Abstract and Applied Analysis

Volume 2014 (2014), Article ID 546046, 6 pages

http://dx.doi.org/10.1155/2014/546046

Research Article

On the Stability of a Functional Equation Associated with the Fibonacci Numbers

Cristinel Mortici,1,2 Michael Th. Rassias,³ and Soon-Mo Jung⁴

¹Valahia University of Târgovişte, Bulevardul Unirii 18, 130082 Târgovişte, Romania

²Academy of Romanian Scientists, Splaiul Independenţei 54, 050094 Bucharest, Romania

³Department of Mathematics, ETH Zürich, Raemistrasse 101, 8092 Zürich, Switzerland

⁴Mathematics Section, College of Science and Technology, Hongik University, Sejong 339-701, Republic of Korea

Received 5 May 2014; Accepted 8 July 2014; Published 20 July 2014

Academic Editor: Chengjian Zhang

Copyright © 2014 Cristinel Mortici et al. 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.

Abstract.

We prove the Hyers-Ulam stability of the generalized Fibonacci functional equation , where and h are given functions.

1. Introduction

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

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

The case of approximately additive functions was solved by Hyers [2] under the assumption that and are Banach spaces. Indeed, he proved the following theorem.

Theorem 1. Let be a function between Banach spaces such that

for some and for all . Then, the limit

exists for each , and is the unique additive function such that

for any . Moreover, if is continuous in , for each fixed , then the function is linear.

Hyers proved that each solution of the inequality can be approximated by an exact solution; say an additive function. In this case, the Cauchy additive functional equation, , is said to have the Hyers-Ulam stability.

Since then, the stability problems of a large variety of functional equations have been extensively investigated by several mathematicians (cf. [3–14]).

In this paper, we investigate the Hyers-Ulam stability of the functional equation

where and are given functions.

In Section 2, we prove that the functional equation (4) has a large class of nontrivial solutions. Section 3 is devoted to the investigation of the Hyers-Ulam stability problems for (4). In the last section, we prove the Hyers-Ulam stability of (4) when is a constant function, which is a generalization of the papers [4, 7, 14]. More precisely, Jung [7] proved the Hyers-Ulam stability of the generalized Fibonacci functional equation

in the class of functions , where is a real (or complex) Banach space.

Theorem 2 (see [7, Theorem 3.1]). Assume that the quadratic equation has real solutions and with . If a function satisfies the inequality

for all and for some , then there exists a unique solution of (5) such that

for all .

A similar case for with was investigated by Brzdęk et al. [4] and Trif [14] who obtained the estimate

If either and or and , then the inequality (7) is sharper than that of (8).

In Section 4 of this paper, we improve the results of papers [4, 7, 14] in the sense that we estimate even when both and are larger or smaller than . Moreover, we deal with a functional equation (4) that is regarded as a more generalized form of the Fibonacci functional equation (5).

In this paper, , , and stand for the sets of real numbers, integers, and positive integers, respectively.

2. Solutions of (4)

Evidently, (4) admits the trivial solution . In order to avoid the trivial case, we search in this section for a class of nontrivial solutions of (4).

Let be a subset of . A function is said to be of disjoint iterated images, shortly (DII)-function, if

(i)there exists a partition

(ii) maps bijectively onto for each integer .

As an example for a (DII)-function, we introduce a function defined by

for all . For every , this function is linear on and it transforms each onto .

We are now in a position to prove that the set of all solutions of (4) is not empty but it is an infinite set.

Theorem 3. Let be a -function and . There is a one-to-one correspondence between the set of all solutions of the functional equation (4) and the set of all functions .

Proof. Given a , we define a function on as

for all . Assume that is defined on for some . If , then and we put

for all . By this inductive procedure, is completely defined.

We now show that is a solution of (4). Let be any point of and let be an integer such that . Put in (12) to get

which is (4).

Conversely, we associate to every solution of (4) the function .

We notice that a (DII)-function is injective as we see the following: if for some with but , then and, hence, because maps bijectively onto , a contradiction. If and for some with , it is then obvious that because , , and . But is not surjective, since .

