Functional Differential and Difference Equations with Applications 2013View this Special Issue
On Linear Difference Equations for Which the Global Periodicity Implies the Existence of an Equilibrium
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).
Let be a set and let be a positive integer. It is said that the transformation is -periodic if where is the identical function on and is the least positive integer with this property. It follows from (1) that is a bijection. If there is a topology on and is continuous, then (1) implies that is a homeomorphism.
The following question was posed by Smith (see ): does any continuous periodic transformation of a Euclidean -space always admit a fixed point? Smith knew that the answer is true if the period of the transformation is a prime number (see ) or a power of a prime number (see ). Moreover, Smith was able to answer the question affirmatively when and for suitably regular transformations, when . But it was shown by Kister (see ) that there exist periodic transformations of a Euclidean space without fixed points. Kister's example is based on the results in the paper .
Special periodic transformations can be derived from difference equations.
Consider the th order difference equation: where, (G) is a positive integer, is a set, and .
We give some basic definitions about the periodicity of (2).
Definition 1. Assume (G).(a)A sequence in is called periodic if there is a positive integer such that is -periodic, which means that for all .(b)We say that (2) is globally periodic if there is a positive integer for which the equation is globally -periodic; that is, every solution of it is -periodic.(c)We say that (2) is globally -periodic with prime period if it is globally -periodic and is the least positive integer with this property.
It is easy to see that (2) is globally -periodic with prime period if and only if the transformation defined by is -periodic.
Even if there is a metric on and 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.
Let stand for either the field of real numbers or the field of complex numbers . Throughout this paper, the term vector space in which the scalar field is not explicitly mentioned will refer to a vector space over or over .
Consider the th order inhomogeneous linear difference equation: where, (A) is a positive integer, is a vector space, is a linear transformation , and is a vector.
The th order homogeneous linear difference equation associated (6) is
Clearly, if that (6) is globally -periodic, the difference of any two solutions of it is also -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 -periodicity of the inhomogeneous equation implies the global -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 -periodic for any .
From this we conclude the following.
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 result is finite dimensional.
Theorem 2. Consider the system of the th order inhomogeneous linear difference equations:
where, (B) is a positive integer, are matrices, and is vector.
If (8) is globally periodic, then it has an equilibrium.
Let be a vector space. and mean the identity and the zero operator on , respectively. If is a linear transformation, we define the kernel and the image of in the usual way:
In the next result first-order equations are investigated.
Theorem 3. Consider the first order inhomogeneous linear difference equation:
where, (C) is a bounded linear operator of the Banach space into itself such that is closed and 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 Theorem 3.
First, we need the following lemma about global periodicity.
Lemma 4. Consider the first order inhomogeneous linear difference equation: where, (D) is a vector space, is a linear transformation, and is a vector.
Let be a positive integer. Equation (11) is globally -periodic if and only if
Remark 5. (a) Condition (12) is equivalent to
The first part of (14) implies that
Since (11) has an equilibrium point exactly if the linear equation has a solution, it follows from the previous establishments that the following two assertions are equivalent. Let be a positive integer.(i)If (11) is globally -periodic, then it has an equilibrium.(ii)If , then
(b) implies that is invertible. If is also invertible, then (16) obviously has a solution (or (17) holds), and therefore the only interesting case is when 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 be a vector space and let be a linear transformation such that for some integer . Either prove that or give an example when is a proper subset of
Henceforth we need some notations (see ).
Definition 6. Let be a Banach space.(a) means its dual space, and let denote the value of the functional at . For a bounded linear operator of into itself, denotes its adjoint operator.(b)Suppose that is a subspace of and is a subspace of . Their annihilators are defined as follows:
In the proof of Theorem 3 the following result will be used, which is related to the Fredholm alternative (see ).
Lemma 7. Let be a Banach space and let be a bounded linear operator of into itself such that is closed. The equation is solvable for given if and only if .
Proof. It is well known (see ) that and is the norm closure of in . Since is closed, which gives the result.
Remark 8. If is finite dimensional, then is closed, since every subspace of is closed. In this case Lemma 7 is exactly the Fredholm alternative.
Proof of Theorem 3. We can obviously suppose that .
Equation (10) has an equilibrium point exactly if the linear equation has a solution. By Lemma 7, it is enough to show that
To prove (24), assume that Recalling Lemma 4, we have gives . Consequently, which means that .
The proof is complete.
By Remark 8, we have the following.
Corollary 9. Consider the first order inhomogeneous linear difference equation: where is a linear operator of the finite dimensional space into itself and is a vector. If (28) is globally periodic, then it has an equilibrium.
We illustrate by an example that the conditions involved in Theorem 3 can be satisfied and not only the finite dimensional case.
Example 10. Let be the Banach space of bounded scalar-valued functions on , with the supremum norm
Define the function by and introduce the following bounded linear operator on :
Then , is not invertible (by Remark 5 (b), this is an interesting case), and is a closed subspace of .
It is easy to see that equation or equivalently, for every is globally -periodic if and only if , and in this case it has the equlibrium point .
The previous example can be extended if the scalars are the complex numbers. Let be an integer, and define the function by where is a primitive th root of unity. Then ; equation is globally -periodic, and it has solutions with prime period .
3. The Proof of Theorem 2
We will use the following notations.
Definition 11. Let be an integer.(a) will mean the -dimensional real vector space of block vectors with entries in .(b)The real vector space of block matrices with entries in will be denoted by ( and can be treated as being identical).(c)The zero matrix and the identity matrix in are denoted by and , respectively.
Let be a given sequence in . Then for any fixed we introduce an -dimensional state vector: defined by .
Another companion matrix is developed in .
Now we prove the first main result.
The authors would like to thank the referees for helpful comments and suggestions which improved the presentation of the paper. This paper is supported by the Hungarian National Foundations for Scientific Research Grant no. K101217.
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