Approximative solutions of difference equations

Asymptotic properties of solutions of difference equations of the form ∆xn = an f (n, xσ(n)) + bn are studied. Using the iterated remainder operator and fixed point theorems we obtain sufficient conditions under which for any solution y of the equation ∆my = b and for any real s ≤ 0 there exists a solution x of the above equation such that ∆kx = ∆ky + o(ns−k) for any nonnegative integer k ≤ m. Using a discrete variant of the Bihari lemma and a certain new technique we give also sufficient conditions under which for a given real s ≤ m − 1 all solutions x of the equation satisfy the condition x = y + o(ns) where y is a solution of the equation ∆my = b. Moreover, we give sufficient conditions under which for a given natural k < m all solutions x of the equation satisfy the condition x = y + u for a certain solution y of the equation ∆my = b and a certain sequence u such that ∆ku = o(1).


Introduction
Let N, Z, R denote the set of positive integers, the set of all integers and the set of real numbers, respectively.Let m ∈ N. In this paper we consider the difference equation of the form We assume there is a given function g : [0, ∞) → [0, ∞) and a sequence w of real numbers such that By a solution of (E) we mean a sequence x : N → R satisfying (E) for all large n.If (E) is satisfied for all n ∈ N we say that x is a full solution of (E).

J. Migda
The purpose of this paper is to study the asymptotic behavior of solutions of equation (E).In the study of solutions with prescribed asymptotic behavior some fixed point theorems are often used.Then there appear multiple sums of the form The reason is shown below.Let Z denote the space of all convergent to zero sequences.Then the operator is bijective (it is a consequence of the equality Z ∩ Ker∆ m = 0).Moreover, if x ∈ ∆ m (Z) then the sum (R) is convergent and we may define a map Then r m (the iterated remainder operator) is a linear operator and (−1) m r m is inverse to ∆ m |Z.
The last equality plays a crucial role in the application of fixed point theorems to the study of solutions of difference equations.Hence the operator r m is very important.In Section 3, we establish some basic properties of this operator.It is easy to see that r m is nondecreasing.This allows us to use the Knaster-Tarski fixed point theorem (see Section 6).The continuity of r m is more subtle.The operator r m is discontinuous (see Remark 4.6) but restrictions r m |S to some important sets S are continuous (see Lemma 4.5).This allows us to use the Schauder fixed point theorem (see Section 5).Multiple sums of the form (R) are used in many papers, see for example [15,20,24,25,43].If the series is convergent, then x ∈ ∆ m (Z) and we may rewrite the sum (R) in the more comfortable form of a single sum (see Lemma 4.2).This is used to obtain fundamental properties of the operator r m .The fact that r m (x) = o(1) is often used to obtain results of type 'for a given sequence y there exists a solution x such that x − y = o(1)', see, for example, [10, Theorem 1], [11,Theorem 1], [28,Theorem 1] or [39,Theorem 2.1].If s ∈ (−∞, 0] and Lemma 4.2).Using this fact one can obtain results of type 'for a given sequence y there exists a solution x such that x − y = o(n s )', see theorems in Section 3 of [29] and theorems in Sections 5 and 6 of this paper.Obviously, if s < t ≤ 0 then the condition x − y = o(n s ) is more restrictive than x − y = o(n t ).Hence we obtain an approximative solution y and we may control the 'degree' of approximation.
