The Ulam-Hyers stability of functional equations is widely studied from various points of view by many authors. The present paper is concerned with local stability of the four Cauchy equations restricted on a bounded domain. These results can be easily adapted to the corresponding Pexiderized equations.

1. Introduction

After this introduction, in Section 2 the local stability of the additive equation

()

and, as a consequence, of the logarithmic equation

()

both restricted on a bounded domain in R² is studied.

It is well known that the problem of stability was posed, for the additive equation, by S. Ulam and was solved by Hyers [1] in 1941, with reference to the equation valid on the whole space. Afterwards, stability was widely studied by many authors, from various points of view, considering further equations on the whole space or putting them in very general settings (see, for instance, [2, 3]).

As for the “local” stability of equations on a restricted domain, first results can be found in [4, 5] (see also [6]) and they concern substantially the set of functions f : D_f ⊂ R → (S, ∥·∥), satisfying the condition of δ-additivity

()

restricted either on the triangular domain

()

for some given r > 0, or the unbounded domain

()

for some given a > 0.

In the present paper (Section 2) the bounded restricted domain of inequality (3) will be assumed to be the triangle

()

for some given (a, b) ∈ R² and r > 0.

It has to be remarked that in the classical paper [1] by Hyers as well as in case of restricted domains studied in [4, 5], the solutions of the “equation” correlated to the inequality (3) on the given domain are either additive functions on the whole space (in cases of the general result by Hyers and of the domain K_a defined in (5)) or the restrictions to D_f = [0, r) of functions F additive on R² (in case of a domain like E₀ defined in (4)).

On the contrary, when a restricted domain like that in (6) is assumed, the local solution of the corresponding additive equation, restricted to the same set, is different from the restriction on the domain of f of some function, which is additive in the whole space R² (see [7]).

Therefore, in order to adhere to the sense of Ulam’s question in case of a restricted domain like E(a, b; r) too, the locally δ-additive function f has to be compared with the local solution of the corresponding exact equation restricted to the same set E(a, b; r).

In this frame, in Section 2, first the local stability of the additive Cauchy equation restricted to E(a, b; r) will be proved (Theorem 1); then, as a consequence of this result, the local stability of the logarithmic Cauchy equation (2) will be proved (Theorem 6).

These results can be easily extended to the Pexiderized forms of the same equations.

Notice that the problem of the “local” stability for the remaining two Cauchy equations,

()

restricted to bounded domains, requires a suitable slightly different approach because of the peculiar properties of the local solutions of such equations when they are restricted on bounded domains (see [8, 9]).

This problem will be the object of Section 3 of the present paper, where results of local stability of (7) and (8) will be proved (Theorems 15 and 16, resp.).

2. About the Additive and the Logarithmic Cauchy’s Equations

2.1. A Result on Local Stability of the Additive Equation

In the set of functions f of a real variable with values in a normed space S = (S, ∥·∥), let us consider the inequality

()

for some δ > 0 and (x, y) ∈ E(a, b; r), defined in (6), for given (a, b) ∈ R² and r > 0.

As usual, the projections of E will be denoted by

()

For E(a, b; r) as defined in (6), the projections are E_x = [a, a + r), E_y = [b, b + r), E_x+y = [a + b, a + b + r), and the domain D_f of functions f satisfying (9) is D_f = E_x ∪ E_y ∪ E_x+y.

We purpose to check whether each f : D_f → (S, ∥·∥) satisfying (9) for (x, y) ∈ E(a, b; r) is uniformly approached on D_f by some φ : D_f → S satisfying the additive equation restricted to E(a, b; r), namely, by a function φ of the following form (see [7]):

()

where h₀ : R → S is additive on the whole space R² and α, β ∈ S are constant.

A positive answer is given by the following.

Theorem 1. Let f : D_f → (S, ∥·∥), (S, ∥·∥) being a Banach space, satisfy (9) for some δ > 0 and every (x, y) ∈ E(a, b; r), defined in (6), for given (a, b) ∈ R² and r > 0; D_f = E_x ∪ E_y ∪ E_x+y.

