A Note on Stability of an Operator Linear Equation of the Second Order
We prove some Hyers-Ulam stability results for an operator linear equation of the second order that is patterned on the difference equation, which defines the Lucas sequences (and in particular the Fibonacci numbers). In this way, we obtain several results on stability of some linear functional and differential and integral equations of the second order and some fixed point results for a particular (not necessarily linear) operator.
Let , , , , and stand, as usual, for the sets of complex numbers, real numbers, integers, nonnegative integers, and positive integers, respectively. Let be a nonempty set, a Banach space over a field , , and , and , denote the complex roots of the equation Clearly we have ,
In what follows, denotes the family of all functions mapping into and is a linear space over with the operations given by for all , , . Throughout this paper, we assume that is a nontrivial subgroup of the group and is an additive operator (i.e., for ).
We investigate the Hyers-Ulam stability of the operator equation for functions with . Namely, we show under suitable assumptions that for every function satisfying (1.4) approximately, that is,
there exists a unique solution of the equation that is “near” to . This kind of issues arise during study of the real-world phenomena, where we very often apply equations; however, in general, those equations are satisfied only with some error. Sometimes that error is neglected and it is believed that this will have only a minor influence on the final outcome. Since it is not always the case, it seems to be of interest to investigate when we can neglect the error, why, and to what extent.
One of the tools for systematic treatment of the problem described above seems to be the notion of Hyers-Ulam stability and some ideas inspired by it. That notion of stability was motivated by a question of Ulam (cf. [1, 2]), and a solution to it published by Hyers in . At the moment, it is a very popular subject of investigation in the areas of, for example, functional, differential, integral equations, but also in other fields of mathematics (for information on this kind of stability and further references see, e.g., [4–8]). Also, the Hyers-Ulam stability is related to the notions of shadowing and controlled chaos (see, e.g., [9–12]).
If for (with and a fixed mapping ), then (1.4) takes the form which is a linear functional equation in a single variable of second order (for some information and further references on the functional equations in single variable, we refer to [13–15]). Stability of (1.6) has been already investigated in [16–23]. A particular case of (1.6), with and , is the difference equation If , then solutions of the difference equation (1.7) are called the Lucas sequences (see, e.g., ); in some special cases they are given specific names; that is, the Fibonacci numbers (, , , and ), the Lucas numbers (, , , and ), the Pell numbers (, , , and ), the Pell-Lucas (or companion Lucas) numbers (, , , and ), and the Jacobsthall numbers (, , , and ).
2. The Main Result
Now we will present a theorem that is the main result of this paper. In this section, we consider only the case Some complementary results for the case where or will be given in the fourth section.
For simplicity, we write in the sequel Next, for a given and , the equality means that for every .
We say that is closed with respect to the uniform convergence (abbreviated in the sequel to c.u.c.) provided the following holds true: if for , and , then , where the symbol means that the sequence tends uniformly to . Moreover, we use in the sequel the following two hypotheses: and for ; for , .
Now, we are in a position to formulate the main result of this paper.
Theorem 2.1. Let , , , and let with satisfying the inequality Suppose that (2.1), , and are valid and one of the following three collections of hypotheses is fulfilled. for , , is c.u.c., and is injective, for , is c.u.c., , and is injective, , , is c.u.c., , and conditions (2.5) and (2.6) hold true. Then there exists a unique function with , that satisfies (1.4) and moreover, is given by (3.30) and where
Remark 2.2. Clearly, if is a linear subspace of and is linear (over ), then and are valid. However, if is “only” additive, is a linear subspace of but over (i.e., actually a divisible subgroup of ), and , then and hold, as well. This shows that it makes sense to assume only instead of linearity of .
Below, before the proof of Theorem 2.1, we provide simple and natural examples of linear operators that satisfy the assumptions of Theorem 2.1 (with suitable ).(i)Let , , and , where is linear and bounded and is fixed for each . Then with . Hence (2.5) holds with . Next, let , , and be bijective, linear, bounded, and . Then where . Clearly, as above, that inequality yields (2.6) with . If additionally is bounded, then analogously as before we obtain that (2.5) holds, as well, with some .(ii)Let , , , the family of all continuous functions mapping the interval into , , , continuous, and for , . Then it is easily seen that is fulfilled with , .(iii)Let , , be the family of all continuously differentiable functions with and . Then is satisfied with , .
3. Proof of Theorem 2.1
The subsequent lemma will be useful in the proof of Theorem 2.1.
Proof. Let for . Then, by (1.2) and (1.4),
for . Consequently, for each ,
if or holds, and
if holds. This means that .
Now, in view of the definition of , which means that So, analogously as before, for each , in the case of , we have and, in the case of or , It is easily seen that in each of those cases the above two inequalities imply that .
