A Note on Stability of an Operator Linear Equation of the Second Order

and Applied Analysis 3 If p, 1/q ∈ Z, then solutions f : N0 → Z of the difference equation 1.7 are called the Lucas sequences see, e.g., 24 ; in some special cases they are given specific names; that is, the Fibonacci numbers p −1, q −1, f 0 0, and f 1 1 , the Lucas numbers p −1, q −1, f 0 2, and f 1 1 , the Pell numbers p −2, q −1, f 0 0, and f 1 1 , the Pell-Lucas or companion Lucas numbers p −2, q −1, f 0 2, and f 1 2 , and the Jacobsthall numbers p −1, q −2, f 0 0, and f 1 1 . 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


Introduction
Let C, R, Z, N 0 , and N stand, as usual, for the sets of complex numbers, real numbers, integers, nonnegative integers, and positive integers, respectively.Let S be a nonempty set, X a Banach space over a field K ∈ {C, R}, p, q ∈ K, q / 0 and p 2 − 4q / 0, and a 1 , a 2 denote the complex roots of the equation qx 2 − px 1 0. 1.1 Clearly we have a 1 / a 2 ,

Abstract and Applied Analysis
In what follows, X S denotes the family of all functions mapping S into X and X S is a linear space over K with the operations given by f h x : f x h x , αf x : αf x 1.3 for all f, h ∈ X S , α ∈ K, x ∈ S. Throughout this paper, we assume that H C is a nontrivial subgroup of the group X S , and L : C → X S is an additive operator i.e., L f h Lf Lh for f, h ∈ X S .
We investigate the Hyers-Ulam stability of the operator equation for functions F ∈ C with LF, L 2 F ∈ C. Namely, we show under suitable assumptions that for every function f ∈ C satisfying 1.4 approximately, that is, there exists a unique solution of the equation that is "near" to f.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 3 .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 Lf f • ξ for f ∈ C with C X S and a fixed mapping ξ : S → S , 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 S Z and ξ x x 1, is the difference equation If p, 1/q ∈ Z, then solutions f : N 0 → Z of the difference equation 1.7 are called the Lucas sequences see, e.g., 24 ; in some special cases they are given specific names; that is, the Fibonacci numbers p −1, q −1, f 0 0, and f 1 1 , the Lucas numbers p −1, q −1, f 0 2, and f 1 1 , the Pell numbers p −2, q −1, f 0 0, and f 1 1 , the Pell-Lucas or companion Lucas numbers p −2, q −1, f 0 2, and f 1 2 , and the Jacobsthall numbers p −1, q −2, f 0 0, and f 1 1 .

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 a 1 / ∈ K or a 2 / ∈ K will be given in the fourth section.
For simplicity, we write in the sequel Next, for a given g ∈ X S and {g n } n∈N ⊂ X S , the equality 3 means that g x lim n → ∞ g n x for every x ∈ S. We say that E ⊂ X S is closed with respect to the uniform convergence abbreviated in the sequel to c.u.c.provided the following holds true: where the symbol f n ⇒ f means that the sequence {f n } n∈N tends uniformly to f.Moreover, we use in the sequel the following two hypotheses: Now, we are in a position to formulate the main result of this paper.

2.4
Suppose that 2.1 , C1 , and L1 are valid and one of the following three collections of hypotheses is fulfilled.

., and
where Remark 2.2.Clearly, if C is a linear subspace of X S and L is linear over K , then C1 and L1 are valid.However, if C is "only" additive, C is a linear subspace of X S but over Q i.e., actually a divisible subgroup of X S , and a 1 , a 2 ∈ Q, then C1 and L1 hold, as well.This shows that it makes sense to assume only L1 instead of linearity of L.
Below, before the proof of Theorem 2.1, we provide simple and natural examples of linear operators L that satisfy the assumptions of Theorem 2.1 with suitable a 1 , a 2 .
i Let C X S , n ∈ N, and Lf with λ i : inf{L ∈ R : Ψ i w ≤ L w , w ∈ X}.Hence 2.5 holds with L 1 : Abstract and Applied Analysis 5 Next, let C X S , Ψ : X → X, and ξ : S → S be bijective, Ψ linear, Ψ −1 bounded, and where L 0 : inf{L ∈ R : Ψ −1 w ≤ L w , w ∈ X}.Clearly, as above, that inequality yields 2.6 with L 2 : L −1 0 .If additionally Ψ is bounded, then analogously as before we obtain that 2.5 holds, as well, with some

Proof of Theorem 2.1
The subsequent lemma will be useful in the proof of Theorem 2.1.

