Banach Limit in the Stability Problem of a Linear Functional Equation

We present some applications of the Banach limit in the study of the stability of the linear functional equation in a single variable.


Introduction
Let us recall that the Banach limit is a linear functional LIM defined on the space ∞ of all bounded real sequences, satisfying the following conditions: inf{a n : n ∈ N} ≤ LIM(a n ) ≤ sup{a n : n ∈ N} (1) and LIM(a n+k ) = LIM(a n ), for all (a n ) ∈ ∞ and k ∈ N (positive integers). The question about stability of the functional equation of group homomorphisms, formulated by S. Ulam in 1940, initiated investigations of the stability of functional equations. Currently, many studies have been published on this subject (see monographs [8,11,12] or survey papers [1,7]), where the reader can learn more about this issue. The problem of stability of a given functional equation is the question whether a function satisfying that equation with a certain accuracy is close to a solution of it. In this work, we focus our attention on various forms of stability of the linear functional equation

Ulam Stability
In this section, we first show that by the Banach limit method we can obtain a modification of the Trif outcome [17, Theorem 2.1]. Next we provide a slightly simplified version of the original proof of [17,Theorem 2.1]. We also state some observations concerning existence of solutions to (3) (see Remark 2). As we have already mentioned, to be more readable, we confine ourselves to the case where functions only take the scalar values.
We start with the following generalization of [17, Theorem 2.1]. be finite for all x ∈ X and k = 1, 2. Suppose that φ : X → R satisfies Then there exists a unique solution Φ : X → R of Eq.
(3) such that Proof. Let us write the assumed inequality (5) as We will show inductively that for all x ∈ X and n ∈ N (positive integers).
To this end observe first that the case n = 1 is just (7). So, assuming that inequalities hold for n, we are to prove that they are true for n + 1. Putting f n (x) instead of x in (7) we get for all x ∈ X. Using the assumed estimates for the expression φ(f n (x)) n−1 i=0 g(f i (x)) , we conclude that the inequalities are true for n + 1.
For x ∈ X the sequences n−1 j=0 as convergent are bounded. So, (8) means that the sequence is bounded for all x ∈ X and we can define the function Φ : X → R by for all x ∈ X, whence inequality (8) makes the function Φ fulfill (6). Next, using shift-invariance (2), the linearity of the Banach limit and condition (1) we get For the proof of the uniqueness of Φ suppose that Φ 0 : X → R is such that and Then We show by induction that, for each n ∈ N 0 := N ∪ {0}, Clearly, the case n = 0 is just (11). So, assume that (12) holds for a fixed n ∈ N 0 . Then replacing x by f (x) in (11), by (3) and (9), for each x ∈ X we obtain Thus we have proved that (12) is valid for each n ∈ N 0 , whence letting n → ∞ we obtain Φ 0 = Φ, which ends the proof.
Note that a very particular case of Theorem 1 (with ε 1 = ε 2 ) is the following simple observation.
Then a function φ : Proof. The necessary condition of the statement follows from Theorem 1 with ε 1 = ε 2 = ε. The converse is easy to verify.
If ε 1 = −ε 2 in Theorem 1, then we get the scalar version of [17, Theorem 2.1], i.e., the next theorem (but only in the case g(X) ⊂ (0, ∞)). We prove Theorem 2 in a somewhat different way than in [17], i.e., by the Banach limit method. Note that the assumption g(X) ⊂ (0, ∞) has been replaced in Theorem 2 by 0 ∈ g(X); moreover, the uniqueness statement has been strengthened a bit.
Then there exists a unique solution Φ : Moreover, Φ is given by (16) Proof. First we show inductively that for all x ∈ X and n ≥ 1. Clearly, the case n = 1 is just (14) divided by |g(x)|. So, assume that the inequality is valid for a fixed n ≥ 1. We are to prove that this is the case for n + 1.
Replacing x by f n (x) in (14) and dividing by whence and by the assumed inequality (18), for all x ∈ X we get Thus we have proved that (18) holds for every n ∈ N, which means that, for every x ∈ X, (19) and consequently the sequence is bounded. So, we can define the function Φ : X → R by (16). The normalization condition (1) of the LIM functional and inequalities (19) make the function Φ fulfill (15). Next, using the linearity of the Banach limit, condition (1) and finally the shift-invariance (2), we get which means that Φ is a solution to (3). It remains to show the uniqueness of Φ. So, assume that also Ψ : X → R satisfies the equation and there is a μ > 0 such that Then where μ 0 := μ + 1. We show by induction that, for each n ∈ N 0 , Clearly, the case n = 0 is just (23). Assume that (24) holds for a fixed n ∈ N 0 .
Then replacing x by f (x) in (24), by (21), we obtain which implies that Thus we have proved that (24) is valid for each n ∈ N 0 , whence letting n → ∞ we obtain Φ = Ψ . This ends the proof.
Remark 1. The statement on the uniqueness of Φ in Theorem 2 can be strengthened a bit. Namely, Φ is the only solution to (3) satisfying the inequality with some function η : X → [0, ∞) such that the sequence η(f n (x)) n∈N0 is bounded for each x ∈ X. It is very easy to modify the end of Theorem 2 proof accordingly [in particular, replacing μ 0 by η(f n (x)) in (24)].
The following observation concerning solutions to Eq. (3) can be easily deduced from the proof of Theorem 2. It shows further advantages of using the Banach limit (see also Remark 4).
Remark 2. Let 0 ∈ g(X) and φ : X → R be such that the sequence is bounded. Then we can define a function Φ : X → R by (16) and (20) shows that Φ is a solution to (3).
On the other hand, (18) with ε(x) ≡ 0 shows that every solution of Φ : X → R of (3) fulfils the condition Thus we obtain a conclusion that Eq. (3) has at least one solution if and only if there exists a function φ : X → R such that the sequence (κ n (x)) n∈N is bounded for each x ∈ X.
The next two remarks show that assumption (13) of Theorem 2 cannot be omitted.

