On the existence of solutions of linear Volterra difference ... - UFPR
Applied Mathematics and Computation 218 (2012) 9310–9320
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence Josef Diblík a,⇑, Ewa Schmeidel b a b
Brno University of Technology, Brno, Czech Republic ´ University of Technology, Poznan ´ , Poland Poznan
a r t i c l e
i n f o
a b s t r a c t Schauder’s fixed point technique is applied to asymptotical analysis of solutions of a linear Volterra difference equation
Keywords: Linear Volterra difference equation Asymptotic formula Asymptotic equivalence
xðn þ 1Þ ¼ aðnÞ þ bðnÞxðnÞ þ
n P Kðn; iÞxðiÞ i¼0
where n 2 N0 , x : N0 ! R, a : N0 ! R, K : N0 N0 ! R, and b : N0 ! R n f0g is x-periodic. In the paper, sufficient conditions are derived for the validity of a property of solutions that, for every admissible constant c 2 R, there exists a solution x ¼ xðnÞ such that
nP 1 xðnÞ c þ
aðiÞ bðnÞ; i¼0 bði þ 1Þ
Q where bðnÞ ¼ n1 j¼0 bðjÞ, for n ! 1 and inequalities for solutions are derived. Relevant comparisons and illustrative examples are given as well. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction In the paper we study the asymptotic behavior of solutions of a linear Volterra difference equation
xðn þ 1Þ ¼ aðnÞ þ bðnÞxðnÞ þ
n P
Kðn; iÞxðiÞ
ð1Þ
i¼0
for n ! 1. In (1) we assume n 2 N0 :¼ f0; 1; 2; . . .g, a : N0 ! R, K : N0 N0 ! R, and the sequence b : N0 ! R n f0g is x-periodic, x 2 N :¼ f1; 2; . . .g. By a solution of Eq. (1) we mean a sequence x : N0 ! R which satisfies (1) for every n 2 N0 . In the last few years, there has been an interest among many authors to study the asymptotic behavior of solutions of Volterra difference equations. The results were published, e.g., by Appleby et al. [2], Elaydi and Murakami [8], Gil and Medina [9], Györi and Horváth [10], Györi and Reynolds [11], Medina [13–16] and Song and Baker [22]. Appleby et al. [2] is concerned with the asymptotic behavior of solutions of difference equations of general linear Volterra non-convolution system of difference equations. Sufficient conditions are given for the asymptotic constancy of solutions to the initial value problem with a formula for the rate of convergence. More detailed results are obtained in the particular case of linear Volterra convolution system.
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (J. Diblík), [email protected] (E. Schmeidel). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.03.010
J. Diblík, E. Schmeidel / Applied Mathematics and Computation 218 (2012) 9310–9320
9311
In [8] the existence of bounded solutions of nonhomogeneous linear Volterra difference equations is studied. For linear Volterra difference equations of a nonconvolution type, the authors characterize a uniform asymptotic stability of the zero solution by the summability of the resolvent matrix. Gil and Medina [9] deals with explicit stability conditions for a class of vector nonlinear discrete-time Volterra equations in the space ‘p by deriving estimates of the norms of solutions. The asymptotic behaviour of solutions of linear Volterra difference equations is analyzed in [10]. Sufficient conditions under which the solutions to a general linear equation converge to limits, given by a limit formula are presented. This result is then used to obtain the exact asymptotic representation of the solutions of a class of convolution scalar difference equations, which have real characteristic roots. In [11], it is shown that a class of linear nonconvolution discrete Volterra equations has asymptotically periodic solutions. Boundedness and stability properties of some classes of equations is studied in [13,14], sufficient conditions for the asymptotic equivalence of some linear and nonlinear equations are derived in [15], and [16] deals with the behavior of a fundamental system of solutions. The theory of admissibility for linear discrete Volterra operators is developed in [22]. Here, several necessary and sufficient conditions for admissibility in various sequence spaces are obtained. Also the existence of solutions (such as bounded, exponential or convergent solutions), of linear or nonlinear discrete Volterra summation equations is studied. For the reader’s convenience, we note that the background for discrete Volterra equations can be found, e.g., in the wellknown books by Agarwal [1], Elaydi [7], and Kocic´ and Ladas [12]. For some further results on Volterra equations and periodic solutions of difference equations see, e.g., [17–19], [21,23,24] and the related references therein. For example, in [17] the convergence of solutions for two delays Volterra integral equations in the critical case is studied. The authors of present paper studied a Volterra difference equation of the form (1) in [3] and prove that Eq. (1) has asymptotically periodic solutions within the meaning of the following definition. Definition 1. Let x be a positive integer. The sequence y : N0 ! R is called x-periodic if yðn þ xÞ ¼ yðnÞ for all n 2 N0 . The sequence y is called asymptotically x-periodic if there exist two sequences u; v : N0 ! R such that u is x-periodic, limn!1 v ðnÞ ¼ 0 and
yðnÞ ¼ uðnÞ þ v ðnÞ for all n 2 N0 . The existence of asymptotically periodic solutions of systems of Volterra equations is studied in recent papers [4,5]. In [6] the authors consider Eq. (1) under the assumption
1 aðiÞ P 0, Eq. (22) has a solution that satisfies inequality (9), i.e.