Then there exists (at least) a function H : R → S, additive on R², such that the function F : D_f → S defined by

()

has both the following properties:

(i)
F is a (local) solution of the additive equation restricted on E(a, b; r);
(ii)
F approaches uniformly f on D_f, and
()
holds for every t ∈ D_f.

In order to prove Theorem 1, let us premise two lemmas.

Lemma 2. Let f : D_f = E_x ∪ E_y ∪ E_x+y → (S, ∥·∥) satisfy the condition (9) restricted to the set E = E(a, b; r) defined in (6). Then the functions γ_i : [0, r) → S, i = 1,2, 3, defined by

()

for t ∈ [0, r) satisfy both the following inequalities:

()

Proof of Lemma 2. Let us prove (15) firstly for i = 1, j = 2. For t ∈ [0, r) the points (a + t, b) and (a, b + t) belong to E(a, b; r); hence, from (9) both the following inequalities hold

()

whence,

()

namely, (15) for i = 1, j = 2.

Similarly we prove (15), for i = 1, j = 3, assuming the pairs (a, b) and (a + t, b) with t ∈ [0, r), from the formulas

()

and for i = 2, j = 3, assuming the pairs (a, b), (a, b + t), and t ∈ [0, r).

In order to prove (16) let us assume (ξ, η) ∈ E₀ = E(0,0; r).

For i = 1, from (15) and (9) we get

()

since (a + ξ + η, b) ∈ E(a, b; r) and (a + ξ, b + η) ∈ E(a, b; r), it follows from (9) that

()

with ∥σ∥ < 2δ, ∥ρ_k∥ < δ for k = 1,2, whence (16) for i = 1.

Similarly, for i = 2, a and b interchanged.

As for γ₃ with (ξ, η) ∈ E₀ = E(0,0; r), from (9)

()

where ∥τ_k∥ < δ, k = 1,2, 3,4, whence ∥Φ₃∥ < 4δ.

Lemma 2 is proved.

Lemma 3. If f : D_f = E_x ∪ E_y ∪ E_x+y → (S, ∥·∥), S being a Banach space, satisfies (9) on E = E(a, b; r), then each of the functions γ_i(t), i = 1,2, 3, defined in (14) for t ∈ [0, r), is uniformly approached on [0, r) by the restriction of a function H_i : R → S additive on R².

The following inequalities hold:

()

Proof of Lemma 3. Since each function γ₁, γ₂, γ₃ is 4δ-additive on E(0,0; r) (from Lemma 2), formula (23) follows immediately from a known result [4, Lemma 2]:

“Let f : [0, d) → (X, ∥·∥), X a Banach space, be δ-additive on E_d = {(x, y) ∈ R² : 0 ≤ x < d, 0 ≤ y < d, x + y < d}.

Then there exists at least one additive function L : R → X such that

()

Moreover, for every i, j = 1,2, 3 and t ∈ [0, r), formula (24) follows from (15) and (23); in fact

()

This means that the restriction to [0, r) of each additive H_j, j = 1,2, 3, approaches uniformly on [0, r) each function γ₁, γ₂, γ₃.

Lemma 3 is proved.

Proof of Theorem 1. According to (24) in Lemma 3, each function γ_i(t), i = 1,2, 3, defined in (14) for t ∈ [0, r), namely,

()

is uniformly approached on [0, r) by each of the additive functions H_j : R → S, j = 1,2, 3.

Let us define F_i : D_f → S, i = 1,2, 3, as follows:

()

Such functions F_i, i = 1,2, 3, satisfy obviously the additive equation restricted to E(a, b; r).

Moreover, thanks to Lemmas 2 and 3, each function F_i approaches uniformly f on D_f as in formula (13); in fact, for arbitrary (x, y) ∈ E(a, b; r),

()

similarly for y ∈ E_y, γ₂(y − b) = f(y) − f(b).

On the projection E_x+y, where x + y = a + b + t and γ₃(x + y − a − b) = f(x + y) − f(a + b), we get from Lemma 3 and formula (9)

()

Therefore, each function F_i : D_f → S, i = 1,2, 3, satisfies (13), and Theorem 1 is proved.

Remark 4. The foregoing study was developed as though the projections E_x, E_y, and E_x+y were pairwise disjoint.

