Stability has many names

A list of 22 definitions of stability of a functional equation and 5 definitions of stability of the alternation of two functional equations is given as well as some simple examples.


Introduction
The notion of stability is presently considered in various senses. The original problem of stability was formulated by Ulam in 1940 as follows [22]: "give conditions in order for a linear mapping near an approximately linear mapping to exist".
Ulam has precised the above in different senses. E.g., in [22] he posed the following question.
Let G 1 be a group and let (G 2 , d) be a metric group. Given δ > 0, does there exist ε > 0 such that if a mapping h : G 1 → G 2 satisfies the inequality d(h(xy), h(x)h(y)) ≤ δ for all x, y ∈ G 1 , then there is a homomorphism a : This question yields the subsequent Definition 4 of uniform b-stability.
In [23] Ulam has formulated the question: Given a metric group (G, d), a number ε > 0 and a mapping f:G → G which satisfies the inequality d(f(xy), f(x)f(y)) < ε for all x, y in G, does there Vol. 90 (2016) Stability has many names 985 δ > 0 such that for every function g : G 1 → G 2 for which (2) is true there exists a solution f of (1) which satisfies (3).
Example 1. By the classical result of Hyers in [9] Eq. (1) for functions f from a Banach space to a Banach space is stable with δ = ε. This equation is not stable for f from a free group F generated by two elements into R [6]. In fact it is proved in [6] that Eq. (1) in this case is not uniformly b-stable (see Definition 4 below). If in the proof of Theorem 8 in [6] p. 220 we consider the function δf (δ > 0) in place of the function f we have the proof that (1) is not stable. Generally if we consider the function f in the equation (1) as a function from the group F into a subgroup (G, +)⊂ (R, +) with the usual metric, then if G is dense Eq. (1) is unstable and it is not b-stable and in the opposite case this equation is stable and it is not b-stable.
The equation of the idempotent function f(f(x))=f(x), for f from a metric space M with a metric d into itself, is stable. Indeed, if for a function g : M → M we have d[g(g(x)), g(x)] ≤ δ for x ∈ M, then f(x)=x for x ∈ g(M) and f(x)=g(x) for x ∈ M\g(M) is the idempotent function and d[g(x), f(x)]≤ δ for x ∈ M.

Definition 2. (normal stability, strong stability). Equation
We say that this stability is strong if there exists a measure of stability of the form Φ(ε) = Kε for some K.
Example 2. Equation (1) for f from an abelian group to a Banach space is normally stable and strongly stable with Φ(ε) = ε [6]. In Example 5 there is given a stable equation which is not normally stable. The above equation of the idempotent function is normally and strongly stable with Φ(ε) = ε. The equation (1) for f from a free group generated by two elements into the additive group of integer numbers with the usual metric is stable (every δ <1 is "good") and this stability is not strong since this equation is not b-stable.
Let I ⊂ R be an interval with a non empty interior. The translation equation F(F(α, x), y) = F(α, x+y) for F:IxR → I in the class of functions from Ix R to I continuous in each variable (i.e. all functions in consideration are in this 986 Z. Moszner AEM class) is strongly stable with δ = ε 10 [18]. Moreover, the solution, which is the approximation, is continuous [18].
Remark. The translation equation F(F(α, x), y) = F(α, x.y), where F : SxG → S, S is an arbitrary set and (G,.) is a groupoid, is in fact the equation of homomorphism of the groupoid G to the semigroup of the family of functions from S to S with the superposition of the functions as the operation. The homomorphism is of the form x → F(., x) : S → S.
The solution of the translation equation for the interval S ⊂ R and G = (R, +) continuous in each variable is continuous. Indeed, since every solution of the translation equation is a solution of the below inequality for every δ > 0, by the above strong stability of the translation equation, every solution of the translation equation continuous in each variable is approximated by a continuous solution. Thus it is continuous too.
The same conclusion is not valid for the solution of the inequality In fact, the function H(α, x) = δ αx α 2 +x 2 for (α, x) = (0, 0) and H(0, 0) = 0 is the solution of the above inequality and it is continuous in each variable but not continuous. (1) is said to be b-stable if for every function g : G 1 → G 2 the function d[g(x+y), g(x) + g(y)] is bounded, then the function d[g(x), f(x)] is bounded for some solution f of (1). (1), if the groupoids G 1 and G 2 are Banach spaces, is bstable. The equation of homomorphism f(x+y) = f(x)f(y), for f : R → (0, +∞), is b-stable [1]. This equation is an example of an equation which is b-stable and unstable. The translation equation F(F(α, x), y) = F(α, xy), for F : ZxG → Z, where Z is the set of integer numbers with the usual metric and G is a free group generated by two elements, is not b-stable [15].  Remark. The uniform b-stability is uniform since ε does not depend on the function g. The Ulam-Hyers stability is in fact already uniform since in Definition 1 the number δ does not depend on the function g. Every equation is non-uniformly stable, i.e., "for ε > 0 and a function g there exists δ > 0 . . . " [17]. Vol. 90 (2016) Stability has many names 987 Definition 5. (normal uniform b-stability, strong b-stability). Equation (1) is said to be normally uniformly b-stable if there exists a function Ψ : (0, +∞) → (0, +∞) (measure of uniform b-stability) for which inf Ψ((0, +∞)) = 0 and such that for every δ > 0 and g : G 1 → G 2 for which (2) is true,

