Abstract
We establish some properties of iterations of the remainder operator which assigns to any convergent series the sequence of its remainders. Moreover, we introduce the spaces of multiple absolute summable sequences. We also present some tests for multiple absolute convergence of series. These tests extend the well-known classical tests for absolute convergence of series. For example we generalize the Raabe, Gauss, and Bertrand tests. Next we present some applications of our results to the study of asymptotic properties of solutions of difference equations. We use the spaces of multiple absolute summable sequences as the measure of approximation.
MSC:39A10.
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1 Introduction
Let r denote the operator which assigns to any convergent series the sequence of its remainders. The purpose of this paper is to study the basic properties of iterations of r and apply these results to the study of asymptotic properties of solutions of difference equations. We also study the spaces of multiple absolute summable sequences. Moreover, we obtain extensions of some classical tests for absolute convergence of series.
The paper is organized as follows. In Section 2, we introduce our notation and terminology. In Section 3, we define the iterations of the remainder operator and the spaces of m-times summable sequences. Moreover, we establish some basic properties of . For example, we show that
for any sequence and any sequence z convergent to zero. This means that the operator is inverse to the restriction where Z denotes the space of all convergent to zero sequences.
In Section 4, we introduce the spaces of absolutely m-times summable sequences and establish the relationships between and the spaces . We also extend some classical tests for absolute convergence of series. For example, the Raabe test states that
In Lemma 4.5 we show that
and, more generally,
On the other hand,
Similarly in Lemma 4.4 we show that
and
For , by definition, we have
If , i.e., , then and
We use (3) to obtain basic properties of the operator . At the end of Section 4, using all possible and , we obtain an infinite stratification of the space of all sequences convergent to zero. This stratification induces a substratification of for any m.
In Section 5, we present some applications of our results to the study of asymptotic properties of solutions of difference equations. We use the spaces as the measure of approximation. For example, we apply our results and fixed point theorems to the study of solutions with prescribed asymptotic behavior. More precisely, using the Schauder fixed point theorem and the Knaster-Tarski fixed point theorem, we establish conditions under which for a given sequence b and a solution y of the equation there exists a solution x of the equation
such that
We also show that if and x is a solution of (4) such that the sequence is bounded, then
Hence x is asymptotically polynomial. The equality
for , which is a consequence of (1), plays a crucial role in the application of fixed point theorems to the study of solutions of difference equations. The value is used, mainly implicitly, in the study of solutions with prescribed asymptotic behavior. In some papers the multiple sums (2) appear explicitly; see for example [1–4] or [5]. This paper is a continuation of the papers [4, 6] and [7]. Our studies were inspired by the papers [8–12] and the papers [13–17], and [18].
Some applications of our results to the study of asymptotic properties of solutions of nonautonomous difference equations are presented in [19].
2 Notation and terminology
Let ℕ, ℤ, ℝ denote the set of positive integers, the set of all integers and the set of real numbers, respectively. The space of all sequences we denote by SQ.
If , , then , denote the sets defined by
If x, y in SQ, then xy denotes the sequence defined by pointwise multiplication
Moreover, denotes the sequence defined by for every n.
We use the symbols ‘big O’ and ‘small o’ in the usual sense but for we also regard and as subspaces of SQ. More precisely, let
and for let
For we define
Then is the space of all polynomial sequences of degree less than m. For , we define numbers by
For we define
We say that a subset U of a metric space X is a uniform neighborhood of a subset Y of X if there exists a positive number ε such that
where denotes an open ball of radius ε about y.
A sequence is called nonoscillatory if for large n.
Assume . We say that a sequence is f-bounded if the sequence is bounded. Note that is f-bounded for any f-bounded sequence x and any sequence .
Example 2.1 If , then f-boundedness of a sequence x is equivalent to the boundedness above of x. Assume for , and let denote the set of limit points of a given sequence x. Then f-boundedness of x is equivalent to the condition .
3 Iterated remainder operator
In this section we introduce the remainder operator r, the spaces of m-times summable sequences and iterations . Next in Lemma 3.1 we establish some basic properties of and the relationships between and . Let
For , we define the sequence by the formula
Then and we obtain the remainder operator
For we define, by induction, the linear space and the linear operator by
The value we denote also by or simply .