If two of them overlap, for instance E_x∩E_y is nonempty, in every common point the values given by the different parts of formulas of approximating function F have to be the same.

More in particular, if the set D_f = E_x ∪ E_y ∪ E_x+y is connected, in (28) the equations

()

hold, whence f(a) − H_i(a) = f(b) − H_i(b) = 0. In this case, the locally δ-additive f is uniformly approached on the whole D_f by the restriction to D_f of a function H : R → S, additive on R² (H = H_i, i either =1, or =2, or =3).

2.2. A Result on Local Stability of the Logarithmic Equation

On the ground of the results in Section 2.1 it is easy to prove the local stability of the logarithmic Cauchy equation (2) restricted to the bounded domain

()

for given (a, b) ∈ R² and r > 0.

The projections J_x, J_y, J_xy of J are given by

()

Since the local stability of (2) depends on the comparison of every f satisfying

()

for (x, y) ∈ J(a, b; r) with some solution φ of the corresponding equation (2) restricted to J(a, b; r), let us premise (Lemma 5) the local solution of (2).

Lemma 5. Let S be a real linear space; if φ : D_f = J_x ∪ J_y ∪ J_xy → S satisfies (2) on the bounded domain J = J(a, b; r) defined in (32), then there exists a function h₀ : R → S, additive on R², such that

()

Proof of Lemma 5. By the usual substitutions

()

the domain J(a, b; r) is transformed into a set E(a, b; r) like the one defined in (6). Let us consider

()

Equation (2) is transformed into

()

Therefore, using Theorem 1, we obtain (35) with additive h₀.

The local stability of the logarithmic equation (2) is stated by the following.

Theorem 6. Let (S, ∥·∥) be a Banach space; if f : D_f = J_x ∪ J_y ∪ J_xy → (S, ∥·∥) satisfies

()

for some δ > 0 and every (x, y) ∈ J(a, b; r), defined in (32), for given (a, b) ∈ R² and r > 0, then there exists (at least) a function H : R → S, additive on R², such that the function L : D_f → S defined by

()

satisfies both the following properties:

(i)
L is a local solution of the logarithmic equation on the restricted domain J(a, b; r);
(ii)
L approaches uniformly fon D_f, and
()
holds for every t ∈ D_f.

Proof of Theorem 6. The usual substitutions x = e^u, y = e^v (like in proof of Lemma 5) transform the inequality (39) restricted to the set J(a, b; r) into

()

restricted to the set E = E(a, b; r), defined in (6).

Now, we can follow the same line of proof as in Section 2.1 by defining the functions γ_i : [0, r) → S, i = 1,2, 3, related to g; namely,

()

then there exist functions H_i : R → S, i = 1,2, 3, additive on R², such that each of the functions G_i, i = 1,2, 3,

()

is a local solution of the equation g(u + v) = g(u) + g(v) restricted to E(a, b; r), and

()

holds for i = 1,2, 3.

Now let us come back to f, by the substitutions which transformed (39) into (42), beginning by the transformation of functions G_i defined in (44); on J_x (from E_u)

()

similarly, for G_i(ln y) on J_y (from E_v) and for G_i(ln xy) on J_xy (from E_u+v).

By the definition

()

formula (44) changes into

()

Obviously each L_i(τ), i = 1,2, 3, satisfies the logarithmic equation restricted to J(a, b; r).

In order to prove the approximation stated in (41), let us begin by the projection J_x: for x ∈ J_x; then u = ln x, g(u) = f(x), and u ∈ [a, a + r); hence, from (44)

()

and from (45)

()

Similarly for f on J_y and on J_xy.

Therefore, (41) is true with L = L₁ or L = L₂ or L = L₃, and Theorem 6 is proved.

Remark 7. Remarks about the consequence of a possible overlapping of the projections of the given restricted domain, like those in Remark 4, could be repeated here.

2.3. About the Pexiderized Forms of the Foregoing Equations

Stability results for the Pexiderized forms of the additive and the logarithmic equations, namely,

()

can be easily stated on the ground of the foregoing Theorems 1 and 6.

In fact, when the inequality

()

is satisfied for every (x, y) ∈ E(a, b; r), the statement of Theorem 1 can be easily adapted to the condition (52), because the functions λ(x), μ(y) and ψ(x + y) play a role like that of the restrictions of f to the projections E_x, E_y, E_x+y in case of a unique function f.

