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

Authors: Janusz Brzdȩk; Soon-Mo Jung
Journal: Abstract and Applied Analysis
 (2011)

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## 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.


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## Body

## 1. Introduction

Letℂ, 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
(1.1)qx2-px+1=0.
Clearly we have a1≠a2,
(1.2)p=1a1+1a2,q=1a1a2.In what follows,XS denotes the family of all functions mapping S into X and XS is a linear space over 𝕂 with the operations given by
(1.3)(f+h)(x)∶=f(x)+h(x),(αf)(x)∶=αf(x)
for all f,h∈XS, α∈𝕂, x∈S. Throughout this paper, we assume that  (H)
𝒞 is a nontrivial subgroup of the group (XS,+) and ℒ:𝒞→XS is an additive operator (i.e., ℒ(f+h)=ℒf+ℒh for f,h∈XS).We investigate the Hyers-Ulam stability of the operator equation(1.4)F=pLF-qL2F
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,
(1.5)sup⁡x∈S‖pLf(x)-qL2f(x)-f(x)‖<∞,there exists a unique solution of the equation that is “near” tof. 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]).Ifℒf=f∘ξ for f∈𝒞 (with 𝒞=XS and a fixed mapping ξ:S→S), then (1.4) takes the form(1.6)f(x)=pf(ξ(x))-qf(ξ2(x)),
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
(1.7)f(x)=pf(x+1)-qf(x+2).
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(2.1)a1,a2∈K.
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(2.2)‖f‖:=sup⁡x∈S‖f(x)‖,f∈XS.
Next, for a given g∈XS and {gn}n∈ℕ⊂XS, the equality
(2.3)g=lim⁡n→∞gn
means that g(x)=limn→∞gn(x) for every x∈S.We say thatℰ⊂XS is closed with respect to the uniform convergence (abbreviated in the sequel to c.u.c.) provided the following holds true:  (E)
iffn∈ℰ for n∈ℕ, f∈XS and fn⇒f, then f∈ℰ, where the symbol fn⇒f means that the sequence {fn}n∈ℕ tends uniformly to f. Moreover, we use in the sequel the following two hypotheses:(C1)
𝒞⊂(a1-a2)𝒞 and aj𝒞=𝒞 for j∈{1,2};(L1)
ℒ(ajf)=ajℒf for f∈𝒞, j∈{1,2}.Now, we are in a position to formulate the main result of this paper.Theorem 2.1.
Letɛ>0, L1>0, L2>0, and let g∈𝒞 with ℒg,ℒ2g∈𝒞 satisfying the inequality
(2.4)‖g-pLg+qL2g‖≤ɛ.
Suppose that (2.1), (C1), and (L1) are valid and one of the following three collections of hypotheses is fulfilled.  (α)
L1<|aj| for j∈{1,2}, ℒ(𝒞)⊂𝒞, 𝒞 is c.u.c., and
(2.5)‖Lf-Lh‖≤L1‖f-h‖,f,h∈C.(β)
ℒ is injective, L2>|aj| for j∈{1,2}, ℒ(𝒞) is c.u.c., 𝒞⊂ℒ(𝒞), and
(2.6)‖L-1f-L-1h‖≤L2-1‖f-h‖,f,h∈L(C).(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
(2.7)‖g-F‖<∞;
moreover, F is given by (3.30) and
(2.8)‖g-F‖≤ɛ|q||a2-a1|(1|L1′-|a1||+1|L2′-|a2||),
where
(2.9)Li′:={L1if(α)holds,L2if(β)holds,Liif(γ)holds,i∈{1,2}.Remark 2.2.
Clearly, if𝒞 is a linear subspace of XS and ℒ is linear (over 𝕂), then (C1) and (L1) are valid. However, if 𝒞 is “only” additive, 𝒞 is a linear subspace of XS but over ℚ (i.e., actually a divisible subgroup of XS), 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 Theorem2.1, we provide simple and natural examples of linear operators ℒ that satisfy the assumptions of Theorem 2.1 (with suitable a1,a2).(i)
Let𝒞=XS, n∈ℕ, and ℒf=∑i=1nΨi∘f∘ξi, where Ψi:X→X is linear and bounded and ξi:S→S is fixed for each i∈{1,…,n}. Then
(2.10)‖Lf(x)-Lh(x)‖≤∑i=1nλi‖f(ξi(x))-h(ξi(x))‖,f,h∈XS,x∈S,
with λi:=inf{L∈ℝ:∥Ψi(w)∥≤L∥w∥,w∈X}. Hence (2.5) holds with L1:=∑i=1nλi. Next, let 𝒞=XS, Ψ:X→X, and ξ:S→S be bijective, Ψ linear, Ψ-1 bounded, and ℒf=Ψ∘f∘ξ. Then
(2.11)‖L-1f(x)-L-1h(x)‖≤L0‖f(ξ-1(x))-h(ξ-1(x))‖,f,h∈XS,x∈S,
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)
Leta,b∈ℝ, a<b, S=[a,b], 𝒞  the family of all continuous functions mapping the interval [a,b] into ℝ, n∈ℕ, A1,…,An∈ℝ, ξ1,…,ξn:S→S continuous, and ℒf(x)=∑i=1n∫axAif∘ξi(t)dt for f∈𝒞, x∈S. Then it is easily seen that (α) is fulfilled with |ai|>L1:=(b-a)∑j=1n|Aj|, i∈{1,2}.(iii)
Leta,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 |ai|<L2:=1/(b-a), i∈{1,2}.


