On the super-stability of exponential Hilbert-valued functional equations

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.

The stability problem of functional equations goes back to a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces (see also [3]). Hyers's theorem was generalized by Aoki [4] for additive mappings and by Rassias [5,6] for linear mappings by considering an unbounded Cauchy difference. Baker et al. [7] have proved the super-stability of the exponential functional equation: If a function f: ℝ → ℝ is approximately exponential function, i.e., there exists a nonnegative number α such that

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 [8] 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.

Theorem 1.1. Let (G, +) be a semigroup and Y be a normed algebra in which the norm is multiplicative. Then, for a function f: G → Y satisfying the inequality

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.

Rassias [5,6] introduced the term mixed stability of the function f: E → ℝ, where E is a Banach space, with respect to two operations addition and multiplication among any two elements of the set {x, y, f(x), f(y)}. Especially, he raised an open problem concerning the behavior of solutions of the inequality:

(see also [9,10]). In connection with this open problem, Gavruta [11] gave an answer to this problem in the spirit of Rassiass approach:

Theorem 1.2 (Gavruta). Let X and Y be a real normed space and a normed algebra with multiplicative norm, respectively. If a function f: X → Y satisfies the inequality

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 [8] 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^xe₁₁ + e^xe₂₂ 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.

Let H be a Hilbert space with a countable orthonormal basis {e_n: n ∈ ℕ}. For two vectors x, y ∈ H, we have the Hadamard product (named after French mathematician Jacques Hadamard), also known as the entrywise product on Hilbert space H as the following:

The Cauchy-Schwartz inequality together with the Parseval identity insure that Hadamard multiplication is well defined. In fact,

In the present paper, we state a super-stability result for the approximately exponential Hilbert-valued functional equation by Hadamard product, see Theorem 2.1 below. As a consequence, we prove if a surjective function f: H → H satisfies the inequality

for some α ≥ 0 and for all x; y ∈ H, then it must be exponential with this product, i.e.,

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.

The function f(x) = a^x is said to be an exponential function, where a > 0 is a fixed real number. The exponent law of exponential functions is well represented by the exponential equation f(x + y) = f(x)f(y). Hence, we call every solution function of the exponential equation as exponential function. A general solution of the exponential equation was introduced in [12]. In fact, a function f: ℝ → ℂ is an exponential function if and only if either f(x) = exp(A(x) + ia(x)) for all x ∈ ℝ or f(x) = 0 for all x ∈ ℝ; where A: ℝ → ℝ is an additive function and a: ℝ → ℝ satisfies

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.

Definition 2.1. For a Hilbert space H and a semi-group (G,.), a function F: G → H is said to be exponential when

for every x, y ∈ G.

The following proposition characterizes the Hilbert-valued function satisfying the exponential equation:

Proposition 2.2. Let H be a separable complex Hilbert space and the mapping F: ℝ → H be exponential then either F ≡ 0 or there exist a positive integer N such that

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.

Proof. For every integer n ≥ 1, consider the function e_n ⊗ F: ℝ → ℂ by

for every h ∈ H. Since F is exponential, so is e_n ⊗ F for every integer n ≥ 1. Indeed, for n ≥ 1 and x, y ∈ H, we see that

This yields the exponential property of e_n ⊗ F for every n ≥ 1. Hence, either

for all x ∈ ℝ or (e_n ⊗ F)(x) = 0 for all x ∈ ℝ; here A_n: ℝ → ℝ is an additive function and a_n is a function satisfying (1). The continuation of proof depend on the dimension of H. In fact, if H is infinite dimensional, since

for every x ∈ H as n → +∞ Equation 2 is not possible for infinitely many positive integer n and hence there exists some positive integer N such that e_n ⊗ F = 0 for every integer n > N. Thus, F can be represented as

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.

Theorem 2.3. Let G be a semigroup and let α > 0 be given. If a function f: G → H satisfies the inequality

for all x; y ∈ G, then either there exists an integer k ≥ 1 such that

for all x ∈ G or

for all x; y ∈ G.

