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

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 [1]).

In 1940, Ulam [2] 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 [5] 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, 5–11] 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).