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.

After this introduction, in Section

It is well known that the problem of stability was posed, for the additive equation, by S. Ulam and was solved by Hyers [

As for the “local” stability of equations on a restricted domain, first results can be found in [

In the present paper (Section

It has to be remarked that in the classical paper [

On the contrary, when a restricted domain like that in (

Therefore, in order to adhere to the sense of Ulam’s question in case of a restricted domain like

In this frame, in Section

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,

This problem will be the object of Section

In the set of functions

As usual, the projections of

For

We purpose to check whether each

A positive answer is given by the following.

Let

Then there exists (at least) a function

In order to prove Theorem

Let

Let us prove (

Similarly we prove (

In order to prove (

For

Similarly, for

As for

Lemma

If

The following inequalities hold:

Since each function

Lemma

According to (

Let us define

Moreover, thanks to Lemmas

On the projection

The foregoing study was developed as though the projections

If two of them overlap, for instance

More in particular, if the set

On the ground of the results in Section

The projections

Since the local stability of (

Let

By the usual substitutions

The local stability of the logarithmic equation (

Let

The usual substitutions

Now, we can follow the same line of proof as in Section

Now let us come back to

By the definition

In order to prove the approximation stated in (

Therefore, (

Remarks about the consequence of a possible overlapping of the projections of the given restricted domain, like those in Remark

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

In fact, when the inequality

Similarly for

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.

As for (

Let us premise the local solutions of the above equations (see papers [

Let

If and only if there exists some _{1}), (P_{2}), (P_{3}) hold:

if

if

if

where

Notice that

Since this equation can be written as
_{3}) of Lemma

The general nowhere vanishing solution

As in Remark

From

The foregoing Remarks

Moreover, it is known (see [

In this frame, the perturbed forms

The stability results which follow are framed in this context.

Here, we will be concerned with the condition

Let us premise a remark about signs of nowhere vanishing functions _{2}), it is known that every nowhere vanishing solution of the exponential Cauchy equation restricted to

We will see that a similar property is true also for every solution of the restricted condition

Moreover, from

As a consequence, from (

This proves the following.

Every nowhere vanishing function

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

Let us consider now the condition

The usual substitutions of variables

Then (

Therefore, from Lemma

Hence, the following result is proved.

Every nowhere vanishing function

In the set of functions

From

On the ground of Theorem

Since

By defining

Moreover, from (

If the function

the values

In the set of nowhere vanishing functions

The usual substitutions

Hence, thanks to Theorem

Formula

From the definition of

for

similarly for

and for

Hence, by defining

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

Let the nowhere vanishing function

the values of

According to the remarks at the end of Section