jyðnÞ ðc0 AðnÞÞbðnÞj 6
ðc0 þ AÞM jbðnÞj 1M
ð23Þ
for every n 2 N0 . In other words, Eq. (1) has a solution x ¼ xðnÞ satisfying inequality
jxðnÞ ðc þ AðnÞÞbðnÞj 6
ðj c j þAÞM jbðnÞj 1M
which is derived from (23) if yðnÞ ¼ xðnÞ. Moreover, from formula (11), it is clear that
yðnÞ ðc0 AðnÞÞbðnÞ
ð24Þ
for n ! 1 where c0 > 0. Substituting xðnÞ ¼ yðnÞ and c ¼ c0 < 0 in (24) we get
xðnÞ ðc þ AðnÞÞbðnÞ for n ! 1 where, due to (10), c þ AðnÞ – 0.
h
Remark 1. For A we assume A 2 ð0; 1Þ in Theorem 1 (see inequality (7)). If A ¼ 0, then aðnÞ ¼ 0 for every n 2 N0 and (1) has a trivial solution xðnÞ ¼ 0; n 2 N0 . Nevertheless, carefully tracing the proof, we state that Theorem 1 remains valid even in this case. That is if A ¼ 0 holds instead of inequality (7) and the rest of the assumptions of Theorem 1 is valid then, for every c – 0, there exists a nontrivial solution of (1), satisfying (9) and (11), i.e.
jxðnÞ cbðnÞj 6
jcjM jbðnÞj 1M
for every n 2 N0 and
xðnÞ cbðnÞ for n ! 1. Remark 2. For M we assume M 2 ð0; 1Þ in Theorem 1 (see inequality (8)). If M ¼ 0, then (1) reduces to a linear difference equation
xðn þ 1Þ ¼ aðnÞ þ bðnÞxðnÞ
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J. Diblík, E. Schmeidel / Applied Mathematics and Computation 218 (2012) 9310–9320
with a general solution
xðnÞ ¼ ðc þ AðnÞÞbðnÞ
ð25Þ
where c is an arbitrary constant. In this case, Theorem 1 remains valid as well (in the proof aðnÞ ¼ 0 for all n 2 N0 and the set S reduces to one sequence zðnÞ ¼ c þ AðnÞ only) and inequality (9) gives equivalent with (25) inequality
jxðnÞ ðc þ AðnÞÞbðnÞj 6 0; n 2 N0 : Example 1. Let us assume that, in Eq. (1),
aðnÞ ¼ 2nþ2 ; bðnÞ ¼ ð1Þn 2; n 2 N0 and
Kð4k; iÞ ¼ ð1Þiði1Þ=2 22k2i2 ; Kð4k þ 1; iÞ ¼ 0; Kð4k þ 2; iÞ ¼ ð1Þiði1Þ=2 22k2i ; Kð4k þ 3; iÞ ¼ 0 where k; i 2 N0 . We show that all assumptions of the Theorem 1 hold. The sequence bðnÞ is x-periodic with x ¼ 2 and
bðnÞ ¼
n1 Q
bðjÞ ¼ ð1Þnðn1Þ=2 2n ; n 2 N0 ;
j¼0
bð4kÞ ¼
4k1 Q
bðjÞ ¼ ð1Þ4kð4k1Þ=2 24k ¼ 24k ; k 2 N0 ;
j¼0
bð4k þ 1Þ ¼ ð1Þð4kþ1Þ4k=2 24kþ1 ¼ 24kþ1 ; k 2 N0 ; bð4k þ 2Þ ¼ ð1Þð4kþ2Þð4kþ1Þ=2 24kþ2 ¼ 24kþ2 ; k 2 N0 ; bð4k þ 3Þ ¼ ð1Þð4kþ3Þð4kþ2Þ=2 24kþ3 ¼ 24kþ3 ; k 2 N0 : Therefore, we get
aðnÞ ¼ ð1Þðnþ1Þn=2 2; n 2 N0 ; bðn þ 1Þ að4kÞ að4k þ 1Þ ¼ 2; ¼ 2; k 2 N0 ; bð4k þ 1Þ bð4k þ 2Þ að4k þ 2Þ að4k þ 3Þ ¼ 2; ¼ 2; k 2 N0 : bð4k þ 3Þ bð4k þ 4Þ This implies that
Að4kÞ ¼
4k1 P i¼0
Að4k þ 1Þ ¼
Að4k þ 2Þ ¼
4k1 P aðiÞ ¼ ð1Þiðiþ1Þ=2 2 ¼ 0; k 2 N0 ; bði þ 1Þ i¼0 4k P
4k P aðiÞ ¼ ð1Þiðiþ1Þ=2 2 ¼ 2; k 2 N0 ; bði þ 1Þ i¼0 i¼0 4kþ1 P i¼0
Að4k þ 3Þ ¼
4kþ2 P i¼0
