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Research Article | Open Access

Volume 2016 |Article ID 6078298 | https://doi.org/10.1155/2016/6078298

Soon-Mo Jung, Young Woo Nam, "On the Hyers-Ulam Stability of the First-Order Difference Equation", Journal of Function Spaces, vol. 2016, Article ID 6078298, 6 pages, 2016. https://doi.org/10.1155/2016/6078298

# On the Hyers-Ulam Stability of the First-Order Difference Equation

Accepted10 Jul 2016
Published04 Aug 2016

#### Abstract

We prove Hyers-Ulam stability of the first-order difference equation of the form , where is a given function with some moderate features. Moreover, we introduce some conditions for the function under which the difference equation is not stable in the sense of Hyers and Ulam.

#### 1. Introduction

The difference equation usually describes the development of a certain phenomenon by recursively defining a sequence, each of whose terms is defined as a function of the preceding terms, once one or more initial terms are known. The difference equation often refers to a specific type of recurrence relation (see ).

In 1940, Ulam  raised an important problem concerning the stability of group homomorphisms (ref. [3, 4]): given a metric group , a positive number , and a function which satisfies inequality for all , do there exist a homomorphism and a constant depending only on and such that for all ?

If the answer to this question is affirmative, the functional equation is said to be stable. A first answer to this question was given by Hyers  in 1941 who proved that the Cauchy additive equation is stable in Banach spaces. In general, a functional equation is said to be stable in the sense of Hyers and Ulam (or the equation has the Hyers-Ulam stability) if for each solution to the perturbed equation, there exists a solution to the equation that differs from the solution to the perturbed equation with a small error. We refer the reader to [2, 511] for the exact definition of Hyers-Ulam stability.

Throughout this paper, we denote by , , , and the set of all positive integers, of all nonnegative integers, and of all real numbers and the set of all complex numbers, respectively.

In this paper, we prove Hyers-Ulam stability of the first-order difference equation of the formfor all integers , where is a given function with some moderate features. More precisely, we prove that if a complex-valued sequence satisfies inequality for all , then there exist a solution to the difference equation (1) and a positive constant depending only on and such that for all . As we know, the stability of the difference equation (1) depends on properties of the map . We show in the last part of this paper which condition of excludes the Hyers-Ulam stability.

#### 2. Conditions for Hyers-Ulam Stability

The principle of recursive definition states that, for any function and any , there exists a unique function such that for all [12, Theorem A.5.6]. This principle assures us of the existence and the uniqueness of the sequence mentioned in Theorem 1.

First, we prove a general theorem that provides us with a powerful tool for proving the Hyers-Ulam stability of a large class of the first-order difference equations.

Theorem 1. Given , let be a function satisfying the conditionfor all and all , where is a monotone increasing function. If a complex-valued sequence satisfies inequalityfor all , then there exists a complex-valued sequence satisfyingfor all , where the function is defined by for all and denotes the value of the th iterate of at ; that is,

Proof. In view of principle of recursive definition, the complex-valued sequence is uniquely determined via formula (7) provided is given.
We will apply an induction on to prove inequality (8). For , it follows from (5), (6), and (7) that which we also obtain by putting in (8). We now assume that inequality (8) is true for some . Then, it follows from (5), (6), (7), and (8) that which proves the validity of (8) for all .

Using Theorem 1, we can prove the Hyers-Ulam stability of a class of the first-order difference equations under the condition, (see the statement of Corollary 2 for and ).

Corollary 2. Let be a monotone increasing mapping such that andfor all , where is a positive real constant less than . Given , let be a function satisfying condition (5) for all and and let be defined by for all . If a complex-valued sequence satisfies inequality (6) for all , then there exists a complex-valued sequence satisfying (7) and for all .

Proof. According to Theorem 1, there exists a complex-valued sequence satisfying (7) andfor all .
It follows from (12) that is also a contraction mapping with the Lipschitz constant . By the contraction mapping theorem, we have for all , where is the unique fixed point of , from which it follows that for each . In view of (14) and the last inequality, we obtainfor all .
We now assert that for all . To prove this assertion, we assume for contradiction that there was such that . Then we would have which is the contradiction to (12). Hence, we obtain the following inequality for the fixed point of :Finally, it follows from (17) and (19) that for all .

Using Corollary 2, we prove the Hyers-Ulam stability of the first-order difference equation (1) under a more explicit condition for and an additional condition, (see the statement of Corollary 3 for and ).

Corollary 3. Given positive real constants and with , let be a function satisfying the conditionfor all and . If a complex-valued sequence satisfies inequality (6) for all , then there exists a complex-valued sequence satisfying (7) and for all .

