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On the stability of pexider functional equation in non-archimedean spaces
Journal of Inequalities and Applications volume 2011, Article number: 17 (2011)
Abstract
In this paper, the Hyers-Ulam stability of the Pexider functional equation
in a non-Archimedean space is investigated, where σ is an involution in the domain of the given mapping f.
MSC 2010:26E30, 39B52, 39B72, 46S10
1.Introduction
The stability problem for functional equations first was planed in 1940 by Ulam [1]:
Let G1 be group and G2 be a metric group with the metric d(·,·). Does, for any ε > 0, there exists δ > 0 such that, for any mapping f : G1 → G2 which satisfies d(f(xy), f(x)f(y)) ≤ δ for all x, y ∈ G1, there exists a homomorphism h : G1 → G2 so that, for any x ∈ G1, we have d(f (x), h(x)) ≤ ε?
In 1941, Hyers [2] answered to the Ulam's question when G1 and G2 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].
Definition 1.1. The function | · | : K → ℝ is called a non-Archimedean valuation or absolute value over the field K if it satisfies following conditions: for any a, b ∈ K,
-
(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|.
In a non-Archimedean field, the triangle inequality is satisfied and so a metric is defined. But an interesting inequality changes the usual Archimedean sense of the absolute value. For any n ∈ ℕ, we have |n · 1| ≤ ℝ. Thus, for any a ∈ K, n ∈ ℕ and nonzero divisor k ∈ ℤ of n, the following inequalities hold:
Definition 1.3. Let V be a vector space over a non-Archimedean field K. A non-Archimedean norm over V is a function || · || : V → R satisfying the following conditions: for any α ∈ K and u, v ∈ V,
-
(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 sequencein a non-Archimedean space is a Cauchy sequence if and only if the sequenceconverges 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.
Proof. Since f(0) = 0, for any ε > 0, there exists δ > 0 that, for any x ∈ V with ||x|| ≤ δ,
and, for any x ∈ V, there exists n ∈ ℕ that and hence
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
Throughout this section, we assume that V1 is a normed space and V2 is a complete non-Archimedean space. Let σ : V1 → V1 be a continuous involution (i.e., σ (x + y) = σ (x) + σ (y) and σ (σ (x)) = x) and φ : V1 × V1 → ℝ be a function with
and define a function ϕ : V1 × V1 → ℝ by
which easily implies
Theorem 2.1. Suppose that φ satisfies the condition 2.1 and let ϕ is defined by Equation 2.2. If f : V1 → V2satisfies the inequality
for all x, y ∈ V1, then there exists a unique solution q : V1 → V2of the functional equation
such that
for all x ∈ V1.
Proof. Replacing x and y in Equation 2.4 with and , respectively, we obtain
Replacing x and y in Equation 2.4 with and , respectively, we obtain
Also, replacing both of x, y in Equation 2.4 with , we get
and so, for any n ∈ ℕ, we get
Similarly, replacing both of x, y in Equation 2.4 with , we get
Replacing x in Equation 2.7 with , we obtain
for all x ∈ V1 and so, by assumption Equation 2.1,
Thus, f(0) = 0 and the inequality Equation 2.10 reduces to
and so,
For any n ∈ ℕ, define
and
Then,
for all x, y ∈ V1.
From Equations (2.9) and (2.11), we get
and so Proposition 1.4 and the hypothesis Equation 2.1 imply that is a Cauchy sequence. Since V2 is complete, the sequence converges to a point of V2 which defines a mapping q : V1 → V2.
Now, we prove
for all n ∈ ℕ. Since Equation 2.7 implies
Assume that ||f(x) -q n (x)|| ≤ ϕ n (x, x) holds for some n ∈ ℕ. Then, we have
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.
For any n ∈ ℕ and x, y ∈ V1, we have
and so, by the continuity of non-Archimedean norm and taking the limit of both sides of the above inequality, we get
Thus, q is a solution of the Equation 2.5 which satisfies Equation 2.6.
Then, by replacing x, y with in Equation 2.5, we obtain the following identities: for any solution g : V1 → V2 of the Equation (2.5),
and
By induction on n, one can show that
and
for all n ∈ ℕ.
Now, suppose that q' : V1 → V2 is another solution of 2.5 that satisfies the Equation 2.6. It follows from Equations 2.14 to 2.16 that
Therefore, since
we have q(x) = q'(x) for all x ∈ V1. This completes the proof.
In the proof of the next theorem, we need a result concerning the Cauchy functional equation
which has been established in [20].
Theorem 2.2. ([20]) Suppose that φ(x, y) satisfies the condition 2.1 and, for a mapping f : V1 → V2,
for all x, y ∈ V1. Then, there exists a unique solution q : V1 → V2of the Equation 2.17 such that
for all x ∈ V1, where
for all x, y ∈ V1
3. Stability of the Pexider functional equation
In this section, we assume that V1 is a normed space and V2 is a complete non-Archimedean space. For any mapping f : V1 → V2, we define two mappings Fe and Fo as follows:
and also define F(x) = f(x) -f(0). Then, we have obviously
Theorem 3.1. Let σ : V1 → V1be a continuous involution and the mappings f i : V1 → V2for i = 1, 2, 3, 4 and δ > 0, satisfy
for all x, y ∈ V1, then there exists a unique solution q : V1 → V2of the Equation 2.5 and a mapping v : V1 → V2which satisfies
for all x, y ∈ V1and exists two additive mappings such that for i= 1, 2 and, for all x ∈ V1,
Proof. It follows from (3.2) that
and so, for all x, y ∈ V1,
then,
Similarly, we have
for all x, y ∈ V1.
Now, first by putting y = 0 in Equation 3.7 and applying Equation 3.2 and second by putting x = 0 in Equation 3.7 and applying Equation 3.2 once again, we obtain
for all x, y ∈ V1 and so these inequalities with Equation 3.7 imply
Replacing y with σ(y) in Equation 3.11, we get
It follows from Equations 3.1, 3.11 and 3.12 that
By Theorem 2.1 of [24], there exists a unique solution q : V1 → V2 of the functional Equation 2.5 such that
for all x ∈ V1.
As a result of the inequalities Equations 3.11 and 3.12, we have
It is easily seen that the mapping v : V1 → V2 defined by
is a solution of the functional equation
for all x, y ∈ V1.
Replacing both of x, y in Equation 3.14 with , We get
for all x ∈ V1. Now, Equations 3.13 and 3.15 imply
and
Similarly, it follows from the inequalities Equations 3.7, 3.10 and 3.13 that
Since Equation 3.8 implies
for all x, y ∈ V1, we have
for all x ∈ V1. Now, from Equations 3.8 and 3.20, we obtain
and so, by interchanging role of x, y in the preceding inequality,
for all x, y ∈ V1. Since y + σ (x) = σ (x + σ (y), it follows from Equations 3.1, 3.24 and 3.25 that
By Theorem 2.2, there exists a unique additive mapping such that
Since
for all x ∈ V1, we deduce for all x ∈ V1.
By a similar deduction, Equations 3.8 and 3.21 imply that there exists a unique additive mapping such that
Moreover, we have for all x ∈ V1. Thus, by Equations 3.16, 3.22, 3.27 and 3.28, we obtain
This proves Equation 3.3. Similarly, one can prove Equations 3.4 to 3.6.
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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.
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All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
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Saadati, R., Vaezpour, S.M. & Sadeghi, Z. On the stability of pexider functional equation in non-archimedean spaces. J Inequal Appl 2011, 17 (2011). https://doi.org/10.1186/1029-242X-2011-17
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DOI: https://doi.org/10.1186/1029-242X-2011-17