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Hyperstability and Superstability
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. [1, 2]) in 1940 in his talk at the University of Wisconsin, have been a very popular subject of investigations for the last nearly fifty years (see, e.g., [3–11]). For example the following definition somehow describes the main ideas of such stability notion for equations in variables ( stands for the set of all nonnegative reals).
Definition 1. Let be a nonempty set, be a metric space, be nonempty, be an operator mapping into , and be operators mapping a nonempty set into . We say that the operator equation is -stable provided for any and with there exists a solution of (1) such that
(As usual, denotes the family of all functions mapping a set into a set .) Roughly speaking, -stability of (1) means that every approximate (in the sense of (2)) solution of (1) is always close (in the sense of (3)) to an exact solution of (1).
The next theorem has been considered to be one of the most classical results on Ulam’s type stability.
Theorem 2. Let and be normed spaces, complete, and and fixed real numbers. If is a mapping satisfying then there exists a unique function such that
This theorem is composed of the outcomes from [1, 12–14] and it is known (see ; cf. also [15, 16]) that for an analogous result is not valid. Moreover, it has been shown in  that estimation (5) is optimal for in the general case.
Theorem 2 has a very nice simple form, but it has been improved in , where it has been shown that, in the case , each satisfying (4) must actually be additive (and the completeness of is not necessary in such a situation). Namely, we have the following result ( stands for the set of all positive integers).
Theorem 3. Let and be normed spaces, nonempty, , and . Assume also that where , and there exists a positive integer with Then every operator such that is additive on ; that is,
Theorem 4. Let and be normed spaces and let and be fixed real numbers. Assume also that is a mapping satisfying (4). If and is complete, then there exists a unique additive function such that (5) holds. If , then is additive.
Following the terminology introduced in  and next used in, for example,  (see also [3, pages 27–29]), we can describe the second statement of Theorem 4, for , as the -hyperstability of the additive Cauchy equation for .
Remark 5. Let , , , and be given by and for . Then clearly
Note that also
In fact, fix and suppose, for instance, that . Then
However, with a somewhat different (though still natural) form of the function , -hyperstability still holds even without (6). Namely, in [21, Theorem 1.3] the subsequent result has been proved.
Theorem 6. Let and be normed spaces, nonempty, , and real numbers with . Assume also that there is an such that Then every operator satisfying the inequality is additive on .
We refer the reader to, for example, [22, Theorem 1.1, Chapter XVIII], [23, Chapter 4], [24, pages 143-144], and [25, Proposition 3.8] for some information on the following natural issue: when for an operator that is additive on , there is an additive with for .
2. Hyperstability Results for the Cauchy Equation
Formally, we can introduce the following definition.
Definition 7. Let be a nonempty set, a metric space, , and operators mapping a nonempty set into . We say that operator equation (1) is -hyperstable provided every satisfying inequality (2) fulfils (1).
The hyperstability results have various interesting consequences. For instance, note that we deduce at once from Theorem 6 a bit surprising conclusion that each function is either additive on or satisfies the condition for any real numbers , where and are normed spaces and is a nonempty subset of fulfilling condition (14) for some .
Corollary 8. Let and be normed spaces, nonempty, , and for some with . Assume also that (14) holds with an and there are and such that and Then the functional equation has no solution in the class of functions .
The hyperstability results that we have presented so far have been obtained through the fixed point theorem from  (see also [35, 36]; cf.  for a survey on similar methods using the fixed point results). Now, we provide some further -hyperstability results (with functions of some other natural forms) for the Cauchy additive equation, proved in  by some other methods.
Theorem 10. Let be a real inner product space with , a normed space, and . If there are positive real numbers and such that then is additive.
If , then does not need to be additive (see ).
Theorem 11. Let and be normed spaces, , and . Suppose also that there are positive real numbers and with If or is not a real inner product space, then is additive.
If is a real inner product space and , then does not need to be additive (see ).
Given a normed space and , we simplify the notations writing and defining the mapping by Moreover, if and , then we put Clearly, if is additive, then we have (with )
Now, we are in a position to present another result from .
Theorem 12. Let and be normed spaces and . Assume that are additive, where . Let, moreover, be such that one of the following two conditions is valid:(a)is injective, and (b), is injective and Then every function for which there exists an such that is additive on .
Remark 13. Observe that condition (25) holds when with a nonnegative integer or for with a rational number (because is assumed to be additive).
Remark 14. For instance, the inequality in (a) holds for , , and
with a . Analogously, the inequality in (b) is valid when , ,
with a .
For similar hyperstability results in some situations where neither condition (a) nor (b) is fulfilled we refer the reader to [25, Corollaries 3.5 and 3.6].
We end this part of the paper with one more hyperstability result (on a restricted domain) from . To do this, let us recall some notions.
Given nonempty sets , and , we say that -almost everywhere (abbreviated to -a.e.) in if there is a set such that for every . If, moreover, is a normed space, then we also write for and .
