A Linear Functional Equation of Third Order Associated with the Fibonacci Numbers

Abstract

Given a vector space X , we investigate the solutions f : R [arrow right] X of the linear functional equation of third order f x = p f x - 1 + q f x - 2 + r f ( x - 3 ) , which is strongly associated with a well-known identity for the Fibonacci numbers. Moreover, we prove the Hyers-Ulam stability of that equation.

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Soon-Mo Jung 1 and Michael Th. Rassias 2

Academic Editor:Jesús G. Falset

1, Mathematics Section, College of Science and Technology, Hongik University, Sejong 339-701, Republic of Korea
2, Department of Mathematics, ETH-Zürich, Ramistraße 101, 8092 Zürich, Switzerland

Received 7 February 2014; Revised 12 June 2014; Accepted 16 June 2014; 30 June 2014

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.

1. Introduction

The problem of stability of functional equations was motivated by a question of Ulam [1] and a solution to it by Hyers [2]. Since then, numerous papers have been published on that subject and we refer to [3-6] for more details, some discussions, and further references; for examples of very recent results, see, for example, [7].

In this paper, as usual, C , R , Z , and N stand for the sets of complex numbers, real numbers, integers, and positive integers, respectively. For a nonempty subset S of a vector space, let ξ : S [arrow right] S be a function. Moreover, ^{ξ 0} ( x ) = x , ^{ξ n + 1} ( x ) = ξ ( ^{ξ n} ( x ) ) , and (only for bijective ξ ) ^{ξ - n - 1} ( x ) = ^{ξ - 1} ( ^{ξ - n} ( x ) ) for x ∈ S and n ∈ _{N 0} : = N ∪ { 0 } .

Jung has proved in [3] (see also [8]) some results on solutions and Hyers-Ulam stability of the functional equation [figure omitted; refer to PDF] in the case where S = R and ξ ( x ) = x - 1 for x ∈ R .

If S : = _{N 0} and p , q ∈ Z , then solutions x : _{N 0} [arrow right] Z of the difference equation f ( x ) = p f ( x - 1 ) - q f ( x - 2 ) are called the Lucas sequences (see, e.g., [9]). In some special cases they are called with specific names, for example, the Fibonacci numbers ( p = 1 , q = - 1 , x ( 0 ) = 0 , and x ( 1 ) = 1 ), the Lucas numbers ( p = 1 , q = - 1 , x ( 0 ) = 2 , and x ( 1 ) = 1 ), the Pell numbers ( p = 2 , q = - 1 , x ( 0 ) = 0 , and x ( 1 ) = 1 ), the Pell-Lucas (or companion Lucas) numbers ( p = 2 , q = - 1 , x ( 0 ) = 2 , and x ( 1 ) = 2 ), and the Jacobsthal numbers ( p = 1 , q = - 2 , x ( 0 ) = 0 , and x ( 1 ) = 1 ).

For some information and further references concerning the functional equations in a single variable, we refer to [10-12]. Let us mention yet that the problem of Hyers-Ulam stability of functional equations is connected to the notions of controlled chaos and shadowing (see [13]).

We remark that if ξ : S [arrow right] S is bijective, then (1) can be written in the following equivalent form: [figure omitted; refer to PDF] where η : = ^{ξ - 1} .

In view of the last remark, the following Hyers-Ulam stability result concerning (1) can be derived from [14, Theorem 2] (see also [15]).

Theorem 1.

Let p , q ∈ R be given with q ...0; 0 and let S be a nonempty subset of a vector space. Assume that _{a 1} , _{a 2} are the complex roots of the quadratic equation ^{x 2} - p x + q = 0 with | _{a i} | ...0; 1 for i ∈ { 1,2 } . Moreover, assume that X is either a real vector space if ^{p 2} - 4 q > 0 or a complex vector space if ^{p 2} - 4 q < 0 . Let ξ : S [arrow right] S be bijective. If a function f : S [arrow right] X satisfies the inequality [figure omitted; refer to PDF] for all x ∈ S and for some [varepsilon] ...5; 0 , then there exists a unique solution F : S [arrow right] X of (1) with [figure omitted; refer to PDF] for all x ∈ S .

