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

Abstract

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.

1. Introduction

Let ℂ, ℝ, ℤ, ℕ₀, 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 𝑝²−4𝑞≠0, and 𝑎₁, 𝑎₂ denote the complex roots of the equation 𝑞𝑥²−𝑝𝑥+1=0.(1.1) Clearly we have 𝑎₁≠𝑎₂, 1𝑝=𝑎₁+1𝑎₂1,𝑞=𝑎₁𝑎₂.(1.2)

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 𝐹=𝑝ℒ𝐹−𝑞ℒ²𝐹(1.4) for functions 𝐹∈𝒞 with ℒ𝐹,ℒ²𝐹∈𝒞. Namely, we show under suitable assumptions that for every function 𝑓∈𝒞 satisfying (1.4) approximately, that is, sup_𝑥∈𝑆𝑝ℒ𝑓(𝑥)−𝑞ℒ²𝑓(𝑥)−𝑓(𝑥)<∞,(1.5)

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 form𝑓(𝑥)=𝑝𝑓(𝜉(𝑥))−𝑞𝑓𝜉²(𝑥),(1.6) 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 𝑓(𝑥)=𝑝𝑓(𝑥+1)−𝑞𝑓(𝑥+2).(1.7) 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 𝑎₁,𝑎₂∈𝕂.(2.1) Some complementary results for the case where 𝑎₁∉𝕂 or 𝑎₂∉𝕂 will be given in the fourth section.

For simplicity, we write in the sequel 𝑓∶=sup_𝑥∈𝑆𝑓(𝑥),𝑓∈𝑋^𝑆.(2.2) Next, for a given 𝑔∈𝑋^𝑆 and {𝑔_𝑛}_𝑛∈ℕ⊂𝑋^𝑆, the equality 𝑔=lim_𝑛→∞𝑔_𝑛(2.3) 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: (𝐸) if 𝑓_𝑛∈ℰ for 𝑛∈ℕ, 𝑓∈𝑋^𝑆 and 𝑓_𝑛⇒𝑓, then 𝑓∈ℰ, 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: (𝐶1)𝒞⊂(𝑎₁−𝑎₂)𝒞 and 𝑎_𝑗𝒞=𝒞 for 𝑗∈{1,2}; (𝐿1)ℒ(𝑎_𝑗𝑓)=𝑎_𝑗ℒ𝑓 for 𝑓∈𝒞, 𝑗∈{1,2}.

Now, we are in a position to formulate the main result of this paper.

Theorem 2.1. Let 𝜀>0, 𝐿₁>0, 𝐿₂>0, and let 𝑔∈𝒞 with ℒ𝑔,ℒ²𝑔∈𝒞 satisfying the inequality 𝑔−𝑝ℒ𝑔+𝑞ℒ²𝑔≤𝜀.(2.4) Suppose that (2.1), (𝐶1), and (𝐿1) are valid and one of the following three collections of hypotheses is fulfilled. (𝛼)𝐿₁<|𝑎_𝑗| for 𝑗∈{1,2}, ℒ(𝒞)⊂𝒞, 𝒞 is c.u.c., and ℒ𝑓−ℒℎ≤𝐿₁𝑓−ℎ,𝑓,ℎ∈𝒞.(2.5)(𝛽)ℒ is injective, 𝐿₂>|𝑎_𝑗| for 𝑗∈{1,2}, ℒ(𝒞) is c.u.c., 𝒞⊂ℒ(𝒞), and ℒ⁻¹𝑓−ℒ⁻¹ℎ≤𝐿₂⁻¹𝑓−ℎ,𝑓,ℎ∈ℒ(𝒞).(2.6)(𝛾)ℒ is injective, 𝐿₁<|𝑎₁|, 𝐿₂>|𝑎₂|, 𝒞 is c.u.c., ℒ(𝒞)=𝒞, and conditions (2.5) and (2.6) hold true. Then there exists a unique function 𝐹∈𝒞 with ℒ𝐹,ℒ²𝐹∈𝒞, that satisfies (1.4) and 𝑔−𝐹<∞;(2.7) moreover, 𝐹 is given by (3.30) and ≤𝜀𝑔−𝐹𝑞𝑎₂−𝑎₁1𝐿^′₁−𝑎₁+1𝐿^′₂−𝑎₂,(2.8) where 𝐿^′_𝑖𝐿∶=¹(𝛼)𝐿ifholds,₂𝛽𝐿ifholds,_𝑖𝛾{}ifholds,𝑖∈1,2.(2.9)

