hence, d defines a complete generalized non-Archimedean metric on (see [27, 28, 33]). Let be defined by for all and all . If for some and ,
for all , then
for all , so
Hence T is a strictly contractive mapping on Ω with the Lipschitz constant L. It follows from (2.14) by using (2.2) that
and
for all , that is, .
Now, by the non-Archimedean alternative contraction principle, T has a unique fixed point in the set , which is defined by
for all . By (2.2),
for all . It follows from (2.1), (2.15) and (2.16) that

for all . This shows that is quadratic. Also,
for all . Therefore, is a quadratic homomorphism. Moreover, by the non-Archimedean alternative contraction principle,
This implies the inequality (2.3) holds. □
Corollary 2.2 Let , be non-Archimedean Banach algebras over , be fixed and θ, r be non-negative real numbers such that . Suppose that a mapping satisfies and

for all , where a, b are positive fixed integers. Then there exists a unique quadratic homomorphism such that
for all , where are integers and .
Proof The proof follows from Theorem 2.1, by taking
for all . Then we choose and we get the desired result. □
Corollary 2.3 Let , be non-Archimedean Banach algebras over , be fixed and δ, s be non-negative real numbers such that . Suppose that a mapping satisfies and

for all , where a, b are positive fixed integers. Then there exists a unique quadratic homomorphism such that

for all , where are integers and .
Theorem 2.4 Let be fixed and let be a mapping with if , for which there exists a function satisfying (2.2) and

for all and nonzero fixed integers a, b. Then there exists a unique quadratic derivation such that
for all , where is defined as in Theorem 2.1.
Proof By (2.2), if , we obtain . Letting in (2.17), we get . So for .
By the same reasoning as that in the proof of Theorem 2.1, there exists a unique quadratic mapping satisfying (2.18). The mapping is given by for all . It follows from (2.17) that

for all . Therefore is a quadratic derivation satisfying (2.18). □
Very recently, J. M. Rassias [43] considered the Cauchy difference controlled by the product and sum of powers of norms, that is, .
Corollary 2.5 Let be a non-Archimedean Banach algebra over , be fixed and ε, p, q be non-negative real numbers such that . Suppose that a mapping satisfies

for all , where a, b are positive fixed integers. Then there exists a unique quadratic derivation such that