In [16, Theorem 1.4], the method presented in [3] was modified so as to prove a theorem which is a complement of Theorem 1. Note that, for bijective ξ , the following theorem improves the estimation (4) in some cases (e.g., _{a 1} = 3 / 2 , _{a 2} = - 3 / 2 , or _{a 1} = 1 / 2 , _{a 2} = - 1 / 2 ). However, in some other situations (e.g., _{a 1} = 3 , _{a 2} = - 3 ), the estimation (4) is better than (5). The following theorem also complements Theorem 1, because ξ can be quite arbitrary in the case of ( α ) .

Theorem 2.

Given p , q ∈ R with q ...0; 0 , assume that the distinct complex roots _{a 1} , _{a 2} of the quadratic equation ^{x 2} - p x + q = 0 satisfy one of the following two conditions:

( α ) : | _{a i} | < 1 for i ∈ { 1,2 } ;

( β ) : | _{a i} | ...0; 1 for i ∈ { 1,2 } and ξ : S [arrow right] S is bijective.

Moreover, assume that X is either a real vector space if ^{p 2} - 4 q > 0 or a complex vector space if ^{p 2} - 4 q < 0 . If a function f : S [arrow right] X satisfies inequality (3), then there exists a solution F : S [arrow right] X of (1) such that [figure omitted; refer to PDF] for all x ∈ S . Moreover, if the condition ( β ) is true, then the F is the unique solution of (1) satisfying (5).

In this paper, we investigate the solutions of the functional equation [figure omitted; refer to PDF] where p , q , r are real constants. Moreover, we also prove the Hyers-Ulam stability of that equation. Equation (6) is a kind of linear functional equations of third order because it is of the form [figure omitted; refer to PDF] for the case of _{a 1} ( x ) = p , _{a 2} ( x ) = q , _{a 3} ( x ) = r , and ξ ( x ) = x - 1 .

2. General Solution

In the following theorem, we apply [16, Theorem 1.1] for the investigation of general solutions of the functional equation (6).

Theorem 3.

Let p , q , r be real constants such that the cubic equation [figure omitted; refer to PDF] has the following properties:

(i) _{α 1} and _{α 2} are two distinct nonzero real roots of the cubic equation (8);

(ii) it holds true that either ( _{α i} + p ^{) 2} + 4 r / _{α i} > 0 for i ∈ { 1,2 } or ( _{α i} + p ^{) 2} + 4 r / _{α i} < 0 for i ∈ { 1,2 } .

Let X be either a real vector space if ( _{α i} + p ^{) 2} + 4 r / _{α i} > 0 for i ∈ { 1,2 } or a complex vector space if ( _{α i} + p ^{) 2} + 4 r / _{α i} < 0 for i ∈ { 1,2 } . Then, a function f : R [arrow right] X is a solution of the functional equation (6) if and only if there exist functions _{h 1} , _{h 2} : [ - 1,1 ) [arrow right] X such that [figure omitted; refer to PDF] where [ x ] denotes the largest integer not exceeding x , and _{U n} , _{V n} are defined in (13) and (23).

Proof.

Assume that f : R [arrow right] X is a solution of (6). If we define an auxiliary function _{g 1} : R [arrow right] X by [figure omitted; refer to PDF] then it follows from (6) that _{g 1} satisfies [figure omitted; refer to PDF] for any x ∈ R . According to [16, Theorem 1.1] or [3, Theorem 2.1], there exists a function _{h 1} : [ - 1,1 ) [arrow right] X such that [figure omitted; refer to PDF] for all x ∈ R , where [figure omitted; refer to PDF] and a , b are the distinct roots of the quadratic equation [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF]

Since a is a root of the quadratic equation (14), we have [figure omitted; refer to PDF] We multiply both sides of (16) with a and make use of (16) and (i) to get [figure omitted; refer to PDF] Similarly, we also obtain [figure omitted; refer to PDF] Using (13), (17), and (18), we have [figure omitted; refer to PDF] for all n ∈ Z .