## 3. Proof of Theorem2.1

The subsequent lemma will be useful in the proof of Theorem2.1.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 ℒi(fj)∈𝒞 for i,j∈{1,2}, be solutions of (1.4) and ∥f1-f2∥<∞. Then f1=f2.Proof.
Lethi:=ℒfi-a2fi for i∈{1,2}. Then, by (1.2) and (1.4),
(3.1)Lhi=L2fi-a2Lfi=a1a2(pLfi-fi)-a2Lfi=a1hi
for i∈{1,2}. Consequently, for each k∈ℕ,
(3.2)‖h1-h2‖=|a1|-k‖Lkh1-Lkh2‖≤|a1|-kL1k‖h1-h2‖
if (α) or (γ) holds, and
(3.3)‖h1-h2‖=|a1|k‖L-kh1-L-kh2‖≤|a1|kL2-k‖h1-h2‖
if (β) holds. This means that h1=h2.
Now, in view of the definition ofhi,
(3.4)Lf1-a2f1-(Lf2-a2f2)=h1-h2=0,
which means that
(3.5)Lf1-Lf2=a2(f1-f2).
So, analogously as before, for each k∈ℕ, in the case of (α), we have
(3.6)‖f1-f2‖=|a2|-k‖Lkf1-Lkf2‖≤|a2|-kL1k‖f1-f2‖,
and, in the case of (β) or (γ),
(3.7)‖f1-f2‖=|a2|k‖L-kf1-L-kf2‖≤|a2|kL2-k‖f1-f2‖.
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 Theorem2.1.To this end, fixi∈{1,2}. Then |ai|>L1 or  |ai|<L2. First consider the situation: L1<|ai|. Clearly this means that (α) or (γ) must be valid, which yields ℒ(𝒞)⊂𝒞. Write (3.8)Aki:=ai-k[Lkg-(p-ai-1)Lk+1g],k∈N0.
Note that, by (C1) and (L1), Aki∈𝒞 for k∈ℕ0. Further, for each k∈ℕ0, from (1.2) we get (3.9)Aki-Ak+1i=ai-k[Lkg-(p-ai-1)Lk+1g]-ai-k-1[Lk+1g-(p-ai-1)Lk+2g]=ai-kLk(g-pLg+qL2g),
whence, according to (2.4) and (2.5),
(3.10)‖Aki-Ak+1i‖≤|ai|-kL1k‖g-pLg+qL2g‖≤|ai|-kL1kɛ,
and consequently
(3.11)‖Aki-Ak+ni‖≤∑j=kk+n-1|ai|-jL1jɛ,n∈N.
This means that, for each x∈S, {𝒜ni(x)}n∈ℕ is a Cauchy sequence, and therefore, there exists the limit Fi(x)=limn→∞𝒜ni(x). Moreover, (3.11) yields 𝒜ni⇒Fi, whence
(3.12)Fi,LFi,L2Fi∈C,
because 𝒞 is c.u.c. and ℒ(𝒞)⊂𝒞.Observe that, for everyn,k∈ℕ0,
(3.13)LnAki=ainAn+ki.
Further, by (2.5), for each n∈ℕ,
(3.14)‖LFi-LAni‖≤L1‖Fi-Ani‖,
which yields
(3.15)LkFi=lim⁡n→∞LkAni,k∈{1,2}.
So, in view of (1.2) and (3.13), we have(3.16)pLFi-qL2Fi=plim⁡n→∞LAni-qlim⁡n→∞L2Ani=pailim⁡n→∞An+1i-qai2lim⁡n→∞An+2i=paiFi-qai2Fi=Fi,
and, by (3.11) with k=0 and n→∞,
(3.17)‖g-(p-ai-1)Lg-Fi‖≤|ai|ɛ|ai|-L1.Now, consider the case when|ai|<L2. Then, according to the assumptions, ℒ is injective, (2.6) holds, and 𝒞⊂ℒ(𝒞), that is,