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 𝑎₁,𝑎₂∈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 𝑎₁,𝑎₂).(i)Let 𝒞=𝑋^𝑆, 𝑛∈ℕ, and ℒ𝑓=∑^𝑛_𝑖=1Ψ_𝑖∘𝑓∘𝜉_𝑖, where Ψ_𝑖∶𝑋→𝑋 is linear and bounded and 𝜉_𝑖∶𝑆→𝑆 is fixed for each 𝑖∈{1,…,𝑛}. Then ℒ𝑓(𝑥)−ℒℎ(𝑥)≤^𝑛_𝑖=1𝜆_𝑖𝑓𝜉_𝑖(𝑥)−ℎ𝜉_𝑖(𝑥),𝑓,ℎ∈𝑋^𝑆,𝑥∈𝑆,(2.10) with 𝜆_𝑖∶=inf{𝐿∈ℝ∶‖Ψ_𝑖(𝑤)‖≤𝐿‖𝑤‖,𝑤∈𝑋}. Hence (2.5) holds with 𝐿₁∶=∑^𝑛_𝑖=1𝜆_𝑖. Next, let 𝒞=𝑋^𝑆, Ψ∶𝑋→𝑋, and 𝜉∶𝑆→𝑆 be bijective, Ψ linear, Ψ⁻¹ bounded, and ℒ𝑓=Ψ∘𝑓∘𝜉. Then ℒ⁻¹𝑓(𝑥)−ℒ⁻¹ℎ(𝑥)≤𝐿₀𝑓𝜉⁻¹(𝑥)−ℎ𝜉⁻¹(𝑥),𝑓,ℎ∈𝑋^𝑆,𝑥∈𝑆,(2.11) where 𝐿₀∶=inf{𝐿∈ℝ∶‖Ψ⁻¹(𝑤)‖≤𝐿‖𝑤‖,𝑤∈𝑋}. Clearly, as above, that inequality yields (2.6) with 𝐿₂∶=𝐿₀⁻¹. If additionally Ψ is bounded, then analogously as before we obtain that (2.5) holds, as well, with some 𝐿₁>0.(ii)Let 𝑎,𝑏∈ℝ, 𝑎<𝑏, 𝑆=[𝑎,𝑏], 𝒞 the family of all continuous functions mapping the interval [𝑎,𝑏] into ℝ, 𝑛∈ℕ, 𝐴₁,…,𝐴_𝑛∈ℝ, 𝜉₁,…,𝜉_𝑛∶𝑆→𝑆 continuous, and ℒ𝑓(𝑥)=∑^𝑛_𝑖=1∫^𝑥_𝑎𝐴_𝑖𝑓∘𝜉_𝑖(𝑡)𝑑𝑡 for 𝑓∈𝒞, 𝑥∈𝑆. Then it is easily seen that (𝛼) is fulfilled with |𝑎_𝑖|>𝐿₁∶=(𝑏−𝑎)∑^𝑛_𝑗=1|𝐴_𝑗|, 𝑖∈{1,2}.(iii)Let 𝑎,𝑏∈ℝ, 𝑎<𝑏, 𝒞 be the family of all continuously differentiable functions 𝑓∶[𝑎,𝑏]→ℝ with 𝑓(𝑎)=0 and ℒ=𝑑/𝑑𝑡. Then (𝛽) is satisfied with |𝑎_𝑖|<𝐿₂∶=1/(𝑏−𝑎), 𝑖∈{1,2}.

3. 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 (𝛼)-(𝛾) is fulfilled with some 𝐿₁,𝐿₂∈(0,∞). Let 𝑓₁,𝑓₂∈𝒞, with ℒ^𝑖(𝑓_𝑗)∈𝒞 for 𝑖,𝑗∈{1,2}, be solutions of (1.4) and ‖𝑓₁−𝑓₂‖<∞. Then 𝑓₁=𝑓₂.