Proof. If we define monotone increasing contraction mappings by and , then, by Corollary 2, we have for all .

Remark 4. If we set , , , and in [13, Theorem 2], then we get , , and , where , while we get from Corollary 3 provided the initial condition is assumed. In general, the result of [13, Theorem 2] is better than that of Corollary 3.

Example 5. Given positive real constants and with , let be a function defined by for all and . Then we have for all and , where we set . Then we have , and hence inequality (21) holds. According to Corollary 3, if a complex-valued sequence satisfies inequality (6) for all , then there exists a complex-valued sequence satisfying (7) and for all .

#### 3. Conditions for Nonstability

In this section, we introduce some conditions for the function , under which the first-order difference equation (1) is not stable in the sense of Hyers and Ulam.

We replace inequality (5) in Theorem 1 with another one and prove the counterpart of Theorem 1 in the following theorem.

Theorem 6. Given , let be a function satisfying the conditionfor all and , where is a monotone increasing function, and let be a monotone increasing function defined by for all . If a complex-valued sequence satisfies inequalityfor all and , then there exists a complex-valued sequence satisfyingfor all .

Proof. We apply the induction on to prove inequality (31). Trivially, (31) is true for . For , we use (27), (29), and (30) to show that that is, inequality (31) holds for . We now assume that inequality is true for all , where is a positive integer. Then, it follows from (27), (29), and (30) that which proves the validity of (31) for all .

Lemma 7. Let be an increasing continuous function for some . Assume that and has fixed points which are greater than ; that is, is nonempty. If , then for all and for all . Moreover,

Proof. The fact that is increasing and continuous implies that for all , where . Denoting for each , assume that there exists satisfying for some . Since is an increasing continuous function, we have the following inequality: which is the contradiction. Consequently, is an upper bound of the sequence for each .
Next, we claim that is an increasing sequence for each . Suppose for contradiction that there exists such that for some . If we define , then the following are satisfied: (i) is continuous.(ii).(iii). Thus, by the intermediate value theorem for , the function has a fixed point in and . It is contrary to the assumption that is the infimum of . Therefore, we conclude that is an increasing sequence for each .
Since the sequence is an increasing sequence bound above, it is convergent. Denote the limit by for each . Then, we conclude that is a fixed point of by considering the relation Hence, we obtain by the minimality of . On the other hand, is an upper bound of ; that is, for all . Thus, we get which implies that for each .

Lemma 8. Let , , and be constants with , , and . Then, the function , defined by , has two distinct fixed points.

Proof. Since , we have and Since , has a fixed point in by the intermediate value theorem for the continuous function . Moreover, if , then we have and hence, ; that is, for each . Hence, has another fixed point in .

Corollary 9. Let , , and be constants with , , and . The function has a fixed point in the interval and another one in .

Proof. We see that andHence, if we apply the intermediate value theorem with the continuous function , then we have and , which implies that has a fixed point in the interval .
Similarly, we have and moreover, We also apply the intermediate value theorem to on the interval . Then has another fixed point in .

Remark 10. The fixed point of in Corollary 9 is contained in the interval independent of which runs over the interval . Since is a fixed point of the mapping and has only two fixed points, converges to as .

In the following theorem, we introduce a condition for the function , under which the first-order difference equation (1) is not stable in the sense of Hyers and Ulam.

Theorem 11. Let and be constants with and . Given , let be a function satisfying the condition for all and . If a complex-valued sequence satisfies inequalityfor all , then for every sequence satisfying and , it holds that

Proof. Let be a constant with . Denote and as follows: We set for all and in view of (45), without loss of generality, we can choose . Then there exists such that Thus we may assume that . By Lemma 8, has two distinct fixed points and with . Since is an increasing continuous function and , there is the point satisfying . Moreover, Corollary 9 implies thatBy the triangular inequality and Theorem 6, we haveInequality (41) implies that Since the functions and are increasing functions, we obtain for all . Then, in view of (49), (50), and (52), is greater than the fixed point ; that is, Moreover, since is increasing, we have By Theorem 6, inequality holds for all and by putting and in Lemma 7 and taking (41) into account, converges to . Hence, by Theorem 6 and (49), we get for all sufficiently large integers .

Remark 12. Theorem 11 can be regarded as a nonstability result because this theorem shows that the difference can be quite large even in the case when is very small. An analogous nonstability result can be deduced from [14, Theorem 1] when and . However, this is not the case when and because [13, Theorem 1] shows that the sequence has Hyers-Ulam stability.

#### Competing Interests

The authors declare that they have no competing interests.

#### Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final paper.

#### Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2015R1D1A1A02061826).

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