Lemma 3.1 Assume , , and . Then
-
(01)
if , then and ,
-
(02)
if and only if ,
-
(03)
if and only if ,
-
(04)
if and only if ,
-
(05)
if , then ,
-
(06)
if , then ,
-
(07)
if , then ,
-
(08)
if , then and ,
-
(09)
, ,
-
(10)
, ,
-
(11)
if and for , then
-
(12)
if and for , then and
-
(13)
if and , then ,
-
(14)
if and y is bounded, then and
-
(15)
if , is nondecreasing and positive, then ,
-
(16)
if and , then .
Proof Assertion (01) is proved in Lemma 1 of [6]. Assertion (02) is proved in Lemma 2 of [6]. (03) is proved in Lemma 3 of [6]. Since we see that (04) is a consequence of (03). (05) is proved in Lemma 2 of [6]. Assertion (06) follows from (05). Assertion (07) is proved in Lemma 5 of [6]. Assertion (08) follows from Lemma 6 of [6]. Assertion (09) is an easy consequence of (08). Assertion (10) is a consequence of (09) and (08). Assertion (11) is obvious for . For it can easily be proved by induction. Assertion (12) is an easy consequence of (11). Assertion (13) is well known for . Assume it is true for certain . Let and . Then and, by assumption, . Moreover, using (11) we have
and, by (10), . Hence and we obtain . Assertion (14) follows from (03) and Lemma 4 of [6]. Assume is nondecreasing and positive and . Then the sequence is bounded and using (04) we have . Moreover, using (05), we have
for any n. Hence . Using (01), we obtain
By (03), . Hence, replacing y by and m by in (5), we obtain . Therefore
and we obtain (15). Using (03) and taking in (15) we obtain (16). □
Remark 3.1 For and , by definition of , we have
Moreover, if , then, by Lemma 3.1(05), we have
Remark 3.2 By Lemma 3.1(07) and (08) the restriction is a bijection with inverse .
Remark 3.3 If X is a linear subspace of and Y is a linear subspace of , then, using Lemma 3.1(08) and (07), we have
For any we have . In the following example we show that for any and any there exists a sequence x such that
and, moreover, .
Example 3.1 Let and . Choose . Let and . Then , and
Hence and .
4 Absolute summable sequences
In this section we introduce the spaces of absolutely m-times summable sequences. We establish the relationships between and the spaces and . There exist many tests for the absolute convergence of series. Most of them may be extended to the case . For example we present five of them in Lemmas 4.3-4.7. At the end of the section, using all possible and we obtain an infinite stratification of the set .
For we define the set by
Moreover, let
Remark 4.1 Note that and the condition is equivalent to the absolute convergence of the series . Moreover, is a linear subspace of for any . Note also that if and , then
Lemma 4.1 Assume , and . Then
-
(a)
,
-
(b)
,
-
(c)
.
Proof Assertion (a) is a consequence of Lemma 3.1(03), (b) is a consequence of (a). Assertion (c) follows from (b) and from the fact that the condition is equivalent to the condition . □
Example 4.1 Let . If we define a sequence x by
then, using Lemma 4.1(a), we obtain .
Example 4.2 Assume and . Choose such that . Let
Then
and for . Hence and we obtain . Let . Then, by Lemma 4.1(a),
Remark 4.2 Note that for real s, t the conditions and are equivalent to the condition .
Lemma 4.2 Assume , and . Then
-
(a)
,
-
(b)
,
-
(c)
,
-
(d)
.
Proof Assertion (a) follows from Lemma 4.1, (b) is a consequence of (a) and Example 4.1, (c) is a consequence of (a) and Example 4.2. Let . If , then, by (b), . Hence for any . Therefore
Let . Choose such that . If , then, by the convergence of the series , we have and so . Hence . Therefore for any and we obtain
□
Lemma 4.3 (Comparison test)
Assume and . Then
-
(a)
if for large n and , then ,
-
(b)
if for large n and , then ,
-
(c)
if and , then ,
-
(d)
if and , then ,
-
(e)
if for large n and , then .
