On the super-stability of exponential Hilbert-valued functional equations
© Rezaei and Sharifzadeh; licensee Springer. 2011
Received: 24 July 2011
Accepted: 21 November 2011
Published: 21 November 2011
We generalize the well-known Baker's super-stability result for exponential mappings with values in the field of complex numbers to the case of an arbitrary Hilbert space with the Hadamard product. Then, we will prove an even more general result of this type.
2000 MSC: primary: 39B72, secondary: 46E40.
for x, y ∈ ℝ, then f is either bounded or exponential. This theorem was the first result concerning the super-stability phenomenon of functional equations. Baker  generalized this famous result to any function f: (G, +) → ℂ where (G, +) is a semigroup. The same result is also true for approximately exponential mappings with values in a normed algebra with the property that the norm is multiplicative.
for all x; y ∈ G and for some α > 0, either for all x ∈ G or f is an exponential function.
In the other world every approximately exponential map f: (G, +) → Y is either bounded or exponential.
for all x; y ∈ X and for some p > 0 and θ > 0, then either ||f(x)|| ≤ δ||x|| p for all x ∈ X with ||x|| ≥ 1 or f is an exponential function, where .
Baker  gave an example to present that the Theorem 1.1 is false if the algebra Y does not have the multiplicative norm: Given δ > 0, choose an ε > 0 with |ε - ε2| = δ. Let M2(ℂ) denote the space of 2 × 2 complex matrices with the usual norm and f: ℝ → M2(ℂ) is defined by f(x) = e x e11 + e x e22 where e ij is defined as the 2 × 2 matrix with 1 in the (i, j) entry and zeroes elsewhere. We will show that such behavior is typical for approximately exponential mappings with values in Hilbert spaces with Hadamard product which is not multiplicative.
Then, we will prove an even more general result of this type. We also generalized Theorem 2.1 concerning the mixed stability for Hilbert-valued functions.
2. Main results
for all x, y ∈ ℝ. Indeed, a function f: ℝ → ℝ continuous at a point is an exponential function if and only if f(x) = a x for all x ∈ ℝ or f(x) = 0 for all x ∈ ℝ, where a > 0 is a constant.
for every x, y ∈ G.
The following proposition characterizes the Hilbert-valued function satisfying the exponential equation:
for all x ∈ H where A n : ℝ → ℝ is an additive function and a n is a function satisfying (1) for n = 1, 2,..., N.
In the case that H is of finite dimensional type, the proof is clear.
In the following theorem, we generalize the well-known Baker's super-stability result for exponential mappings with values in the field of complex numbers to the case of an arbitrary Hilbert space with the Hadamard product.
for all x; y ∈ G.
for all x, y ∈ G and any n ∈ ℕ. Letting n → +∞, we conclude that h(x, y) = 0 and so f(x.y) = f(x) * f(y) for all x, y ∈ G.
Notice that if f: H → H is a surjection function, then every component function e n ⊗ f is unbounded. In fact, for every positive integer n, there exists some x n ∈ H such that f(x n ) = ne n , and so (e n ⊗ f)(x n ) = n. This led to the following corollary:
for some α ≥ 0 and for all x; y ∈ G, then f(x * y) = f(x) * f(y) for all x; y ∈ G.
In the next theorem, we generalize the Gavruta Theorem on mixed stability for Hilbert-valued function with Hadamard product:
for all x; y ∈ X. where .
for all x, y ∈ X and any n ∈ ℕ. Letting n → +∞, we conclude that h(x, y) = 0 and so f(x + y) = f(x) * f(y) for all x, y ∈ X.
Hence, it is interesting to study and to phrase the super-stability phenomenon for functions with values in (H, •). For instance, it is desirable to have a sufficient condition for approximately exponential mappings with values in (H, •) to be exponential with the convolution product.
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