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A Functional equation related to inner product spaces in non-archimedean normed spaces
Advances in Difference Equations volume 2011, Article number: 37 (2011)
Abstract
In this paper, we prove the Hyers-Ulam stability of a functional equation related to inner product spaces in non-Archimedean normed spaces.
2010 Mathematics Subject Classification: Primary 46S10; 39B52; 47S10; 26E30; 12J25.
1. Introduction and preliminaries
One of the most interesting questions in the theory of functional analysis concerning the Ulam stability problem of functional equations is as follows: When is it true that a mapping satisfying a functional equation approximately must be close to an exact solution of the given functional equation?
The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940 and affirmatively solved by Hyers [2]. The result of Hyers was generalized by Aoki [3] for approximate additive mappings and by Rassias [4] for approximate linear mappings by allowing the difference Cauchy equation || f(x 1 + x 2) - f(x 1) - f(x 2) || to be controlled by ε (|| x 1 || p + || x 2 || p ). Taking into consideration a lot of influence of Ulam, Hyers and Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Rassias is called Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. In 1994, a generalization of the Rassias' theorem was obtained by Gǎvruta [5], who replaced ε (|| x 1 || p + || x 2 || p ) by a general control function φ(x 1, x 2).
Quadratic functional equations were used to characterize inner product spaces [6]. A square norm on an inner product space satisfies the parallelogram equality ||x 1 + x 2||2 + ||x 1 - x 2||2 = 2(||x 1||2 + ||x 1||2). The functional equation
is related to a symmetric bi-additive mapping [7, 8]. It is natural that this equation is called a quadratic functional equation, and every solution of the quadratic equation (1.1) is said to be a quadratic mapping.
It was shown by Rassias [9] that the norm defined over a real vector space X is induced by an inner product if and only if for a fixed integer n ≥ 2
for all x 1,⋯, x n ∈ X.
Let be a field. A non-Archimedean absolute value on is a function such that for any , we have
-
(i)
|a| ≥ 0 and equality holds if and only if a = 0,
-
(ii)
|ab| = |a||b|,
-
(iii)
|a + b| ≤ max{|a|, |b|}.
The condition (iii) is called the strict triangle inequality. By (ii), we have |1| = | - 1| = 1. Thus, by induction, it follows from (iii) that |n| ≤ 1 for each integer n. We always assume in addition that | | is non-trivial, i.e., that there is an such that |a 0| ≠ 0, 1.
Let X be a linear space over a scalar field with a non-Archimedean non-trivial valuation |·|. A function || · || : X → ℝ is a non-Archimedean norm (valuation) if it satisfies the following conditions:
(NA1) ||x|| = 0 if and only if x = 0;
(NA2) ||rx|| = |r|||x|| for all ℝ and x ∈ X;
(NA3) the strong triangle inequality (ultrametric); namely,
Then (X, || · ||) is called a non-Archimedean space.
Thanks to the inequality
a sequence {x m } is Cauchy in X if and only if {x m+1- x m } converges to zero in a non-Archimedean space. By a complete non-Archimedean space, we mean a non-Archimedean space in which every Cauchy sequence is convergent.
In 1897, Hensel [10] introduced a normed space which does not have the Archimedean property.
During the last three decades, the theory of non-Archimedean spaces has gained the interest of physicists for their research in particular in problems coming from quantum physics, p-adic strings and superstrings [11]. Although many results in the classical normed space theory have a non-Archimedean counterpart, their proofs are essentially different and require an entirely new kind of intuition [12–16].
The main objective of this paper is to prove the Hyers-Ulam stability of the following functional equation related to inner product spaces
(n ∈ ℕ, n ≥ 2) in non-Archimedean normed spaces. Interesting new results concerning functional equations related to inner product spaces have recently been obtained by Najati and Rassias [17] as well as for the fuzzy stability of a functional equation related to inner product spaces by Park [18] and Eshaghi Gordji and Khodaei [19]. During the last decades, several stability problems for various functional equations have been investigated by many mathematicians (see [20–49]).
2. Hyers-Ulam stability in non-Archimedean spaces
In the rest of this paper, unless otherwise explicitly stated, we will assume that G is an additive group and that X is a complete non-Archimedean space. For convenience, we use the following abbreviation for a given mapping f : G → X:
for all x 1,⋯, x n ∈ G, where n ≥ 2 is a fixed integer.
Lemma 2.1. [17]. Let V 1 and V 2 be real vector spaces. If an odd mapping f : V 1 → V 2 satisfies the functional equation (1.2), then f is additive.