(3.18)L-k(C)⊂C⊂L(C),k∈N.
Write (3.19)Aki:=aik[L-kg-(p-ai-1)L-k+1g],k∈N0.
Then, for each k∈ℕ, we have 𝒜ki∈𝒞 (because g is such that ℒg∈𝒞),
(3.20)Aki-Ak-1i=aik[L-kg-(p-ai-1)L-k+1g]-aik-1[L-k+1g-(p-ai-1)L-k+2g]=aikL-k(g-pLg+qL2g),
and next, by (2.6),
(3.21)‖Aki-Ak-1i‖≤|ai|kL2-k‖g-pLg+qL2g‖≤|ai|kL2-kɛ.
This yields
(3.22)‖Aki-Ak+ni‖≤∑j=k+1k+n|ai|jL2-jɛ,k,n∈N0,n>0.
So, for each x∈S, {𝒜ni(x)}n∈ℕ is a Cauchy sequence, and consequently there exists the limit Fi(x)=limn→∞𝒜ni(x). Note that, by (3.22), 𝒜ni⇒Fi, whence
(3.23)Fi∈L(C)
(because 𝒜ni∈𝒞⊂ℒ(𝒞) for n∈ℕ and ℒ(𝒞) is c.u.c.), and again by (3.22), with k=0 and n→∞,
(3.24)‖g-(p-ai-1)Lg-Fi‖≤ɛL2-1|ai|1-L2-1|ai|=|ai|ɛL2-|ai|.It is easy to observe that(3.25)L-nAki=ai-nAk+ni,k,n∈N0.
Further, by (2.6), for each k∈ℕ, (3.26)L-kFi=lim⁡n→∞⁡L-kAni.
So, by (3.18), (3.23), and (3.25), we have
(3.27)pL-1Fi-qFi=plim⁡n→∞L-1Ani-qlim⁡n→∞Ani=pai-1lim⁡n→∞An+1i-qlim⁡n→∞Ani=pai-1Fi-qFi=ai-2Fi=lim⁡n→∞ai-2Ani=lim⁡n→∞L-2An-2i=L-2Fi.
This and (3.23) yield qFi=pℒ-1Fi-ℒ-2Fi∈𝒞, that is, Fi∈𝒞. Repeating yet that reasoning twice, we get
(3.28)Fi∈L-2(C)∩L-1(C)
(i.e., (3.12) holds) and consequently
(3.29)pLFi-qL2Fi=L2(pL-1Fi-qFi)=L2(L-2Fi)=Fi.Thus we have proved that, fori∈{1,2}, in either case inequalities (3.17) or (3.24), respectively, hold and Fi is a solution to (1.4), with (3.12) fulfilled. Define F:S→X by
(3.30)F:=a2a2-a1F1-a1a2-a1F2.
Then, by (3.12),
(3.31)F,LF,L2F∈C,
and it follows from (3.16) and (3.29), respectively, that
(3.32)pLF-qL2F=a2a2-a1[pLF1-qL2F1]-a1a2-a1[pLF2-qL2F2]=F.
Moreover, (3.33)a2(p-a1-1)-a1(p-a2-1)=0,
and consequently (3.34)‖g-F‖=1|a2-a1|‖(a2-a1)g-a2F1+a1F2‖=1|a2-a1|‖(a2-a1)g-a2F1+a1F2-[a2(p-a1-1)-a1(p-a2-1)]Lg‖≤|a2||a2-a1|‖g-F1-(p-a1-1)Lg‖+|a1||a2-a1|‖g-F2-(p-a2-1)Lg‖,whence, by (3.17) and (3.24), respectively, we obtain (2.8).For the proof of the statement concerning uniqueness ofF, take F0∈𝒞 with ℒF0,ℒ2F0∈𝒞. Suppose that F0 is a solution of (1.4) such that ∥g-F0∥<∞. Then we have (3.35)‖F-F0‖≤‖F-g‖+‖g-F0‖<∞,