In the study of asymptotic behavior of solutions of difference equations, asymptotically polynomial solutions play an important role.It is related to the fact that the solutions of the 'simplest' difference equation ∆ m x = 0 are polynomial sequences.Analogously, if the difference ∆ m x is 'sufficiently small', then x is asymptotically polynomial.This effect is used in many papers, see for example Theorems 2 and 3 in [34] or Theorem 5 in [28].The 'method of small difference' has been developed in [29].It is based on [29,Theorem 2.1] In Lemma 3.11 we extend this result and, in Section 7, we use the 'method of small difference' to establish sufficient conditions under which all solutions of (E) are asymptotically polynomial.
Asymptotically polynomial solutions appear in the theory of both differential and difference equations.In particular, in the theory of second order equations, so called asymptotically linear solutions are considered.In the theory of differential equations, asymptotic linearity of solution x, usually means one of the following two conditions In [32] the condition of the form x(t) = at + o(t d ) for certain d ∈ (0, 1) is also considered.
In some papers in addition to (1.3), some properties of derivative x are also considered.For example, in [32] Mustafa and Rogovchenko consider solutions x such that Ehrnström in [13] considers solutions x such that A discrete analog of (1.4) may be written in the form We generalize (1.5) as follows.We say that a sequence x is regularly asymptotically polynomial When k = m we obtain a special case.By Lemma 3.8 the condition x ∈ Pol(m) + ∆ −m o(1) is equivalent to the convergence of the sequence ∆ m x n and to the condition for certain fixed real λ and any p ∈ {0, 1, . . ., m}.Convergence of the sequence ∆ m x n is comparatively easy to verify and condition (1.7) appears in many papers, see for example [9,14,25,30,42] or the proof of Theorem 3.1 in [41].Our 'small difference method' covers both the case of usually asymptotically polynomial sequences (1.2) and the case of regularly J. Migda asymptotically polynomial sequences (1.6).Moreover, we extend the method to the case of 'forced' equations (see Lemma 3.11).
The paper is organized as follows.In Section 2 we introduce notation and terminology.In Sections 3 and 4 we present some preliminary results.Section 3 is devoted to asymptotically polynomial sequences.We establish some fundamental properties of the spaces of asymptotically polynomial sequences and regularly asymptotically polynomial sequences.At the end of Section 3 we obtain Lemma 3.11 which is the base of our 'small difference method'.This method will be used in the proofs of Theorems 7.4 and 7.5.
In Section 4 we establish some properties of the iterated remainder operator.These results will be used in the proofs of Theorems 5.1, 5.2, 6.1 and 6.2.Moreover, using the Schauder's fixed point theorem we obtain a certain fixed point lemma (Lemma 4.7) which will be used in the proofs of Theorems 5.1 and 5.2.Using the Knaster-Tarski fixed point theorem we obtain another fixed point lemma (Lemma 4.9) which will be used in the proofs of Theorems 6.1 and 6.2.
The main results appear in Sections 5, 6 and 7.In Section 5, assuming the function f is continuous, we establish conditions under which for any y ∈ ∆ −m b such that (y • σ)w = O(1) there exists a solution (or full solution) x of (E) such that x = y + o(n s ).In Section 6 we obtain analogous results under the assumption that the function f is monotonic with respect to the second variable.In Section 7 we obtain the conditions under which all solutions of (E) are asymptotically polynomial or 'translated' asymptotically polynomial.then N(p), N(p, k) denote the sets defined by

