Abstract
We study perturbations of homogeneous linear difference systems over infinite fields with absolute values. The coefficient matrices of the treated systems belong to commutative groups which do not need to be bounded. We present a general limit periodic transformation of an arbitrarily given system such that the obtained system has non-almost periodic solutions. We also formulate corollaries which show how the presented construction of the perturbed system improves and extends known results.
1. Introduction
In this paper, we study non-almost periodic solutions of homogeneous linear difference systems (1) (1) where the coefficient matrices are taken from a commutative group for all considered k. In addition, we consider these systems over a field F with an absolute value . The research of non-almost periodic solutions of general systems in the form (Equation1(1) (1) ) is typically based on special iterative constructions of limit periodic and almost periodic sequences. To prove our results, we use a construction of limit periodic sequences as well. Nevertheless, the presented construction is original and differs from all constructions applied in previous papers. For other constructive methods in this research area, we refer to strongly relevant papers [Citation9, Citation10, Citation21–25, Citation41, Citation43, Citation45] (see also [Citation20] for a different usage of constructions of almost periodic sequences). We point out that the basic motivation comes from articles [Citation9, Citation10, Citation24, Citation25] (see Section 4 below).
Now, we mention a short literature overview. We begin with monographs [Citation5, Citation13, Citation29, Citation35], where the fundamental properties of limit periodic and almost periodic sequences and functions are presented. Concerning almost periodicity of solutions of (almost periodic) linear difference equations, we refer to [Citation1, Citation6–8, Citation11, Citation17–19, Citation46, Citation48]. We remark that [Citation17] is the first paper, where a construction is applied to prove results about non-almost periodic solutions of homogeneous linear difference equations. Concerning the special complex case of Equation (Equation1(1) (1) ), see, e.g. [Citation3, Citation26].
If one replaces the commutativity of by its boundedness, then one obtains the almost periodic theory of transformable and weakly transformable groups (see [Citation21, Citation43]). This theory extends results from [Citation36, Citation37, Citation39, Citation40], where Equation (Equation1(1) (1) ) is treated for the unitary (or orthogonal) group . We highlight that, in [Citation39], there is proved that, in any neighbourhood of any almost periodic unitary Equation (Equation1(1) (1) ), there exists an almost periodic unitary Equation (Equation1(1) (1) ) whose fundamental matrix is not almost periodic. In a certain sense, this result is improved in this paper (see Corollary 5.2 below). We add that the process from [Citation39] cannot be used for a commutative group .
The corresponding research about almost periodic and non-almost periodic solutions of homogeneous linear differential systems of the form (2) (2) is realized as well. We mention papers [Citation27, Citation28, Citation38], where the almost periodicity of solutions of Equation (Equation2(2) (2) ) with skew-Hermitian and skew-symmetric coefficient matrices A is studied, together with [Citation42, Citation44] (and also [Citation37]), where non-almost periodic solutions of skew-Hermitian and skew-symmetric Equation (Equation2(2) (2) ) are analysed. Constructions of homogeneous linear differential systems with almost periodic coefficients are presented, e.g. in [Citation30, Citation31, Citation33, Citation34].
This paper is organized as follows. Section 2 is devoted to definitions and basic properties of all considered generalizations of periodicity in metric spaces. In Section 3, we state complete notations with respect to the studied difference systems over F with . The motivation of our research is explicitly described in Section 4. In the last (and, at the same time, main) section, there are formulated and proved our new results.
2. Periodic, limit periodic, almost periodic, and asymptotically almost periodic sequences
In this section, we present the notion of periodicity, limit periodicity, almost periodicity, and asymptotic almost periodicity in a metric space . We mention the definitions together with only the properties of considered sequences which are used in the proofs of our results. Note that we put .
Definition 2.1
A sequence is called periodic if there exists such that for all . A sequence is called periodic if there exists such that for all .
Definition 2.2
A sequence or is called limit periodic if there exists a sequence of periodic sequences or for such that uniformly with respect to or , respectively.
Remark 2.1
In Definition 2.2, the periods of sequences do not need to be the same. Thus, limit periodic sequences coincide with the so-called semi-periodic sequences. We refer to [Citation4] (and to [Citation2] in the continuous case).
Remark 2.2
In the literature, there is possible to find another definition of limit periodicity which is not equivalent. The different definition leads to a larger class of sequences. See, e.g. [Citation14, Citation32]. We consider Definition 2.2, because this version is the standard one and we also obtain the strongest results in this case.
Definition 2.3
A sequence is called almost periodic if for any , there exists a positive integer such that any set consisting of consecutive integers contains at least one integer l with the property that The integer l is called an ε-translation number of . A sequence is called almost periodic if there exists an almost periodic sequence such that for all .
