# On Linear Difference Equations for Which the Global Periodicity Implies the Existence of an Equilibrium

Authors: István Győri; László Horváth
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

It is proved that any first-order globally periodic linear inhomogeneous
autonomous difference equation defined by a linear operator with closed
range in a Banach space has an equilibrium. This result is extended for
higher order linear inhomogeneous system in a real or complex Euclidean
space. The work was highly motivated by the early works of Smith (1934,
1941) and the papers of  Kister (1961) and Bas (2011).


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## Body

## 1. Introduction

LetX be a set and let p be a positive integer. It is said that the transformation T:X→X is p-periodic if
(1)Tp:=T∘⋯∘T=idX,
where idX is the identical function on X and p is the least positive integer with this property. It follows from (1) that T is a bijection. If there is a topology on X and T is continuous, then (1) implies that T is a homeomorphism.The following question was posed by Smith (see [1]): does any continuous periodic transformation of a Euclidean n-space always admit a fixed point? Smith knew that the answer is true if the period p of the transformation is a prime number (see [2]) or a power of a prime number (see [1]). Moreover, Smith was able to answer the question affirmatively when n≤3 and for suitably regular transformations, when n=4. But it was shown by Kister (see [3]) that there exist periodic transformations of a Euclidean space without fixed points. Kister's example is based on the results in the paper [4].Special periodic transformations can be derived from difference equations.Consider thesth order difference equation:
(2)x(n)=h(x(n-1),…,x(n-s))n≥0,
where, (G)s is a positive integer, X is a set, and h:Xs→X.It is clear that the solutions of (2) are uniquely determined by their initial values:
(3)x(n)=φ(n),-s≤n≤-1,
where φ(n)∈X. The unique solution of (2) and (3) is denoted by x(φ)=(x(φ)(n))n≥-s, where φ:=(φ(-s),…,φ(-1))T∈Xs.We give some basic definitions about the periodicity of (2).Definition 1.
Assume (G).(a)
A sequencev=(v(n))n≥-s in X is called periodic if there is a positive integer p such that v is p-periodic, which means that v(n+p)=v(n) for all n≥-s.(b)
We say that (2) is globally periodic if there is a positive integer p≥s for which the equation is globally p-periodic; that is, every solution of it is p-periodic.(c)
We say that (2) is globally p-periodic with prime period p if it is globally p-periodic and p is the least positive integer with this property.It is easy to see that (2) is globally p-periodic with prime period p if and only if the transformation T:Xs→Xs defined by
(4)T(x-s,…,x-1):=(x-s+1,…,x-2,h(x-s,…,x-1))
is p-periodic.About periodicity of general difference equations, see [5, 6]. Periodicity of linear difference equations is considered in [7].We recall that the solution(x(n))n≥-s of (2) is a steady state solution if x(n)=v(n≥-s), where v∈X is an equilibrium of (2); that is, v obeys
(5)v=h(v,…,v).It is obvious thatv∈X is an equilibrium of (2) exactly if (v,…,v) is a fixed point of the transformation T given in (4).Even if there is a metric onX and h is continuous, it is still an open problem to determine whether (2) has or not an equilibrium point, or equivalently, the transformation (4) has a fixed point, if (2) is globally periodic.In this paper we solve this problem for some linear equations.Let𝕂 stand for either the field of real numbers ℝ or the field of complex numbers ℂ. Throughout this paper, the term vector space in which the scalar field is not explicitly mentioned will refer to a vector space over ℝ or over ℂ.Consider thesth order inhomogeneous linear difference equation:
(6)x(n)=∑i=1sLi(x(n-i))+b,n≥0,
where, (A)s is a positive integer, X is a vector space, Li:X→X is a linear transformation (1≤i≤s), and b∈X is a vector.Thesth order homogeneous linear difference equation associated (6) is
(7)x(n)=∑i=1sLi(x(n-i)),n≥0.Clearly, if that (6) is globally p-periodic, the difference of any two solutions of it is also p-periodic. On the other hand, the general solution of the inhomogeneous equation (6) can be written as the sum of the general solution of the homogeneous equation (7) and an arbitrarily fixed particular solution of the inhomogeneous equation. Thus the global p-periodicity of the inhomogeneous equation implies the global p-periodicity of the related homogeneous equation. One can easily see that the opposite statement is also true if the inhomogeneous equation has a steady state solution which is obviously p-periodic for any p≥1.From this we conclude the following.Conclusion. If (6) has an equilibrium, then (6) and (7) both behave in the same way regarding the global periodicity; that is, they both are globally periodic or both are not globally periodic.The crux in the application of the above self-evident statement is that not all autonomous inhomogeneous linear difference equations have an equilibrium. But this crux is eliminated by the main theorems of this work in two special cases of (6).In the first resultX is finite dimensional.Theorem 2.
Consider the system of thesth order inhomogeneous linear difference equations:
(8)x(n)=∑i=1sAix(n-i)+b,n≥0,
where, (B)  s is a positive integer, Ai∈𝕂d×d(1≤i≤s) are matrices, and b∈𝕂d is vector.
If (8) is globally periodic, then it has an equilibrium.LetX be a vector space. I and O mean the identity and the zero operator on X, respectively. If L:X→X is a linear transformation, we define the kernel and the image of L in the usual way:
(9)ker(L):={x∈X∣L(x)=0},im(L):={L(x)∣x∈X}.In the next result first-order equations are investigated.Theorem 3.
Consider the first order inhomogeneous linear difference equation:(10)x(n)=L(x(n-1))+b,n≥0,
where, (C)  L is a bounded linear operator of the Banach space X into itself such that im(I-L) is closed and b∈X is a vector.
If (10) is globally periodic, then it has an equilibrium.