In the following theorem, we prove the Hyers-Ulam stability of the functional equation (1.2) in non-Archimedean spaces for an odd case.
Theorem 2.2. Let φ : G n → [0, ∞) be a function such that
for all x, x 1, x 2,⋯, x n ∈ G, and
exists for all x ∈ G, where
for all x ∈ G. Suppose that an odd mapping f : G → X satisfies the inequality
for all x 1, x 2,⋯, x n ∈ G. Then there exists an additive mapping A : G → X such that
for all x ∈ G, and if
then A is a unique additive mapping satisfying (2.5).
Proof. Letting x 1 = nx 1, in (2.4) and using the oddness of f, we obtain that
for all . Interchanging x 1 with in (2.7) and using the oddness of f, we get
for all . It follows from (2.7) and (2.8) that
for all . Setting x 1 = nx 1, , x i = 0 (i = 3,..., n) in (2.4) and using the oddness of f, we get
for all . It follows from (2.9) and (2.10) that
for all . Putting , x i = 0 (i = 2,..., n) in (2.4), we obtain
for all . It follows from (2.11) and (2.12) that
for all . Replacing x 1 and by and in (2.13), respectively, we obtain
for all x ∈ G. Hence,
for all x ∈ G. Replacing x by 2m-1 x in (2.14), we have
for all x ∈ G. It follows from (2.1) and (2.15) that the sequence is Cauchy. Since X is complete, we conclude that is convergent. So one can define the mapping A : G → X by for all x ∈ G. It follows from (2.14) and (2.15) that
for all m ∈ ℕ and all x ∈ G. By taking m to approach infinity in (2.16) and using (2.2), one gets (2.5). By (2.1) and (2.4), we obtain
for all x 1, x 2,⋯, x n ∈ G. Thus, the mapping A satisfies (1.2). By Lemma 2.1, A is additive.
If A' is another additive mapping satisfying (2.5), then
for all x ∈ G, Thus A = A'. □
Corollary 2.3. Let ρ : [0, ∞) → [0, ∞) be a function satisfying
-
(i)
ρ (|2|t) ≤ ρ(|2|)ρ(t) for all t ≥ 0,
-
(ii)
ρ(|2|) < |2|.
Let ε > 0 and let G be a normed space. Suppose that an odd mapping f : G → X satisfies the inequality
for all x 1,⋯, x n ∈ G. Then there exists a unique additive mapping A : G → X such that
for all x ∈ G.
Proof. Defining φ : G n → [0, ∞) by , we have
for all x 1,⋯, x n ∈ G. So we have
and
for all x ∈ G. It follows from (2.3) that
Applying Theorem 2.2, we conclude that
for all x ∈ G. □
Lemma 2.4. [17]. Let V 1 and V 2 be real vector spaces. If an even mapping f : V 1 → V 2 satisfies the functional equation (1.2), then f is quadratic.
In the following theorem, we prove the Hyers-Ulam stability of the functional equation (1.2) in non-Archimedean spaces for an even case.
Theorem 2.5. Let φ : G n → [0, ∞) be a function such that
for all x, x 1, x 2,⋯, x n ∈ G, and
exists for all x ∈ G, where
and
for all x ∈ G. Suppose that an even mapping f : G → X with f(0) = 0 satisfies the inequality (2.4) for all x 1, x 2,⋯, x n ∈ G. Then there exists a quadratic mapping Q : G → X such that
for all x ∈ G, and if
then Q is a unique quadratic mapping satisfying (2.21).