In some books (see, e.g. [Citation16]), the almost periodicity is introduced using the so-called Bochner concept, which is equivalent with the so-called Bohr one (presented in Definition 2.3). This equivalence is the content of the next theorem.
Theorem 2.1
A sequence is almost periodic if and only if any sequence has a subsequence such that, for all , there exists for which the inequality (3) (3) holds for all , .
Proof.
See, e.g. [Citation41, Theorem 2.3].
The result below follows directly from Theorem 2.1.
Corollary 2.1
Let be metric spaces and let be sequences in , respectively. The sequence given by is almost periodic if and only if all sequences are almost periodic.
We will also use the following simple results.
Theorem 2.2
Any almost periodic sequence is bounded.
Proof.
The theorem follows directly from Definition 2.3.
Theorem 2.3
Any uniform limit of almost periodic sequences is almost periodic.
Proof.
The proof of the theorem can be easily obtained by a modification of the proof of [Citation12, Theorem 6.4].
Finally, we define the asymptotic almost periodicity.
Definition 2.4
A sequence is asymptotically almost periodic if for any , there exist positive integers and such that any set consisting of consecutive positive integers contains at least one number l satisfying
Remark 2.3
Evidently, any periodic sequence is limit periodic. From Theorem 2.3, one can see that any limit periodic sequence is almost periodic. It is also seen that any almost periodic sequence is asymptotically almost periodic. We add that any opposite implication is not generally true.
Remark 2.4
To clarify the asymptotic almost periodicity, we mention the following result which is valid in any Banach space. A sequence is asymptotically almost periodic if and only if it can be expressed as the sum of an almost periodic sequence and a sequence vanishing at infinity (see, e.g. [Citation47, Chapter 5]).
Remark 2.5
The Bochner concept of asymptotic almost periodicity is presented in [Citation15].
3. Considered systems
In this section, we collect the used notation concerning the studied homogeneous linear difference systems over general infinite fields with absolute values.
Let be a field with a zero and a unit . Of course, with respect to the studied problems, we assume that F is infinite. Let be an absolute value on F, i.e. let
and if and only if ,
,
,
where . Let a positive integer m be arbitrarily given as the dimension of systems under consideration. Then, denotes the set of all matrices with entries from F and the set of all vectors with elements from F. In addition, ·, + denote the multiplication and the addition on the spaces and . We also denote the identity matrix as , the zero matrix as , and the zero vector as .
The absolute value on F gives the norm on and as the sum of m and absolute values of elements, respectively. These norms are denoted by . We also have
and if and only if M = O,
,
,
where , . The absolute value on F and the norms induce the metrics. For simplicity, each one of these metrics is denoted by ϱ and the ε-neighbourhoods are denoted by in all considered metric spaces. In contrast with many research papers, we do not assume that the valued field F (with ) is separable or that the metric space is complete, i.e. our main result is valid without such limitations (although it is new for separable and, at the same time, complete spaces).
In the whole paper, will be an arbitrary infinite commutative group and we will study the homogeneous linear difference systems (4) (4) where . Let , , and denote the set of all Equation (Equation4(4) (4) ) such that the sequence of matrices is periodic, limit periodic, and almost periodic, respectively. One can identify the sequence with Equation (Equation4(4) (4) ) which is given by . Thus, in (consider also Theorem 2.2), there is introduced the metric (5) (5) Analogously, one can consider Equation (Equation4(4) (4) ) for and define The symbol means the ε-neighbourhood of or in .
4. Previous results
Non-almost periodic and non-asymptotically almost periodic solutions of perturbations of linear difference equations are studied in many papers. Concerning homogeneous systems given by matrices from a commutative group, we point out at least the following two results about the non-almost periodicity of solutions of initial problems.
Theorem 4.1
Let be a commutative group having the property that there exists such that, for any , there exists with the property that, for any , , there exist matrices satisfying Then, for any , , and for any sequence of , there exists such that the solution of is not almost periodic for any
Proof.
See [Citation9, Theorem 5.1].
Theorem 4.1 was improved as follows (see Definitions 2.3 and 2.4).
Theorem 4.2
Let be a commutative group having the property that there exists such that, for any , there exists with the property that, for any , , there exist matrices satisfying Then, for any , , and for any sequence of , there exists such that the solution of is not asymptotically almost periodic or for any .
Proof.
See [Citation25, Theorem 4.1].