Proof. Let ℎ_𝑖∶=ℒ𝑓_𝑖−𝑎₂𝑓_𝑖 for 𝑖∈{1,2}. Then, by (1.2) and (1.4), ℒℎ_𝑖=ℒ²𝑓_𝑖−𝑎₂ℒ𝑓_𝑖=𝑎₁𝑎₂𝑝ℒ𝑓_𝑖−𝑓_𝑖−𝑎₂ℒ𝑓_𝑖=𝑎₁ℎ_𝑖(3.1) for 𝑖∈{1,2}. Consequently, for each 𝑘∈ℕ, ℎ₁−ℎ₂=𝑎₁^−𝑘ℒ^𝑘ℎ₁−ℒ^𝑘ℎ₂≤𝑎₁^−𝑘𝐿^𝑘₁ℎ₁−ℎ₂(3.2) if (𝛼) or (𝛾) holds, and ℎ₁−ℎ₂=𝑎₁^𝑘ℒ^−𝑘ℎ₁−ℒ^−𝑘ℎ₂≤𝑎₁^𝑘𝐿₂^−𝑘ℎ₁−ℎ₂(3.3) if (𝛽) holds. This means that ℎ₁=ℎ₂.
Now, in view of the definition of ℎ_𝑖, ℒ𝑓₁−𝑎₂𝑓₁−ℒ𝑓₂−𝑎₂𝑓₂=ℎ₁−ℎ₂=0,(3.4) which means that ℒ𝑓₁−ℒ𝑓₂=𝑎₂𝑓₁−𝑓₂.(3.5) So, analogously as before, for each 𝑘∈ℕ, in the case of (𝛼), we have 𝑓₁−𝑓₂=𝑎₂^−𝑘ℒ^𝑘𝑓₁−ℒ^𝑘𝑓₂≤𝑎₂^−𝑘𝐿^𝑘₁𝑓₁−𝑓₂,(3.6) and, in the case of (𝛽) or (𝛾), 𝑓₁−𝑓₂=𝑎₂^𝑘ℒ^−𝑘𝑓₁−ℒ^−𝑘𝑓₂≤𝑎₂^𝑘𝐿₂^−𝑘𝑓₁−𝑓₂.(3.7) It is easily seen that in each of those cases the above two inequalities imply that 𝑓₁=𝑓₂.

Now, we have all tools to prove Theorem 2.1.

To this end, fix 𝑖∈{1,2}. Then |𝑎_𝑖|>𝐿₁ or |𝑎_𝑖|<𝐿₂. First consider the situation: 𝐿₁<|𝑎_𝑖|. Clearly this means that (𝛼) or (𝛾) must be valid, which yields ℒ(𝒞)⊂𝒞. Write 𝒜^𝑖_𝑘∶=𝑎_𝑖^−𝑘ℒ^𝑘𝑔−𝑝−𝑎_𝑖⁻¹ℒ^𝑘+1𝑔,𝑘∈ℕ₀.(3.8) Note that, by (𝐶1) and (𝐿1), 𝐴^𝑖_𝑘∈𝒞 for 𝑘∈ℕ₀. Further, for each 𝑘∈ℕ₀, from (1.2) we get 𝒜^𝑖_𝑘−𝒜^𝑖_𝑘+1=𝑎_𝑖^−𝑘ℒ^𝑘𝑔−𝑝−𝑎_𝑖⁻¹ℒ^𝑘+1𝑔−𝑎_𝑖^−𝑘−1ℒ^𝑘+1𝑔−𝑝−𝑎_𝑖⁻¹ℒ^𝑘+2𝑔=𝑎_𝑖^−𝑘ℒ^𝑘𝑔−𝑝ℒ𝑔+𝑞ℒ²𝑔,(3.9) whence, according to (2.4) and (2.5), 𝒜^𝑖_𝑘−𝒜^𝑖_𝑘+1≤𝑎_𝑖^−𝑘𝐿^𝑘₁𝑔−𝑝ℒ𝑔+𝑞ℒ²𝑔≤𝑎_𝑖^−𝑘𝐿^𝑘₁𝜀,(3.10) and consequently 𝒜^𝑖_𝑘−𝒜^𝑖_𝑘+𝑛≤^{𝑘+𝑛−1}_𝑗=𝑘𝑎_𝑖^−𝑗𝐿^𝑗₁𝜀,𝑛∈ℕ.(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 𝐹_𝑖(𝑥)=lim_𝑛→∞𝒜^𝑖_𝑛(𝑥). Moreover, (3.11) yields 𝒜^𝑖_𝑛⇒𝐹_𝑖, whence 𝐹_𝑖,ℒ𝐹_𝑖,ℒ²𝐹_𝑖∈𝒞,(3.12) because 𝒞 is c.u.c. and ℒ(𝒞)⊂𝒞.