## 2. Existence of an Equilibrium in an Abstract First-Order Inhomogeneous Linear Equation

In this section we prove Theorem3.First, we need the following lemma about global periodicity.Lemma 4.
Consider the first order inhomogeneous linear difference equation:(11)x(n)=L(x(n-1))+b,n≥0,
where, (D)  X is a vector space, L:X→X is a linear transformation, and b∈X is a vector.Letp be a positive integer. Equation (11) is globally p -periodic if and only if
(12)Lp=I,∑i=0p-1Li(b)=0.Proof.
It is easy to check that (11) is globally p-periodic if and only if
(13)(Lp-I)φ+∑i=0p-1Li(b)=0,
for every φ∈X, but this condition and (12) are equivalent.Remark 5.
(a) Condition (12) is equivalent to
(14)(∑i=0p-1Li)(I-L)=O,b∈ker(∑i=0p-1Li).
The first part of (14) implies that
(15)im(I-L)⊂ker(∑i=0p-1Li).
Since (11) has an equilibrium point exactly if the linear equation
(16)(I-L)x=b
has a solution, it follows from the previous establishments that the following two assertions are equivalent. Let p be a positive integer.(i)
If (11) is globally p-periodic, then it has an equilibrium.(ii)
IfLp=I, then
(17)im(I-L)=ker(∑i=0p-1Li).
(b)Lp=I implies that L is invertible. If I-L is also invertible, then (16) obviously has a solution (or (17) holds), and therefore the only interesting case is when I-L is not invertible.We can see that if (11) is globally periodic, then the problem of the existence or nonexistence of an equilibrium leads to a pure linear algebraic problem.Problem. Let X be a vector space and let L:X→X be a linear transformation such that Lp=I for some integer p≥2. Either prove that
(18)im(I-L)=ker(∑i=0p-1Li),
or give an example when im(I-L) is a proper subset of
(19)ker(∑i=0p-1Li).IfL is a linear operator of the Banach space X into itself such that im(I-L) is closed, then Theorem 3 shows that (18) holds.Henceforth we need some notations (see [8]).Definition 6.
LetX be a Banach space.(a)
X* means its dual space, and let (w,u) denote the value of the functional w∈X* at u∈X. For a bounded linear operator L of X into itself, L*:X*→X* denotes its adjoint operator.(b)
Suppose thatM is a subspace of X and N is a subspace of X*. Their annihilators are defined as follows:
(20)M⊥:={w∈X*∣(w,u)=0,u∈M},N⊥:={u∈X∣(w,u)=0,w∈N}.
In the proof of Theorem3 the following result will be used, which is related to the Fredholm alternative (see [9]).Lemma 7.
LetX be a Banach space and let L be a bounded linear operator of X into itself such that im(I-L) is closed. The equation (I-L)x=b is solvable for given b∈X if and only if b∈⊥(ker(I-L*)).Proof.
It is well known (see [8]) that
(21)(ker(I-L*))⊥=⊥(im(I-L)⊥),
and ⊥(im(I-L)⊥) is the norm closure of im(I-L) in X. Since im(I-L) is closed,
(22)(ker(I-L*))⊥=im(I-L),
which gives the result.Remark 8.
IfX is finite dimensional, then im(I-L) is closed, since every subspace of X is closed. In this case Lemma 7 is exactly the Fredholm alternative.Proof of Theorem3.
We can obviously suppose thatp≥2.
Equation (10) has an equilibrium point exactly if the linear equation
(23)(I-L)x=b
has a solution. By Lemma 7, it is enough to show that
(24)b∈⊥(ker(I-L*)).
To prove (24), assume that
(25)w∈ker(I-L*).
Recalling Lemma 4, we have
(26)(w,b)=(L*(w),-∑i=1p-1Li(b))=-(w,∑i=1p-1Li+1(b))=-(w,b+∑i=2p-1Li(b))=-〈w,b〉-(∑i=2p-1(L*)iw,b).w=L*(w) gives w=(L*)i(w). Consequently,
(27)(w,b)=-(p-1)(w,b),
which means that (w,b)=0.
The proof is complete.By Remark8, we have the following.Corollary 9.
Consider the first order inhomogeneous linear difference equation:(28)x(n)=L(x(n-1))+b,n≥0,
where L is a linear operator of the finite dimensional space X into itself and b∈X is a vector. If (28) is globally periodic, then it has an equilibrium.We illustrate by an example that the conditions involved in Theorem3 can be satisfied and not only the finite dimensional case.Example 10.
LetB([0,1]) be the Banach space of bounded scalar-valued functions on [0,1], with the supremum norm
(29)∥f∥∞:=sup{|f(t)|∣t∈[0,1]}.
Define the functionα∈B([0,1]) by
(30)α(t):={1,iftisrational-1,iftisirrational,
and introduce the following bounded linear operator L on B([0,1]):
(31)L(f):=αf,f∈B([0,1]).
ThenL2=I, I-L is not invertible (by Remark 5 (b), this is an interesting case), and
(32)im(I-L)={g∈B([0,1])∣g(t)=0iftisrational}
is a closed subspace of B([0,1]).
It is easy to see that equation(33)x(n)=L(x(n-1))+b,hhx,b∈B([0,1]),n≥0,
or equivalently, for every t∈[0,1](34)x(n)(t)=α(t)x(n-1)(t)+b(t),hhhhhhhhhx,b∈B([0,1]),n≥0,
is globally 2-periodic if and only if b∈im(I-L), and in this case it has the equlibrium point (1/2)b.The previous example can be extended if the scalars are the complex numbers. Letp≥3 be an integer, and define the function α∈B([0,1]) by
(35)α(t)≔{1,iftisrationalɛp,iftisirrational,
where
(36)ɛp:=e(2π/p)i
is a primitive pth root of unity. Then Lp=I; equation
(37)hx(n)=L(x(n-1)),hx∈B([0,1]),n≥0,
is globally p-periodic, and it has solutions with prime period p.