Proof. Letting x 1 = nx 1, x i = nx 2 (i = 2,⋯, n) in (2.4) and using the evenness of f, we obtain
for all x 1, x 2 ∈ G. Interchanging x 1 with x 2 in (2.23) and using the evenness of f, we obtain
for all x 1, x 2 ∈ G. It follows from (2.23) and (2.24) that
for all x 1, x 2 ∈ G. Setting x 1 = nx 1, x 2 = -nx 2, x i = 0 (i = 3,⋯, n) in (2.4) and using the evenness of f, we obtain
for all x 1, x 2 ∈ G. So we obtain from (2.25) and (2.26) that
for all x 1, x 2 ∈ G. Setting x 1 = x, x 2 = 0 in (2.27), we obtain
for all x ∈ G. Putting x 1 = nx, x i = 0 (i = 2,⋯, n) in (2.4), one obtains
for all x ∈ G. It follows from (2.28) and (2.29) that
for all x ∈ G. Letting x 2 = - (n - 1) x 1 and replacing x 1 by in (2.26), we get
for all x ∈ G. It follows from (2.28) and (2.31) that
for all x ∈ G. It follows from (2.30) and (2.32) that
for all x ∈ G. Setting x 1 = x 2 = n x , x i = 0 (i = 3,⋯, n) in (2.4), we obtain
for all x ∈ G. It follows from (2.33) and (2.34) that
for all x ∈ G. Thus,
for all x ∈ G. Replacing x by 2 m - 1 x in (2.36), we have
for all x ∈ G. It follows from (2.17) and (2.37) that the sequence is Cauchy. Since X is complete, we conclude that is convergent. So one can define the mapping Q : G → X by for all x ∈ G. By using induction, it follows from (2.36) and (2.37) that
for all n ∈ ℕ and all x ∈ G. By taking m to approach infinity in (2.38) and using (2.18), one gets (2.21).
The rest of proof is similar to proof of Theorem 2.2. □
Corollary 2.6. Let η : [0, ∞) → [0, ∞) be a function satisfying
-
(i)
η(|l|t) ≤ η(|l|)η(t) for all t ≥ 0,
-
(ii)
η(|l|) < |l|2 for l ∈ {2, n - 1, n}.
Let ε > 0 and let G be a normed space. Suppose that an even mapping f : G → X with f(0) = 0 satisfies the inequality
for all x 1,⋯, x n ∈ G. Then there exists a unique quadratic mapping Q : G → X such that
for all x ∈ G.
Proof. Defining φ : G n → [0, ∞) by , we have
for all x 1,⋯, x n ∈ G. We have
and
for all x ∈ G. It follows from (2.20) that
Hence, by using (2.19), we obtain
for all x ∈ G.
Applying Theorem 2.5, we conclude the required result. □
Lemma 2.7. [17]. Let V 1 and V 2 be real vector spaces. A mapping f : V 1 → V 2 satisfies (1.2) if and only if there exist a symmetric bi-additive mapping B : V 1 × V 1 → V 2 and an additive mapping A : V 1 → V 2 such that f(x) = B(x, x) + A(x) for all x ∈ V 1.
Now, we prove the main theorem concerning the Hyers-Ulam stability problem for the functional equation (1.2) in non-Archimedean spaces.
Theorem 2.8. Let φ : G n → [0, ∞) be a function satisfying (2.1) and (2.17) for all x, x 1, x 2,⋯, x n ∈ G, and and exist for all x ∈ G, where and are defined as in Theorems 2.2 and 2.5. Suppose that a mapping f : G → X with f(0) = 0 satisfies the inequality (2.4) for all x 1, x 2,⋯, x n ∈ G. Then there exist an additive mapping A : G → X and a quadratic mapping Q : G → X such that
for all x ∈ G. If
then A is a unique additive mapping and Q is a unique quadratic mapping satisfying (2.39).
Proof. Let for all x ∈ G. Then
for all x 1, x 2,⋯, x n ∈ G. By Theorem 2.5, there exists a quadratic mapping Q : G → X such that
for all x ∈ G. Also, let for all x ∈ G. By Theorem 2.2, there exists an additive mapping A : G → X such that
for all x ∈ G. Hence (2.39) follows from (2.40) and (2.41).
The rest of proof is trivial. □
Corollary 2.9. Let γ : [0, ∞) → [0, ∞) be a function satisfying
(i) γ(|l|t) ≤ γ(|l|) γ(t) for all t ≥ 0,
(ii) γ(|l|) < |l|2 for l ∈ {2, n - 1, n}.
Let ε > 0, G a normed space and let f : G → X satisfy
for all x 1,⋯, x n ∈ G and f (0) = 0. Then there exist a unique additive mapping A : G → X and a unique quadratic mapping Q : G → X such that
for all x ∈ G.
Proof. The result follows by Corollaries 2.6 and 2.3. □
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Acknowledgements
Dong Yun Shin was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).
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Gordji, M.E., Khodabakhsh, R., Khodaei, H. et al. A Functional equation related to inner product spaces in non-archimedean normed spaces. Adv Differ Equ 2011, 37 (2011). https://doi.org/10.1186/1687-1847-2011-37
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DOI: https://doi.org/10.1186/1687-1847-2011-37
Keywords
- non-Archimedean spaces
- additive and quadratic functional equation
- Hyers-Ulam stability