4kþ1 P aðiÞ ¼ ð1Þiðiþ1Þ=2 2 ¼ 0; k 2 N0 ; bði þ 1Þ i¼0 4kþ2 P aðiÞ ¼ ð1Þiðiþ1Þ=2 2 ¼ 2; k 2 N0 : bði þ 1Þ i¼0
From the above,
A ¼ supfjAðnÞjg ¼ 2 < 1: n2N0
Now, we calculate that
Kð4k; iÞbðiÞ 1 Kð4k þ 1; iÞbðiÞ ¼ iþ2kþ3 ; ¼ 0; bð4k þ 1Þ bð4k þ 2Þ 2 Kð4k þ 2; iÞbðiÞ 1 Kð4k þ 3; iÞbðiÞ ¼ iþ2kþ3 ; ¼ 0: bð4k þ 3Þ bð4k þ 4Þ 2
J. Diblík, E. Schmeidel / Applied Mathematics and Computation 218 (2012) 9310–9320
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Hence
0<M¼ ¼
2 1 1 ! j Kðj; iÞbðiÞ 1 P 1 1 1 1 1 1 P P1 1 1 6 P PKðj; iÞbðiÞ ¼ P PKð2j; iÞbðiÞ ¼ PP 1 ¼ 1 P 1 ¼ iþjþ3 j 8 i¼0 2i 8 1 1=2 j¼0 i¼0 bðj þ 1Þ j¼0 i¼0 bðj þ 1Þ j¼0 i¼0 bð2j þ 1Þ j¼0i¼0 2 j¼0 2
1 1 4 ¼ < 1; 8 2
i.e., 0 < M 6 1=2 and M=ð1 MÞ 6 1. By virtue of Theorem 1 (inequality (9)), for any constant c, there exists a solution x : N0 ! R of (1) such that
jxð4kÞ c 24k j 6 ð2 þ jcjÞ 24k ; jxð4k þ 1Þ ðc þ 2Þ 24kþ1 j 6 ð2 þ jcjÞ 24kþ1 ; jxð4k þ 2Þ þ c 24kþ2 j 6 ð2 þ jcjÞ 24kþ2 ; jxð4k þ 3Þ þ ðc 2Þ 24kþ3 j 6 ð2 þ jcjÞ 24kþ3 where k 2 N0 . Moreover, for any constant c such that c þ AðnÞ – 0, if n ! 1, i.e., for c–0 and c– 2, there exists a solution x : N0 ! R of (1) such that (11) holds, i.e.,
xð4kÞ c 24k ; xð4k þ 1Þ ðc þ 2Þ 24kþ1 ; xð4k þ 2Þ c 24kþ2 ; xð4k þ 3Þ ðc 2Þ 24kþ3 for k ! 1. 3. Corollaries We set
B :¼ bðxÞ:
ð26Þ
Moreover, we define
n1 n :¼ n 1 x
ð27Þ
x
where b c is the floor function (the greatest-integer function) and n is the ‘‘remainder’’ of dividing n 1 by x. Obviously, fbðn Þg, n 2 N0 is an x-periodic sequence. We denote
A :¼ lim AðnÞ ¼ n!1
1 P
aðiÞ þ 1Þ
i¼0 bði
provided that the series 1 P
aðiÞ bði þ 1Þ i¼0
ð28Þ
is convergent. The following corollary is a consequence of Theorem 1. Corollary 1. Let x be a positive integer, b : N0 ! R n f0g be x-periodic, a : N0 ! R and K : N0 N0 ! R, series (28) be convergent and inequality (8) hold. Then, for any constant c – A there exists a weighted asymptotically x-periodic solution x ¼ xðnÞ; x : N0 ! R of (1) with u; v : N0 ! R and w : N0 ! R n f0g in representation (4) such that n1
wðnÞ ¼ Bb x c ; uðnÞ :¼ ðc þ A Þbðn þ 1Þ; lim v ðnÞ ¼ 0; n!1
i.e,
xðnÞ n1
Bb x c
¼ ðc þ A Þbðn þ 1Þ þ v ðnÞ; n 2 N0 :
Proof. By virtue of Theorem 1, for any constant c satisfying (10), there exists a solution x : N0 ! R of (1) that fulfills (11). Since c – A , (10) is satisfied for n ! 1. Then
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J. Diblík, E. Schmeidel / Applied Mathematics and Computation 218 (2012) 9310–9320
lim
xðnÞ
n!1 bðnÞ
¼ c þ A
or
xðnÞ ¼ c þ A þ oð1Þ bðnÞ for n ! 1. Hence
xðnÞ ¼ bðnÞðc þ A þ oð1ÞÞ; n ! 1:
ð29Þ
From (3), (26) and (27) follows
bðnÞ ¼
n1 Q
n1