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 Theorem2.1, 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,whereℜ(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 XS, then (C2) and (L2) are fulfilled.The next theorem complements Theorem2.1 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.
Let𝕂=ℝ, ɛ>0, L>0, and g∈𝒞, with ℒg,ℒ2g∈𝒞, satisfy (2.4). Suppose that 2L<|aj| for j∈{1,2}, ℒ(𝒞)⊂𝒞, and
(4.1)‖Lf-Lh‖≤L‖f-h‖,f,h∈C.
Then there exists a unique function F∈𝒞, with ℒF,ℒ2F∈𝒞, that satisfies (1.4) and inequality (2.7); moreover,
(4.2)‖g-F‖≤ε|q||a2-a1|(1|2L-|a1||+1|2L-|a2||).Proof.
We apply a well-known method of complexification of the real Banach spaceX. 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
(4.3)(x,y)+(z,w):=(x+z,y+w)forx,y,z,w∈X,(α+iβ)(x,y):=(αx-βy,βx+αy)forx,y∈X,α,β∈R,‖(x,y)‖T:=sup⁡0≤θ≤2π‖(cos⁡θ)x+(sinθ)y‖forx,y∈X.Note that
(4.4)max⁡{‖x‖,‖y‖}≤‖(x,y)‖T≤‖x‖+‖y‖,x,y∈X.
Analogously as before we write(4.5)‖μ‖T:=sup⁡x∈S‖μ(x)‖T
for each function μ:S→X2. Next,
(4.6)pi(w1,w2):=wi,i∈{1,2};w1,w2∈X.
Let(4.7)C0:={μ:S⟶X2:pi∘μ∈C,i∈{1,2}},χ:S→X2 be given by χ(x):=(g(x),0) for x∈S, and
(4.8)L0μ(x):=(L(p1∘μ)(x),L(p2∘μ)(x))
for every μ∈𝒞0 and x∈S. Since ℒ(𝒞)⊂𝒞 and 𝒞 is a subgroup of the group (XS,+) (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,ℒ02χ∈𝒞0 and
(4.9)L0(C0)⊂C0.
Next, for eachf∈𝒞0, we have f1:=p1∘f,f2:=p2∘f∈𝒞, whence, in view of (C2) and (L2), for each j∈{1,2},(4.10)L(R(aj)f1-I(aj)f2)=R(aj)Lf1-I(aj)Lf2,L(I(aj)f1+R(aj)f2)=I(aj)Lf1+R(aj)Lf2,
and consequently
(4.11)L0(ajf)=L0(R(aj)f1-I(aj)f2,R(aj)f2+I(aj)f1)=(L(R(aj)f1-I(aj)f2),L(R(aj)f2+I(aj)f1))=(R(aj)Lf1-I(aj)Lf2,I(aj)Lf1+R(aj)Lf2)=aj(Lf1,Lf2)=ajL0f.
Thus, we have obtained that  (L1′)
ℒ0(ajf)=ajℒ0f for f∈𝒞0 and j∈{1,2}.Analogously, for every μ∈𝒞0, i∈{1,2}, we get
(4.12)aiμ=(R(ai)p1∘μ-I(ai)p2∘μ,R(ai)p2∘μ+I(ai)p1∘μ),1aiμ=ai¯|ai|2μ=(R(ai)|ai|2p1∘μ+I(ai)|ai|2p2∘μ,R(ai)|ai|2p2∘μ-I(ai)|ai|2p1∘μ)
which means that aiμ,ai-1μ∈𝒞0 (because 𝒞0 is a group and (C2) holds). Moreover,
(4.13)1a1-a2C0=a1¯-a2¯|a1-a2|2C0⊂C0.
Thus we have proved that  (C1′)
aj𝒞0=𝒞0 and 𝒞0⊂(a1-a2)𝒞0 for j∈{1,2}.