Remark 3.
If X ⊂ R, f (x) = x and g(x) = 1 for x ∈ X, and h is a non-zero bounded function, then for every function φ : X → R we have where ε 0 = sup x∈X |h(x)|, but the set of solutions of the equation Remark 4. Assume that 0 ∈ g(X) and there is an x 0 ∈ X such that On the other hand, according to (27), k j=0 g(f j (x 0 )) ∈ {1, 2} for all k ∈ N and the sequence (φ(f n (x 0 ))) n∈N is bounded for each function φ : X → R. Consequently The next remark provides an example of a very simple application of Theorem 2.
Remark 5. If α ∈ (1, ∞) and g(x) = α for x ∈ X, then and, by Theorem 2, for every function φ : X → R fulfilling there exists a unique solution Φ : X → R of the equation As Forti [10] noted, in the case where (X, +) is a commutative semigroup, α = 2 and f (x) = 2x, this result allows us to obtain the classical stability result for the Cauchy equation (see also [5, pp. 11-14]). Analogously, we may obtain the stability of the quadratic functional equation and several other similar equations.

Iterative Stability
As we have already mentioned, Brydak [6] introduced the concept of stability, which was later referred to as iterative stability (see Turdza [18]). Below we show that the Banach limit method allows to prove the following two generalizations of Brydak's theorem (but without the regularity properties).

Theorem 3.
Assume that f is bijective and 0 ∈ g(X). If a function φ : X → R satisfies inequality for all x ∈ X and n ∈ N, where then there exists a solution Φ : X → R of Eq.
(3) such that Proof. First we observe that, for all x ∈ X and n ∈ N, whence by our assumption (28), If we replace in the last inequality x by f −n (x), then we obtain for all x ∈ X and n ∈ N. But, for all x ∈ X and n ∈ N, whence (31) can be rewritten as Therefore, for all x ∈ X and n ∈ N, which means that the sequence (a n (x)) n∈N defined, for a fixed x ∈ X, by a n (x) = φ(f −n (x)) is bounded (belongs to the space ∞ ). So, we may define a function Φ : X → R by the following formulae Φ(x) = LIM a n (x) n∈N , x ∈ X.