Observe that, for every 𝑛,𝑘∈ℕ₀, ℒ^𝑛𝒜^𝑖_𝑘=𝑎^𝑛_𝑖𝒜^𝑖_𝑛+𝑘.(3.13) Further, by (2.5), for each 𝑛∈ℕ, ℒ𝐹_𝑖−ℒ𝒜^𝑖_𝑛≤𝐿₁𝐹_𝑖−𝒜^𝑖_𝑛,(3.14) which yields ℒ^𝑘𝐹_𝑖=lim_𝑛→∞ℒ^𝑘𝒜^𝑖_𝑛,𝑘∈{1,2}.(3.15) So, in view of (1.2) and (3.13), we have𝑝ℒ𝐹_𝑖−𝑞ℒ²𝐹_𝑖=𝑝lim_𝑛→∞ℒ𝒜^𝑖_𝑛−𝑞lim_𝑛→∞ℒ²𝒜^𝑖_𝑛=𝑝𝑎_𝑖lim_𝑛→∞𝒜^𝑖_𝑛+1−𝑞𝑎²_𝑖lim_𝑛→∞𝒜^𝑖_𝑛+2=𝑝𝑎_𝑖𝐹_𝑖−𝑞𝑎²_𝑖𝐹_𝑖=𝐹_𝑖,(3.16) and, by (3.11) with 𝑘=0 and 𝑛→∞, 𝑔−𝑝−𝑎_𝑖⁻¹ℒ𝑔−𝐹_𝑖≤𝑎_𝑖𝜀𝑎_𝑖−𝐿₁.(3.17)

Now, consider the case when |𝑎_𝑖|<𝐿₂. Then, according to the assumptions, ℒ is injective, (2.6) holds, and 𝒞⊂ℒ(𝒞), that is, ℒ^−𝑘(𝒞)⊂𝒞⊂ℒ(𝒞),𝑘∈ℕ.(3.18) Write 𝒜^𝑖_𝑘∶=𝑎^𝑘_𝑖ℒ^−𝑘𝑔−𝑝−𝑎_𝑖⁻¹ℒ^−𝑘+1𝑔,𝑘∈ℕ₀.(3.19) Then, for each 𝑘∈ℕ, we have 𝒜^𝑖_𝑘∈𝒞 (because 𝑔 is such that ℒ𝑔∈𝒞), 𝒜^𝑖_𝑘−𝒜^𝑖_𝑘−1=𝑎^𝑘_𝑖ℒ^−𝑘𝑔−𝑝−𝑎_𝑖⁻¹ℒ^−𝑘+1𝑔−𝑎_𝑖^𝑘−1ℒ^−𝑘+1𝑔−𝑝−𝑎_𝑖⁻¹ℒ^−𝑘+2𝑔=𝑎^𝑘_𝑖ℒ^−𝑘𝑔−𝑝ℒ𝑔+𝑞ℒ²𝑔,(3.20) and next, by (2.6), 𝒜^𝑖_𝑘−𝒜^𝑖_𝑘−1≤𝑎_𝑖^𝑘𝐿₂^−𝑘𝑔−𝑝ℒ𝑔+𝑞ℒ²𝑔≤𝑎_𝑖^𝑘𝐿₂^−𝑘𝜀.(3.21) This yields 𝒜^𝑖_𝑘−𝒜^𝑖_𝑘+𝑛≤^𝑘+𝑛_𝑗=𝑘+1𝑎_𝑖^𝑗𝐿₂^−𝑗𝜀,𝑘,𝑛∈ℕ₀,𝑛>0.(3.22) 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 𝐹_𝑖∈ℒ(𝒞)(3.23) (because 𝒜^𝑖_𝑛∈𝒞⊂ℒ(𝒞) for 𝑛∈ℕ and ℒ(𝒞) is c.u.c.), and again by (3.22), with 𝑘=0 and 𝑛→∞, 𝑔−𝑝−𝑎_𝑖⁻¹ℒ𝑔−𝐹_𝑖≤𝜀𝐿₂⁻¹𝑎_𝑖1−𝐿₂⁻¹𝑎_𝑖=𝑎_𝑖𝜀𝐿₂−𝑎_𝑖.(3.24)