Lemma 3.1. Assume that 2.1 , C1 , and L1 are valid and one of the collections of hypotheses
Proof.Let h i : Lf i − a 2 f i for i ∈ {1, 2}.Then, by 1.2 and 1.4 , if α or γ holds, and So, analogously as before, for each k ∈ N, in the case of α , we have and, in the case of β or γ , It is easily seen that in each of those cases the above two inequalities imply that f 1 f 2 .
Now, we have all tools to prove Theorem 2.1.To this end, fix i ∈ {1, 2}.Then First consider the situation: Clearly this means that α or γ must be valid, which yields L C ⊂ C. Write Note that, by C1 and L1 , A i k ∈ C for k ∈ N 0 .Further, for each k ∈ N 0 , from 1.2 we get whence, according to 2.4 and 2.5 , and consequently

3.11
This means that, for each x ∈ S, {A i n x } n∈N is a Cauchy sequence, and therefore, there exists the limit Observe that, for every n, k ∈ N 0 , Further, by 2.5 , for each n ∈ N, So, in view of 1.2 and 3.13 , we have and, by 3.11 with k 0 and n → ∞, 3.17 Now, consider the case when |a i | < L 2 .Then, according to the assumptions, L is injective, 2.6 holds, and C ⊂ L C , that is,

3.18
Write Then, for each k ∈ N, we have and next, by 2.6 ,

3.21
This yields So, for each x ∈ S, {A i n x } n∈N is a Cauchy sequence, and consequently there exists the limit F i x lim n → ∞ A i n x .Note that, by 3.22 , A i n ⇒ F i , whence . , and again by 3.22 , with k 0 and n → ∞, It is easy to observe that Further, by 2.6 , for each k ∈ N,

3.27
This and 3.23 yield . Repeating yet that reasoning twice, we get i.e., 3.12 holds and consequently

3.29
Thus we have proved that, for i ∈ {1, 2}, in either case inequalities 3.17 or 3.24 , respectively, hold and F i is a solution to 1.4 , with 3.12 fulfilled.Define F : S → X by

3.30
Then, by 3.12 , and it follows from 3.16 and 3.29 , respectively, that
For the proof of the statement concerning uniqueness of F, take and therefore, by Lemma 3.1, F F 0 .This completes the proof of Theorem 2.1.

Complementary Results
In this section, we consider the cases that are complementary to those of Theorem 2.1, that is, when K R and 2.1 may not be fulfilled.We will use the following assumptions: Abstract and Applied Analysis where R z and I z denote the real and imaginary parts of the complex number z if z is a real number, then simply R z z and I z 0 .Observe that, in the case a 1 , a 2 ∈ K R, C2 and L2 become just C1 and L1 .Note also that if C is a real linear subspace of X S , then C2 and L2 are fulfilled.
The next theorem complements Theorem 2.1 when α is valid however, with a bit stronger assumption on L 1 .The cases of β and γ are more complicated, and some results concerning them will be published separately.
Then there exists a unique function F ∈ C, with LF, L 2 F ∈ C, that satisfies 1.4 and inequality 2.7 ; moreover, Proof.We apply a well-known method of complexification of the real Banach space X.Namely, see, e.g., 25, page 39 , 26 , or 27, 1.9.6, page 66 X 2 is a complex Banach space with the linear structure and the Taylor norm • T given by x, y z, w : x z, y w for x, y, z, w ∈ X, α iβ x, y : αx − βy, βx αy for x, y ∈ X, α, β ∈ R, x, y T : sup 0≤θ≤2π cos θ x sin θ y for x, y ∈ X.