Example 3. Equation
is satisfied for some solution f of (1). If there exists a measure of uniform b-stability of the form Ψ(δ) = Kδ for some K, then the b-stability is called strong.  [15] and stable (Φ(ε) = 3ε 2 ) but it is not strongly b-stable. Indeed, assume that there exists a K such that for every δ > 0 and every function g :  (1) is restrictedly uniformly b-stable if there exists a δ 0 > 0 such that for every δ > 0 and δ < δ 0 and for every g : G 1 → G 2 for which (2) is true there exist an ε > 0 and a solution f of (1) which satisfies (3). If ε does not depend on the function g in (2) this restricted b-stability is said to be uniform. The measure of restricted uniform b-stbility is defined here as the function Ψ : (0, δ 0 ) → (0, +∞). This restricted uniform b-stability is normal if there exists a measure Ψ such that inf Ψ((0, δ 0 )) = 0. Example 6. Every equation (normally) uniformly b-stable is evidently (normally) restrictedly uniformly b-stable.
The equation for f : R → R, has f(x) = x as the only solution and it is restrictedly uniformly b-stable and not b-stable. Indeed, for y=0 we have f(x) = x+f(0) and from x−y − 1| is bounded and the function |0 − x| is unbounded, thus Eq. (4) is not b-stable.
Assume that for a function g : R → R and 0 < δ < 1. Therefore the function g is thus strictly increasing. Let y ∈ R be arbitrarily fixed. Since g is increasing, there exists lim x→y−0 g(x) ≤ g(y). For lim x→y−0 g(x) < g(y), we have thus a contradiction to (5). Therefore lim x→y−0 g(x) = g(y) and analogously lim x→y+0 g(x) = g(y). The function g is thus continuous. This function is unbounded from above and from below. For the indirect proof assume that g is bounded from above. The sequence g(n) is increasing and bounded in this case, thus there exists limg(n). We have lim g(n+1)−g(n) (n+1)−n = 0, thus a contradiction to (5). The function g is thus unbounded from above. We have, by analogous proof, that g is unbounded from below. Since g is continuous we obtain that g(R) = R. Thus for every x ∈ R there exists a y such that g(y)=x and we have |g(x) − x| = |g(g(y)) − g(y)| ≤ δ. Our equation is thus restrictedly uniformly b-stable. It is also stable since for every ε > 0 every positive number δ < 1 is "good" for the stability. This stability is not normal since normal stability implies b-stability [17]. (1) is called uniquely stable (uniquely normally stable, uniquely uniformly b-stable, uniquely normally uniformly bstable, uniquely restrictedly uniformly b-stable) if the solution of (1) in (3) or in (3') is unique.