Remark 4.1
Theorem 4.2 extends also the main result of [Citation22]. For other relevant results, we can refer to [Citation10] as well. Concerning solutions vanishing at infinity in the statement of Theorem 4.2, we add the fact that any non-trivial solution of any almost periodic homogeneous linear difference system cannot be almost periodic if See [Citation43, Lemma 3.10].
Now, we recall the most relevant results whose relevance is demonstrated by corollaries at the end of this paper (see Corollaries 5.1 and 5.2). In these results, arbitrarily small changes of limit periodic and almost periodic systems are considered in such a way that the obtained systems have at least one non-almost periodic solution. We add that these results follow from [Citation24, Theorems 10 and 11], where non-asymptotically almost periodic solutions are analysed (analogously as in Theorem 4.2).
Theorem 4.3
Let the unit ball be compact. Let be a commutative group having the property that there exists such that, for any , there exist matrices satisfying Then, for any and any , there exists with the property that the fundamental matrix of is not almost periodic.
Proof.
See [Citation24, Corollary 1].
Theorem 4.4
Let the unit ball be compact. Let be a commutative group having the property that there exists such that, for any , there exist matrices satisfying Then, for any and any , there exists with the property that the fundamental matrix of is not almost periodic.
Proof.
See [Citation24, Corollary 2].
5. Results
We begin with a known result which covers a special case of our main theorem, i.e. Theorem 5.1 formulated below.
Lemma 5.1
Let be a commutative group and let be arbitrarily given. If for any N>0 and , there exist matrices satisfying then, for any , there exists a limit periodic sequence with the property that the fundamental matrix of is not almost periodic.
Proof.
The lemma follows from the proof of [Citation22, Lemma 5.2], where it suffices to put
Concerning the statement of the main result below, we recall the usual notation (6) (6) In addition, in its proof, for , we consider the notation and i.e. denotes the smallest positive integer divisible by which is greater than or equal to k.
Theorem 5.1
Let be a commutative group and let be arbitrarily given. Let there exist and such that, for any , there exist matrices satisfying (7) (7) Then, for any , there exists a limit periodic sequence with the property that the fundamental matrix of (8) (8) is not almost periodic, i.e. Equation (Equation8(8) (8) ) has non-almost periodic solutions.
Proof.
If the fundamental matrix (denoted as) of (9) (9) is not almost periodic, then one can consider . Hence, we assume that the sequence is almost periodic, i.e. all solutions of Equation (Equation9(9) (9) ) are almost periodic. Note that the system has infinitely many non-almost periodic solutions if it has at least one non-almost periodic solution (it suffices to consider its non-zero multiples).
For an arbitrarily given number , we define (10) (10) As for , we denote the integer (from the statement of the theorem) which corresponds to . Of course, we can assume that and that , . For any , we consider the product satisfying (Equation7(7) (7) ) given by matrices Based on Lemma 5.1, we can assume the existence of K>0 with the property that (11) (11)
For the given from the statement of the theorem, we define (see also (Equation11(11) (11) )) (12) (12) It holds (13) (13) if (14) (14) Indeed, if (Equation14(14) (14) ) is valid, then we have (see (Equation7(7) (7) ), (Equation11(11) (11) ), and (Equation12(12) (12) )) which yields (Equation13(13) (13) ).
We introduce the required sequence by a construction, where we consider auxiliary sequences for . Note that we put for all and .
We begin with the construction. Since the fundamental matrix of Equation (Equation9(9) (9) ) is almost periodic and , there exists , , such that (15) (15) Indeed, it follows directly from Definition 2.3. We recall that, for the given (see (Equation10(10) (10) )), we consider matrices satisfying (16) (16) We put . We define the periodic sequence with the period by if and by if Introducing , we have finished the first step of our construction.
Considering (Equation13(13) (13) ) ⇐ (Equation14(14) (14) ) and (Equation15(15) (15) ), we know that the fundamental matrix of (17) (17) satisfies (18) (18) because and if and because if The second step of our construction has two parts. At first, we repeat that, for (see (Equation10(10) (10) )), there exist matrices satisfying (19) (19) We put In the first part of the second step, we consider the fundamental matrix of Equation (Equation17(17) (17) ). If the sequence is not almost periodic, then we have the resulting sequence given by the choice . Thus, we consider the almost periodicity of which implies the existence of an integer with the property that (20) (20) We define the periodic sequence with the period in the following way.
If then we put If then we put Using (Equation13(13) (13) ) ⇐ (Equation14(14) (14) ) and (Equation20(20) (20) ), we obtain that the fundamental matrix of (21) (21) has the property that (22) (22) Indeed, and if and if Now we proceed with the second part of the second step. We consider the fundamental matrix of Equation (Equation21(21) (21) ). Again, we can assume that the sequence is almost periodic. Therefore, there exists an integer such that (23) (23) We define the periodic sequence with the period in the following way.
