On the Characterization of a Class of Difference Equations

We focus on the behavior of solutions of the difference equation xn b1xn−1 b2xn−2 · · · bnx0 yn, n 1, 2, . . ., where bk is a fixed sequence of complex numbers, and yk is a fixed sequence in a complex Banach space. We give the general solution of this difference equation. To examine the asymptotic behavior of solutions, we compute the spectra of operators which correspond to such type of difference equations. These operators are represented by upper triangular or lower triangular infinite banded Toeplitz matrices.


Introduction
The theory of difference equations is one of the most important representations of real world problems.The situation of an event at a fixed time usually depends on the situations of the event in the history.One of the ways to mathematically model such an event is to find a difference equation that directly or asymptotically describes the dependence of the situation at a time to the situations of the event in the history.
Let X be a complex Banach space.In this work, we are interested in a difference equation of the form 1.1 n 1, 2, . .., where b k and y k are fixed sequences in C and X, respectively, and x 0 y 0 .The difference equation 1.1 is a generalization of the difference equations investigated by Copson 1 , Popa 2 , and Stević 3 .We will give the general solution of the system of equations 1.1 and examine the different types of stability conditions by determining the spectra of related matrix operators.We will also examine the system of equations which correspond to the transpose of these matrices.
Let X and Y be Banach spaces and T : X → Y be a bounded linear operator.By R T , we denote the range of T , that is, R T {y ∈ Y : y Tx; x ∈ X}.By B X , we denote the set of all bounded linear operators on X into itself.If X is any Banach space and T ∈ B X , then the adjoint T * of T is a bounded linear operator on the dual X * of X defined by T * φ x φ Tx for all φ ∈ X * and x ∈ X.Let X / {θ} and T : D T → X be a linear operator with domain D T ⊂ X.With T , we associate the operator where λ is a complex number, and I is the identity operator on D T .If T λ has an inverse, which is linear, we denote it by T −1 λ , that is, and call it the resolvent operator of T .Many properties of T λ and T −1 λ depend on λ, and spectral theory is concerned with those properties.For instance, we will be interested in the set of all λ in the complex plane such that T −1 λ exists.Boundedness of T −1 λ is another property that will be essential.We will also ask for what λ's the domain of T −1 λ is dense in X.For our investigation of T , T λ , and T −1 λ , we need some basic concepts in spectral theory which are given as follows see 4, page 370-371 .
Let X / {θ} be a complex normed space, and let T : D T → X be a linear operator with domain D T ⊂ X.A regular value λ of T is a complex number such that λ is defined on a set which is dense in X.
The resolvent set ρ T of T is the set of all regular values λ of T .Its complement σ T C \ ρ T in the complex plane C is called the spectrum of T .Furthermore, the spectrum σ T is partitioned into three disjoint sets as follows.The point spectrum σ p T is the set such that T −1 λ does not exist.A λ ∈ σ p T is called an eigenvalue of T .The continuous spectrum σ c T is the set such that T −1 λ exists and satisfies R3 but not R2 .The residual spectrum σ r T is the set such that T −1 λ exists but does not satisfy R3 .We will write ∞ , c and c 0 for the spaces of all bounded, convergent, and null sequences, respectively.By p , we denote the space of all p-absolutely summable sequences, where 1 ≤ p < ∞.Let μ and γ be two sequence spaces and A a nk be an infinite matrix of real or complex numbers a nk , where n, k ∈ N.Then, we say that A defines a matrix mapping from μ into γ, and we denote it by writing A : μ → γ, if for every sequence x x k ∈ μ, the sequence Ax { Ax n }, the A-transform of x, is in γ, where By μ, γ , we denote the class of all matrices A such that A : μ → γ.Thus, A ∈ μ, γ if and only if the series on the right side of 1.4 converges for each n ∈ N and every x ∈ μ, and we have Ax { Ax n } n∈N ∈ γ for all x ∈ μ.
Several authors have studied the spectrum and fine spectrum of linear operators defined by matrices over some sequence spaces.Rhoades 5 examined the fine spectra of the weighted mean operators.Reade 6 worked on the spectrum of the Cesàro operator over the sequence space c 0 .Gonzáles 7 studied the fine spectrum of the Cesàro operator over the sequence space p .Yıldırım 8 examined the fine spectra of the Rhally operators over the sequence spaces c 0 and c.Akhmedov and Bas ¸ar 9 have determined the fine spectrum of the difference operator Δ over p .Later, Bilgic ¸et al. 10 worked on the spectrum of the operator B r, s, t , defined by a triple-band lower triangular matrix, over the sequence spaces c 0 and c.Recently, Altun and Karakaya 11 determined the fine spectra of Lacunary operators. Let which is the left shift operator, and let the transpose of L be which is the right shift operator.Let D be the unit disc {z ∈ C : |z| ≤ 1}.Let a a 0 , a 1 , a 2 , . . . .A lower triangular Toeplitz matrix corresponding to a is in the form And an upper triangular Toeplitz matrix corresponding to a is in the form L a R a t .
Proof.Let us do the proof for μ c.The proof for μ c 0 or ∞ is similar.Let • denote the norm of c.Firstly, we have Now, fix n ∈ N, and let a a k be a sequence such that Hence, R a c a 1 .Now, let μ 1 , and let • denote the norm of 1 .We have

