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Hyers-Ulam Stability of Generalized Tribonacci Functional Equation

S. Arolkar*, Y.S. Valaulikar

Department of Mathematics, D.M.'s College and Research Centre, Assagao, Goa 403 507- India

Abstract: In this paper we study Hyers-Ulam stability of the generalized Tribonacci functional equation, where a, b and c are non- zero constants. The functional equation is solved and its stability is established in the class of functions where X is a Banach space.

Keywords: Fibonacci functional equation, Hyers-Ulam stability, Tribonacci functional equation

Citation: S. Arolkar, Y.S. Valaulikar, Hyers-Ulam Stability of Generalized Tribonacci Functional Equation. Turkish Journal of Analysis and Number Theory, 5 (3). 80-85. January 2017. doi:10.12691/tjant-5-3-1

1. Introduction

Stability problems of functional equations have been extensively studied. (see [1,5,6] and references therein). The importance of the topic lies in the fact that stability of functional equation is associated with notions of Controlled Chaos [11] and Shadowing [14]. In [7] and [8] Jung discusses the stability problem for Fibonacci functional equation and its extension in Banach Space respectively whereas in [3] the problem is discussed in Modular Functional space. In [10] k-Fibonacci functional equation is discussed. In [2] and [4] solution and stability of Tribonacci functional equation in non-Archimedean Banach spaces and 2-normed spaces have been discussed respectively. Stability of Tribonacci and k-Tribonacci functional equations in Modular spaces is discussed in [13]. In [9], authors investigate the solution of generalized linear Tribonacci functional equation in terms of Fibonacci numbers. In this paper we show that the generalized linear Tribonacci functional equation is associated with generalized Tribonacci sequence and also obtain its stability in the class of functions where is a real (or complex) Banach space. Like Fibonacci numbers, Tribonacci numbers also play an important role in problems of combinatorics [12] and also in evaluation of determinants of circulant matrices [15].

Definition 1. The generalized Tribonacci sequence is defined by

[picture] | (1.1)

where a, b and c are non-zero fixed real numbers and is the term of (1.1) given by Binet type formula,

[picture] | (1.2)

where α, β and γ are distinct roots of the characteristics equation corresponding to (1.1).

Definition 2. Let X be a real (or complex) Banach space. A function is called a generalized Tribonacci function if it satisfies generalized Tribonacci functional equation

[picture] | (1.3)

2. Existence of General Solution to the Equation (1.3)

In this section we solve equation (1.3) and prove that its solution is in terms of (1.2). To prove this result, we require the following lemma.

Lemma 3. If α, β and γ are distinct roots of the characteristics equation corresponding to (1.1), then the generalized Tribonacci function satisfies

[picture] | (2.1)

Proof. Characteristics equation corresponding to (1.1) is

[picture] | (2.2)

Since α, β and γ are distinct roots of (2.2), we get and

Substituting a, b and c in (1.3), we have

which implies

[picture] | (2.3)

Replacing by in (2.3), we get

and hence (2.3) yields

By induction on n, we get

[picture] | (2.4)

Similarly, we have

[picture] | (2.5)

and

[picture] | (2.6)

Now replacing x by x + 1 in (2.3), we get

Therefore,

Thus by induction on n, we get

Similarly, we have

and

Therefore equations (2.4), (2.5) and (2.6) are true and

Now multiplying the equations (2.4), (2.5) and (2.6) by respectively, and adding we get

Using (1.2), this gives

We use Lemma 3 to prove the following result.

Theorem 4. A function is a solution of functional equation (1.3) if and only if there exists a function such that

[picture] | (2.7)

where is given by (1.2).

Proof. If is a solution of (1.3), then by lemma 3 satisfies (2.1). Putting in (2.1), we get

Since and , we define a function by , then f is of the form (2.7).

Now we assume that f is a function of the form (2.7) and prove that it is a solution of (1.3).

Consider

Therefore (2.7) is a solution of (1.3). Hence the theorem is proved.

We next illustrate this result.

Example 5. Consider the following functional equation

[picture] | (2.8)

Define the function by

[picture] | (2.9)

Note tha and 5 are distinct roots of the characteristic equation corresponding to Tribonacci sequence defined by

[picture] | (2.10)

Therefore (2.7) implies

[picture] | (2.11)

where h(x) is given by (2.9).

3. Hyers-Ulam Stability

In this section we prove the Hyers-Ulam stability of functional equation (1.3) by assuming that roots and are distinct and We first prove the lemma required to prove Hyers-Ulam stability of functional equation (1.3).

Lemma 6. If a function satisfies,

for some and and are distinct roots of (2.2) such that then there exists Tribonacci functions defined by

and

such that

[picture] | (3.1)

[picture] | (3.2)

and

[picture] | (3.3)

Proof. Using (2.4) with n = 1 in given condition on f, we have

Replacing x by we have

Multiplying both side by

[picture] | (3.4)

Further,

Therefore

[picture] | (3.5)

Thus for and for any (3.5) implies that the sequence is a cauchy sequence.

Therefore, since X is Banach space, we can define a function by

We now prove that satisfies (1.3).

Consider,

Hence is a Tribonacci function.

Now taking (3.5) implies

Similarly since using equation (2:6) with we can prove that there exists a Tribonacci function given by

such that

From equation (2.5) with n = 1, it follows that

Replacing by and multiplying both sides by we get

[picture] | (3.6)

For

Thus, for all and ,

[picture] | (3.7)

Equation (3.7) implies that

Banach space, we can define a function by

Now Consider,

Therefore is a Tribonacci function. So as we have

We prove the following theorem using Lemma 6.

Theorem 7. If a function satisfies the inequality,

[picture] | (3.8)

and for some then there exists a unique solution function of the functional equation (1.3) such that

Proof. From (3.1), (3.2) and (3.3), we see that

We now define a function by

Consider,

Therefore F is a solution of (1.3). Now we prove the uniqueness of

Assume that are solutions of (1.3) and that there exist positive constants C1 and C2 such that and for all

Therefore by Theorem 4, there exist such that

[picture] | (3.9)

and

[picture] | (3.10)

for any

Fix t with It then follows from (3.9) and (3.10) that

That is,

This implies

[picture] | (3.11)

Dividing both sides by and by letting we obtain

[picture] | (3.12)

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

[picture] | (3.13)

and

[picture] | (3.14)

Rewritting equations (3.12), (3.13) and (3.14) in matrix form, we get

[picture] | (3.15)

Note that since and are distinct roots,

Therefore (3.15) has only trivial solution and we have,

That is, for all Therefore, we conclude that for all

We illustrate this result.

Example 8. Consider the functional equation

and Tribonacci recurrence relation associated to it.

Roots of the equation are and . Let and Note that roots are distinct and and

Hence the solution is given by

where

Therefore

and

4. Conclusion

Hyers-Ulam stability of generalized Tribonacci functional equation is discussed after obtaining the solution of generalized Tribonacci functional equation in terms of generalized Tribonacci numbers.

References

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