If we define an auxiliary function _{g 2} : R [arrow right] X by [figure omitted; refer to PDF] then it follows from (6) that _{g 2} satisfies [figure omitted; refer to PDF] for any x ∈ R . According to [16, Theorem 1.1] or [3, Theorem 2.1], there exists a function _{h 2} : [ - 1,1 ) [arrow right] X such that [figure omitted; refer to PDF] for all x ∈ R , where [figure omitted; refer to PDF] and c , d are the distinct roots of the quadratic equation [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] As in the first part, we verify that [figure omitted; refer to PDF] for all n ∈ Z .

We now multiply (12) with _{α 2} and (22) with _{α 1} , we subtract the former from the latter, and we then divide the resulting equation by ( _{α 1} - _{α 2} ) to get (9).

We assume that a function f : R [arrow right] X is given by (9), where _{h 1} , _{h 2} : [ - 1,1 ) [arrow right] X are arbitrarily given functions and _{U n} , _{V n} are given by (13) and (23), respectively. Then, by (9), (19), and (26), we have [figure omitted; refer to PDF] for all x ∈ R , which implies that f is a solution of (6).

According to [17, p. 92], the Fibonacci numbers _{F n} satisfy the identity [figure omitted; refer to PDF] for all integers n > 3 . We can easily notice that the linear equation of third order [figure omitted; refer to PDF] is strongly related to identity (28).

Corollary 4.

Let X be a real vector space. A function f : R [arrow right] X is a solution of the functional equation (29) if and only if there exist functions _{h 1} , _{h 2} : [ - 1,1 ) [arrow right] X such that [figure omitted; refer to PDF] where _{U n} and _{V n} are defined in (33).

Proof.

If we set p = 2 , q = 2 , and r = - 1 in (8), then the cubic equation [figure omitted; refer to PDF] has three distinct nonzero roots including [figure omitted; refer to PDF] Moreover, it holds that ( _{α 1} + p ^{) 2} + 4 r / _{α 1} > 0 and ( _{α 2} + p ^{) 2} + 4 r / _{α 2} > 0 . By (13), (15), (23), and (25), we have [figure omitted; refer to PDF] where we make use of (15) and (25) to calculate [figure omitted; refer to PDF]

Finally, in view of Theorem 3, we conclude that the assertion of our corollary is true.

Corollary 5.

If a function f : R [arrow right] R is a solution of functional equation (29), then there exist real constants _{μ 1} , _{μ 2} , _{ν 1} , and _{ν 2} such that [figure omitted; refer to PDF] for all n ∈ Z , where _{U n} and _{V n} are defined in (33).

3. Hyers-Ulam Stability

We apply the classical direct method to the proof of the following theorem. The classical direct method was first proposed by Hyers [2].

Theorem 6.

Let p , q , r be real constants with r ...0; 0 , let α be a nonzero root of the cubic equation (8), and let a , b be the roots of the quadratic equation ^{x 2} - ( α + p ) x - r / α = 0 with | a | > 1 and 0 < | b | < 1 . Let X be either a real Banach space if ^{( α + p ) 2} + 4 r / α > 0 or a complex Banach space if ^{( α + p ) 2} + 4 r / α < 0 . If a function f : R [arrow right] X satisfies the inequality [figure omitted; refer to PDF] for all x ∈ R and for some [varepsilon] ...5; 0 , then there exists a solution G : R [arrow right] X of (6) such that [figure omitted; refer to PDF] for all x ∈ R .

Proof.

If we define an auxiliary function g : R [arrow right] X by [figure omitted; refer to PDF] then, as we did in (11), it follows from (36) that g satisfies the inequality [figure omitted; refer to PDF] or [figure omitted; refer to PDF] for any x ∈ R .

If we replace x with x - k in the last inequality, then we have [figure omitted; refer to PDF] for all x ∈ R . Furthermore, we get [figure omitted; refer to PDF] for all x ∈ R and k ∈ Z . By (42), we obviously have [figure omitted; refer to PDF] for x ∈ R and n ∈ N .