## 3. The Proof of Theorem2

We will use the following notations.Definition 11.
Letm≥1 be an integer.(a)
BVm,d will mean the md-dimensional real vector space of block vectors with entries in 𝕂d.(b)
The real vector space ofm×m block matrices with entries in 𝕂d×d will be denoted by BMm,d (BMm,d and 𝕂md×md can be treated as being identical).(c)
The zero matrix and the identity matrix in𝕂d×d are denoted by Od and Id, respectively.Let(x(n))n≥-s be a given sequence in 𝕂d. Then for any fixed n≥0 we introduce an sd-dimensional state vector:
(38)xn=(xn(-s),…,xn(-1))T∈BVs,d,
defined by xn(i):=x(n+i)(-s≤i≤-1).As it is well known (see [10]), by using the state vector notation, (8) may be written as an sd-dimensional system of first order difference equations.Lemma 12.
For anyφ=(φ(-s),…,φ(-1))T∈BVs,d, x(φ)=(x(φ)(n))n≥-s is the solution of (8) and (3) exactly if
(39)(xk(φ))k≥1=((xk(φ)(-s),…,xk(φ)(-1))T)k≥1
is the solution of
(40)xk=𝒞xk-1+ℬ,k≥1,x0=φ,
where the companion matrix  𝒞∈BM𝕂s,d  and the block vector ℬ can be written in the forms
(41)𝒞=(OdIdOd…OdOdOdId…Od⋮⋮⋮⋱⋮OdOdOd…IdAsAs-1As-2…A1),(42)ℬ=(00⋮0b).Another companion matrix is developed in [11].There is a one-to-one correspondence between the global periodicity of (8) and that of (40) and also between equilibrium of (8) and that of (40).Lemma 13.
(a)  Letp≥s be an integer. Equation (8) is globally p -periodic if and only if (40) is also globally p-periodic.
(b)c∈𝕂d is an equilibrium of (8) exactly if c=(c,…,c)T∈BVs,d is an equilibrium of (40).Now we prove the first main result.Proof of Theorem2.
We can apply Theorem3 and Lemma 13.



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