Qualitative approximation of solutions to difference equations of various types

In this paper we study the asymptotic behavior of solutions to difference equations of various types. We present sufficient conditions for the existence of solutions with prescribed asymptotic behavior, and establish some results concerning approximations of solutions, extending some of our previous results. Our approach allows us to control the degree of approximation. As a measure of approximation we use o(un) where u is an arbitrary fixed positive nonincreasing sequence.


Introduction
Let N, R denote the set of positive integers and real numbers respectively.The space of all sequences x : N → R we denote by R N .Assume m, k ∈ N, a, b : N → R. In this paper we will examine the asymptotic properties of the solutions of various specific cases of the following equations ∆ m x n = a n F(x)(n) + b n , F : R N → R N , (E) In particular, we will examine the properties of solutions to equations of the form and discrete Volterra equations of the form Email: migda@amu.edu.plBy a solution of (E) we mean a sequence x : N → R satisfying (E) for all large n.Analogously we define a solution of (QE).
In recent years the author presented a new theory of the study of asymptotic properties of the solutions to difference equations.This theory is based mainly on the examination of the behavior of the iterated remainder operator and on the application of asymptotic difference pairs.This approach allows us to control the degree of approximation.The properties of the iterated remainder operator are presented in [15].Asymptotic difference pairs were introduced and used in [17].They were also used in [18] and [21].
In this paper, in Lemma 2.1, we present a new type of asymptotic difference pair.Using Lemma 2.1 and some earlier results, we get a number of theorems about the asymptotic properties of the solutions.Let u be a positive and nonincreasing sequence.Lemma 2.1 allows us to use o(u n ) as a measure of approximation of solutions.Asymptotic pair technique does not work in the case of equations of type (QE).In this case, instead of Lemma 2.1, we use Lemma 2.3.
The paper is organized as follows.In Section 2, we introduce some notation and terminology.Moreover, in Lemma 2.1 and Lemma 2.3 we present the basic tools that will be used in the main part of the paper.In Section 3, we present our main results concerning the existence of solutions with prescribed asymptotic behavior.We essentially use here a fixed point theory which is frequently used in literature, see for example [1-7, 11-31, 35-37].This section is divided into four parts devoted to various types of equations.In Section 4, we establish some results concerning approximations of solutions.

Preliminaries
If x, y : N → R, then xy and |x| denote the sequences defined by xy(n) = x n y n and |x|(n) = |x n | respectively.Moreover We say that a subset B of R N is bounded if there exists a constant M such that a − b ≤ M for any a, b ∈ B. We regard any bounded subset of R N as a metric space with metric d defined by d(a, b) = a − b .Assume Y ⊂ X ⊂ R N and Y is bounded.We say that an operator F : X → R N , is mezocontinuous on Y if for any fixed index n the function ϕ n : Y → R defined by ϕ n (y) = F(y)(n) is uniformly continuous.
Let m ∈ N. We will use the following notations For any a ∈ S(m) we define the sequence r m (a) by Then S(m) is a linear subspace of c 0 , r m (a) ∈ c 0 for any a ∈ S(m) and is a linear operator which we call the remainder operator of order m.If a ∈ A(m), then a ∈ S(m) and for any a ∈ A(m) and any n ∈ N.For more information about the remainder operator see [15].We say that a pair (A, Z) of linear subspaces of R N is an asymptotic difference pair of order m or, simply, m-pair if A ⊂ ∆ m Z, w + z ∈ Z for any eventually zero sequence w and any z ∈ Z, and ba ∈ A for any bounded sequence b and any a ∈ A. We say that an m-pair (A, Z) Then (A, Z) is an evanescent m-pair.
Proof.It is clear that ba ∈ A for any bounded sequence b and any a ∈ A. Obviously w + z ∈ Z for any eventually zero sequence w and any z ∈ Z.Let a ∈ A. Since u is nonincreasing, we have a ∈ A(m).Define sequences w, a + , a − by Then 0 ≤ a + ≤ |a|.Hence a + ∈ A(m) and using (2.2) we get Hence r m A ⊂ Z. Now, using (2.3), we obtain Then a ∈ A(m) and r m (a Proof.The assertion is a consequence of the proof of Lemma 2.1.
Proof.Define sequences z, w by By assumption, z n = o(1).Moreover

Solutions with prescribed asymptotic behavior
Assume b, u ∈ R N and u is positive and nonincreasing.In this section we present sufficient conditions for the existence of solution x with the asymptotic behavior where y is a given solution of the equation ∆ m y n = b n or the equation ∆(r n ∆y n ) = b n .