For any x ∈ R , (42) implies that the sequence { ^{b n} [ g ( x - n ) - a g ( x - n - 1 ) ] } is a Cauchy sequence (note that 0 < | b | < 1 ). Therefore, we can define a function _{G 1} : R [arrow right] X by [figure omitted; refer to PDF] since X is complete. In view of the definition of _{G 1} and using the relations, a + b = α + p and a b = - r / α , we obtain [figure omitted; refer to PDF] for all x ∈ R . Since α is a nonzero root of the cubic equation (8), it follows from (45) that [figure omitted; refer to PDF] for all x ∈ R . Hence, we conclude that _{G 1} is a solution of (6).

If n tends to infinity, then (43) yields that [figure omitted; refer to PDF] for every x ∈ R .

On the other hand, it also follows from (36) that [figure omitted; refer to PDF] for all x ∈ R . Analogously to (42), replacing x by x + k in the last inequality and then dividing by ^{| a | k} both sides of the resulting inequality, then we have [figure omitted; refer to PDF] for all x ∈ R and k ∈ Z . By using (49), we further obtain [figure omitted; refer to PDF] for x ∈ R and n ∈ N .

On account of (49), we see that the sequence { ^{a - n} [ g ( x + n ) - b g ( x + n - 1 ) ] } is a Cauchy sequence for any fixed x ∈ R (note that | a | > 1 ). Hence, we can define a function _{G 2} : R [arrow right] X by [figure omitted; refer to PDF] Due to the definition of _{G 2} and the relations, a + b = α + p and a b = - r / α , we get [figure omitted; refer to PDF] for any x ∈ R . Similarly as in the first part, we can show that _{G 2} is a solution of (6).

If we let n tend to infinity, then it follows from (50) that [figure omitted; refer to PDF] for x ∈ R .

It follows from (47) and (53) that [figure omitted; refer to PDF] for any x ∈ R .

Finally, if we define a function G : R [arrow right] X by [figure omitted; refer to PDF] for all x ∈ R , then G is also a solution of (6). Moreover, the validity of (37) follows from the last inequality.

The following theorem is the main theorem of this paper.

Theorem 7.

Given real constants p , q , r with r ...0; 0 , let _{α 1} and _{α 2} be distinct nonzero roots of cubic equation (8) and let _{a i} , _{b i} be the roots of the quadratic equation ^{x 2} - ( _{α i} + p ) x - r / _{α i} = 0 with | _{a i} | > 1 and 0 < | _{b i} | < 1 for i ∈ { 1,2 } . Assume that either ^{( _{α i} + p ) 2} + 4 r / _{α i} > 0 for all i ∈ { 1,2 } or ^{( _{α i} + p ) 2} + 4 r / _{α i} < 0 for all i ∈ { 1,2 } . Let X be either a real Banach space if ^{( _{α i} + p ) 2} + 4 r / _{α i} > 0 or a complex Banach space if ^{( _{α i} + p ) 2} + 4 r / _{α i} < 0 . If a function f : R [arrow right] X satisfies inequality (36) for all x ∈ R and for some [varepsilon] ...5; 0 , then there exists a solution F : R [arrow right] X of (6) such that [figure omitted; refer to PDF] for all x ∈ R .

Proof.

According to Theorem 6, there exists a solution _{F i} : R [arrow right] X of (6) such that [figure omitted; refer to PDF] for any x ∈ R and i ∈ { 1,2 } . In view of the last inequalities, we have [figure omitted; refer to PDF] for all x ∈ R .

If we define a function F : R [arrow right] X by [figure omitted; refer to PDF] for each x ∈ R , then F is also a solution of (6), and inequality (56) follows from the last inequality.

Acknowledgments

This research paper was completed while Soon-Mo Jung was a visiting scholar at National Technical University of Athens during February 2014. He would like to express his cordial thanks to Professor Themistocles M. Rassias for his hospitality and kindness. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557). The authors would like to express their cordial thanks to the referees for useful remarks.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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