Proof. Assume that the first conclusion (i.e., (4)) is not true. Hence, for every integer k ≥ 1, there exists a a_k ∈ G such that

Let , f_k(x) = 〈f(x), e_k〉, and g_k = 2^-k f_k. Then, β² - 2β = α, β > 2 and |f_k(a_k)| > 2^kβ whence |g_k(a_k)| > β. By applying the Parseval identity and definition of Hadamard product with together relation (3), we find that each scalar-valued function f_k is approximately exponential, i.e.,

for every integer k ≥ 1 and x, y ∈ G. Let

then γ_k > 1 for every integer k ≥ 1. It follows from (5) for x = y = a_k that

and so

Now, make the induction hypothesis

Then, by using (5) for and (6), we observe that

and (6) is established for all n ∈ ℕ. Hence, by definition of f_k and g_k, we see that

On the other hand, for every x, y, z ∈ G, we have

Consequently, for h(x, y) = f(x.y) - f(x) * f(y), one can see

Now, by using Parseval identity for h(x, y) * f(z) observe that

Applying the last relation for and relation (7) to deduce that

It follows that

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:

Corollary 2.4. If a surjective function f: H → H satisfies the inequality

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:

Theorem 2.5. Let X be a normed space and H be a separable Hilbert space. If a function f: X → H satisfies the inequality

for all x; y ∈ X and for some p > 0 and θ > 0, then either there exists an integer k ≥ 1 such that

for all x ∈ X with ||x|| ≥ 1 or

for all x; y ∈ X. where .

Proof. Assume that for every integer k ≥ 1 there exists an x_k ∈ X with ||x_k|| ≥ 1 such that

If we set f_k(x): = 〈f(x), e_k〉 and g_k := 2^-kf_k, this is equivalent with

It follows from Parseval identity, definition of Hadamard product and relation (8) that

for every x, y ∈ X and k ≥ 1. In particular, for x = y = x_k

Since β² = 2^{p + 1} β + 2θ, hence

Now, make the induction hypothesis

Then, by using (10) for x = y = 2ⁿx_k and (11), we get

which in turn proves that the inequality (11) is true for all n ∈ ℕ. Hence, by definition of f_k and g_k, we see that

Choose x; y; z ∈ X with f(z) ≠ 0. It then follows from (8) that

and again by (8) we get

which together with the last relation yields

where

Let h(x, y) = f(x + y) - f(x) * f(y), then by (13)

and so

In particular, by using the last relation for z_k = 2ⁿx_k and by considering (12) we deduce that

and consequently,

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.

At the end of this paper, let us consider the other type multiplication in a Hilbert space. In fact, for a separable Hilbert space H and two elements and of H, one can define the convolution product by

where the numbers can be obtained by discrete convolution:

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.

The authors declare that they have no competing interests.

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.

1. Ulam, SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience. New York (1960)

2. Hyers, DH: On the stability of the linear functional equation. Proc Nat Acad Sci USA. 27, 222–224 (1941). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

3. Hyers, DH, Rassias, ThM: Approximate homomorphisms. Aequat Math. 44, 125–153 (1992). Publisher Full Text

4. Aoki, T: On the stability of the linear transformation in Banach spaces. J Math Soc Japan. 2, 64–66 (1950). Publisher Full Text

5. Rassias, ThM: Problem 18, in Report on the 31st ISFE. Aequat Math. 47, 312–313 (1994)

6. Rassias, ThM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc. 72(2), 297–300 (1978). Publisher Full Text

7. Baker, JA, Lawrence, J, Zorzitto, F: The stability of the equation f(x + y) = f(x)f(y). Proc Am Math Soc. 74, 242–246 (1979)

8. Baker, JA: The stability of the cosine equation. Proc Am Math Soc. 80, 411–416 (1980). Publisher Full Text

9. Jung, SM: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York (2011)

10. Kuczma, M: An Introduction to the Theory of Functional Equations and Inequalities. PWN, Warszawa, Krak ow, and Katowice (1985)

11. Gavruta, P: An answer to a question of Th.M. Rassias and J. Tabor on mixed stability of mappings. Bul Stiint Univ Politeh Timis Ser Mat Fiz. 42, 1–6 (1997)

12. Czerwik, S: Functional Equations and Inequalities in Several Variables. World Scientific, Hackensacks (2002)