Abstract equations
and F is bounded and mezocontinuous on U. Then there exists a solution x of the equation Proof.The assertion is a consequence of Lemma 2.1 and [18, Corollary 4.3].

Functional equations
and f is continuous and bounded on N × Y k .Then there exists a solution x of the equation Proof.Define an operator F : R N → R N and a subset U of R N by Then F is bounded on U. By [18, Example 3.4] F is mezocontinuous on U. Using Theorem 3.1 we obtain the result.
and f is continuous and locally equibounded.Then for any bounded solution y of the equation ∆ m y n = b n , there exists a solution x of the equation Proof.Assume y is a bounded solution of the equation ∆ m y n = b n , c > 0, and Then Y k is a bounded subset of R k .For any t ∈ Y k there exist a neighborhood U t of t and a positive constant . Now, using Theorem 3.2 we obtain the result.
and f is continuous and bounded.Then for any solution y of the equation ∆ m y n = b n , there exists a solution x of the equation Proof.The assertion is an immediate consequence of Theorem 3.2.
and f is continuous and bounded on the set Then there exists a solution x of the equation Proof.Define an operator F : R N → R N and a subset U of R N by Assume x ∈ U, n ∈ N, and j ∈ {1, . . ., k}.Then Therefore F is bounded on U. By [18, Example 3.5] F is mezocontinuous on U. Using Theorem 3.1 we obtain the result.
and f is continuous and locally equibounded.Then for any bounded solution y of the equation ∆ m y n = b n , there exists a solution x of the equation Proof.Assume y is a bounded solution of the equation ∆ m y n = b n , c > 0, and As in the proof of Corollary 3.3 one can show that f is bounded on N × Y k .Now, using Theorem 3.5 we obtain the result.
and f is continuous and bounded.Then for any solution y of the equation ∆ m y n = b n , there exists a solution x of the equation Proof.The assertion is an immediate consequence of Theorem 3.5.

Discrete Volterra equations
y is a solution of the equation ∆ m y n = b n , and there exists a uniform neighborhood U of the set y(N) such that the restriction f |N × U is continuous and bounded.Then there exists a solution x of the equation Proof.The assertion is a consequence of Lemma 2.1 and [21, Theorem 3.1].

Quasi-difference equations
Asymptotic pair technique does not work in the case of equations of type (QE).Therefore, in this subsection we will use Lemma 2.3.Moreover, we will need the following two lemmas.
for any n ∈ N. Then any continuous map H : X → X has a fixed point.
Theorem 3.11.Assume a, b, r, u : N → R, r > 0, u > 0, ∆u ≤ 0, y is a solution of the equation and f : R → R is continuous and bounded on U. Then there exists a solution x of the equation Proof.In the proof we use the methods analogous to the methods from previous papers [22] and [23].For n ∈ N and x ∈ R N let Therefore HX ⊂ X.Let x ∈ X, and ε > 0. Using (3.4) and Lemma 3.9 we get Choose an index m ≥ p and a positive constant γ such that Hence the map H : X → X is continuous with respect to the metric defined by (3.1).By Lemma 3.10 there exists a point x ∈ X such that x = Hx.Then for n ≥ p we have

F
Hence, for n ≥ p we get∆(r n ∆x n ) = ∆(r n ∆y n ) + ∆ r n (x)(j) = F(x)(n) + b n = a n f (x σ(n) ) + b n for large n.Since x ∈ X and ρ n = o(u n ), we get x n = y n + o(u n ).