4.3
Note that max x , y ≤ x, y T ≤ x y , x,y ∈ X.

4.4
Analogously as before we write for each function μ : S → X 2 .Next, χ : S → X 2 be given by χ x : g x , 0 for x ∈ S, and for every μ ∈ C 0 and x ∈ S. Since L C ⊂ C and C is a subgroup of the group X S , i.e., the function μ 0 : S → X defined by μ 0 x 0 for x ∈ S, is in C , it is easily seen that χ, L 0 χ, L 2 0 χ ∈ C 0 and Next, for each f ∈ C 0 , we have f 1 : and consequently

4.11
Thus, we have obtained that L1 L 0 a j f a j L 0 f for f ∈ C 0 and j ∈ {1, 2}.
Analogously, for every μ ∈ C 0 , i ∈ {1, 2}, we get which means that a i μ, a −1 i μ ∈ C 0 because C 0 is a group and C2 holds .Moreover, Thus we have proved that C1 a j C 0 C 0 and C 0 ⊂ a 1 − a 2 C 0 for j ∈ {1, 2}.Now, we show that C 0 is c.u.c. with regard to the Taylor norm.To this end, take μ ∈ X S and μ n ∈ C 0 for n ∈ N such that μ n ⇒ μ with respect to the Taylor norm .Then, by 4.4 , Note yet that, according to 4.1 and 4.4 , for every μ ∈ C 0 , we have 4.15 because p 2 • χ x 0 for each x ∈ S. In this way, we have shown that the assumptions of Theorem 2.1 α are satisfied with g, L, L 1 , and C replaced by χ, L 0 , 2L, and C 0 , resp.and consequently there is a solution H ∈ C 0 of the equation Hence, by Lemma 3.1 with L and C replaced by L 0 and C 0 , resp., H H 0 , which yields

Final Remarks on Fixed Points and Open Problems
Theorems 2.1 and 4.1 can be actually expressed in the terms of fixed points.Namely, they may be reformulated as follows.
5 If L is linear, then Theorems 2.1 and 4.1 can also be expressed in the following way cf.7 .
Theorem 5.2.Let K : pL − qL 2 − I, where I : X S → X S is the identity operator given by If f for f ∈ X S .Suppose that C1 , L1 , and one of conditions a , b are valid with some L 1 > 0, L 2 > 0.Then, for every g ∈ C with Lg, L 2 g ∈ C and ε : Kg < ∞, 5.3 there exists a unique F ∈ C with LF, L 2 F ∈ C and such that F ∈ ker K i.e., Kf x 0 for x ∈ S and g − F < ∞; moreover, if a is valid, then 2.8 holds and if b is valid, then 4.2 holds with L L 1 .
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 17 .It seems that this issue deserves a more systematic treatment.
Another question concerns the case where L 1 ≥ |a i | for some i ∈ {1, 2} when α is valid and analogous situations for β and γ .In general, the assumption L 1 ≤ |a i | for i ∈ {1, 2} is necessary in the case of α , as it follows from nonstability results in 18 .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 g pLg qL 2 g H x 5.4 with a nontrivially given function H : S → X.Also, it is interesting if these methods can be applied for a higher-order operator linear equation, for example, for the third-order equation g pLg qL 2 g rL 3 g.

5.5
For related results, in some particular situations and obtained with different methods, we refer to 19, 28 .
17 Observe that F : p 1 • H is a solution of 1.4 and, by 4.4 , 4.2 holds.It remains to prove the statement concerning uniqueness of F. So, let F 0 Suppose that C1 , L1 , and one of the following two conditions are valid: a Condition 2.1 and one of the collections of hypotheses α -γ are fulfilled; b the collection of hypotheses α is fulfilled and 2L 1 < |a j | for j ∈ {1, 2}.
.1 Then there exists a unique F ∈ C with LF, L 2 F ∈ C such that F is a fixed point of T and g − F < ∞; 5.2 moreover, if a is valid, then 2.8 holds and if b is valid, then 4.2 holds with L L 1 .