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On the stability of pexider functional equation in nonarchimedean spaces
Journal of Inequalities and Applications volume 2011, Article number: 17 (2011)
Abstract
In this paper, the HyersUlam stability of the Pexider functional equation
in a nonArchimedean 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 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 HyersUlamRassias stability of functional equations (for more details, see [5] where a discussion on definitions of the HyersUlam stability is provided by Moszner, also [6–12]).
In this paper, we give a modification of the approach of Belaid et al. [13] in nonArchimedean spaces. Recently, Ciepliński [14] studied and proved stability of multiadditive mappings in nonArchimedean normed spaces, also see [15–22].
Definition 1.1. The function  ·  : K → ℝ is called a nonArchimedean 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 nonArchimedean valuation is called a nonArchimedean 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 nonArchimedean 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 nonArchimedean field K. A nonArchimedean 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 nonArchimedean norm  ·  : V → ℝ is called a nonArchimedean space. If the metric d(u, v) = u  v is induced by a nonArchimedean norm  ·  : V → ℝ on a vector space V which is complete, then (V,  · ) is called a complete nonArchimedean space.
Proposition 1.4. ([23]) A sequencein a nonArchimedean space is a Cauchy sequence if and only if the sequenceconverges to zero.
Since any nonArchimedean norm satisfies the triangle inequality, any nonArchimedean norm is a continuous function from its domain to real numbers.
Proposition 1.5. Let V be a normed space and E be a nonArchimedean 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 nonArchimedean spaces, that is, every continuous function from a connected space to a nonArchimedean space is constant. This is a consequence of totally disconnectedness of every nonArchimedean space (see [23]).
2. Stability of quadratic and Cauchy functional equations
Throughout this section, we assume that V_{1} is a normed space and V_{2} is a complete nonArchimedean space. Let σ : V_{1} → V_{1} be a continuous involution (i.e., σ (x + y) = σ (x) + σ (y) and σ (σ (x)) = x) and φ : V_{1} × V_{1} → ℝ be a function with
and define a function ϕ : V_{1} × V_{1} → 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 : V_{1} → V_{2}satisfies the inequality
for all x, y ∈ V_{1}, then there exists a unique solution q : V_{1} → V_{2}of the functional equation
such that
for all x ∈ V_{1}.
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 ∈ V_{1} 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 ∈ V_{1}.
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 V_{2} is complete, the sequence converges to a point of V_{2} which defines a mapping q : V_{1} → V_{2}.
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 ∈ V_{1}, we have
and so, by the continuity of nonArchimedean 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 : V_{1} → V_{2} of the Equation (2.5),
and
By induction on n, one can show that
and
for all n ∈ ℕ.
Now, suppose that q' : V_{1} → V_{2} 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 ∈ V_{1}. 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 : V_{1} → V_{2},
for all x, y ∈ V_{1}. Then, there exists a unique solution q : V_{1} → V_{2}of the Equation 2.17 such that
for all x ∈ V_{1}, where
for all x, y ∈ V_{1}
3. Stability of the Pexider functional equation
In this section, we assume that V_{1} is a normed space and V_{2} is a complete nonArchimedean space. For any mapping f : V_{1} → V_{2}, we define two mappings F^{e} and F^{o} as follows:
and also define F(x) = f(x) f(0). Then, we have obviously
Theorem 3.1. Let σ : V_{1} → V_{1}be a continuous involution and the mappings f_{ i } : V_{1} → V_{2}for i = 1, 2, 3, 4 and δ > 0, satisfy
for all x, y ∈ V_{1}, then there exists a unique solution q : V_{1} → V_{2}of the Equation 2.5 and a mapping v : V_{1} → V_{2}which satisfies
for all x, y ∈ V_{1}and exists two additive mappings such that for i= 1, 2 and, for all x ∈ V_{1},
Proof. It follows from (3.2) that
and so, for all x, y ∈ V_{1},
then,
Similarly, we have
for all x, y ∈ V_{1}.