Now we are in a position to present [25, Theorem 4.1] (which actually is a consequence of some previous results).
Theorem 15. Let and be normed spaces, , and a -ideal such that Assume also that one of the following two conditions is fulfilled:(i)there exist , , , and such that (ii)there exist , with for and positive reals and such that Then there is a unique additive operator with -a.e. in .
A hyperstability result for the multi-Cauchy equation (which actually is a system of Cauchy equations) can be found in [37, Corollary 4].
Finally, we would like to call the reader’s attention to a general theorem in  which yields numerous other hyperstability results for the Cauchy additive equation.
3. Hyperstability of the Linear Functional Equation
Now, we present the hyperstability results for the linear functional equation of the form in the class of functions , where is a linear space over a field , is a linear space over a field , , and . Clearly, for , and (35) is the well-known (additive) Cauchy functional equation and with , and it is the Jensen equation If , , , , , and , then (35) has the form and its solution is called a -affine function. For further information and references on (36)–(38) we refer the reader to [22, 39].
The subsequent theorem has been proved in .
Theorem 16. Let be a normed space over , be a Banach space over , , , , , and satisfy Then
Theorem 17. Let be a normed space, be a nonempty subset of such that there exists a positive integer with let be a Banach space, , , , and satisfy Then is Jensen on ; that is,
Theorem 18. Let be a normed space, a nonempty subset of such that there exists a positive integer with a Banach space, , , , and satisfy (42). Then is Jensen on .
Theorem 19. Let be a normed space, a nonempty subset of such that there exists a positive integer with a Banach space, , , , and satisfy (42). Then is Jensen on .
We finish this section of the paper by proving one more very simple hyperstability result for (35).
Theorem 20. Let be a normed space over a field , a normed space over a field , , , , , , and let one of the following two conditions be valid:(i) and ;(ii) and . Then every function satisfying the inequality is a solution of the equation
Proof. First, observe that in the case when , inequality (46) with implies .
Put It is easily seen that whence with we get .
We consider only case (i) (case (ii) is analogous). First, assume that . Then (49) with gives
We show by induction that, for each , The case follows immediately from (49). So take an and assume that (51) holds true with . Then, by (50), Thus we have proved that (51) is valid for each .
Letting in (51) we see that which implies (47).
If , then and from (49) with we obtain Analogously as before we show that, for each , Letting in (55) we get (53), and consequently (47) holds.
Remark 22. Let , , , and for . Then Thus is an example of a function which satisfies (46) with , and but is not a solution of (47). This proves that assumptions (i) and (ii) of Theorem 20 are not superfluous.
4. Hyperstability of Some Other Equations
In this part of the paper we present the hyperstability results for some other equations. The first two theorems have been proved in .
Theorem 23. Let be a solution of the functional equation and for some . Assume also that a function satisfy the inequality for some . Then
Theorem 24. Let be a semigroup and pairwise distinct automorphisms of such that the set is a group with the operation of composition of mappings. Let, moreover, be a function for which there exists a sequence of elements of satisfying one of the following two conditions: If a function , mapping into a real normed space , fulfils the inequality then is a solution of the functional equation
The following result, concerning the parametric fundamental equation of information, has been obtained in .
Theorem 25. Let and be a function such that where Then
Theorem 26. Let and be real normed spaces. If a function satisfies the inequality with some and , then
Theorem 27. Let and be real normed spaces and a positive integer. If a function satisfies the inequality with some and , then
The next theorem from  contains a hyperstability result for the Drygas equation.
Theorem 28. Assume that is a nonempty subset of a normed space such that and there exists an with Let be a Banach space and fulfill the inequality for some and . Then satisfies the conditional Drygas equation
Theorem 28 yields at once the following characterization of the inner product spaces.
Corollary 29. Let be a normed space and for some . Then is an inner product space.
Proof. Write for . Then from Theorem 28 we easily derive that It is easy to see that this implies which yields the statement.
The next hyperstability result has been proved in [45, Corollary 2.9] and is actually a particular consequence of two more general theorems proved there.
Theorem 30. Let be a normed space, be a Banach space, and . Assume also that a mapping satisfies the inequality Then
A result on hyperstability of the equation of -Wright affine functions has been obtained in  and it reads as follows.
Theorem 31. Let be a normed space over a field , a Banach space, , , and satisfy Then is a -Wright affine function; that is,
The next result has been proved in  and concerns the homogeneity equation.
Theorem 32. Let and be normed spaces over , , , and satisfy for any and such that is defined. Assume also that or . Then Moreover, if one of the following two conditions is valid:(a) and ;(b) and , then
Some further hyperstability (but also superstability) results for the homogeneity equation can be found in [48, 49]. Unfortunately, they are too involved to be presented here. Therefore, we only give below the following simple corollary (see [48, Corollary 3]).