Now, we show that𝒞0 is c.u.c. with regard to the Taylor norm. To this end, take μ∈XS and μn∈𝒞0 for n∈ℕ such that μn⇒μ (with respect to the Taylor norm). Then, by (4.4),(4.14)max⁡{‖p1∘μn-p1∘μ‖,‖p2∘μn-p2∘μ‖}≤‖μn-μ‖T,n∈N,
which means that p1∘μn⇒p1∘μ and p2∘μn⇒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(4.15)∥L0μ∥T=‖(L(p1∘μ),L(p2∘μ))‖T≤‖L(p1∘μ)‖+‖L(p2∘μ)‖≤L‖p1∘μ‖+L‖p2∘μ‖≤2Lmax⁡{‖p1∘μ‖,‖p2∘μ‖}≤2L‖μ‖T,‖χ-pL0χ+qL02χ‖T=‖(g-pLg+qL2g,0)‖T≤‖g-pLg+qL2g‖≤ε,
because p2∘χ(x)=0 for each x∈S.
In this way, we have shown that the assumptions of Theorem2.1(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 equation(4.16)H=pL0H-qL02H
such that
(4.17)‖χ-H‖T≤ɛ|q||a1-a2|(1||a1|-2L|+1||a2|-2L|).
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 ofF. 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-qℒ02H0. Moreover,(4.18)‖H-H0‖T=‖F-F0‖≤‖F-g‖+‖g-F0‖<∞.
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

Theorems2.1 and 4.1 can be actually expressed in the terms of fixed points. Namely, they may be reformulated as follows.Theorem 5.1.
Let𝒯:=pℒ-qℒ2, ɛ>0, L1>0, L2>0, and g∈𝒞, with ℒg,ℒ2g∈𝒞, satisfying the inequality
(5.1)‖g-Tg‖≤ɛ.
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<|aj| 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
(5.2)‖g-F‖<∞;
moreover, if (a) is valid, then (2.8) holds and if (b) is valid, then (4.2) holds with L=L1.Ifℒ is linear, then Theorems 2.1 and 4.1 can also be expressed in the following way (cf. [7]).Theorem 5.2.
Let𝒦:=pℒ-qℒ2-ℐ, where ℐ:XS→XS is the identity operator (given by ℐf=f for f∈XS). 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
(5.3)ε:=‖Kg‖<∞,
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 whereL1≥|ai| for some i∈{1,2} when (α) is valid (and analogous situations for (β) and (γ)). In general, the assumption L1≤|ai| 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(5.4)g=pLg+qL2g+H(x)
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
(5.5)g=pLg+qL2g+rL3g.
For related results, in some particular situations and obtained with different methods, we refer to [19, 28].



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