Notation and terminology
The space of all sequences x : N → R we denote by SQ.For any x ∈ SQ we denote by x the sequence defined by xn = f (n, x σ(n) ). (2.1) The Banach space of all bounded sequences x ∈ SQ with the norm For m ∈ N(0) we define Then Pol(m − 1) is the space of all polynomial sequences of degree less than m.Note that Pol(−1) = Ker∆ 0 = 0 is the zero space.For a subset X of SQ let denote respectively the image and the inverse image of X under the map ∆ m : SQ → SQ.If b ∈ SQ, then ∆ −m {b} we also denote simply by ∆ −m b.Now, we can define spaces of asymptotically polynomial sequences and regularly asymptotically polynomial sequences where s ∈ (−∞, m − 1] and k ∈ N(0, m − 1).Moreover, we will also use the sets of 'translated' asymptotically polynomial sequences and so 1) are affine subsets of the space SQ.Now we define the spaces S(m) of m-times summable sequences and the remainder operator.Let x n is convergent .
For x ∈ S(1), we define the sequence r(x) by the formula Then r(x) ∈ S(0) and we obtain the remainder operator r : S(1) → S(0).

J. Migda
Obviously the operator r is linear.If m ∈ N then, by induction, we define the linear space S(m + 1) and the operator r m+1 : S(m + 1) → S(0) by S(m The value r m (x)(n) we denote also by r m n (x) or simply r m n x.Note that for any x ∈ S(m) and any n ∈ N.
A function h : X → Y of a topological space X to metric space Y is called locally bounded if for any x ∈ X there exists a neighborhood U of x such that h| U is bounded.
Remark 2.2.If X ⊂ R, then every continuous, every monotonic and every bounded function h : X → R is locally bounded.Moreover, if X is closed, then h is locally bounded if and only if it is bounded on every bounded subset of X.
For k ∈ N(1) we use the factorial notation We say that the equation (E) is of continuous type if the function f is continuous (we regard N × R as a metric subspace of the plane R 2 ).If f is monotonic with respect to the second variable we say that (E) is of monotone type.

Asymptotically polynomial sequences
In this section we establish some basic properties of the spaces of asymptotically polynomial sequences.The main result of this section is Lemma 3.11 which will be used in the proofs of Theorems 7.4 and 7.5.
, then choosing a sequence w such that ∆ m−k w = u we obtain and .
Again by the Stolz-Cesàro theorem ) and so on.
Inverse implication is obvious.
is proper for any m ∈ N(1).For example, if a n = (−1) n , then and so a / ∈ ∆ −m o(1).In the next two lemmas we describe elements of the spaces of asymptotically polynomial sequences and elements of the spaces of regularly asymptotically polynomial sequences.
Moreover, the constants c m , . . ., c k and the sequence w are unique.
Proof.If Pol(m, k) denotes the subspace of Pol(m) generated by sequences and we obtain the result.

J. Migda
Lemma 3.5.Assume m ∈ N(0), k ∈ N(0, m) and x ∈ SQ.Then if and only if there exist constants c m , . . ., c k and a sequence w ∈ o(n k ) such that Proof.The result is a consequence of Lemma 3.4 and Lemma 3.1 (b).
In the next lemma we describe the elements of the space Lemma 3.8.Assume z ∈ SQ and m ∈ N(0).The following conditions are equivalent.

(b)
The sequence ∆ m z is convergent.
The next two lemmas are used in the proof of Lemma 3.11.

J. Migda
Lemma 3.10.Assume u is a positive and nondecreasing sequence, m ∈ N(1) and Then there exists a sequence w ∈ o(u −1 ) such that ∆ m w = a.
The following lemma is a base of our 'small difference method' which we use in Section 7.This lemma extends Theorem 2.1 of [29].
Let u n = n −s .By Lemma 3.10, there exists a sequence w = o(n s ) such that and by Lemma 3.10 there exists and by (a) we obtain By Lemma 3.10, there exists a sequence and and by (c) we obtain The proof is complete.

