Open Access

Functional Equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq1_HTML.gif and Its Hyers-Ulam Stability

Journal of Inequalities and Applications20092009:181678

DOI: 10.1155/2009/181678

Received: 2 July 2009

Accepted: 5 November 2009

Published: 8 November 2009

Abstract

We solve the functional equation, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq2_HTML.gif , and prove its Hyers-Ulam stability in the class of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq3_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq4_HTML.gif is a real (or complex) Banach space.

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq5_HTML.gif be a group and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq6_HTML.gif be a metric group with a metric https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq7_HTML.gif . Given any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq8_HTML.gif , does there exist a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq9_HTML.gif such that if a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq10_HTML.gif satisfies the inequality https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq11_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq12_HTML.gif , then there exists a homomorphism https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq13_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq14_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq15_HTML.gif ?

In the following year, Hyers affirmatively answered in his paper [2] the question of Ulam for the case where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq16_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq17_HTML.gif are Banach spaces.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq18_HTML.gif be a groupoid and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq19_HTML.gif be a groupoid with the metric https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq20_HTML.gif . The equation of homomorphism

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ1_HTML.gif
(1.1)

is stable in the Hyers-Ulam sense (or has the Hyers-Ulam stability) if for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq21_HTML.gif there exists an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq22_HTML.gif such that for every function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq23_HTML.gif satisfying

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ2_HTML.gif
(1.2)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq24_HTML.gif there exists a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq25_HTML.gif of the equation of homomorphism with

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ3_HTML.gif
(1.3)

for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq26_HTML.gif (see [3, Definition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq27_HTML.gif ]).

This terminology is also applied to the case of other functional equations. It should be remarked that we can find in the books [47] a lot of references concerning the stability of functional equations (see also [818]).

Throughout this paper, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq28_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq29_HTML.gif be fixed real numbers with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq31_HTML.gif . By https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq32_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq33_HTML.gif we denote the distinct roots of the equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq34_HTML.gif . More precisely, we set

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ4_HTML.gif
(1.4)

Moreover, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq35_HTML.gif , we define

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ5_HTML.gif
(1.5)

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq37_HTML.gif are integers, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq38_HTML.gif is called the Lucas sequence of the first kind. It is not difficult to see that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ6_HTML.gif
(1.6)

for any integer https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq39_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq40_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq41_HTML.gif stands for the largest integer that does not exceed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq42_HTML.gif .

In this paper, we will solve the functional equation

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ7_HTML.gif
(1.7)

and prove its Hyers-Ulam stability in the class of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq43_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq44_HTML.gif is a real (or complex) Banach space.

2. General Solution to (1.7)

In this section, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq45_HTML.gif be either a real vector space if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq46_HTML.gif or a complex vector space if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq47_HTML.gif . In the following theorem, we investigate the general solution of the functional equation (1.7).

Theorem 2.1.

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq48_HTML.gif is a solution of the functional equation (1.7) if and only if there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq49_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ8_HTML.gif
(2.1)

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq50_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq51_HTML.gif , it follows from (1.7) that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ9_HTML.gif
(2.2)
By the mathematical induction, we can easily verify that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ10_HTML.gif
(2.3)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq52_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq53_HTML.gif . If we substitute https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq54_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq55_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq56_HTML.gif in (2.3) and divide the resulting equations by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq57_HTML.gif , respectively, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq58_HTML.gif , and if we substitute https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq59_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq60_HTML.gif in the resulting equations, then we obtain the equations in (2.3) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq61_HTML.gif in place of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq62_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq63_HTML.gif . Therefore, the equations in (2.3) are true for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq65_HTML.gif .

We multiply the first and the second equations of (2.3) by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq66_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq67_HTML.gif , respectively. If we subtract the first resulting equation from the second one, then we obtain

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ11_HTML.gif
(2.4)

for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq68_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq69_HTML.gif .

If we put https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq70_HTML.gif in (2.4), then

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ12_HTML.gif
(2.5)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq71_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq73_HTML.gif , if we define a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq74_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq75_HTML.gif , then we see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq76_HTML.gif is a function of the form (2.1).

Now, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq77_HTML.gif is a function of the form (2.1), where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq78_HTML.gif is an arbitrary function. Then, it follows from (2.1) that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ13_HTML.gif
(2.6)
for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq79_HTML.gif . Thus, by (1.6), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ14_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq80_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq81_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq82_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq84_HTML.gif the distinct roots of the equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq85_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq86_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq87_HTML.gif . Moreover, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq88_HTML.gif be either a real Banach space if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq89_HTML.gif or a complex Banach space if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq90_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq91_HTML.gif satisfies the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ15_HTML.gif
(3.1)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq92_HTML.gif and for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq93_HTML.gif , then there exists a unique solution function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq94_HTML.gif of the functional equation (1.7) such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ16_HTML.gif
(3.2)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq95_HTML.gif .