Definition 7. (unique stability). Equation
Example 7. Hyers in [9] proved that Eq. (1) is uniquely stable. The translation equation for functions from I x R to R is stable in the class of continuous functions and this stability is not unique [18].
Remark. The uniqueness of the stability of Eq. (1) allows us to study the properties of the Hyers operator which assigns the solution f of (1), which satisfies (3), to the function g so that (2) is satisfied [14]. Vol. 90 (2016) Stability has many names 989 Definition 8. (iterative stability). Let S be an arbitrary set and M be a metric space with metric d. Let k, l be functions from SxS to S and F, K be the functions from SxSxMxMxMxM to M. We define for g:S→M k 1 (x, y) = k(x, y), k n+1 (x, y) = k[k n (x, y), y] for n ∈ N, F 1 (x, y, g) = F[x, y, g(x), g(y), g(k(x, y)), g(l(x, y))], F n+1 (x, y, g) = F[k n (x, y), y, F n (x, y, g), g(y), g(k n+1 (x, y)), g(l(x, y))] for n ∈ N and functions K n analogously.
The functional equation is said to be iteratively stable if for every ε > 0 there exists a δ > 0 such that for every function g : S → M for which If there exists a constant L such that δ = Lε, then this stability is called strong.
where α, β, γ are countinuous functions from an interval I to R and f is the unknown function, is strongly iteratively stable in the class of continuous functions under an adequate hypothesis [4]. The equation f(x+c)=f(x) for the periodic function is not stable and it is iteratively stable. (1) is said to be superstable if every function g : G 1 → G 2 for which the function d[g(x+y), g(x)+g(y)] is bounded, is the solution of (1) or it is bounded.

Definition 9. (superstability). Equation
Remark. The term "superstability" is not "good" since a superstable equation may be unstable. E.g., the equation f(x+y) = f(x)f(y) for f: R → (0, +∞) with the usual metric is superstable ( [1]; the first paper on the theme of superstability) and it is not stable (see Example 1).  for f from an abelian group uniquely divisible by 2 to C, is superstable and it is not uniformly superstable [5].
Example 11. If every function f : G 1 → G 2 is a homomorphism, then Eq. (1) is evidently hyperstable. Also if, e.g., x+y=x for every x, y ∈ G 1 and for every x, y ∈ G 2 .
The equation for f from a normed algebra to a complete normed algebra, is hyperstable [3].
The derivation equation where f:A → B(E) and E is a Banach space, A is an algebra of operators on E, B(E) is the algebra of bounded operators on E, is hyperstable (the corollary from the theorem in the paper [20]). The sine equation (see Example 10) is superstable and not hyperstable (g(x) = √ δ is the solution of the inequality |g(x+y)g(x − y) − g 2 (x) + g 2 (y)| ≤ δ and it is not the solution of the sine equation).
Analogously to Ulam's problem one may pose the following one: Is a function, which is approximated by a solution of a functional equation, an approximate solution of this equation? More exactly Definition 12. (inverse stability). Equation (1) is said to be inversely stable if for every δ > 0 there exists a ε > 0 such that for every function g : (3) is true for a solution f of (1), then g satisfies (2).
Example 12. The equation of homomorphism f(xy)=f(x)f(y) for f from a groupoid to a metric groupoid, is inversely stable with ε = δ/3 [14]. The equation f(x+y)=f(x)f(y) for f from R to R, is not inversely stable (for g(x) = expx + δ we have |g(x) − expx| ≤ δ and the function g(x+y)-g(x)g(y) is unbounded). (1) is said to be absolutely uniformly b-stable if it is uniformly b-stable and inversely uniformly b-stable.
The b-stability is generalised in [7] in the following way. Let G 2 be a group in which there is defined a family B of subsets of G 2 such that (a) B contains all singletons, A set in B is called B-bounded and functions whose range is in B are called B-bounded.