The iterated remainder operator and fixed point lemmas
In the first three lemmas we present some basic properties of the iterated remainder operator.
Next we obtain some fixed point lemmas (Lemma 4.7 and Lemma 4.9).These lemmas will be used in Sections 5 and 6.Proof.The assertion (a) is proved in Lemma 1 of [28] and (b) is proved in Lemma 3 of [28].The assertion (c) follows from Lemma 2 of [28] and from the proof of Lemma 3 in [28], while (d) is proved in Lemma 5 of [28].The assertion (e) follows from Lemma 6 of [28], while (f) is an easy consequence of (e) and (d).The assertion (g) is a consequence of (f) and (e).The assertion (h) is obvious for m = 1.For m > 1 it can be easily proved by induction.The assertion (i) is an easy consequence of (h).
Then x ∈ S(m), r m x = o(n s ) and Proof.Let u n = n −s .Then by Lemma 3.10 there exists a sequence w = o(n s ) such that ∆ m w = x.By Lemma 4.1 (f) we obtain By Lemma 4.1 (e) we obtain r m x = r m ∆ m w = (−1) m w ∈ o(n s ).The assertion (4.2) follows from Lemma 2 of [28].
Hence, by the first part of the proof we obtain r m−k x = o(n s−k ).On the other hand and we obtain Then S ⊂ S(m) and the map r m |S is continuous.
Remark 4.6.The operator r m : S(m) → S(0) is discontinuous for any m ∈ N(1).For example if Hence r m is a linear unbounded operator.Therefore it is discontinuous.
Then G is a finite ε-net for T. Hence T is a complete and totally bounded metric space and so, T is compact.Hence T is a convex and compact subset of the Banach space BS.Let F : T → S be a map given by F(x The proof is complete. The following lemma is a version of the Knaster-Tarski fixed point theorem.This theorem may be found in [1] or in [16] but we use a simpler version.For the convenience of the reader we cite the proof from [3].Lemma 4.8 (Knaster-Tarski).If X is a complete partially ordered set and a map T : X → X is nondecreasing then there exists x 0 ∈ X such that T(x 0 ) = x 0 .
Lemma 4.9 (Knaster-Tarski fixed point lemma).Let y, ρ ∈ SQ and let S denote the set {x ∈ SQ : |x − y| ≤ ρ} with natural order defined by: x ≤ z if x n ≤ z n for any n ∈ N(1).Then every nondecreasing map T : S → S has a fixed point.
Proof.By Lemma 4.8 it follows that it is sufficient to show that the set S is complete; i.e. for every B ⊂ S there exists a sup B ∈ S. Let B ⊂ S. For n ∈ N let B n = {x n : x ∈ B}.Then B n is a subset of the complete partially ordered set By compactness of W, the function f is uniformly continuous on W. Hence, there exists δ > 0 such that if (n, s), (n, (5.4) Hence (n, x σ(n) ) ∈ W. Analogously (n, z σ(n) ) ∈ W and |u n | ≤ α for n ≤ q.By (5.3) and (5.4) we get Thus the map A is continuous and, by Lemma 4.7, there exists a sequence x ∈ S such that Ax = x.Then Then there exists a full solution x of (E) such that Proof.By (2.1) and (G), there exists a constant M such that | x| ≤ M for every x ∈ SQ.Let ρ = r m (|a|M + |b|), Similarly as in the proof of Theorem 5.1 it can be shown that there exists a sequence x ∈ S such that Ax = x.Then x = y + (−1) m r m (a x). Hence This means that x is a full solution of (E).Moreover, by Lemma 4.2, we have Then there exists a solution x of (E) such that Moreover, if g is bounded, then we may assume x is full.
Let k ∈ N(0, m).By Lemma 4.2 we have ∆ k u = o(n s−k ) and by Theorem 5.1 there exists a solution x of (E) such that If g is bounded, then by Theorem 5.2, we can assume x is full.
Hence ∆u n is convergent to zero and nondecreasing for n ≥ p. Therefore ∆u n ≤ 0 for n ≥ p. Hence the sequence u n is convergent to zero and nonincreasing for n ≥ p. Therefore u n ≥ 0 for n ≥ p.Moreover, for n ≥ p. Hence for n ≥ p we obtain Since λt 2 − t ≥ −1/4λ for every t ∈ R, we obtain for n ≥ p.If 2α > p 4 then 4α 2 > p 8 .Hence α 2 /p 3 − p 5 /4 > 0 and we obtain 0 ≥ ∆u p+1 > 0 which is impossible.Hence if 2α > p 4 then the sequence ϕ(n) = αn is not asymptotically equivalent to any sequence (x n ) fulfilling the equation (E1) for every n ≥ p.In particular if α > 1/2 then the sequence ϕ(n) = αn is not asymptotically equivalent to any full solution of (E1).

Approximative solutions of monotone type equations
In this section we obtain results analogous to that obtained in Section 5. We replace the continuity of f by monotonicity of f with respect to second variable.
We will use the following conditions (a) f is nondecreasing with respect to the second variable and (−1) m a n ≥ 0 for all large n, (b) f is nonincreasing with respect to the second variable and (−1) m a n ≤ 0 for all large n.
(C) one of the following conditions is satisfied: (c) f is nondecreasing with respect to the second variable and (−1) m a n ≥ 0 for all n, (d) f is nonincreasing with respect to the second variable and (−1) m a n ≤ 0 for all n.Then there exists a solution x of (E) such that for any k ∈ N(0, m).
Proof.Assume that the condition (a) is fulfilled.The proof in the case (b) is analogous.We define the sets T, S, the index p and the operator A as in the proof of Theorem 5.1.We may assume (−1) m a n ≥ 0 for n ≥ p. Similarly as in the proof of Theorem 5.1 it can be shown that A(S) ⊂ S. Assume x, z ∈ S and x ≤ z.Then