Proof.

Analogously to the first equation of (2.2), it follows from (3.1) that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ17_HTML.gif
(3.3)
for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq96_HTML.gif . If we replace https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq97_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq98_HTML.gif in the last inequality, then we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ18_HTML.gif
(3.4)
and further
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ19_HTML.gif
(3.5)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq99_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq100_HTML.gif . By (3.5), we obviously have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ20_HTML.gif
(3.6)

for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq101_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq102_HTML.gif .

For any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq103_HTML.gif , (3.5) implies that the sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq104_HTML.gif is a Cauchy sequence (note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq105_HTML.gif .) Therefore, we can define a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq106_HTML.gif by

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ21_HTML.gif
(3.7)
since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq107_HTML.gif is complete. In view of the previous definition of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq108_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ22_HTML.gif
(3.8)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq109_HTML.gif , since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq110_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq111_HTML.gif goes to infinity, then (3.6) yields that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ23_HTML.gif
(3.9)

for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq112_HTML.gif .

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ24_HTML.gif
(3.10)
(see the second equation in (2.2)). Analogously to (3.5), replacing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq113_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq114_HTML.gif in the previous inequality and then dividing by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq115_HTML.gif both sides of the resulting inequality, then we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ25_HTML.gif
(3.11)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq116_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq117_HTML.gif . By using (3.11), we further obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ26_HTML.gif
(3.12)

for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq118_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq119_HTML.gif .

On account of (3.11), we see that the sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq120_HTML.gif is a Cauchy sequence for any fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq121_HTML.gif (note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq122_HTML.gif .) Hence, we can define a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq123_HTML.gif by

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ27_HTML.gif
(3.13)
Using the previous definition of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq124_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ28_HTML.gif
(3.14)
for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq125_HTML.gif . If we let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq126_HTML.gif go to infinity, then it follows from (3.12) that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ29_HTML.gif
(3.15)

for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq127_HTML.gif .

By (3.9) and (3.15), we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ30_HTML.gif
(3.16)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq128_HTML.gif . We now define a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq129_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ31_HTML.gif
(3.17)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq130_HTML.gif . Then, it follows from (3.8) and (3.14) that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ32_HTML.gif
(3.18)

for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq131_HTML.gif ; that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq132_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq133_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq134_HTML.gif are solutions of (1.7) and that there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq135_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq136_HTML.gif with

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ33_HTML.gif
(3.19)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq137_HTML.gif . According to Theorem 2.1, there exist functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq138_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ34_HTML.gif
(3.20)

for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq139_HTML.gif , since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq140_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq141_HTML.gif are solutions of (1.7).

Fix a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq142_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq143_HTML.gif . It then follows from (3.19) and (3.20) that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ35_HTML.gif
(3.21)
for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq144_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ36_HTML.gif
(3.22)
for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq145_HTML.gif . Dividing both sides by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq146_HTML.gif yields that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ37_HTML.gif
(3.23)
and by letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq147_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ38_HTML.gif
(3.24)
Analogously, if we divide both sides of (3.22) by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq148_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq149_HTML.gif , then we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ39_HTML.gif
(3.25)

By (3.24) and (3.25), we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ40_HTML.gif
(3.26)
Because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq150_HTML.gif (where both https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq151_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq152_HTML.gif are nonzero and so https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq153_HTML.gif ), it should hold that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ41_HTML.gif
(3.27)

for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq154_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq155_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq156_HTML.gif . Therefore, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq157_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq158_HTML.gif . (The presented proof of uniqueness of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq159_HTML.gif is somewhat long and involved. Indeed, the referee has remarked that the uniqueness can be obtained directly from [21, Proposition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq160_HTML.gif ].)

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq161_HTML.gif ]. 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq162_HTML.gif satisfy the inequality (3.1) for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq163_HTML.gif and for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq164_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq165_HTML.gif be the distinct roots of the equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq166_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq167_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq169_HTML.gif , then there exists a solution function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq170_HTML.gif of (1.7) such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ42_HTML.gif
(3.28)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq171_HTML.gif .

If either https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq172_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq173_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq174_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq175_HTML.gif , then

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_Equ43_HTML.gif
(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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq176_HTML.gif is the Fibonacci sequence. So if we set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq177_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq178_HTML.gif in Theorems 2.1 and 3.1, then we obtain the same results as in [24, Theorems https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq179_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq180_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F181678/MediaObjects/13660_2009_Article_1909_IEq181_HTML.gif ].

Declarations

Acknowledgments

The author would like to express his cordial thanks to the referee for useful remarks which have improved the first version of this paper. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (no. 2009-0071206).

Authors’ Affiliations

(1)
Mathematics Section, College of Science and Technology, Hongik University

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© Soon-Mo Jung. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.