1. Introduction

In this paper , , , and stand, as usual, for the sets of complex numbers, real numbers, integers, and positive integers, respectively. Let be a nonempty set, , be a Banach space over a field , , , and denote the complex roots of the equation

(1.1)

Moreover, , , and (only for bijective ) for and .

The problem of stability of functional equations was motivated by a question of Ulam asked in 1940 and a solution to it by Hyers published in [1]. Since then numerous papers have been published on that subject and we refer to [2–7] for more details, some discussions and further references; for examples of very recent results see, for example, [8–12]. Jung has proved in [5] (see also [13]) some results on solutions and stability of the functional equation

(1.2)

in the case where and for . The result on stability (see [5, Theorem ]) can be stated as follows.

Theorem 1.1.

Let , , , , , and satisfy the inequality

(1.3)

Then there is a unique solution of the functional equation

(1.4)

with

(1.5)

If and , then solutions of the difference equation (1.4) are called the Lucas sequences (see, e.g., [14]); in some special cases they are called with specific names; for example; the Fibonacci numbers (, , and ), the Lucas numbers (, , and ), the Pell numbers (, , and ), the Pell-Lucas (or companion Lucas) numbers (, , and ), and the Jacobsthal numbers (, , and ).

For some information and further references concerning the functional equations in single variable we refer to [15–17]; for an ample survey on stability results for those equations see [2]. Let us mention yet that the problem of stability of functional equations is connected to the notions of controlled chaos (see [18]) and shadowing (see [19–21]).

Remark 1.2.

If is bijective, then, with , (1.2) can be written in the following equivalent form:

(1.6)

Clearly, (1.1) is the characteristic equation of (1.6).

In view of Remark 1.2, from [22,Theorem ] (see also [23]) the following stability result, concerning (1.2), can be derived.

Theorem 1.3.

Let for , be bijective, , and satisfy the inequality

(1.7)

Then there is a unique solution of (1.2) with

(1.8)

Theorem 1.3 appears to be much more general than Theorem 1.1 (obtained by a different method of proof). But on the other hand, estimation (1.5) is significantly sharper than (1.8) in numerous cases (take, e.g., and , with some large ). Therefore, there arises a natural question if the method applied in [5] can be modified so as to prove a more general equivalent of Theorem 1.3, but with an estimation better than (1.8). In this paper, we show that this is the case. Namely, we prove the following.

Theorem 1.4.

Let and satisfy inequality (1.7). Suppose that and one of the following two conditions is valid:

() for ;

() for and is bijective.

Then there exists a solution of (1.2) such that

(1.9)

Moreover, if condition is valid, then there exists exactly one solution of (1.2) with .

Remark 1.5.

Note that, for bijective , Theorem 1.4 improves estimation (1.8) in some cases (take, e.g., , , or , ); however, in some other situations (e.g., , ), estimation (1.8) is better. Theorem 1.4 also complements Theorem 1.3 because can be quite arbitrary in the case of .

2. Proof of Theorem 1.4

Clearly, and . We start with the case .

Fix and first assume that . Write

(2.1)

Then, for each and ,

(2.2)

whence

(2.3)

and consequently

(2.4)

This means that, for each , is a Cauchy sequence and therefore there exists the limit . Further, for every ,

(2.5)

and, by (2.4) with and ,

(2.6)

Now, assume that . This means that is bijective. Let

(2.7)

Then, for each and ,

(2.8)

and next, by (1.7),

(2.9)

Hence,

(2.10)

So, for each , is a Cauchy sequence and consequently there exists the limit . Note that, for every , (2.5) holds and, by (2.10) with and ,

(2.11)

Thus, we have proved that, for , inequality (2.6) holds and is a solution to (1.2). Define by

(2.12)

Then, for , it follows from (2.5) that

(2.13)

and, by (1.1) and (2.6),

(2.14)

In the case where is bijective, the uniqueness of results from [22, Proposition ], in view of Remark 1.2.

Now, assume that . Then (see, e.g., [24, page 39], [25], or [26, 27]) is a complex Banach space with the linear structure and the Taylor norm given by

(2.15)

Clearly, for all .

Define by for . Then,

(2.16)

So, by the previous part of the proof, there exists a solution of (1.2) such that

(2.17)

Write for , . Clearly, , given by for , is a solution of (1.2), and (1.9) holds.

It remains to prove the statement concerning uniqueness of . So, let be a solution of (1.2) with . Let for . It is easily seen that is a solution of (1.2). Moreover, for every ,

(2.18)

Hence, by [22, Proposition ], , which yields .

3. Consequences of Theorem 1.4

Now we present some consequences of Theorem 1.4 and some results from [22, 28, 29].

Theorem 3.1.

Let and satisfy (1.7). Suppose that one of the following three conditions is valid:

(i) for and ;

(ii) for and is bijective;

(iii)(ii) holds and .

Then there exists a solution of (1.2) such that

(3.1)

where

(3.2)

Moreover, if for , then there exists exactly one solution of (1.2) such that .

Proof.

If (i) is valid, then Theorem 1.4 yields (3.1) with . Further, by (1.7),

(3.3)

and for are roots of the equation

(3.4)

Hence, by [22, Theorem ], there is a solution of the functional equation

(3.5)

such that

(3.6)

(The last equality is due to the fact that .) It is easily seen that is a solution to (1.2).

Next, consider the case of (ii). Then, in view of Theorem 1.4, there is a solution of (1.2) satisfying (3.1) with . Further,

(3.7)

with . Hence, according to [22, Theorem ], there exists a function satisfying (1.6) and inequality (3.1), with . Now, it is enough to note that is a solution to (1.2), as well.

Finally, if (iii) holds, then it is enough to use [22, Theorem ] and Theorem 1.4 (the case of ).

The statement concerning uniqueness results from [22, Proposition ].

Remark 3.2.

If for some (or, equivalently, for some ), then (1.2) can be nonstable, by which we mean that there is a function such that (1.7) holds with some real and for each solution of (1.2) (see, e.g., [28], [22, Example ], or [29]).

Remark 3.3.

Note that, in the case where are real numbers, we have

(3.8)

with

(3.9)

4. Some Critique and Final Remarks

Functional equation (1.2) has been patterned on difference equation (1.4). However, if we want to apply the results of Theorems 1.1–3.1 to the Lucas sequences we come across two obstacles. The first one concerns the domain of and arises from the difference equation (1.4) being written in "wrong" historical form, inconsistent with the general concept of functional equations. Actually it should be written as the functional equation

(4.1)

which corresponds to (1.6). The second obstacle is connected with the restrictions on . For some interesting cases (Fibonacci, Lucas, or Pell numbers), we have (or, if somebody prefers, ) and such case is not covered if is not bijective (which is the case when and or, equivalently, ). All these obstacles can be overcome if, instead of Theorems 1.1–3.1, we use the following result derived from [29, Theorem ].

Proposition 4.1.

Let , , and for , , and

(4.2)

Then there is a solution of (4.1) that satisfies (1.8).

For instance, if and (the case of the Fibonacci and Lucas numbers), we have the following.

Corollary 4.2.

Let , , , and be a sequence in with

(4.3)

Then there is a sequence in such that

(4.4)

Proof.

Note that . Thus, by Proposition 4.1, there is a sequence in such that (4.4) is valid.

Remark 4.3.

If and (the case of Jacobsthal numbers), then one of the roots of (1.1) is equal to and therefore (4.1) is not stable (see [28]), by which we mean that, for each , there is such that and for every solution of (1.6); for such function can be chosen with, for example, and (in [28, the proofs of Lemma and Theorem ] take ).