It is easy to observe that ℒ^−𝑛𝒜^𝑖_𝑘=𝑎_𝑖^−𝑛𝒜^𝑖_𝑘+𝑛,𝑘,𝑛∈ℕ₀.(3.25) Further, by (2.6), for each 𝑘∈ℕ, ℒ^−𝑘𝐹_𝑖=lim_𝑛→∞ℒ^−𝑘𝒜^𝑖_𝑛.(3.26) So, by (3.18), (3.23), and (3.25), we have 𝑝ℒ⁻¹𝐹_𝑖−𝑞𝐹_𝑖=𝑝lim_𝑛→∞ℒ⁻¹𝒜^𝑖_𝑛−𝑞lim_𝑛→∞𝒜^𝑖_𝑛=𝑝𝑎_𝑖⁻¹lim_𝑛→∞𝒜^𝑖_𝑛+1−𝑞lim_𝑛→∞𝒜^𝑖_𝑛=𝑝𝑎_𝑖⁻¹𝐹_𝑖−𝑞𝐹_𝑖=𝑎_𝑖⁻²𝐹_𝑖=lim_𝑛→∞𝑎_𝑖⁻²𝒜^𝑖_𝑛=lim_𝑛→∞ℒ⁻²𝒜^𝑖_𝑛−2=ℒ⁻²𝐹_𝑖.(3.27) This and (3.23) yield 𝑞𝐹_𝑖=𝑝ℒ⁻¹𝐹_𝑖−ℒ⁻²𝐹_𝑖∈𝒞, that is, 𝐹_𝑖∈𝒞. Repeating yet that reasoning twice, we get 𝐹_𝑖∈ℒ⁻²(𝒞)∩ℒ⁻¹(𝒞)(3.28) (i.e., (3.12) holds) and consequently 𝑝ℒ𝐹_𝑖−𝑞ℒ²𝐹_𝑖=ℒ²𝑝ℒ⁻¹𝐹_𝑖−𝑞𝐹_𝑖=ℒ²ℒ⁻²𝐹_𝑖=𝐹_𝑖.(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 𝑎𝐹∶=²𝑎₂−𝑎₁𝐹₁−𝑎₁𝑎₂−𝑎₁𝐹₂.(3.30) Then, by (3.12), 𝐹,ℒ𝐹,ℒ²𝐹∈𝒞,(3.31) and it follows from (3.16) and (3.29), respectively, that 𝑝ℒ𝐹−𝑞ℒ²𝑎𝐹=²𝑎₂−𝑎₁𝑝ℒ𝐹₁−𝑞ℒ²𝐹₁−𝑎₁𝑎₂−𝑎₁𝑝ℒ𝐹₂−𝑞ℒ²𝐹₂=𝐹.(3.32) Moreover, 𝑎₂𝑝−𝑎₁⁻¹−𝑎₁𝑝−𝑎₂⁻¹=0,(3.33) and consequently =1𝑔−𝐹𝑎₂−𝑎₁𝑎₂−𝑎₁𝑔−𝑎₂𝐹₁+𝑎₁𝐹₂=1𝑎₂−𝑎₁𝑎₂−𝑎₁𝑔−𝑎₂𝐹₁+𝑎₁𝐹₂−𝑎₂𝑝−𝑎₁⁻¹−𝑎₁𝑝−𝑎₂⁻¹≤𝑎ℒ𝑔₂𝑎₂−𝑎₁𝑔−𝐹₁−𝑝−𝑎₁⁻¹+𝑎ℒ𝑔₁𝑎₂−𝑎₁𝑔−𝐹₂−𝑝−𝑎₂⁻¹ℒ𝑔,(3.34)whence, by (3.17) and (3.24), respectively, we obtain (2.8).