If then we put If then we put Finally, in the second step, we introduce Evidently, the sequence is periodic. Applying (Equation13(13) (13) ) ⇐ (Equation14(14) (14) ) and (Equation23(23) (23) ), we have that the fundamental matrix of satisfies (24) (24) Indeed, and if and if We continue in the construction in the same manner. Before the step no. i for an arbitrary integer , we have the sequence which is periodic with the period . We point out that the step no. i has i parts.
For (see (Equation10(10) (10) )), we have matrices satisfying (25) (25) We put At first, in the step no. i, we consider the fundamental matrix of If is not almost periodic, then it suffices to choose . Let be almost periodic. Then, there exists an integer such that (26) (26) We put We define the periodic sequence with the period in the following way.
If then we put If then we put From (Equation13(13) (13) ) ⇐ (Equation14(14) (14) ) and (Equation26(26) (26) ), we know that the fundamental matrix of (27) (27) has the property that (28) (28) Indeed, and if and if Next, we consider the second part of step no. i, where we consider the fundamental matrix of Equation (Equation27(27) (27) ). We assume that the sequence is almost periodic which gives an integer such that (29) (29) We define the periodic sequence with the period in the following way.
If then we put If then we put Considering (Equation13(13) (13) ) ⇐ (Equation14(14) (14) ) and (Equation29(29) (29) ), we obtain that the fundamental matrix of satisfies (30) (30) Indeed, and if and if We continue in the step no. i. Before its last part, we consider the fundamental matrix of Again, we assume that the sequence is almost periodic. Therefore, there exists an integer such that (31) (31) We define the periodic sequence with the period in the following way.
If then we put
If then we put Finally, in this step, we define Considering (Equation13(13) (13) ) ⇐ (Equation14(14) (14) ) and (Equation31(31) (31) ), we obtain that the fundamental matrix of satisfies (32) (32) Indeed, and if and if We continue in the construction for We obtain the resulting sequence of by the limit This definition is correct, because for all and for all , . In addition, from the construction, one can see that (see also (Equation16(16) (16) ), (Equation19(19) (19) ), …, (Equation25(25) (25) ), …) (33) (33) and that, for any , there exist uniquely determined , for which (see (Equation10(10) (10) )) (34) (34) where we put , . In particular, from (Equation33(33) (33) ), we obtain that is the uniform limit of the sequences for ; and, from (Equation34(34) (34) ), we have that . Since all sequences are periodic, the sequence is limit periodic (see Definition 2.2).
It remains to show that the fundamental matrix of Equation (Equation8(8) (8) ) is not almost periodic. Using (Equation18(18) (18) ) from the first step, (Equation22(22) (22) ) and (Equation24(24) (24) ) from the second step, …, (Equation28(28) (28) ), (Equation30(30) (30) ), …, (Equation32(32) (32) ) from the step no. i, …, we have (35) (35) (36) (36) (37) (37) (38) (38) (39) (39) (40) (40) We apply Theorem 2.1 (together with the definition of almost periodicity for variable in Definition 2.3), where we put Evidently, (Equation35(35) (35) )–(Equation37(37) (37) ), …, (Equation38(38) (38) ), (Equation39(39) (39) ), …, (Equation40(40) (40) ), … imply a contradiction with (Equation3(3) (3) ) for . Therefore, we have proved that cannot be almost periodic.
Remark 5.1
Let us briefly comment the main condition in Theorem 5.1, i.e. the validity of (Equation7(7) (7) ). This assumption is motivated by the statement of the theorem, where we obtain non-almost periodic solutions using arbitrarily small changes of the given system . In addition, we consider limit periodic transformations, because the result is not valid for only periodic ones (see, e.g. and the unitary group in the scalar case together with [Citation23, Theorem 5]).
Example 5.1
Theorems 4.3 and 4.4 can be used, e.g. for any transformable commutative group (see [Citation21, Citation23]). Let us consider the complex or real case (i.e. or with the usual absolute value) and an arbitrary non-singular matrix S. Then, the groups are transformable and commutative, i.e. they satisfy all conditions of Theorems 4.3 and 4.4. In particular, in the complex case, we can consider the matrix group which is the intersection of the group of unitary matrices and the set of circulant matrices Indeed, it is well-known that is a special case of . In the real case, we point out that the group of all orthogonal matrices with determinant 1 is transformable. Therefore, for m = 2 and , all matrices in the form determine the transformable and commutative group , i.e. satisfies all conditions of Theorems 4.3 and 4.4. Concerning other concrete examples, we refer again to [Citation21, Citation23].