We now study the set of solutions of (4) under the assumption that is a bijection. For any pair of points , we use the notation if there exists a with . Since “” is an equivalence relation in , let

be the corresponding partition in “-equivalence classes” ; that is,

Theorem 4. Given a subset of , let be a bijective function and . Assume that

is a partition of corresponding to the equivalence relation with the property (15). Then, there exists a one-to-one correspondence between the set of all solutions of the functional equation (4) and the set of all real sequences .

Proof. For any real sequence , we define for all , where is the index set for the partition corresponding to the equivalence relation with the property (15). We further define the function by

In general, if is defined at and , then is defined at and by

For each , we can use such an inductive procedure to define the function on and we see that is uniquely determined by the value of .

Conversely, every solution of (4) can be associated to the real sequence .

Corollary 5. Given a subset of , let be a bijective function and . Assume that

is a partition of corresponding to the equivalence relation with the property (15). Then there exists a one-to-one correspondence between the set of all solutions of the functional equation

and the set of all real numbers .

3. Hyers-Ulam Stability of (4)

The above conditions imposed on the function were necessary for showing that the functional equations (4) and (20) have large classes of nontrivial solutions. The stability results presented in the sequel are valid also under weaker conditions as we shall see in the following theorems.

Theorem 6. Given real numbers and with , let and be given functions, where is an interval of length . Assume that a bounded function satisfies the inequality

for all and for some . Then, for every , there exists a solution of (20) such that

for any , where .

Proof. First, we prove that

for all , where we set . Indeed, it follows from (21) that

for every .

By replacing with and then multiplying with both sides of (23), we get

for all and . Since

we have

for any and .

The inequality (25) shows that the sequence is a Cauchy sequence for every . Thus, we can define a function by

for all . Hence, it follows from (23) that

for each , which implies that is a solution of (20).

Finally, inequality (22) is an immediate consequence of (27) if we take the limit as .

4. When Is Constant

In the case of for all , we investigate the Hyers-Ulam stability of the functional equation

where and are functions and is a real Banach space.

Theorem 7. Let be a real Banach space and let be a real number with . If a function satisfies the inequality

for all and for some , then there exists a solution of (30) such that

for all .

Proof. By replacing with and multiplying with both sides of (31), we get

for all and . By (33), we have

for all and . Hence, we get

for any and .

The inequality (33) shows that the sequence is a Cauchy sequence for any fixed . Thus, since is a complete space, we can define a function by

for all .

It follows from (31) that

which implies that is a solution of (30).

Finally, the inequality (32) immediately follows from (35) provided that we take the limit as .

Assume now that is bijective. A similar theorem can be proved when .

Theorem 8. Let be a real Banach space, let be a bijective function, and let be given with . If a function satisfies the inequality

for all and for some , then there exists a solution of (30) such that

for all .

Proof. By replacing with and dividing by both sides of (38), we get

for any . Since the constant is less than , our assertion follows from Theorem 7. In particular, we have

for each .

Corollary 9. Let be a real Banach space, let be a bijective function, and let be given with . If a function satisfies the inequality

for any and for some , then there exists a solution of (30) such that

for all .

By combining the results of Theorems 7 and 8, we can present a stability result of the following functional equation

where is bijective and the range space of the function is a real Banach space.

Theorem 10. Let and be given real numbers such that the quadratic equation has distinct real solutions and with and . Assume that a bijective function is given and is a real Banach space. If a function satisfies

for all and for some , then there exists a solution of (44) such that

for all .

As we mentioned in the Introduction, our result extends Jung’s result in [7], since

when . Moreover, Jung’s result is a particular case of Theorem 10 when we set in (45).

Proof of Theorem 10. If we set

then the inequality (45) yields

for all . According to Corollary 9, there exists a solution of

with

for any .

If we set

then the inequality (45) yields

for any . In view of Corollary 9 again, there exists a solution of

with

for all .

We now define a function by

for any , where is a real number. Then, it follows from (50) and (54) that

for all , which implies that is a solution of (44) for every fixed real number .

We now set

and assert that the function

satisfies the requirements of this theorem. Indeed, it follows from (51) and (55) that

for any .

Conflict of Interests

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

Acknowledgments

Soon-Mo Jung was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557). The work of Cristinel Mortici was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI Project no. PN-II-ID-PCE-2011-3-0087. Cristinel Mortici’s final remarks for improving this paper were made during his visit at CERN laboratories in Geneva, Switzerland.

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