- Research
- Open Access
On the stability of pexider functional equation in non-archimedean spaces
- Reza Saadati^{1}Email author,
- Seiyed Mansour Vaezpour^{2} and
- Zahra Sadeghi^{1}
https://doi.org/10.1186/1029-242X-2011-17
© Saadati et al; licensee Springer. 2011
- Received: 7 January 2011
- Accepted: 24 June 2011
- Published: 24 June 2011
Abstract
Keywords
- Hyers-Ulam stability of functional equation
- Non-Archimedean space
- Quadratic
- Cauchy and Pexider functional equations
1.Introduction
The stability problem for functional equations first was planed in 1940 by Ulam [1]:
Let G_{1} be group and G_{2} be a metric group with the metric d(·,·). Does, for any ε > 0, there exists δ > 0 such that, for any mapping f : G_{1} → G_{2} which satisfies d(f(xy), f(x)f(y)) ≤ δ for all x, y ∈ G_{1}, there exists a homomorphism h : G_{1} → G_{2} so that, for any x ∈ G_{1}, we have d(f (x), h(x)) ≤ ε?
In 1941, Hyers [2] answered to the Ulam's question when G_{1} and G_{2} are Banach spaces. Subsequently, the result of Hyers was generalized by Aoki [3] for additive mappings and Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [4] has provided a lot of influences in the development of the Hyers-Ulam-Rassias stability of functional equations (for more details, see [5] where a discussion on definitions of the Hyers-Ulam stability is provided by Moszner, also [6–12]).
In this paper, we give a modification of the approach of Belaid et al. [13] in non-Archimedean spaces. Recently, Ciepliński [14] studied and proved stability of multi-additive mappings in non-Archimedean normed spaces, also see [15–22].
- (1)
|a| ≥ 0;
- (2)
|a| = 0 if and only if a = 0;
- (3)
|ab| = |a| |b|
- (4)
|a + b| ≤ max{|a|, |b|};
- (5)
there exists a member a _{0} ∈ K such that |a _{0}| ≠ 0, 1.
A field K with a non-Archimedean valuation is called a non-Archimedean field.
Corollary 1.2. |-1| = |1| = 1 and so, for any a ∈ K, we have |-a| = |a|. Also, if |a| < |b| for any a, b ∈ K, then |a + b| = |b|.
- (1)
||u|| = 0 if and only if u = 0;
- (2)
||αu|| = |α| ||u||;
- (3)
||u + v|| ≤ max{||u||, ||v||}.
Since 0 = ||0|| = ||v - v|| ≤ max{||v||, ||-v||} = ||v|| for any v ∈ V, we have ||v|| ≥ 0. Any vector space V with a non-Archimedean norm || · || : V → ℝ is called a non-Archimedean space. If the metric d(u, v) = ||u - v|| is induced by a non-Archimedean norm || · || : V → ℝ on a vector space V which is complete, then (V, || · ||) is called a complete non-Archimedean space.
Proposition 1.4. ([23]) A sequence in a non-Archimedean space is a Cauchy sequence if and only if the sequence converges to zero.
Since any non-Archimedean norm satisfies the triangle inequality, any non-Archimedean norm is a continuous function from its domain to real numbers.
Proposition 1.5. Let V be a normed space and E be a non-Archimedean space. Let f : V → E be a function, continuous at 0 ∈ V such that, for any × ∈ V, f(2x) = 2f(x) (for example, additive functions). Then, f = 0.
Since this inequality holds for all ε > 0, it follows that, for any x ∈ V, f(x) = 0. This completes the proof.
The preceding fact is a special case of a general result for non-Archimedean spaces, that is, every continuous function from a connected space to a non-Archimedean space is constant. This is a consequence of totally disconnectedness of every non-Archimedean space (see [23]).
2. Stability of quadratic and Cauchy functional equations
for all x ∈ V_{1}.
for all x, y ∈ V_{1}.
and so Proposition 1.4 and the hypothesis Equation 2.1 imply that is a Cauchy sequence. Since V_{2} is complete, the sequence converges to a point of V_{2} which defines a mapping q : V_{1} → V_{2}.
Therefore, by induction on n, Equation 2.13 follows from Equation 2.12. Taking the limit of both sides of Equation 2.13, we prove that q satisfies Equation 2.6.
Thus, q is a solution of the Equation 2.5 which satisfies Equation 2.6.
for all n ∈ ℕ.
we have q(x) = q'(x) for all x ∈ V_{1}. This completes the proof.
which has been established in [20].
for all x, y ∈ V_{1}
3. Stability of the Pexider functional equation
for all x, y ∈ V_{1}.
for all x ∈ V_{1}.
for all x, y ∈ V_{1}.
for all x ∈ V_{1}, we deduce for all x ∈ V_{1}.