In addition, we can consider direct sums. We recall that, for groups , , the direct sum is defined as where if it has the form One can use Theorems 4.3 and 4.4 for the direct sum of commutative groups if at least one of the groups satisfies the conditions of the theorems.
Since Theorem 5.1 can be used for any group which satisfies the conditions of Theorems 4.3 and 4.4, we can consider examples mentioned above. We highlight at least the group for which one cannot apply Theorems 4.1 and 4.2 (consider ).
The main difference between Theorem 5.1 and Theorems 4.3 and 4.4 is given by the fact Theorem 5.1 covers perturbations of any system , while only limit periodic and almost periodic systems are treated in Theorems 4.3 and 4.4. Nevertheless, Theorem 5.1 is new also for and , which is commented in Example 5.2 below as well.
Theorem 5.1 has several new corollaries. Concerning the motivation in the previous section, we explicitly mention the following two new consequences.
Corollary 5.1
Let be a commutative group having the property that there exists such that, for any , there exist matrices satisfying Then, for any and any , there exists with the property that the fundamental matrix of is not almost periodic.
Proof.
Since is limit periodic (consider Theorem 2.2 and Remark 2.3), there exists such that (41) (41) From [Citation22, Lemma 4.4], it follows the uniform continuity of the multiplication of matrices on the set for any . Thus, for any , there exists with the property that (42) (42)
We use Theorem 5.1, where we put for the considered . Then, (Equation7(7) (7) ) reduces to (see (Equation6(6) (6) )) Applying Theorem 5.1 for introduced in the previous paragraph, we have a limit periodic sequence such that the fundamental matrix of is not almost periodic. Therefore, we consider the homogeneous linear difference system given by whose fundamental matrix is not almost periodic. Evidently (see (Equation5(5) (5) ), (Equation41(41) (41) ), and (Equation42(42) (42) )), we have that , because . Moreover, (43) (43)
Considering the limit periodicity of and , there exist periodic sequences and for all such that which yield (see also (Equation41(41) (41) ) and (Equation43(43) (43) )) These estimations together with the periodicity of all sequences prove that is limit periodic if and are limit periodic. The proof is complete.
Remark 5.2
It is evident that the statement of Corollary 5.1 does not change if one replaces the system by a periodic one, i.e. if . Indeed, it follows directly from Definition 2.2.
Corollary 5.2
Let be a commutative group having the property that there exists such that, for any , there exist matrices satisfying Then, for any and any , there exists with the property that the fundamental matrix of is not almost periodic.
Proof.
We can proceed as in the proof of Corollary 5.1 for (44) (44) where the boundedness of (see (Equation41(41) (41) )) remains valid. Note that the sequences , are almost periodic ( is even limit periodic, see Remark 2.3). Thus, the almost periodicity of the sequence (introduced in (Equation44(44) (44) )) follows from Theorem 2.1 (consider also Corollary 2.1), because any linear combination of almost periodic sequences is almost periodic and the product of two almost periodic sequences is almost periodic in the considered spaces.
Remark 5.3
Corollary 5.2 does not follow from Corollary 5.1. In [Citation4], there is proved that there exist sequences which are almost periodic and, at the same time, which are not limit periodic. It suffices to consider the almost periodic sequence for the unitary group in the scalar case. Then, the system determined by has a neighbourhood (in ) without any limit periodic system.
Remark 5.4
In fact, considering the statements of our results above and their proofs, one can see that it is not necessary to introduce the absolute value on the whole field. It suffices to have it on a neighbourhood of and for all elements of matrices from the matrix group . For details, we refer to [Citation23, Remark 15].
Example 5.2
To show that Corollaries 5.1 and 5.2 do not follow from Theorems 4.3 and 4.4, it suffices to consider a field F for which the unit ball is not compact. For example, let us consider the field of all meromorphic functions defined on an arbitrarily given non-empty open connected set T, where meromorphic functions can be added, subtracted, multiplied, and the quotient can be formed applying the analytic continuation to eliminate removable singularities. For simplicity, we use Remark 5.4 and we put m = 1 (i.e. we consider the scalar case, when all groups are commutative). Let be the set of all bounded functions on T. For any , we introduce where is the absolute value of complex number . Note that , .
Now, for any group whose all elements are from , one can apply Corollaries 5.1 and 5.2 if for all , where 1 denotes a constant function. Indeed, it suffices to consider only powers of matrices from . We remark that does not need to be bounded (cf. [Citation23, Example 4]).
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The authors would like to thank the referees.
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References
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