This is a survey paper concerning the notions of hyperstability and superstability, which are connected to the issue of Ulam’s type stability. We present the recent results on those subjects.

In this paper we provide some recent results concerning hyperstability and superstability of functional equations. Those two notions are very similar but somewhat different. They are connected with the issue of Ulam’s type stability.

Let us mention that various aspects of Ulam’s type stability, motivated by a problem raised by Ulam (cf. [

Let

(As usual,

The next theorem has been considered to be one of the most classical results on Ulam’s type stability.

Let

This theorem is composed of the outcomes from [

Theorem

Let

Clearly, since (

On account of Theorem

Let

Following the terminology introduced in [

It is interesting that the hyperstability result, described in Theorem

Let

However, with a somewhat different (though still natural) form of the function

Let

We refer the reader to, for example, [

Formally, we can introduce the following definition.

Let

The hyperstability results have various interesting consequences. For instance, note that we deduce at once from Theorem

Theorem

Let

Let

The hyperstability results that we have presented so far have been obtained through the fixed point theorem from [

Let

If

Let

If

Given a normed space

Now, we are in a position to present another result from [

Let

Observe that condition (

For instance, the inequality in (a) holds for

For similar hyperstability results in some situations where neither condition (a) nor (b) is fulfilled we refer the reader to [

We end this part of the paper with one more hyperstability result (on a restricted domain) from [

Given nonempty sets

Now we are in a position to present [

Let

there exist

there exist

A hyperstability result for the multi-Cauchy equation (which actually is a system of Cauchy equations) can be found in [

Finally, we would like to call the reader’s attention to a general theorem in [

Now, we present the hyperstability results for the linear functional equation of the form

The subsequent theorem has been proved in [

Let

Similar results, for Jensen equation (

Let

Let

Let

We finish this section of the paper by proving one more very simple hyperstability result for (

Let

First, observe that in the case when

Put

We consider only case (i) (case (ii) is analogous). First, assume that

We show by induction that, for each

Letting

If

Let

Let

In this part of the paper we present the hyperstability results for some other equations. The first two theorems have been proved in [

Let

Let

The following result, concerning the parametric fundamental equation of information, has been obtained in [

Let

Let us recall (see [

The next two theorems have been proved in [

Let

Let

The next theorem from [

Assume that

Theorem

Let

Write

The next hyperstability result has been proved in [

Let

A result on hyperstability of the equation of

Let

The next result has been proved in [

Let

Some further hyperstability (but also superstability) results for the homogeneity equation can be found in [

Let

Pexiderized hyperstability of the functional equation of biadditivity, of the form

Let

Let

For some further results, related somehow to the issue of hyperstability, we refer the reader to:

[

[

[

[

[

In this part of the paper we present several recent results on superstability of some functional equations. For numerous earlier results as well as the historical background of the subject we refer the reader to [

The following definition explains how the notion of superstability for functional equations (in

Let

Let us start with the results that Moszner has proved in [

Let

Let

an operation

an element

a metric

Let

Let

The next theorem, proved by Moszner in [

Let

Using the method from the proof of Theorem

Let

The above theorem generalizes (to some extent) the following result, which has been obtained in [

Let

Chahbi in [

Let

The form of (

The following result comes from [

Assume that

Let

With this notations, we have the following theorem (proved in [

Assume that

The next result has been proved in [

Assume that

In [

Let

An immediate consequence of Theorem

Let

The below superstability outcomes for the functional equations

Let

if

if

if

Let

if

if

if

The next two theorems have been obtained in [

Let

Let

The next three theorems do not actually provide superstability results in the sense of Definition

The subsequent theorem, proved in [

Let

The next result has been proved in [

Let

The last presented theorem is the main result of [

Assume that