Abstract
We deal with a Pexider difference
where f and g map be a given abelian group (G, +) into a sequentially complete Hausdorff topological vector space. We also investigate the Hyers-Ulam stability of the following Pexiderized functional equation
in topological vector spaces.
Mathematics subject classification (2000): Primary 39B82; Secondary 34K20, 54A20.
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1. Introduction and preliminaries
In 1940, Ulam [1] proposed the general stability problem: Let G1 be a group, G2 be a metric group with the metric d. Given ε > 0, does there exists δ > 0 such that if a function h: G1→ G2 satisfies the inequality
then there is a homomorphism H: G 1 → G 2 with
Hyers [2] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces. In 1950, Aoki [3] extended the theorem of Hyers by considering the unbounded cauchy difference inequality
In 1978, Rassias [4] also generalized the Hyers' theorem for linear mappings under the assumption t ↦ f (tx) is continuous in t for each fixed x.
Recently, Adam and Czerwik [5] investigated the problem of the Hyers-Ulam stability of a generalized quadratic functional equation in linear topological spaces. Najati and Moghimi [6] investigated the Hyers-Ulam stability of the functional equation
in quasi-Banach spaces. In this article, we prove that the Pexiderized functional equation
is stable for functions f, g defined on an abelian group and taking values in a topological vector space.
Throughout this article, let G be an abelian group and X be a sequentially complete Hausdorff topological vector space over the field ℚ of rational numbers.
A mapping f: G → X is said to be quadratic if and only if it satisfies the following functional equation
for all x y ∈ G. A mapping f G → X is said to be additive if and only if it satisfies f (x + y) = f (x) + f (y) for all x y ε G. For a given f: G → X, we will use the following notation
For given sets A B ⊆ X and a number k ∈ ℝ, we define the well known operations
We denote the convex hull of a set U ⊆ X by conv(U) and by the sequential closure of U. Moreover it is well know that:
-
(1)
If A ⊆ X are bounded sets, then conv(A) and are bounded subsets of X.
-
(2)
If A, B ⊆ X and α β ∈ ℝ, then α conv(A) + β conv(B) = conv(αA + βB).
-
(3)
Let X1 and X2 be linear spaces over ℝ. If f: X1→ X2 is a additive (quadratic) function, then f (rx) = rf (x) (f (rx) = r2f (x)), for all x ∈ X1 and all r ∈ ℚ.
2. Main results
We start with the following lemma.
Lemma 2.1. Let G be a 2-divisible abelian group and B ⊆ X be a nonempty set. If the functions f, g: G → X satisfy
for all x, y ∈ G, then
for all x, y ∈ G.
Proof. Putting y = 0 in (2.1), we get
for all x ∈ G. If we replace x by in (2.4), then we have
for all x ∈ G. It follows from (2.5) and (2.1) that
Moreover, we have
Theorem 2.2. Let G be a 2-divisible abelian group and B ⊆ X be a bounded set. Suppose that the odd functions f, g: G → X satisfy (2 1) for all x, y ∈ G. Then there exists exactly one additive function such that
for all x ∈ G. Moreover the function is given by
for all x ∈ G. Moreover, the convergence of the sequences are uniform on G.
Proof. By Lemma 2.1, we get (2.2). Setting y = x, y = 3x and y = 4x in (2.2), we get
for all x ∈ G. It follows from (2.7), (2.8), and (2.9) that
for all x ∈ G. So
for all x ∈ G. Using (2.7) and (2.10), we obtain
for all x ∈ G. Therefore
for all x ∈ G and all integers n > m ≥ 0. Since B is bounded, we conclude that conv(B - B) is bounded. It follows from (2.11) and boundedness of the set conv(B - B) that the sequence is (uniformly) Cauchy in X for all x ∈ G. Since X is a sequential complete topological vector space, the sequence is convergent for all x ∈ G, and the convergence is uniform on G. Define
Since conv(B - B) is bounded, it follows from (2.2) that
for all x y ∈ G. So is additive (see [6]). Letting m = 0 and n →∞ in (2.11), we get
for all x ∈ G. Similarly as before applying (2.3) we have an additive mapping defined by which is satisfying
for all x ∈ G. Since B is bounded, it follows from (2.5) that . Letting , we obtain (2.6) from (2.12) and (2.13).
To prove the uniqueness of , suppose that there exists another additive function : G → X satisfying (2.6). So
for all x ∈ G. Since and are additive, replacing x by 2nx implies that
for all x ∈ G and all integers n. Since is bounded, we infer . This completes the proof of theorem.
Theorem 2.3 Let G be a 2, 3-divisible abelian group and B ⊆ X be a bounded set. Suppose that the even functions f, g: G → X satisfy (2 1) for all x, y ∈ G. Then there exists exactly one quadratic function such that
for all x ∈ G. Moreover, the function is given by
for all x ∈ G. Moreover, the convergence of the sequences are uniform on G.
Proof. By replacing y by x + y in (2.2), we get
for all x, y ∈ G. Replacing y by - y in (2.14), we get
for all x, y ∈ G. It follows from (2.2), (2.14), and (2.15) that
for all x, y ∈ G. By letting y = 0 and y = 3x in (2.16), we get
for all x ∈ G. Using (2.17) and (2.18), we obtain
for all x ∈ G. If we replace x by in (2.19), then
for all x ∈ G. Therefore
for all x ∈ G and all integers n. So
for all x ∈ G and all integers n >m ≥ 0. It follows from (2.21) and boundedness of the set conv(B - B) that the sequence is (uniformly) Cauchy in X for all x ∈ G. The rest of the proof is similar to proof of of Theorem 2.2.
Remark 2.4. If the functions f, g: G → X satisfy (2.1), where f is even (odd) and g is odd (even), then it is easy to show that f and g are bounded.
References
Ulam SM: Problem in Modern Mathematics, Science edn. Wiley, New York; 1960.
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Aoki T: On the stability of linear trasformation in Banach spaces. J Math Soc Japan 1950, 2: 64–66. 10.2969/jmsj/00210064
Rassias ThM: On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1
Adam M, Czerwik S: On the stability of the quadratic functional equation in topological spaces. Banach J Math Anal 2007, 1: 245–251.
Najati A, Moghimi MB: Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces. J Math Anal Appl 2008, 337: 399–415. 10.1016/j.jmaa.2007.03.104
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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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Najati, A., Ostadbashi, S., Kim, G.H. et al. A pexider difference for a pexider functional equation. Adv Differ Equ 2012, 26 (2012). https://doi.org/10.1186/1687-1847-2012-26
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DOI: https://doi.org/10.1186/1687-1847-2012-26