Definition 18. (B-stability).
We say that Eq. (1) is B-stable if for every function g : G 1 → G 2 for which the function g(x+y)-g(x)-g(y) is B-bounded there exists a solution f of (1) such that the function g(x)-f(x) is B-bounded. Uniform B-stability (for every B-bounded set A there exists a set B in B such that . . . ) B-superstability etc. are defined analogously (see also [13]). The equation of homomorphism from a groupoid G 1 to the abelian group G 2 is B-stable for every family B of B-bounded sets in G 2 , i.e., for every set A ∈ B there exists a set B ∈ B such that for every function g : G 1 → G 2 if g(x+y)-g(x)-g(y) ∈ B for x, y ∈ G 1 , there exists a homomorphism f : G 1 → G 2 for which g(x) − f(x) ∈ A for x in G 1 . Indeed, for A=∅ it is sufficient to put B=∅. If A = ∅, let a ∈ A. Assume that g(x+y) − g(x) − g(y) ∈ {−a} for x, y ∈ G 1 . Then the function f(x)=g(x)-a is the homomorphism from G 1 to G 2 and g(x) − f(x) = a ∈ A for every x in G 1 . F (x, y) for x, y in G 1 be a sentential function. The conditional functional equation

Definition 19. (stability of conditional equation). Let
is called stable if for every ε > 0 there exists δ > 0 such that for every function g : there is a function f : The conditional equation x | y ⇒ f(x+y) = f(x) + f(y) if f is a mapping from an orthogonality space onto a real Banach space, is stable with δ = 3 16 ε [8]. Let X and Y be a real normed space and a real Banach space, respectively. Let d> 0 be a given real number. The conditional equation [11]. Remark. We consider the stability of the conditional functional equation in which the condition depends on the unknown function. E.g, we have the following result [3].
Let (S, +) be an Abelian semigroup and let (X, |.|) be a Banach space. If, for some ε 1 , ε 2 ≥ 0 and all x, y ∈ S, a function f:S → X satisfies

Examples of relations between these stabilities
We have here all possibilities. 1. Strong stability and strong b-stability are equivalent. 2. Normal stability implies uniform b-stability and the normal uniform b-stability implies normal stability but not conversely [17].
3. Stability and b-stability are independent, i.e., stability does not imply b-stability and vice versa. Stability implies the restricted b-stability but not conversely. This implication is not true for restricted uniform b-stability. 4. B-stability and the conditional stability are generalizations of stability.

Examples of the dynamical system
A dynamical system is defined as a continuous solution F:IxR → I, where I is an interval in R with a nonempty interior, of the equation This definition is equvalent to the one: a dynamical system is a solution F:IxR → I of the equation where F'(x, 0) means the derivative of the function F(., 0) : I → I at the point x. Equation (6) in the class of continuous functions from IxR to I is (see [16]) -stable (normally, strongly) only for I=R, -b-stable (normally, strongly), uniformly b-stable and restrictedly uniformly b-stable (normally) only for I being bounded or I=R, -inversely b-stable, absolutely b-stable, inversely uniformly b-stable, absolutely uniformly b-stable, superstable, uniformly superstable and inversely superstable only for I bounded, -inversely stable, absolutely stable and hyperstable for no I, -inversely hyperstable for every I. Equation (7) in the class of continuous functions from IxR to I is (see [16] and [17]) -stable, restrictively uniformly b-stable (normally) and inversely hyperstable for every I, -normally stable, b-stable 1 , uniformly b-stable (normally), superstable and uniformly superstable only for I being bounded, -inversely stable, absolutely stable, inversely b-stable, absolutely b-stable, inversely uniformly b-stable, absolutely uniformly b-stable, inversely superstable and hyperstable for no I.
Remark. The stability of Eq. (7) is not strong for the interval I unbouded.
Indeed, assume there is a K> 0 such that for every ε > 0 and for every function G: then there exists a solution F of (7) such that |G(x, t) − F(x, t)| ≤ ε for (x, t) ∈ IxR. We have a contradiction since for ε = K −1 and G(x, t) = c ∈ I the function |G(x, 0) − F(x, 0)| = |c − x| is unbounded for x ∈ I. For a bounded I this stability is strong with e.g. K = min{ 1 10 , 2 5|I| } by the Corollary 3.8 in [15] and by the fact that |F 1 (x, t)−F 2 (x, t)| ≤ |I| for all functions F 1 , F 2 : IxR → I. The situation is the same for b-stability.
Vol. 90 (2016) Stability has many names 995 Let E 1 and E 2 be Banach spaces. If f : E 1 → E 2 satisfies the inequality for some Q ≥ 0 and some 0≤ p <1, then there exists a unique additive mapping A :

