On the stability of pexider functional equation in non-archimedean spaces
- Reza Saadati^{1}Email author,
- Seiyed Mansour Vaezpour^{2} and
- Zahra Sadeghi^{1}
DOI: 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 equations1.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
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