Similarly for

()

restricted to J(a, b; r), by use of Theorem 6.

Remark 8. In case of a Pexiderized equation on restricted domain, overlapping of the projections of the given bounded domain obviously produces no changes in the result.

3. About the Remaining Two Cauchy Equations (7) φ(x + y) = φ(x)φ(y) and (8) φ(xy) = φ(x)φ(y) on a Bounded Restricted Domain

3.1. Preliminaries

As for (7) φ(x + y) = φ(x)φ(y) the restricted domain is assumed to be E(a, b; r) defined in (6); the domain of (8) φ(xy) = φ(x)φ(y) is J(a, b; r) defined in (32) for fixed real a, b and r > 0.

Let us premise the local solutions of the above equations (see papers [8, 9] and [10], resp.).

Lemma 9. Let φ : D_φ = E_x ∪ E_y ∪ E_x+y → R satisfy (7) restricted to E(a, b; r) defined in (6).

If and only if there exists some (x^′, y^′) ∈ E(a, b; r), (x^′ ∈ E_x, y^′ ∈ E_y) such that φ(x^′) ≠ 0 and φ(y^′) ≠ 0, the following properties (P₁), (P₂), (P₃) hold:

(P₁)
φ(t) ≠ 0 for every t ∈ E_x+y (hence for every t ∈ D_φ);
(P₂)
sgn φ(t) is constant on each projection E_x, E_y, E_x+y (not necessarily the same in different projections);
(P₃)
φ(t) is given on D_φ by the following formulas:
- (i)
  if t ∈ E_x, φ(t) = Ae^G(t)−G(a),
- (ii)
  if t ∈ E_y, φ(t) = Be^G(t)−G(b),
- (iii)
  if t ∈ E_x+y, φ(t) = ABe^{G(t)−G(a+b)},
- where G : R → R is additive on R²; A ≠ 0, B ≠ 0 are constant.

Remark 10. Notice that φ restricted to each of the projections E_x, E_y, E_x+y is the restriction of a solution Φ : R → R of the equation

()

valid on the whole R², for suitable K ≠ 0.

Since this equation can be written as

()

we get

, for some additive G(t), whence formulas in (P₃) of Lemma 9 for

()

respectively, in E_x, E_y, E_x+y.

Lemma 11 (see [10].)The general nowhere vanishing solution φ : D_φ = J_x ∪ J_y ∪ J_xy → R of (8) restricted to the set J(a, b; r) defined in (32) is given by the following formulas:

()

where h : R → R is additive on R².

Remark 12. As in Remark 10, we can see that the local solution φ of (8), restricted to each of the projections J_x, J_y, J_xy, is the restriction of a solution Ψ : R⁺ → R of a more general equation

()

for suitable K ≠ 0.

From KΨ(xy) = KΨ(x)KΨ(y), it follows that Ψ(t) = (1/K)e^h(ln t), with

()

3.2. How the Question of Local Stability of (7) or (8) Has to Be Properly Formulated?

The foregoing Remarks 10 and 12, which point out a connection of the restricted equation under consideration with more general equations, namely,

()

suggest the following forms of perturbation of such equations:

for −1 < θ(x, y) < 1 and some fixed δ > 0.

Moreover, it is known (see [8, 9]) that the local solutions of the restricted equations (7) or (8), which vanish somewhere, are expressed by formulas containing arbitrary functions; therefore, the problem of the local stability seems to be significant in the set of nowhere vanishing functions f.

In this frame, the perturbed forms (7)δ and (8)δ can be written equivalently as

The stability results which follow are framed in this context.

3.3. A Sign Property concerning the Perturbed Forms of the Exponential Equation and the Power Equation

Here, we will be concerned with the condition (7)δ′, (x, y) ∈ E(a, b; r) defined in (6) for some fixed δ > 0, in the set of functions f : D_f = E_x ∪ E_y ∪ E_x+y → R, such that f(t) ≠ 0 for every t ∈ D_f.

