Mezocontinuous Operators and Solutions of Difference Equations

We attempt to unify and extend the theory of asymptotic properties of solutions to difference equations of various types. Usually in difference equations some functions are used which generate transformations of sequences. We replace these functions by abstract operators and investigate some properties of such operators. We are interested in properties of operators which correspond to continuity or boundedness or local boundedness of functions. Next we investigate asymptotic properties of the set of all solutions to 'abstract' and 'functional' difference equations. Our approach is based on using the iterated remainder operator and the asymptotic difference pair. Moreover , we use the regional topology on the space of all real sequences and the 'regional' version of the Schauder fixed point theorem.


Introduction
Let N, R denote the set of positive integers and the set of real numbers, respectively.Moreover, let SQ = R N denote the space of all real sequences x : N → R. In the paper we assume that m ∈ N, F : SQ → SQ and consider difference equations of the form where a n , b n ∈ R. We say that (E) is an abstract difference equation of order m.Let p ∈ N. We say that a sequence x ∈ SQ is a p-solution of equation (E) if equality (E) is satisfied for any n ≥ p.We say that x is a solution if it is a p-solution for certain p ∈ N. If x is a p-solution for any p ∈ N, then we say that x is a full solution.
Email: migda@amu.edu.plAs a special case of (E) we get equations of type: where k ∈ N, σ 1 , . . ., σ k : N → N, or where k is an arbitrary natural number (the case k > m is not excluded).
In the series of papers [15][16][17][18][19][20][21][22][23] a new method in the study of asymptotic properties of solutions to difference equations is presented.This method, based on using the iterated remainder operator, and the regional topology on the space of all real sequences, allows us to control the degree of approximation.In the paper [22], summarizing some earlier results, the notion of a difference asymptotic pair was introduced and the theory of such pairs was used to study the asymptotic properties of solutions to autonomous difference equations of the form In this paper we extend the results from [22] to more general classes of equations.Our approach to the study of asymptotic properties of solutions were inspired by the papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and [24][25][26][27][28][29][30][31][32][33].
The paper is organized as follows.In Section 2, we introduce notation and terminology.In Section 3 we introduce the notion of mezocontinuous operator and we give some examples to show that the mezocontinuity of the operator F in equation (E) corresponds to the continuity of the function f in equations (E1) and (E2).Main results are obtained in Section 4. In Section 5 we apply our results to 'functional' equations (E1) and (E2).

Notation and terminology
If p, k ∈ N, p ≤ k, then N(p), N(p, k) denote the sets defined by We use the symbols to denote the set of all full solutions of (E), the set of all p-solutions of (E), and the set of all solutions of (E) respectively.If x, y in SQ, then xy and |x| denote the sequences defined by We will use the following notations For a subset Y of a metric space X and c > 0 let where B * (y, c) denotes the closed ball of radius c centered at y.For y, ρ ∈ SQ and p ∈ N we define B * (y, ρ, p) = {x ∈ SQ : |x − y| ≤ |ρ| and x n = y n for n < p}.
Assume that Z is a linear subspace of a linear space X.We say that a subset W of X is

Regional topology
Let X be a real vector space.We say that a function • : X → [0, ∞] is a regional norm if the condition x = 0 is equivalent to x = 0 and for any x, y ∈ X and α ∈ R we have Hence, the notion of a regional norm generalizes the notion of a usual norm.Note that a regional norm may take the value ∞.If a regional norm on X is given, then we say that X is a regional normed space, if there exists a vector x ∈ X such that x = ∞, then we say that X is extraordinary.
Assume X is a regional normed space.We say that a subset Z of X is ordinary if x − y < ∞ for any x, y ∈ Z.We regard any ordinary subset Z of X as a metric space with metric defined by d(x, y) = x − y .
We say that a subset The family of all regionally open subsets is a topology on X which we call the regional topology.We regard any subset of X as a topological space with topology induced by the regional topology.Let Reg(0) = {x ∈ X : x < ∞}.
Obviously Reg(0) is a linear subspace of X.Moreover, the regional norm induces a usual norm on Reg(0).We say that X is a Banach regional space if Reg(0) is complete.Let x ∈ X.We say that the set Reg(x) = x + Reg(0) is a region of x.If y ∈ X and x − y < ∞, then Reg(x) = Reg(y).Any region is ordinary and open in X.Moreover, any region is connected and is metrically equivalent to the normed space Reg(0).From a topological point of view, the space X is a disjoint union of all regions.Note that if x ∈ X, then the region Reg(x) is the ordinary component of x and the connected component of x.Hence any ordinary subset of X is a subset of a certain region, and any connected subset of X is ordinary.Moreover, if H : SQ → SQ is continuous and x ∈ SQ, then We say that a subset Y of X is regional if Reg(y) ⊂ Y for any y ∈ Y.The basic properties of the regional topology are presented in [21].We will use the following theorem (see [21,Theorem 3.1]).