1.11
On the other hand, 12 So, we have the same norm R a 1 a 1 .
Remark 1.2.We have B μ μ, μ for the sequence spaces in Lemma 1.1 since these spaces are BK spaces.
We also have an L a version of the last lemma, for which we leave the proof to the reader.
For any sequence a, let us associate the function f a z

Spectra of the Operators
Theorem 2.1.Let a ∈ 1 .Then R a f a R and L a f a L .
Proof.Let us do the proof for R a .The proof for L a can be done similarly.Let a n a − a 0 , a 1 , . . ., a n , 0, 0, . . . .Then

2.1
So by Lemma 1.1, and so R a f a R .
Theorem 2.2.Let μ be one of the sequence spaces c 0 , c, ∞ , or p with 1 ≤ p < ∞.Then L is a bounded linear operator over μ with L μ 1 and σ L, μ D.
Theorem 2.3.Let μ be one of the sequence spaces c 0 , c, ∞ or p with 1 ≤ p < ∞.Then R is a bounded linear operator over μ with R μ 1 and σ R, μ D.
Proof.The boundedness of the operator can be proved as in the proof of Theorem 2.2.Now, we will use the fact that the spectrum of a bounded operator over a Banach space is equal to the spectrum of the adjoint operator.The adjoint operator is the transpose of the matrix for c 0 and p with 1 ≤ p < ∞.Hence, It is known by Cartlidge 12 that if a matrix operator A is bounded on c, then σ A, c σ A, ∞ .Then we also have

2.10
Theorem 2.4.Let a be a sequence such that f a is holomorphic in a region containing D. Then σ R a , μ σ L a , μ f a D .
Proof.By the spectral mapping theorem for holomorphic functions see, e.g., 13, page 569 we have

2.11
Theorem 2.5.Let μ be a sequence space.If R a is not a multiple of the identity mapping, then σ p R a , μ ∅.

2.12
Proof.Suppose λ is an eigenvalue and x x 0 , x 1 , . . .∈ μ is an eigenvector which corresponds to λ.Then R a x λx and so we have the following linear system of equations.

2.13
Let x k be the first nonzero entry of x.Then the system of equations reduces to

2.14
From the first equation we get λ a 0 , and using the other equations in the given order we get a k 0 for k > 0. This means there is no solution if there exists any k > 0 with a k / 0.

Applications to the System of Equations
Let us consider the evolutionary difference equation Here a n is a sequence of complex numbers.The condition a 0 / 0 is needed to make the system of equations 3.1 solvable.By solvability of a system of equations we mean that for any given sequence y n of complex numbers there exists a unique sequence x n of complex numbers satisfying the equations.Equation 3.1 may be written in the form for n 1, 2, . .., and x 0 u 0 , by the change of variables b k −a k /a 0 for k ∈ {1, 2, . ..} and where u n is a function of n variables with u n u n x 0 , x 1 , . . ., x n−1 y n /a 0 .Let 0 0 1 for the following theorem.
Theorem 3.1.The solution of the difference equation 3.2 is

, where the summation is over nonnegative integers satisfying
k 2, 3, . . . .We can see that the left side of 3.5 can be written in the form where the summation is over nonnegative integers satisfying m 1 2m    Then for the coefficient a m 1 , m 2 , . . ., m k , we have two cases.