For the proof of the statement concerning uniqueness of 𝐹, take 𝐹₀∈𝒞 with ℒ𝐹₀,ℒ²𝐹₀∈𝒞. Suppose that 𝐹₀ is a solution of (1.4) such that ‖𝑔−𝐹₀‖<∞. Then we have 𝐹−𝐹₀≤𝐹−𝑔+𝑔−𝐹₀<∞,(3.35) and therefore, by Lemma 3.1, 𝐹=𝐹₀. This completes the proof of Theorem 2.1.

4. 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: (𝐶2)𝑎𝒞⊂𝒞 for 𝑎∈{ℜ(𝑎₁),ℜ(𝑎₂),ℑ(𝑎₁),ℑ(𝑎₂),|𝑎₁−𝑎₂|⁻²,|𝑎₁|⁻²,|𝑎₂|⁻²}, (𝐿2)ℒ(𝑎𝑓)=𝑎ℒ𝑓 for 𝑎∈{ℜ(𝑎₁),ℜ(𝑎₂),ℑ(𝑎₁),ℑ(𝑎₂)}, 𝑓∈𝒞,

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 𝑎₁,𝑎₂∈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 𝐿₁). The cases of (𝛽) and (𝛾) are more complicated, and some results concerning them will be published separately.

Theorem 4.1. Let 𝕂=ℝ, 𝜀>0, 𝐿>0, and 𝑔∈𝒞, with ℒ𝑔,ℒ²𝑔∈𝒞, satisfy (2.4). Suppose that 2𝐿<|𝑎_𝑗| for 𝑗∈{1,2}, ℒ(𝒞)⊂𝒞, and ℒ𝑓−ℒℎ≤𝐿𝑓−ℎ,𝑓,ℎ∈𝒞.(4.1) Then there exists a unique function 𝐹∈𝒞, with ℒ𝐹,ℒ²𝐹∈𝒞, that satisfies (1.4) and inequality (2.7); moreover, ≤𝜀𝑔−𝐹𝑞𝑎₂−𝑎₁12𝐿−𝑎₁+12𝐿−𝑎₂.(4.2)