Theorem 2.1 (Generalized Schauder theorem).
Assume Q is a closed and convex subset of a regional Banach space X, a map H : Q → Q is continuous and the set HQ is ordinary and totally bounded.Then there exists a point x ∈ Q such that Hx = x.

J. Migda
We will use the standard regional norm on SQ defined by Moreover, we will use the following fixed point theorem.

Remainder operator
Then r m is a linear operator which we call the iterated remainder operator of order m.
For more information about the remainder operator see [20].

Asymptotic difference pairs
We say that a pair (A, Z) of linear subspaces of SQ is an asymptotic difference pair of order m or, simply, m- We say that an m-pair (A, Z) is evanescent if Z ⊂ o(1).The following lemma is a consequence of [22, Lemma 3.5 and Lemma 3.7].
are evanescent m-pairs.
For more information about difference pairs see [22].

Mezocontinuous operators
Assume W ⊂ X ⊂ SQ and H : X → SQ.We define H ∈ [0, ∞] by We say, that H is bounded if H < ∞.Let P be a property of operators.We say that H has the property P on W if the restriction H|W has the property P. Recall that we regard any subset of SQ as a topological space with topology induced by the regional topology.Hence, the continuity of H is defined by a standard way.We say, that H is: paracontinuous if for any ε > 0 and any n ∈ N there exists a δ = δ(n, ε) > 0 such that if x, z ∈ X, and mezocontinuous if it is paracontinuous on any bounded subset of X, regionally bounded if it is bounded on X ∩ Reg(x) for any x ∈ X.
We say that a map G : X → Y from a subset X of SQ to a metric space Y is uniformly continuous if it is uniformly continuous on any ordinary subset Z of X.
Remark 3.1.For n ∈ N let ev n denote the evaluation (projection operator) defined by is paracontinuous.

J. Migda
Example 3.3.Assume ϕ n : R → R is a sequence of uniformly continuous functions.Then the operator H : SQ → SQ defined by is mezocontinuous.
Justification.Assume S is a bounded subset of SQ.Choose a positive ε and an index n.Let Since S is bounded, the set S n is a bounded subset of R k .Choose a compact interval I such that S n ⊂ I k .The function is uniformly continuous on I k .Choose δ > 0 such that if α, β ∈ I k , and α − β < δ, then |g(α) − g(β)| < ε.Now, assume x, z ∈ S and x − z < δ.Then Therefore H is paracontinuous on S.
Example 3.5.If k ∈ N and f : N × R k+1 → R is continuous, then the operator Justification.Assume S is a bounded subset of SQ.Choose a positive ε and an index n.Let Since S is bounded, the set S n is a bounded subset of R k+1 .Choose a compact interval I such that S n ⊂ I k+1 .The function x, z ∈ S and x − z < δ.Then Therefore H is paracontinuous on S.
Example 3.6.Assume B is a bounded subset of R, f : R → R, the restriction f |B is continuous but not uniformly continuous, p ∈ N, W = {x ∈ SQ : x(N) ⊂ B}, and Then H is continuous but not mezocontinuous.
Hence H is not paracontinuous.Since W is bounded, H is not mezocontinuous.
Example 3.7.Let f : R → R and H : SQ → SQ is given by H Justification.The assertion (a) is obvious, and (b) is a consequence of Example 3.4.Assume f is not uniformly continuous.Then there exists a positive ε such that for any n ∈ N there exist Choose k ∈ N such that δ < 1/k and define y ∈ SQ by Then x − y < δ and Hx − Hy ≥ ε.Hence H is discontinuous at x.