Stability of the alternation of two functional equations
Below we consider, in the real case, the relation betwen different stabilities of the alternation "(E 1 ) or (E 2 )" of two functional equations (E 1 ) and (E 2 ) and the stability of these equations. This alternation is in fact the implication "not (E 1 ) ⇒ (E 2 )", i.e., it is the conditional functional equation.
Let E i (f) = 0 for i = 1, 2, ... be functional equations of one real variable x (described also as E i (f)(x)=0) and f: R → D ⊂ R.
For the equations f(x)-f(x)=0 and f(x)-f(1)x=0 for f : R → R\{0}, the first is evidently stable, the second is not stable (see (b)) and their alternation is evidently stable.
Since the alternation of the same two stable (unstable) equations is evidently stable (ustable) we have Remark. Every alternation (10) is not-uniformly stable in the sense that "for ε > 0 and for every function g there exists δ > 0 . . . ". Indeed, if g is a solution Vol. 90 (2016) Stability has many names 997 of the alternation, then every δ > 0 is "good". If g is not a solution of the alternation, then there exists an x 0 ∈ S such that E 1 (g)(x 0 ) = 0 and E 2 (g)(x 0 ) = 0.

Definition 22.
(b-stability and the uniform b-stability of the alternation). The alternation (10) is called b-stable if for every function g: R → D ⊂ R for which this alternation is bounded there exists a solution f of this alternation (10) such that |g(x) − f(x)| is bounded. The b-stability is called uniform if the constant which bounds |g(x) − f(x)| depends not on the function g, i.e., if for every δ > 0 there exsits an ε > 0 such that for every function g: R → D ⊂ R for which (11) for x ∈ R is satisfied, there exists a solution f of (10) such that Definition 23. (inverse stability of the alternation). The alternation (10) is called inversely stable if for every ε > 0 there exsits δ > 0 such that for every function g: R → D ⊂ R for which there exists a solution f of (10) such that |g(x) − f(x)| ≤ δ for x ∈ R, we have |E 1 (g)| ≤ ε or |E 2 (g)| ≤ ε for x ∈ R.
Definition 24. (inverse b-stability of the alternation). The alternation (10) is said to be inversely b-stable if for every function g the boundedness of the function |g(x) − f(x)| for a solution f of (10) implies (11) for some δ > 0. The inverse uniform b-stability, absolute stability, absolute b-stability, superstability, inverse superstability, absolute superstability for the alternation of two functional equations are defined analogously as for a single functional equation.
Remark. For the above stabilities there is the same situation as in Conclusion 1 except for inverse superstability.
We have here Theorem 2. If in the alternation of two equations at least one is inversely superstable, then this alternation is inversely superstable, too.
Proof. If the function g is the solution of the alternation, then the alternation for this function g is evidently bounded. If g is bounded and E(f)=0 is the equation in the alternation which is inversely superstable, then E(g) is bounded, thus the alternation for the function g is bounded too.
Therefore, the stability of the alternation (10) implies the stability of the product equation (12).
A consideration of the functional equations without the solutions is not very interesting (but it is perhaps simpler) therefore the following problem arises. Problem. Improve the above example and examples in the proof of Theorem 1 replacing those functional equations which do not have solutions by equations having their solutions.
The condition (11) is equivalent to the condition