for all , where are integers and .
References
Pólya G, Szegő G I. In Aufgaben und Lehrsätze aus der Analysis. Springer, Berlin; 1925.
Ulam SM: A Collection of Mathematical Problems. Interscience, New York; 1960.
Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222
Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 64–66. 10.2969/jmsj/00210064
Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1
Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431–436. 10.1006/jmaa.1994.1211
Rassias JM: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 1982, 46: 126–130. 10.1016/0022-1236(82)90048-9
Rassias JM: Solution of a problem of Ulam. J. Approx. Theory 1989, 57(3):268–273. 10.1016/0021-9045(89)90041-5
Rassias JM: On the stability of the Euler-Lagrange functional equation. Chin. J. Math. 1992, 20(2):185–190.
Rassias JM: Complete solution of the multi-dimensional problem of Ulam. Discuss. Math. 1994, 14: 101–107.
Gǎvruta P: An answer to a question of John M. Rassias concerning the stability of Cauchy equation. Hardronic Math. Ser. Advances in Equations and Inequalities 1999, 67–71.
Bourgin DG: Approximately isometric and multiplicative transformations on continuous function rings. Duke Math. J. 1949, 16: 385–397. 10.1215/S0012-7094-49-01639-7
Badora R: On approximate ring homomorphisms. J. Math. Anal. Appl. 2002, 276: 589–597. 10.1016/S0022-247X(02)00293-7
Šemrl P: The functional equation of multiplicative derivation is superstable on standard operator algebras. Integral Equ. Oper. Theory 1994, 18: 118–122. 10.1007/BF01225216
Badora R: On approximate derivations. Math. Inequal. Appl. 2006, 9: 167–173.
Arriola LM, Beyer WA: Stability of the Cauchy functional equation over p -adic fields. Real Anal. Exch. 2005/2006, 31: 125–132.
Cho YJ, Park C, Saadati R: Functional inequalities in non-Archimedean Banach spaces. Appl. Math. Lett. 2010, 23: 1238–1242. 10.1016/j.aml.2010.06.005
Cho YJ, Saadati R: Lattice non-Archimedean random stability of ACQ functional equation. Adv. Differ. Equ. 2011., 2011: Article ID 31
Cho YJ, Saadati R, Vahidi J:Approximation of homomorphisms and derivations on non-Archimedean Lie -algebras via fixed point method. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 373904
Eshaghi Gordji M, Khodabakhsh R, Khodaei H, Park C, Shin DY: A functional equation related to inner product spaces in non-Archimedean normed spaces. Adv. Differ. Equ. 2011., 2011: Article ID 37
Eshaghi Gordji M, Khodaei H, Khodabakhsh R: General quartic-cubic-quadratic functional equation in non-Archimedean normed spaces. U. P. B. Sci. Bull., Ser. A 2010, 72: 69–84.
Eshaghi Gordji M, Savadkouhi MB: Stability of cubic and quartic functional equations in non-Archimedean spaces. Acta Appl. Math. 2010, 110: 1321–1329. 10.1007/s10440-009-9512-7
Eshaghi Gordji M, Savadkouhi MB: Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces. Appl. Math. Lett. 2010, 23: 1198–1202. 10.1016/j.aml.2010.05.011
Najati A, Cho YJ: Generalized Hyers-Ulam stability of the pexiderized Cauchy functional equation in non-Archimedean spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 309026
Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4: 91–96.
Diaz JB, Margolis B: A fixed point theorem of the alternative for contractions on the generalized complete metric space. Bull. Am. Math. Soc. 1968, 126: 305–309.
Cădariu L, Radu V: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 2003., 4: Article ID 4
Cădariu L, Radu V: On the stability of the Cauchy functional equation: A fixed point approach. Grazer Math. Ber. 2004, 346: 43–52.
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
Cho YJ, Kang JI, Saadati R: Fixed points and stability of additive functional equations on the Banach algebras. J. Comput. Anal. Appl. 2012, 14: 1103–1111.
Gǎvruta P, Gǎvruta L: A new method for the generalized Hyers-Ulam-Rassias stability. Int. J. Nonlinear Anal. Appl. 2010, 1(2):11–18.
Jung S, Kim T: A fixed point approach to the stability of the cubic functional equation. Bol. Soc. Mat. Mexicana 2006, 12: 51–57.
Mirmostafaee AK: Hyers-Ulam stability of cubic mappings in non-Archimedean normed spaces. Kyungpook Math. J. 2010, 50: 315–327. 10.5666/KMJ.2010.50.2.315
Eshaghi Gordji M, Khodaei H: On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations. Abstr. Appl. Anal. 2009., 2009: Article ID 923476
Eshaghi Gordji M: Nearly ring homomorphisms and nearly ring derivations on non-Archimedean Banach algebras. Abstr. Appl. Anal. 2010., 2010: Article ID 393247
Hensel K: Uber eine neue Begrundung der Theorie der algebraischen Zahlen. Jahresber. Dtsch. Math.-Ver. 1897, 6: 83–88.
Khrennikov A: p-adic Valued Distributions in Mathematical Physics. Kluwer Academic, Dordrecht; 1994.
Khrennikov A: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic, Dordrecht; 1997.
Gouvêa FQ: p-adic Numbers. Springer, Berlin; 1997.
Vladimirov VS, Volovich IV, Zelenov EI: p-adic Analysis and Mathematical Physics. World Scientific, Singapore; 1994.
Murphy, GJ: Non-Archimedean Banach algebras. PhD thesis, University of Cambridge (1977)
Narici L: Non-Archimedian Banach spaces and algebras. Arch. Math. 1968, 19: 428–435. 10.1007/BF01898426
Ravi K, Arunkumar M, Rassias JM: Ulam stability for the orthogonally general Euler-Lagrange type functional equation. Int. J. Math. Stat. 2008, 3: 36–46.
Bouikhalene B, Elqorachi E, Rassias JM: The superstability of d’Alembert’s functional equation on the Heisenberg group. Appl. Math. Lett. 2010, 23: 105–109. 10.1016/j.aml.2009.08.013
Cao H-X, Lv J-R, Rassias JM: Superstability for generalized module left derivations and generalized module derivations on a Banach module. I. J. Inequal. Appl. 2009., 2009: Article ID 718020
Cao H-X, Lv J-R, Rassias JM: Superstability for generalized module left derivations and generalized module derivations on a Banach module. II. J. Inequal. Pure Appl. Math. 2009., 10(3): Article ID 85
Author information
Authors and Affiliations
Corresponding authors
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Eshaghi Gordji, M., Khodaei, H., Khodabakhsh, R. et al. Fixed points and quadratic equations connected with homomorphisms and derivations on non-Archimedean algebras. Adv Differ Equ 2012, 128 (2012). https://doi.org/10.1186/1687-1847-2012-128
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2012-128
Keywords
- fixed point approach
- non-Archimedean algebra
- quadratic homomorphism
- quadratic derivation
- stability