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]) 𝑋² is a complex Banach space with the linear structure and the Taylor norm ‖⋅‖_𝑇 given by 𝑥,𝑦+(𝑧,𝑤)∶=𝑥+𝑧,𝑦+𝑤for𝑥,𝑦,𝑧,𝑤∈𝑋,𝛼+𝑖𝛽𝑥,𝑦∶=𝛼𝑥−𝛽𝑦,𝛽𝑥+𝛼𝑦for𝑥,𝑦∈𝑋,𝛼,𝛽∈ℝ,𝑥,𝑦_𝑇∶=sup_{0≤𝜃≤2𝜋}(cos𝜃)𝑥+(sin𝜃)𝑦for𝑥,𝑦∈𝑋.(4.3)Note that max‖𝑥‖,𝑦≤𝑥,𝑦_𝑇≤‖𝑥‖+𝑦,𝑥,𝑦∈𝑋.(4.4)
Analogously as before we write𝜇_𝑇∶=sup_𝑥∈𝑆𝜇(𝑥)_𝑇(4.5) for each function 𝜇∶𝑆→𝑋². Next, 𝑝_𝑖𝑤₁,𝑤₂∶=𝑤_𝑖,𝑖∈{1,2};𝑤₁,𝑤₂∈𝑋.(4.6)
Let𝒞₀∶=𝜇∶𝑆⟶𝑋²∶𝑝_𝑖∘𝜇∈𝒞,𝑖∈{1,2},(4.7)𝜒∶𝑆→𝑋² be given by 𝜒(𝑥)∶=(𝑔(𝑥),0) for 𝑥∈𝑆, and ℒ₀𝜇(𝑥)∶=ℒ𝑝₁∘𝜇(𝑥),ℒ𝑝₂∘𝜇(𝑥)(4.8) for every 𝜇∈𝒞₀ and 𝑥∈𝑆. Since ℒ(𝒞)⊂𝒞 and 𝒞 is a subgroup of the group (𝑋^𝑆,+) (i.e., the function 𝜇₀∶𝑆→𝑋 defined by 𝜇₀(𝑥)=0 for 𝑥∈𝑆, is in 𝒞), it is easily seen that 𝜒,ℒ₀𝜒,ℒ²₀𝜒∈𝒞₀ and ℒ₀𝒞₀⊂𝒞₀.(4.9)
Next, for each 𝑓∈𝒞₀, we have 𝑓₁∶=𝑝₁∘𝑓,𝑓₂∶=𝑝₂∘𝑓∈𝒞, whence, in view of (𝐶2) and (𝐿2), for each 𝑗∈{1,2},ℒℜ𝑎_𝑗𝑓₁−ℑ𝑎_𝑗𝑓₂=ℜ𝑎_𝑗ℒ𝑓₁−ℑ𝑎_𝑗ℒ𝑓₂,ℒℑ𝑎_𝑗𝑓₁+ℜ𝑎_𝑗𝑓₂=ℑ𝑎_𝑗ℒ𝑓₁+ℜ𝑎_𝑗ℒ𝑓₂,(4.10) and consequently ℒ₀𝑎_𝑗𝑓=ℒ₀ℜ𝑎_𝑗𝑓₁−ℑ𝑎_𝑗𝑓₂,ℜ𝑎_𝑗𝑓₂+ℑ𝑎_𝑗𝑓₁=ℒℜ𝑎_𝑗𝑓₁−ℑ𝑎_𝑗𝑓₂,ℒℜ𝑎_𝑗𝑓₂+ℑ𝑎_𝑗𝑓₁=ℜ𝑎_𝑗ℒ𝑓₁−ℑ𝑎_𝑗ℒ𝑓₂,ℑ𝑎_𝑗ℒ𝑓₁+ℜ𝑎_𝑗ℒ𝑓₂=𝑎_𝑗ℒ𝑓₁,ℒ𝑓₂=𝑎_𝑗ℒ₀𝑓.(4.11) Thus, we have obtained that (𝐿1^′)ℒ₀(𝑎_𝑗𝑓)=𝑎_𝑗ℒ₀𝑓 for 𝑓∈𝒞₀ and 𝑗∈{1,2}. Analogously, for every 𝜇∈𝒞₀, 𝑖∈{1,2}, we get 𝑎_𝑖𝜇=ℜ𝑎_𝑖𝑝₁∘𝜇−ℑ𝑎_𝑖𝑝₂∘𝜇,ℜ𝑎_𝑖𝑝₂∘𝜇+ℑ𝑎_𝑖𝑝₁,1∘𝜇𝑎_𝑖𝜇=𝑎_𝑖𝑎_𝑖²ℜ𝑎𝜇=^𝑖𝑎_𝑖²𝑝₁ℑ𝑎∘𝜇+^𝑖𝑎_𝑖²𝑝₂ℜ𝑎∘𝜇,^𝑖𝑎_𝑖²𝑝₂ℑ𝑎∘𝜇−^𝑖𝑎_𝑖²𝑝₁∘𝜇(4.12) which means that 𝑎_𝑖𝜇,𝑎_𝑖⁻¹𝜇∈𝒞₀ (because 𝒞₀ is a group and (𝐶2) holds). Moreover, 1𝑎₁−𝑎₂𝒞₀=𝑎₁−𝑎₂𝑎₁−𝑎₂²𝒞₀⊂𝒞₀.(4.13) Thus we have proved that (𝐶1^′)𝑎_𝑗𝒞₀=𝒞₀ and 𝒞₀⊂(𝑎₁−𝑎₂)𝒞₀ for 𝑗∈{1,2}.
Now, we show that 𝒞₀ is c.u.c. with regard to the Taylor norm. To this end, take 𝜇∈𝑋^𝑆 and 𝜇_𝑛∈𝒞₀ for 𝑛∈ℕ such that 𝜇_𝑛⇒𝜇 (with respect to the Taylor norm). Then, by (4.4),max𝑝₁∘𝜇_𝑛−𝑝₁∘𝜇,𝑝₂∘𝜇_𝑛−𝑝₂∘𝜇≤𝜇_𝑛−𝜇_𝑇,𝑛∈ℕ,(4.14) which means that 𝑝₁∘𝜇_𝑛⇒𝑝₁∘𝜇 and 𝑝₂∘𝜇_𝑛⇒𝑝₂∘𝜇. Consequently, 𝑝₁∘𝜇,𝑝₂∘𝜇∈𝒞. Hence, 𝜇∈𝒞₀.
Note yet that, according to (4.1) and (4.4), for every 𝜇∈𝒞₀, we have‖ℒ₀𝜇‖_𝑇=ℒ𝑝₁∘𝜇,ℒ𝑝₂∘𝜇_𝑇≤ℒ𝑝₁∘𝜇+ℒ𝑝₂𝑝∘𝜇≤𝐿₁∘𝜇+𝐿𝑝₂∘𝜇≤2𝐿max𝑝₁∘𝜇,𝑝₂𝜇∘𝜇≤2𝐿_𝑇,𝜒−𝑝ℒ₀𝜒+𝑞ℒ²₀𝜒_𝑇=𝑔−𝑝ℒ𝑔+𝑞ℒ²𝑔,0_𝑇≤𝑔−𝑝ℒ𝑔+𝑞ℒ²𝑔≤𝜀,(4.15) because 𝑝₂∘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, 𝐿₁, and 𝒞 replaced by 𝜒, ℒ₀, 2𝐿, and 𝒞₀, resp.) and consequently there is a solution 𝐻∈𝒞₀ of the equation𝐻=𝑝ℒ₀𝐻−𝑞ℒ²₀𝐻(4.16) such that 𝜒−𝐻_𝑇≤𝜀𝑞𝑎₁−𝑎₂1𝑎₁+1−2𝐿𝑎₂.−2𝐿(4.17) Observe that 𝐹∶=𝑝₁∘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 𝐹₀∈𝒞, with ℒ𝐹₀,ℒ²𝐹₀∈𝒞, be a solution of (1.4) such that ‖𝑔−𝐹₀‖<∞. Write 𝐻₀(𝑥)∶=(𝐹₀(𝑥),𝑝₂(𝐻(𝑥))) for 𝑥∈𝑆. It is easily seen that 𝐻₀∈𝒞₀ and 𝐻₀=𝑝ℒ₀𝐻₀−𝑞ℒ²₀𝐻₀. Moreover,𝐻−𝐻_0𝑇=𝐹−𝐹₀≤𝐹−𝑔+𝑔−𝐹₀<∞.(4.18) Hence, by Lemma 3.1 (with ℒ and 𝒞 replaced by ℒ₀ and 𝒞₀, resp.), 𝐻=𝐻₀, which yields 𝐹₀=𝑝₁∘𝐻₀=𝑝₁∘𝐻=𝐹.