Let us premise a remark about signs of nowhere vanishing functions f satisfying (7)δ′ on E(a, b; r). From Lemma 9, Property (P₂), it is known that every nowhere vanishing solution of the exponential Cauchy equation restricted to E(a, b; r) keeps a constant sign in each of the projections E_x, E_y, E_x+y of E(a, b; r).

We will see that a similar property is true also for every solution of the restricted condition (7)δ′, which is rewritten here as follows:

From (7)δθ′, assuming (x, y) = (a, b + ξ) and (x, y) = (a + ξ, b) with 0 ≤ ξ < r, we get

()

whence

()

Moreover, from (7)δθ′ for (x, y) = (a + ξ, b + ξ), 0 ≤ ξ < r/2, θ^′′′ = θ(a + ξ, b + ξ), f(a + b + 2ξ) = ; hence, f(t) has constant sign in E_x+y.

As a consequence, from (62), (64) it follows that f has constant signs also in E_x and in E_y (the signs of f(a) and of f(b), resp.).

This proves the following.

Lemma 13. Every nowhere vanishing function f : D_f = E_x ∪ E_y ∪ E_x+y → R satisfying (7)δ in E(a, b; r) keeps constant sign in each of the projections E_x, E_y, E_x+y of E(a, b; r).

Similarly, we can prove a sign property concerning the perturbed form of the power equation.

Let us consider now the condition (8)δ′; namely,

()

for some fixed δ > 0, assuming f(t) ≠ 0 for every t ∈ D_f = J_x ∪ J_y ∪ J_xy.

The usual substitutions of variables x, y allow us to use the foregoing results about the exponential equation. Put

()

whence f(xy) = :g(u + v), a + b ≤ u + v < a + b + r.

Then (65) is transformed into

()

namely

Therefore, from Lemma 13, it follows that g(u) has constant sign (=sgn g(a)) in [a, a + r), whence f(x) has constant sign (=sgn f(e^a)) in [e^a, e^a+r); similarly for g(v) in [b, b + r), namely for f(y) in [e^b, e^b+r) and for g(u + v) in [a + b, a + b + r), namely f(xy) in [e^a+b, e^a+b+r).

Hence, the following result is proved.

Lemma 14. Every nowhere vanishing function f : D_f = J_x ∪ J_y ∪ J_xy → R satisfying (65) restricted to J(a, b; r) has constant signs in each of the projections J_x, J_y, J_xy of J(a, b; r).

3.4. A Result of Local Stability for the Exponential Cauchy Equation

In the set of functions f : D_f = E_x ∪ E_y ∪ E_x+y → R such that f(t) ≠ 0 for every t ∈ D_f, let us consider the inequality (7)δ′, with (x, y) ∈ E(a, b; r) for some fixed δ > 0.

From (7)δ′

()

since

()

f satisfies

()

namely for λ(t): = ln |f(t)|,

()

On the ground of Theorem 1, there exists (at least) one additive function H : R → R, such that the function L : D_f → S defined by

()

is a local solution of the additive Cauchy equation restricted to E(a, b; r), such that

()

Since |f(t)| = e^λ(t), whence,

()

we get

()

substitution of L(t) by its explicit formulas gives the following:

()

and similarly

()

By defining

()

it is easily proved that

()

Moreover, from (78),

()

This means that the values of f(t) in D_f are “near” (in dependence on δ) the values of a local solution F(t) of the corresponding equation restricted to the same domain E(a, b; r) and give the following theorem of local stability.

Theorem 15. If the function f : D_f = E_x ∪ E_y ∪ E_x+y → R, is nowhere vanishing in its domain D_f and satisfies (7)δ′, for some given δ > 0 and every (x, y) ∈ E(a, b; r) defined in (6) for given (a, b) ∈ R² and r > 0, then there exists (at least) an additive function H : R → R such that the function F : D_f → R

()

has both the properties:

(i)
F is a nowhere vanishing local solution of the exponential Cauchy equation F(x + y) = F(x)F(y) restricted to E(a, b; r);
(ii)
the values F(t) are near the values f(t) on D_f; more exactly
()

3.5. A Result on Local Stability of the Power Cauchy Equation

In the set of nowhere vanishing functions f : D_f = J_x ∪ J_y ∪ J_xy → R, let us consider the inequality (65), (x, y) ∈ J(a, b; r) ⊂ R² defined in (32), for some given δ > 0.