Theorem 33. Let be a real linear space, a Banach space, and satisfy with some . Then
Pexiderized hyperstability of the functional equation of biadditivity, of the form has been considered in  (actually, for some reason, it has been called the bi-Jensen functional equation by the authors), where the following two theorems have been presented.
Theorem 34. Let be a normed space, a Banach space, , and a function satisfy the inequality with some mappings . Then is biadditive; that is, (88) holds for all .
Theorem 35. Let be a normed space, a Banach space, , and a function satisfy the inequality with some mappings . Then is biadditive.
For some further results, related somehow to the issue of hyperstability, we refer the reader to:(i)[51, Theorem ] (for a generalization of the quadratic equation);(ii) (for the equations of homomorphism and derivation in proper -triples);(iii)[53, Theorem ] (for the equations of homomorphism for square symmetric groupoids, considered in a class of set-valued mappings);(iv)[54, Theorem ] (for a functional equation in one variable in a class of set-valued mappings);(v) (for functional equations of trigonometric forms in hypergroups).
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 [6, 7, 9, 10].
The following definition explains how the notion of superstability for functional equations (in variables) is understood nowadays.
Definition 36. Let be a nonempty set, a metric space, and operators mapping a nonempty set into . We say that operator equation (1) is superstable if every that is unbounded (i.e., ) and satisfies the inequality is a solution of (1).
Theorem 37. Let be a uniquely -divisible commutative group and a finite-dimensional commutative normed algebra without the zero divisors. Then every unbounded function such that is a solution of the sine equation
Theorem 38. Let be a commutative semigroup and a groupoid equipped with(i)an operation such that (ii)an element such that for and for ;(iii)a metric satisfying the condition Moreover, assume that each nonzero element of is cancellative on the left or on the right, the groupoid operation in is continuous, and the unit sphere is compact in . Then every unbounded function such that is a homomorphism; that is,
Theorem 39. Let be a uniquely -divisible commutative monoid and a finite-dimensional commutative normed algebra without the zero divisors. Then every unbounded function such that is a solution of the Lobachevski equation
Theorem 40. Let be a commutative group and a finite-dimensional unital normed algebra without the zero divisors. Then every unbounded function such that is a solution of the cosine equation
Theorem 41. Let be a groupoid and a finite-dimensional normed algebra without the zero divisors. Then every unbounded function such that is a solution of the Dhombres equation
Using the method from the proof of Theorem 41, Moszner also got the superstability of the Mikusiński equation This result reads as follows.
Theorem 42. Let be a group and a finite-dimensional normed algebra without the zero divisors. Then every unbounded function such that is a solution of (104).
The above theorem generalizes (to some extent) the following result, which has been obtained in , by another method of proof and under stronger assumptions.
Theorem 43. Let and be a commutative group. If a function satisfies then is additive or
Chahbi in  has dealt with the equation where , and showed the following result on its superstability.
The form of (108) has been motivated by the Gołąb-Schinzel equation and some other equations related to it. A survey on superstability results for such equations can be found in [3, pages 29–32] (for more information and further references on those equations see also ).
The following result comes from .
Theorem 45. Assume that and . If a function fulfills the inequality then either is bounded or it is a solution of the equation
Let be a ring (not necessarily commutative) uniquely divisible by 2 and . Then , where is a noncommutative group, which in the case when is isomorphic to the Heisenberg group. Denote by a selfmap of given by
With this notations, we have the following theorem (proved in ) on the superstability of a functional equation connected with the d’Alembert and Stetkær equations.
Theorem 46. Assume that . If a function fulfills the inequality then either or
The next result has been proved in .
Theorem 47. Assume that is a normed space over and is a Banach algebra over in which the norm is multiplicative, that is, If a mapping fulfills then either or
In , Kim dealt with the pexiderized Lobachevski equation and proved the following theorem.
Theorem 48. Let and be a uniquely 2-divisible commutative semigroup. If nonzero and nonconstant functions satisfy the inequality then either there exist such that or both and satisfy (99).
An immediate consequence of Theorem 48 is the following corollary.
Corollary 49. Let and be a uniquely 2-divisible commutative semigroup. If nonzero and nonconstant functions satisfy the inequality then either there exist such that or both and satisfy (99).
Theorem 50. Let be a linear space and let functions be such that Then the following three statements hold:(i)if , then is arbitrary;(ii)if is nonzero and bounded or , then is also bounded;(iii)if is unbounded, then , is unbounded, and (127) holds for all .
Theorem 51. Let be a linear space. If functions are such that then the following three statements hold:(i)if , then is arbitrary;(ii)if is nonzero and bounded or , then is also bounded;(iii)if is unbounded, then , is unbounded, and (128) holds for all .
The next two theorems have been obtained in . It is assumed in them that is a commutative group, is a finite subgroup of the group of automorphisms of (the action of on is denoted by ), and is the cardinality of .
Theorem 52. Let . If , the function is bounded, and the function is unbounded, then the function