3.8
So, by Cases 1 and 2, we have 3.9 Now, we will prove the theorem by induction over n. x 1 u 1 b 1 u 0 u 1 c 1 u 0 and so 3.3 is true for n 1. Suppose 3.3 is true for n ≤ s for a positive integer s.We have

3.10
Hence, 3.3 is true for n s 1.
A special case of 3.1 is the one where a n consists of finitely many nonzero terms.So there exists a fixed k ∈ N such that the equations turn into the form We give the following theorem, which is a direct consequence of Lemma 1.1, to compare the results of it with the results of the next theorem.Theorem 3.2.Let a n be a sequence of complex numbers such that the system of difference equations 3.1 hold for the complex number sequences x x n and y y n .Then the following are equivalent: Theorem 3.3.Suppose f a is a nonconstant holomorphic function on a region containing D. Let a n be a sequence of complex numbers such that the system of difference equations 3.1 hold for the complex number sequences x x n and y y n .Then the following are equivalent: i boundedness of y n always implies boundedness of x n , ii convergence of y n always implies convergence of x n , iii y n → 0 always implies v f a z has no zero in the unit disc D.
Proof.Let us prove only i ⇔ v .We will omit the proofs of ii ⇔ v , iii ⇔ v , iv ⇔ v since they are similarly proved.Since f a is a holomorphic function on a region containing D, we have a ∈ 1 which means R a is bounded by Lemma 1.1.Suppose boundedness of y n implies boundedness of x n .Then the operator R a ∈ ∞ , ∞ is onto.We have R a x y and since f a is not constant R a ∈ ∞ , ∞ is one to one by Theorem 2.5.Hence, R a is bijective and by the open mapping theorem R −1 a is continuous.This means that λ 0 is not in the spectrum σ R a , ∞ , so 0 / ∈ f a D .For the inverse implication, suppose f a z has no zero on the unit disc D. So, λ 0 is in the resolvent set ρ R a , ∞ .Hence by Lemma 7.2-3 of 4 R −1 a is defined on the whole space ∞ , which means that the boundedness of y n implies the boundedness of x n .a j x n j , 3.12 n 0, 1, 2, . . . .Here a n is a sequence of complex numbers.A special case of 3.12 is the one where a n consists of finitely many nonzero terms.So there exists a fixed k ∈ N such that the equations turn into the form y n k j 0 a j x n j 3.13 n 0, 1, 2, . . . .Now, we again give a theorem, which is a direct consequence of Lemma 1.3, to compare the results of it with the results of the next theorem.Theorem 3.5.Let a n be a sequence of complex numbers such that the system of difference equations 3.12 hold for the complex number sequences x x n and y y n .Then the following are equivalent: Theorem 3.6.Suppose that f a is a holomorphic function on a region containing D. Let a n be a sequence of complex numbers such that the system of difference equations 3.12 hold for the complex number sequences x x n and y y n .Then the following are equivalent: i boundedness of y n always implies a unique bounded solution x n , ii convergence of y n always implies a unique convergent solution x n , iii y n → 0 always implies a unique solution x n with x n → 0, iv |y n | p < ∞ always implies a unique solution x n with |x n | p < ∞, v f a z has no zero on the unit disc D.
Proof.Let us prove only i ⇔ v .We will omit the proofs of ii ⇔ v , iii ⇔ v , iv ⇔ v since they are similarly proved.Suppose boundedness of y n implies a unique bounded solution x n .Then the operator L a ∈ ∞ , ∞ is bijective.Since f a is a holomorphic function on a region containing D, we have a ∈ 1 which means L a is bounded by Lemma 1. 3. By the open mapping theorem L −1 a is continuous.This means that λ 0 is not in the spectrum σ L a , ∞ , so 0 / ∈ f a D .For the inverse implication, suppose that f a z has no zero on the unit disc D. So, λ 0 is in the resolvent set ρ L a , ∞ .Hence by Lemma 7.2-3 of 4 L −1 a is defined on the whole space ∞ , which means that the boundedness of y n implies a bounded unique solution x n .Corollary 3.7.Let a polynomial P z a 0 a 1 z • • • a k z k be given such that the system of difference equations 3.13 hold for the complex number sequences x x n and y y n .Then the following are equivalent: i boundedness of y n always implies a bounded unique solution x n , ii convergence of y n always implies a convergent unique solution x n , iii y n → 0 always implies a unique solution x n with x n → 0, iv |y n | p < ∞ always implies a unique solution x n with |x n | p < ∞, v all zeros of P z are outside the unit disc D.
Remark 3.8.We see that the unit disc also has an important role in examining the different types of stability conditions of difference equations or system of equations related with holomorphic functions.

Corollary 3 . 4 .
Let a polynomial P z a 0 a 1 z • • • a k z k be given such that the system of difference equations 3.11 hold for the complex number sequences xx n and y y n .Then the following are equivalent: i boundedness of y n always implies boundedness of x n , ii convergence of y n always implies convergence of x n , iii y n → 0 always implies x n → 0, iv |y n | p < ∞ always implies |x n | p < ∞, v all zeros of P z are outside the unit disc D. Now consider the system of equations y n ∞ j 0 2 • • • km k k.To show 3.5 , let us fix k ≥ 2 and the numbers m 1 , m 2 , . . ., m k such that m 1 2m 2