This proves Equation 3.3. Similarly, one can prove Equations 3.4 to 3.6.
Declarations
Acknowledgements
The authors would like to thank the referee and area editor Professor Ondrĕj Došlý for giving useful suggestions and comments for the improvement of this paper.
Authors’ Affiliations
References
- Ulam SM: Problems in Modern Mathematics, Chapter IV, Science Editions. Wiley, New York; 1960.Google Scholar
- Hyers DH: On the stability of the linear functional equation. Proc Nat Acad Sci USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar
- Aoki T: On the stability of the linear transformation in Banach spaces. J Math Soc Jpn 1950, 2: 64–66. 10.2969/jmsj/00210064View ArticleGoogle Scholar
- Rassias THM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View ArticleGoogle Scholar
- Moszner Z: On the stability of functional equations. Aequationes Math 2009, 77: 33–88. 10.1007/s00010-008-2945-7MathSciNetView ArticleGoogle Scholar
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ; 2002.Google Scholar
- Hyers DH, Isac G, Rassias THM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.View ArticleGoogle Scholar
- Jung SM: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor; 2001.Google Scholar
- Rassias TM: On the stability of functional equations and a problem of Ulam. Acta Appl Math 2000, 62: 23–130. 10.1023/A:1006499223572MathSciNetView ArticleGoogle Scholar
- Rassias THM: Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht; 2003.View ArticleGoogle Scholar
- Ciepliński K: Generalized stability of multi-additive mappings. Appl Math Lett 2010,23(10):1291–1294. 10.1016/j.aml.2010.06.015MathSciNetView ArticleGoogle Scholar
- Ciepliński K: Stability of the multi-Jensen equation. J Math Anal Appl 2010,363(1):249–254. 10.1016/j.jmaa.2009.08.021MathSciNetView ArticleGoogle Scholar
- Bouikhalene B, Elqorachi E, Rassias THM: On the Hyers-Ulam stability of approximately Pexider mappings. Math Inequal Appl 2008, 11: 805–818.MathSciNetGoogle Scholar
- Ciepliński K: Stability of multi-additive mappings in non-Archimedean normed spaces. J Math Anal Appl 2011, 373: 376–383. 10.1016/j.jmaa.2010.07.048MathSciNetView ArticleGoogle Scholar
- Cho YJ, Park C, Saadati R: Functional inequalities in non-Archimedean in Banach spaces. Appl Math Lett 2010, 60: 1994–2002.MathSciNetGoogle Scholar
- Mirmostafaee AK: Stability of quartic mappings in non-Archimedean normed spaces. Kyungpook Math J 2009, 49: 289–297.MathSciNetView ArticleGoogle Scholar
- Moslehian MS, Sadeghi GH: A Mazur-Ulam theorem in non-Archimedean normed spaces. Nonlinear Anal 2008, 69: 3405–3408. 10.1016/j.na.2007.09.023MathSciNetView ArticleGoogle Scholar
- Moslehian MS, Sadeghi GH: Stability of two types of cubic functional equations in non-Archimedean spaces. Real Anal Exch 2008, 33: 375–384.MathSciNetGoogle Scholar
- Moslehian MS, Rassias THM: Stability of functional equations in non-archimedean spaces. Appl Anal Discrete Math 2007, 1: 325–334. 10.2298/AADM0702325MMathSciNetView ArticleGoogle Scholar
- Moslehian MS, Rassias THM: Stability of functional equations in non-Archimedean spaces. Appl Anal Discrete Math 2007, 1: 325–334. 10.2298/AADM0702325MMathSciNetView ArticleGoogle Scholar
- Najati A, Moradlou F: Hyers-Ulam-Rassias stability of the Apollonius type quadratic mapping in non-Archimedean spaces. Tamsui Oxf J Math Sci 2008, 24: 367–380.MathSciNetGoogle Scholar
- Saadati R, Cho YJ, Vahidi J: The stability of the quartic functional equation in various spaces. Comput Math Appl 2010, 60: 1994–2002. 10.1016/j.camwa.2010.07.034MathSciNetView ArticleGoogle Scholar
- Schneider P: Non-Archimedean Functional Analysis. Springer, New York; 2002.View ArticleGoogle Scholar
- Bouikhalene B, Elqorachi E, Rassias THM: On the generalized Hyers-Ulam stability of the quadratic functional equation with a general involution. Nonlinear Funct Anal Appl 2007,12(2):247–262.MathSciNetGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.