x
= sup{|x n | : n ∈ N} we denote by BS.If x, y in SQ, then xy denotes the sequence defined by pointwise multiplication xy(n) = x n y n .Moreover, |x| denotes the sequence defined by |x|(n) = |x n | for every n.We use the symbols 'big O' and 'small o' in the usual sense but for a ∈ SQ we also regard o(a) and O(a) as subspaces of SQ.More precisely, let o(1) = {x ∈ SQ : x is convergent to zero}, O(1) = {x ∈ SQ : x is bounded} and for a ∈ SQ let o(a) = ao(1) = {ax : x ∈ o(1)}, O(a) = aO(1) = {ax : x ∈ O(1)}.

Lemma 4 . 1 .
Assume x, y ∈ SQ, m ∈ N(1) and p ∈ N(0).Then (a) if |x| ∈ S(m), then x ∈ S(m) and |r m x| ≤ r m |x|, (b) |x| ∈ S(m) if and only e) if x = o(1), then ∆ m x ∈ S(m) and r m ∆ m x = (−1) m x, ( f ) ∆ m (S(0)) = S(m), r m (S(m)) = S(0), (g) ∆ p (S(m)) = S(m + p), r p (S(m + p)) = S(m), (h) if x, y ∈ S(m) and x n ≤ y n for n ≥ p, then r m n x ≤ r m n y for n ≥ p, (i) if x ∈ S(m) and y n = x n for n ≥ p, then y ∈ S(m) and r m n y = r m n x for n ≥ p.

Lemma 4 . 7 (
Schauder's fixed point lemma).Assume y, ρ ∈ SQ, ρ ≥ 0, and lim ρ n = 0.In the set S = {x ∈ SQ : |x − y| ≤ ρ} we define the metric by the formula d(x, z) = sup n∈N |x n − z n |.Then every continuous map H : S → S has a fixed point.Proof.Assume H : S → S is continuous and let T = {x ∈ BS : |x| ≤ ρ}.Obviously T is a convex and closed subset of BS.Choose an ε > 0. Then there exists p ∈ N(1) such that ρ n ≤ ε for any n ≥ p.For n = 1, . . ., p let G n denote a finite ε-net for the interval [−ρ n , ρ n ] and let

Proof. By Lemma 4 . 2 ,
b ∈ S(m) and r m b = o(n s ).Let u = (−1) m r m b and y

(− 1 )
m a n f (n, x σ(n) ) + b n ≤ (−1) m a n f (n, z σ(n) ) + b nfor n ≥ p.Since the operator r m is nondecreasing, A(x) ≤ A(z).By Lemma 4.9, there exists x ∈ S such that A(x) = x.The rest of the proof is analogous to the proof of the Theorem 5.1.Proof.For n ∈ N(p, m) lets n = b +

Lemma 7 . 2 . 1 ∑Lemma 7 . 3 .
the function g is strictly positive on [b, ∞), we obtain s n ≤ c.Hence u n ≤ s n ≤ c.Assume a, u are nonnegative sequences, p N, λ, µ > 0, and b≥ 0. Let g : [0, ∞) → [0, ∞) be nondecreasing, g(b) > 0, = ∞,and u n ≤ b + λ n−k=p a k g(µu k ) for n ≥ p. Then the sequence u is bounded.Proof.For n ≥ p we have µu n ≤ µb + Lemma 7.1, we obtain µu n ≤ c for n ≥ p.If x is a sequence of real numbers, m ∈ N(1) and p ∈ N(m) then there exists a positive constant L = L(x, p, m) such that |x n | ≤ n (m−1) L + n−1 ∑ i=p |∆ m x i | for n ≥ p.
Then∆s n = s n+1 − s n = a n g(u n ) ≤ a n g(s n ),∆s n g(s n ) ≤ a n for n ∈ N(p, m − 1).