Proof Assertion (a) follows from Lemma 3.1(13). Assertions (b), (c), (d), and (e) are consequences of (a). □
Lemma 4.4 (Generalized logarithmic test)
Assume , and
Then
-
(a)
if , then ,
-
(b)
if for large n, then ,
-
(c)
if , then ,
-
(d)
if , then .
Proof If , then there exists a number such that for large n. Then for large n. Hence, using Lemma 4.1 and Lemma 4.3, we obtain (a). If for large n, then for large n and (b) follows from Lemma 4.3 and the fact that . Assertion (c) follows immediately from (b), and (d) is a consequence of (a). □
Lemma 4.5 (Generalized Raabe test)
Assume , and
Then
-
(a)
if , then ,
-
(b)
if for large n, then ,
-
(c)
if , then ,
-
(d)
if , then .
Proof For assertion (a) follows from the usual Raabe test. Assume it is true for certain and . Let
Then
Hence and, by inductive hypothesis, . Hence, by Lemma 4.1(b), and we obtain (a). Similarly we may obtain (b), by taking , using the usual Raabe test and Lemma 4.1(b). Assertion (c) follows from (b), and (d) is a consequence of (a). □
Lemma 4.6 (Generalized Gauss test)
Let , , , and
Then
-
(a)
if , then ,
-
(b)
if , then ,
-
(c)
if and , then ,
-
(d)
if and , then .
Proof Note that
Hence (a) and (b) follow from the d’Alembert ratio test. For assertions (c) and (d) follow from the usual Gauss test. Assume they are true for certain . Let , . Then
for certain . If , then and, by inductive hypothesis, . Similarly, if , then . Now, assertions (c) and (d) follow from Lemma 4.1(b). □
Lemma 4.7 (Generalized Bertrand test)
Assume , and
Then
-
(a)
if , then ,
-
(b)
if for large n, then ,
-
(c)
if , then .
Proof Let . Then
where
Let
Then
and
Moreover,
For assertion (a) follows from the classical Bertrand test. Assume it is true for certain . Then using the inductive hypothesis, (8), and (7), we have . Hence, by Lemma 4.1, and we obtain (a). Analogously, using (9), we obtain (b). Assertion (c) is a consequence of (b). The proof is complete. □
Remark 4.3 Computing from (6) we have
Replacing (6) by (10) in Lemma 4.7 one can obtain another form of the Bertrand test.
Lemma 4.8 Assume . Then
-
(a)
,
-
(b)
,
-
(c)
,
-
(d)
.
Proof For let be defined by
Let . By Lemma 3.1(09) there exists such that . Obviously and, by induction, . Moreover, . Hence
Now assume . Then and . Hence
By Lemma 3.1(13) we have . Hence
From we have . Moreover, . Hence . Therefore and we obtain
The proof is complete. □
Example 4.3 Let , , . By Theorem 2.2 in [7], we have . Hence, by Lemma 4.2, . On the other hand and we obtain . Therefore .
Example 4.4 Let , , . Then, by Lemma 4.1, . Assume . Choose such that . Then, using Lemma 3.1(07), we obtain . Since x is alternating we have . Hence, by Lemma 4.3, . This contradiction shows that .
Remark 4.4 Using Lemma 4.8 we obtain the following infinite diagram, where arrows denote inclusions:
Using Remark 3.2 and Example 4.3 we can see that any vertical arrow represents a proper inclusion. Analogously, using Remark 3.2 and Example 4.4 we can see that any horizontal arrow represents a proper inclusion.
Remark 4.5 Introducing new notation
for and we can extend the diagram from Remark 4.4 in the following way:
Note that if and , then
Moreover,
5 Approximative solutions of difference equations
In this section we present some applications of our previous results in the study of asymptotic properties of solutions of difference equations. We use the spaces to measure the ‘degree of approximation’.
Assume , , , and . We consider the equation
By a solution of (E) we mean a sequence satisfying (E) for all large n. Note that the assumption does not exclude the case of equations with delayed argument. For example we may define for and for .