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 ∈ V_{1} 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 : V_{1} → V_{2} of the functional Equation 2.5 such that
for all x ∈ V_{1}.
As a result of the inequalities Equations 3.11 and 3.12, we have
It is easily seen that the mapping v : V_{1} → V_{2} defined by
is a solution of the functional equation
for all x, y ∈ V_{1}.
Replacing both of x, y in Equation 3.14 with , We get
for all x ∈ V_{1}. 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 ∈ V_{1}, we have
for all x ∈ V_{1}. 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 ∈ V_{1}. 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 ∈ V_{1}, we deduce for all x ∈ V_{1}.
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 ∈ V_{1}. 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.
References
Ulam SM: Problems in Modern Mathematics, Chapter IV, Science Editions. Wiley, New York; 1960.
Hyers DH: On the stability of the linear functional equation. Proc Nat 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 THM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297–300. 10.1090/S00029939197805073271
Moszner Z: On the stability of functional equations. Aequationes Math 2009, 77: 33–88. 10.1007/s0001000829457
Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ; 2002.
Hyers DH, Isac G, Rassias THM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.
Jung SM: HyersUlamRassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor; 2001.
Rassias TM: On the stability of functional equations and a problem of Ulam. Acta Appl Math 2000, 62: 23–130. 10.1023/A:1006499223572
Rassias THM: Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht; 2003.
Ciepliński K: Generalized stability of multiadditive mappings. Appl Math Lett 2010,23(10):1291–1294. 10.1016/j.aml.2010.06.015
Ciepliński K: Stability of the multiJensen equation. J Math Anal Appl 2010,363(1):249–254. 10.1016/j.jmaa.2009.08.021
Bouikhalene B, Elqorachi E, Rassias THM: On the HyersUlam stability of approximately Pexider mappings. Math Inequal Appl 2008, 11: 805–818.
Ciepliński K: Stability of multiadditive mappings in nonArchimedean normed spaces. J Math Anal Appl 2011, 373: 376–383. 10.1016/j.jmaa.2010.07.048
Cho YJ, Park C, Saadati R: Functional inequalities in nonArchimedean in Banach spaces. Appl Math Lett 2010, 60: 1994–2002.
Mirmostafaee AK: Stability of quartic mappings in nonArchimedean normed spaces. Kyungpook Math J 2009, 49: 289–297.
Moslehian MS, Sadeghi GH: A MazurUlam theorem in nonArchimedean normed spaces. Nonlinear Anal 2008, 69: 3405–3408. 10.1016/j.na.2007.09.023
Moslehian MS, Sadeghi GH: Stability of two types of cubic functional equations in nonArchimedean spaces. Real Anal Exch 2008, 33: 375–384.
Moslehian MS, Rassias THM: Stability of functional equations in nonarchimedean spaces. Appl Anal Discrete Math 2007, 1: 325–334. 10.2298/AADM0702325M
Moslehian MS, Rassias THM: Stability of functional equations in nonArchimedean spaces. Appl Anal Discrete Math 2007, 1: 325–334. 10.2298/AADM0702325M
Najati A, Moradlou F: HyersUlamRassias stability of the Apollonius type quadratic mapping in nonArchimedean spaces. Tamsui Oxf J Math Sci 2008, 24: 367–380.
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.2010.07.034
Schneider P: NonArchimedean Functional Analysis. Springer, New York; 2002.
Bouikhalene B, Elqorachi E, Rassias THM: On the generalized HyersUlam stability of the quadratic functional equation with a general involution. Nonlinear Funct Anal Appl 2007,12(2):247–262.
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 nonarchimedean spaces. J Inequal Appl 2011, 17 (2011). https://doi.org/10.1186/1029242X201117
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DOI: https://doi.org/10.1186/1029242X201117
Keywords
 HyersUlam stability of functional equation
 NonArchimedean space
 Quadratic
 Cauchy and Pexider functional equations