Solutions of abstract equations
Let W ⊂ SQ, a ∈ A(m), and p ∈ N. We say that W is (F, a, p)-regular if for any y ∈ W there exists a positive constant M such that F is paracontinuous on B = B * (y, Mr m |a|, p) and F|B ≤ M, F-regular if for any y ∈ W there exist a positive constant c and an index q such that F|B * (y, c, q) is paracontinuous and bounded, F-optimal if W is o(1)-invariant and F|W is mezocontinuous and regionally bounded, and F is paracontinuous or continuous on B. Then y ∈ Sol p (E) + Z.
Proof.If x ∈ B, then the sequence Fx is bounded.Hence aFx ∈ O(a) ⊂ A(m).Define If n ≥ p, then using Lemma 2.3 we have Hence HB ⊂ B. Let ε > 0. Assume that F is paracontinuous on B. There exist q ≥ p and α > 0 such that For any n ∈ {p, . . ., q} there exists δ n > 0 such that if x, z ∈ B and x − z < δ n , then Let δ = min(δ p , δ p+1 , . . ., δ q ).If x, z ∈ B and x − z < δ, then using Lemma 2.3, we obtain Hence H is continuous.Now assume that F is continuous on B and x ∈ B. There exists a δ(x, ε) > 0 such that the condition z − x < δ(x, ε) implies |Fx − Fz| < ε.If z ∈ B, z − x < δ(x, ε), and n ≥ p, then, we obtain Let X ⊂ SQ.We say that an operator H : X → SQ is unbounded at a point p ∈ [−∞, ∞] if there exists a sequence x ∈ X and an increasing sequence α : The Hence, by (4.10), lim n→∞ x(α(n)) = p.Moreover, using (4.9), we get Therefore p ∈ U(F).Since y is a subsequence of x, we have p ∈ L(x).Thus By Lemma 2.4 (b) we get x ∈ ∆ −m b + Z.

Solutions of functional equations
For a subset V of R we denote by V the closure of V in the extended line [−∞, ∞].Assume f is bounded on U. Choose y ∈ W and ε ∈ (0, c/2).It is easy to see that the set y(N) \ B * (V, ε) is finite.Hence there exists an index p 1 ≥ p such that y n ∈ B * (V, ε) for any n ≥ p 1 .Choose q ∈ N such that σ i (n) ≥ p 1 for any n ≥ q and any i ∈ {1, . . ., k}.Let σ i (N(1, q)), Y = y(N(1, q 1 )), C = N(1, q) × B * (Y, ε) k .
Then C is compact and f is bounded on C. Define Let x ∈ B * (y, ε, q).Then |x n − y n | ≤ ε for any n.If n ≥ q and i ∈ N(1, k), then σ i (n) ≥ p 1 and Hence there exists u ∈ V such that |u − y σ i (n) | ≤ ε.Then Hence (x σ 1 (n) , . . ., x σ k (n) ) ∈ B * (V, c) k and for n ≥ q we get If n ≤ q and i ∈ N(1, k), then
then an operator H : X → SQ is paracontinuous if and only if for any n the function ev n • H : SQ → R is uniformly continuous.If X ⊂ SQ, H 1 , H 2 , . . ., H k : X → SQ are paracontinuous, and a function ϕ : R k → R is uniformly continuous, then the operator H : X → SQ defined by then Hz − Hx < ε.Hence H is continuous.Choose positive ε and δ.Since f |B is not uniformly continuous, there exist s, t ∈ B such that |s − t| < δ and | f (s) − f (t)| ≥ ε.Define sequences x, z by: following theorem extends [22, Theorem 4.2].
Theorem 4.8.Assume (A, Z) is an m-pair, a ∈ A, and x