5. 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.

Theorem 5.1. Let 𝒯∶=𝑝ℒ−𝑞ℒ², 𝜀>0, 𝐿₁>0, 𝐿₂>0, and 𝑔∈𝒞, with ℒ𝑔,ℒ²𝑔∈𝒞, satisfying the inequality 𝑔−𝒯𝑔≤𝜀.(5.1) 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 2𝐿₁<|𝑎_𝑗| for 𝑗∈{1,2}. Then there exists a unique 𝐹∈𝒞 with ℒ𝐹,ℒ²𝐹∈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=𝐿₁.

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−𝑞ℒ²−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 𝐿₁>0, 𝐿₂>0. Then, for every 𝑔∈𝒞 with ℒ𝑔,ℒ²𝑔∈𝒞 and 𝜀∶=𝒦𝑔<∞,(5.3) there exists a unique 𝐹∈𝒞 with ℒ𝐹,ℒ²𝐹∈𝒞 and such that 𝐹∈ker𝒦 (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=𝐿₁.

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 𝐿₁≥|𝑎_𝑖| for some 𝑖∈{1,2} when (𝛼) is valid (and analogous situations for (𝛽) and (𝛾)). In general, the assumption 𝐿₁≤|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+𝑞ℒ²𝑔+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+𝑞ℒ²𝑔+𝑟ℒ³𝑔.(5.5) For related results, in some particular situations and obtained with different methods, we refer to [19, 28].

Acknowledgment

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).