In Theorem 5.1, using fixed point theorems and the iterated remainder operator, we establish conditions under which there exist solutions of (E) with prescribed asymptotic behavior. In Theorem 5.2 we show that in many cases some assumptions of Theorem 5.1 are necessary. In Theorem 5.3 we establish conditions under which all f-bounded solutions of (E) are asymptotically polynomial.
In the proof of our first theorem we will use the Schauder fixed point theorem and the following version of the Knaster-Tarski fixed point theorem.
Lemma 5.1 If X is a complete partially ordered set and a map is nondecreasing then there exists such that .
A simple proof of this result can be found in [20] (or in [19]).
Theorem 5.1 Assume , , , , f is bounded on some uniform neighborhood U of the set and one of the following conditions is satisfied:
-
(a)
f is nondecreasing on U and for large n,
-
(b)
f is nonincreasing on U and for large n,
-
(c)
f is continuous on U.
Then there exists a solution x of (E) such that
Proof For let . Choose and such that
Let
Choose numbers and k such that for and for . Let
It is easy to see that S, with natural order defined by if for every , is a complete partially ordered set. If and , then . Hence
for any and . If , then, by (11) and (12), the sequence is bounded and so . We define an operator by
If , then, using Lemma 3.1(01) and Lemma 3.1(14), we obtain
Hence .
Assume now that the condition (a) is satisfied. We can assume that for . Let and . Since f is nondecreasing on U we have for . Hence
for . By Lemma 3.1(11), we have for . Moreover, for . Hence . Now, using the Knaster-Tarski fixed point theorem we obtain such that . Analogously if condition (b) is satisfied then for certain .
Assume (c) and let d be a metric on S defined by
Let BS denote the Banach space of all bounded sequences with the norm and let
It is easy to see that T is a convex and closed subset of BS. Choose an . Then there exists such that for . For let denote a finite ε-net for the interval and let
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 and, by the Schauder fixed point theorem, any continuous map has a fixed point. Let be a map given by . Then F is an isometry of T onto S. Assume is a continuous map and let . Then is continuous and there exists a point such that . Let . Then
Hence any continuous map has a fixed point. Let . Choose and such that
Let
Then . By compactness of W, f is uniformly continuous on W. Hence there exists such that if and , then . Assume , . Let . Then
Hence H is continuous and there exists such that . Then
for . Hence
Since the sequence is bounded, we have . Hence
Moreover, for , by Lemma 3.1(07), we have
Hence x is a solution of (E). The proof is complete. □
Remark 5.1 The conclusion of Theorem 5.1 may be written in the form
We can say that y is an approximative solution of (E) with ‘degree of approximation’ .
Remark 5.2 Using Lemma 4.5 we can replace the assumption of Theorem 5.1 by
Analogously, the conclusion of this theorem may be replaced: there exist a solution x of (E) and a sequence such that and
Similarly, using other tests, we can obtain many formulations of this theorem.
Theorem 5.2 Assume , , , , , , a is nonoscillatory, there exists a uniform neighborhood U of the set such that or , and one of the following conditions is satisfied:
-
(a)
there exists a solution x of (E) such that ,
-
(b)
there exists a solution x of (E) such that .
Then .
Proof By Lemma 4.2 we have . Hence, there exists a solution x of (E) and a sequence such that . Let . For large n we have
By Lemma 3.1(10), . Hence, by Lemma 3.1(12), we have . Since , we have for large n. Hence the sequence is nonoscillatory and we obtain , which means . Moreover,
for large n. Hence, by Lemma 3.1(04), . The proof is complete. □
Remark 5.3 Theorem 5.2 extends Theorem 1 of [8] and parts of Theorems 6 and 7 of [6].
Lemma 5.2 Assume , , and . Then
Proof Using Lemma 4.8(c) we obtain
Let . Then . Hence
By (13) we have . Hence
□
Theorem 5.3 Assume , and x is an f-bounded solution of (E). Then
Proof The sequence is bounded. Hence and we obtain . Therefore and, by Lemma 5.2, we have
□
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Migda, J. Iterated remainder operator, tests for multiple convergence of series, and solutions of difference equations. Adv Differ Equ 2014, 189 (2014). https://doi.org/10.1186/1687-1847-2014-189
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DOI: https://doi.org/10.1186/1687-1847-2014-189