Now, we have all tools to prove Theorem 2.1.
To this end, fix . Then or . First consider the situation: . Clearly this means that () or () must be valid, which yields . Write Note that, by and , for . Further, for each , from (1.2) we get whence, according to (2.4) and (2.5), and consequently This means that, for each , is a Cauchy sequence, and therefore, there exists the limit . Moreover, (3.11) yields , whence because is c.u.c. and .
Now, consider the case when . Then, according to the assumptions, is injective, (2.6) holds, and , that is, Write Then, for each , we have (because is such that ), and next, by (2.6), This yields So, for each , is a Cauchy sequence, and consequently there exists the limit . Note that, by (3.22), , whence (because for and is c.u.c.), and again by (3.22), with and ,
It is easy to observe that Further, by (2.6), for each , So, by (3.18), (3.23), and (3.25), we have This and (3.23) yield , that is, . Repeating yet that reasoning twice, we get (i.e., (3.12) holds) and consequently
Thus we have proved that, for , in either case inequalities (3.17) or (3.24), respectively, hold and is a solution to (1.4), with (3.12) fulfilled. Define by Then, by (3.12), and it follows from (3.16) and (3.29), respectively, that Moreover, and consequently whence, by (3.17) and (3.24), respectively, we obtain (2.8).
For the proof of the statement concerning uniqueness of , take with . Suppose that is a solution of (1.4) such that . Then we have and therefore, by Lemma 3.1, . This completes the proof of Theorem 2.1.
4. Complementary Results
where and denote the real and imaginary parts of the complex number (if is a real number, then simply and ). Observe that, in the case , and become just and . Note also that if is a real linear subspace of , then and are fulfilled.
The next theorem complements Theorem 2.1 when is valid (however, with a bit stronger assumption on ). The cases of and are more complicated, and some results concerning them will be published separately.
Proof. We apply a well-known method of complexification of the real Banach space . Namely, (see, e.g., [25, page 39], , or [27, 1.9.6, page 66]) is a complex Banach space with the linear structure and the Taylor norm given by
Analogously as before we write for each function . Next,
Let be given by for , and for every and . Since and is a subgroup of the group (i.e., the function defined by for , is in ), it is easily seen that and
Next, for each , we have , whence, in view of and , for each , and consequently Thus, we have obtained that for and . Analogously, for every , , we get which means that (because is a group and holds). Moreover, Thus we have proved that and for .
Now, we show that is c.u.c. with regard to the Taylor norm. To this end, take and for such that (with respect to the Taylor norm). Then, by (4.4), which means that and . Consequently, . Hence, .
Note yet that, according to (4.1) and (4.4), for every , we have because for each .
In this way, we have shown that the assumptions of Theorem 2.1() are satisfied (with , , , and replaced by , , , and , resp.) and consequently there is a solution of the equation such that Observe that is a solution of (1.4) and, by (4.4), (4.2) holds.
It remains to prove the statement concerning uniqueness of . So, let , with , be a solution of (1.4) such that . Write for . It is easily seen that and . Moreover, Hence, by Lemma 3.1 (with and replaced by and , resp.), , which yields .
5. Final Remarks on Fixed Points and Open Problems
Theorem 5.1. Let , , , , and , with , satisfying the inequality Suppose that , , and one of the following two conditions are valid: (a)Condition (2.1) and one of the collections of hypotheses are fulfilled; (b)the collection of hypotheses is fulfilled and for . Then there exists a unique with such that is a fixed point of and moreover, if is valid, then (2.8) holds and if is valid, then (4.2) holds with .
Theorem 5.2. Let , where is the identity operator given by for . Suppose that , , and one of conditions , are valid with some , . Then, for every with and there exists a unique with and such that i.e., for and ; moreover, if is valid, then (2.8) holds and if is valid, then (4.2) holds with .
In connection with the results presented in this paper, there arise several natural questions (apart from those regarding the situation where (2.1) is not fulfilled). We mention here some of them.
The first one concerns optimality of estimations (2.8) and (4.2). It is known that in general they are not the best possible, and for suitable comments and examples, see . It seems that this issue deserves a more systematic treatment.
Another question concerns the case where for some when is valid (and analogous situations for and ). In general, the assumption for is necessary in the case of , as it follows from nonstability results in . But maybe in some particular situation some partial stability results are possible.
One more question is whether methods similar to those used in this paper can be applied for a bit more general equation of the form with a nontrivially given function . Also, it is interesting if these methods can be applied for a higher-order operator linear equation, for example, for the third-order equation For related results, in some particular situations and obtained with different methods, we refer to [19, 28].
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2011-0004919).
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