1. Introduction

In 1940, Ulam gave a wide-ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems [1]. Among those was the question concerning the stability of homomorphisms.

Let be a group and let be a metric group with a metric . Given any , does there exist a such that if a function satisfies the inequality for all , then there exists a homomorphism with for all ?

In the following year, Hyers affirmatively answered in his paper [2] the question of Ulam for the case where and are Banach spaces.

Let be a groupoid and let be a groupoid with the metric . The equation of homomorphism

(1.1)

is stable in the Hyers-Ulam sense (or has the Hyers-Ulam stability) if for every there exists an such that for every function satisfying

(1.2)

for all there exists a solution of the equation of homomorphism with

(1.3)

for any (see [3, Definition ]).

This terminology is also applied to the case of other functional equations. It should be remarked that we can find in the books [4–7] a lot of references concerning the stability of functional equations (see also [8–18]).

Throughout this paper, let and be fixed real numbers with and . By and we denote the distinct roots of the equation . More precisely, we set

(1.4)

Moreover, for any , we define

(1.5)

If and are integers, then is called the Lucas sequence of the first kind. It is not difficult to see that

(1.6)

for any integer . For any , stands for the largest integer that does not exceed .

In this paper, we will solve the functional equation

(1.7)

and prove its Hyers-Ulam stability in the class of functions , where is a real (or complex) Banach space.

2. General Solution to (1.7)

In this section, let be either a real vector space if or a complex vector space if . In the following theorem, we investigate the general solution of the functional equation (1.7).

Theorem 2.1.

A function is a solution of the functional equation (1.7) if and only if there exists a function such that

(2.1)

Proof.

Since and , it follows from (1.7) that

(2.2)

By the mathematical induction, we can easily verify that

(2.3)

for all and . If we substitute for in (2.3) and divide the resulting equations by , respectively, , and if we substitute for in the resulting equations, then we obtain the equations in (2.3) with in place of , where . Therefore, the equations in (2.3) are true for all and .

We multiply the first and the second equations of (2.3) by and , respectively. If we subtract the first resulting equation from the second one, then we obtain

(2.4)

for any and .

If we put in (2.4), then

(2.5)

for all .

Since and , if we define a function by , then we see that is a function of the form (2.1).

Now, we assume that is a function of the form (2.1), where is an arbitrary function. Then, it follows from (2.1) that

(2.6)

for any . Thus, by (1.6), we obtain

(2.7)

which completes the proof.

Remark 2.2.

It should be remarked that the functional equation (1.7) is a particular case of the linear equation with and . Moreover, a substantial part of proof of Theorem 2.1 can be derived from theorems presented in the books [19, 20]. However, the theorems in [19, 20] deal with solutions of the linear equation under some regularity conditions, for example, the continuity, convexity, differentiability, analyticity and so on, while Theorem 2.1 deals with the general solution of (1.7) without regularity conditions.

3. Hyers-Ulam Stability of (1.7)

In this section, we denote by and the distinct roots of the equation satisfying and . Moreover, let be either a real Banach space if or a complex Banach space if .

We can prove the Hyers-Ulam stability of the functional equation (1.7) as we see in the following theorem.

Theorem 3.1.

If a function satisfies the inequality

(3.1)

for all and for some , then there exists a unique solution function of the functional equation (1.7) such that

(3.2)

for all .

Proof.

Analogously to the first equation of (2.2), it follows from (3.1) that

(3.3)

for each . If we replace by in the last inequality, then we have

(3.4)

and further

(3.5)

for all and . By (3.5), we obviously have

(3.6)

for and .

For any , (3.5) implies that the sequence is a Cauchy sequence (note that .) Therefore, we can define a function by

(3.7)

since is complete. In view of the previous definition of , we obtain

(3.8)

for all , since . If goes to infinity, then (3.6) yields that

(3.9)

for every .

On the other hand, it also follows from (3.1) that

(3.10)

(see the second equation in (2.2)). Analogously to (3.5), replacing by in the previous inequality and then dividing by both sides of the resulting inequality, then we have

(3.11)

for all and . By using (3.11), we further obtain

(3.12)

for and .

On account of (3.11), we see that the sequence is a Cauchy sequence for any fixed (note that .) Hence, we can define a function by

(3.13)

Using the previous definition of , we get

(3.14)

for any . If we let go to infinity, then it follows from (3.12) that

(3.15)

for .

By (3.9) and (3.15), we have

(3.16)

for all . We now define a function by

(3.17)

for all . Then, it follows from (3.8) and (3.14) that

(3.18)

for each ; that is, is a solution of (1.7). Moreover, by (3.16), we obtain the inequality (3.2).

Now, it only remains to prove the uniqueness of . Assume that are solutions of (1.7) and that there exist positive constants and with

(3.19)

for all . According to Theorem 2.1, there exist functions such that

(3.20)

for any , since and are solutions of (1.7).

Fix a with . It then follows from (3.19) and (3.20) that

(3.21)

for each , that is,

(3.22)

for every . Dividing both sides by yields that

(3.23)

and by letting , we obtain

(3.24)

Analogously, if we divide both sides of (3.22) by and let , then we get

(3.25)

By (3.24) and (3.25), we have

(3.26)

Because (where both and are nonzero and so ), it should hold that

(3.27)

for any , that is, for all . Therefore, we conclude that for any . (The presented proof of uniqueness of is somewhat long and involved. Indeed, the referee has remarked that the uniqueness can be obtained directly from [21, Proposition ].)

Remark 3.2.

The functional equation (1.7) is a particular case of the linear equations of higher orders and the Hyers-Ulam stability of the linear equations has been proved in [21, Theorem ]. Indeed, Brzdęk et al. have proved an interesting theorem, from which the following corollary follows (see also [22, 23]):

Corollary 3.3.

Let a function satisfy the inequality (3.1) for all and for some and let be the distinct roots of the equation . If , and , then there exists a solution function of (1.7) such that

(3.28)

for all .

If either and or and , then

(3.29)

Hence, the estimation (3.2) of Theorem 3.1 is better in these cases than the estimation (3.28).

Remark 3.4.

As we know, is the Fibonacci sequence. So if we set and in Theorems 2.1 and 3.1, then we obtain the same results as in [24, Theorems , , and ].