The usual substitutions

()

transform the condition (65) into

Hence, thanks to Theorem 15 (referred to g(t) instead of f(t)), there exists (at least) one additive function H : R → R such that the function G : D_g → R defined by

()

satisfies the exponential equation G(u + v) = G(u)G(v) restricted to E(a, b; r) and approaches g on D_g = E_u ∪ E_v ∪ E_u+v as follows:

Formula (82)g can be rewritten as

for t ∈ E_u ∪ E_v ∪ E_u+v.

From the definition of G,

for u ∈ E_u = [a, a + r) then g(u) = f(x), G(u) = G(ln x) = f(e^a) · e^−H(a) · e^H(ln x),
similarly for v ∈ E_v = [b, b + r) then g(v) = f(y), G(ln y) = f(e^b) · e^−H(b) · e^H(ln y),
and for u + v ∈ E_u+v = [a + b, a + b + r): G(ln (xy)) = f(e^a)f(e^b) · e^{−H(a)−H(b)} · e^H(ln xy).

Hence, by defining Φ : D_f → R as follows:

()

we get

()

namely,

()

This proves the following property of local stability of the “power” Cauchy equation.

Theorem 16. Let the nowhere vanishing function f : D_f ⊂ R⁺ → R satisfy the condition (65) for some given δ > 0 and every (x, y) ∈ J(a, b; r) defined in (32), for given (a, b) ∈ R² and r > 0; D_f = J_x ∪ J_y ∪ J_xy; then there exists (at least) an additive function H : R → R such that the function Φ : D_f → R defined by

()

has both the following properties:

(i)
Φ is a local solution of the Cauchy equation Φ(xy) = Φ(x)Φ(y) restricted to J(a, b; r);
(ii)
the values of Φ(t) are near the values f(t) in D_f; more exactly
()

3.6. Remark about the Pexiderized Forms of the Foregoing Equations

According to the remarks at the end of Section 2, the stability results given by Theorems 15 and 16 can be easily adapted to the Pexiderized forms of the corresponding equations, namely, to

()

for nowhere vanishing functions

()

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About the Local Stability of the Four Cauchy Equations Restricted on a Bounded Domain and of Their Pexiderized Forms

Abstract

1. Introduction

2. About the Additive and the Logarithmic Cauchy’s Equations

2.1. A Result on Local Stability of the Additive Equation

2.2. A Result on Local Stability of the Logarithmic Equation

2.3. About the Pexiderized Forms of the Foregoing Equations

3. About the Remaining Two Cauchy Equations (7) φ(x + y) = φ(x)φ(y) and (8) φ(xy) = φ(x)φ(y) on a Bounded Restricted Domain

3.1. Preliminaries

3.2. How the Question of Local Stability of (7) or (8) Has to Be Properly Formulated?

3.3. A Sign Property concerning the Perturbed Forms of the Exponential Equation and the Power Equation

3.4. A Result of Local Stability for the Exponential Cauchy Equation

3.5. A Result on Local Stability of the Power Cauchy Equation

3.6. Remark about the Pexiderized Forms of the Foregoing Equations

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About the Local Stability of the Four Cauchy Equations Restricted on a Bounded Domain and of Their Pexiderized Forms

Abstract

1. Introduction

2. About the Additive and the Logarithmic Cauchy’s Equations

2.1. A Result on Local Stability of the Additive Equation

2.2. A Result on Local Stability of the Logarithmic Equation

2.3. About the Pexiderized Forms of the Foregoing Equations

3. About the Remaining Two Cauchy Equations (7) φ(x + y) = φ(x)φ(y) and (8) φ(xy) = φ(x)φ(y) on a Bounded Restricted Domain

3.1. Preliminaries

3.2. How the Question of Local Stability of (7) or (8) Has to Be Properly Formulated?

3.3. A Sign Property concerning the Perturbed Forms of the Exponential Equation and the Power Equation

3.4. A Result of Local Stability for the Exponential Cauchy Equation

3.5. A Result on Local Stability of the Power Cauchy Equation

3.6. Remark about the Pexiderized Forms of the Foregoing Equations

References

References

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