bðjÞ ¼ Bb x c bðn þ 1Þ
j¼0
for n 2 N0 Then, utilizing (29), n1
xðnÞ ¼ Bb x c bðn þ 1Þðc þ A þ oð1ÞÞ; n ! 1 or
xðnÞ n1
Bb x c
¼ ðc þ A Þbðn þ 1Þ þ bðn þ 1Þoð1Þ; n ! 1:
We will complete the proof. The sequence fbðn þ 1Þg is x-periodic, hence bounded and, due to the properties of Landau order symbols, we have
bðn þ 1Þoð1Þ ¼ oð1Þ as n ! 1. It is easy to see that the choice n1
uðnÞ :¼ ðc þ A Þbðn þ 1Þ; wðnÞ :¼ Bb x c ; and an appropriate function
n 2 N0
v : N0 ! R such that
lim v ðnÞ ¼ 0
n!1
finishes the proof.
h
Tracing the proof of Corollary 1, we conclude that, without loss of generality, it can be reformulated as follows. Corollary 2. Let x be a positive integer, b : N0 ! R n f0g be x-periodic, a : N0 ! R and K : N0 N0 ! R, series (28) be convergent and inequality (8) hold. Then, for any constant c – 0 there exists a weighted asymptotically x-periodic solution x ¼ xðnÞ; x : N0 ! R of (1) with u; v : N0 ! R and w : N0 ! R n f0g in representation (4) such that n1
wðnÞ ¼ Bb x c ; uðnÞ :¼ cbðn þ 1Þ; lim v ðnÞ ¼ 0; n!1
i.e,
xðnÞ n1
Bb x c
¼ cbðn þ 1Þ þ v ðnÞ; n 2 N0 :
In [6], the following theorem is proved. Theorem 2 [6, Theorem 2.2]. Let x be a positive integer, b : N0 ! R n f0g be x-periodic, a : N0 ! R and K : N0 N0 ! R. Assume that
1 aðiÞ P < 1; i¼0 bði þ 1Þ
ð30Þ
j Kðj; iÞbðiÞ 1 P P bðj þ 1Þ < 1;
ð31Þ
j¼0 i¼0
and that at least one of the real numbers in the left-hand sides of inequalities (30), (31) is positive. Then, for any nonzero constant c, there exists a weighted asymptotically x-periodic solution x : N0 ! R of (1) with u; v : N0 ! R and w : N0 ! R n f0g in representation (4) such that n1
wðnÞ ¼ Bb x c ; uðnÞ :¼ cbðn þ 1Þ; lim v ðnÞ ¼ 0; n!1
J. Diblík, E. Schmeidel / Applied Mathematics and Computation 218 (2012) 9310–9320
9319
i.e,
xðnÞ n1
Bb x c
¼ cbðn þ 1Þ þ v ðnÞ; n 2 N0 :
Comparing Theorem 2 with Corollary 2, we see that Corollary 2 significantly generalizes Theorem 2 because (30) implies the convergence of the series (28). In the following corollary, we use the notation LCMðp; qÞ for the lowest common multiple of two positive integers p and q, i.e., for the smallest positive integer that is a multiple of both p and q. Corollary 3. Let the assumptions of Theorem 1 be true,
bðxÞ ¼ 1;
ð32Þ
and the sequence AðnÞ, defined by (6), be x0 -periodic where x0 is a positive integer. Then, there exists a solution x : N0 ! R of (1) which is asymptotically LCMðx; x0 Þ-periodic. Proof. Since AðnÞ is x0 -periodic, we state that the sequence c þ AðnÞ is also x0 -periodic for an arbitrary c 2 R. By virtue of Theorem 1, the sequences
xðnÞ ; c þ AðnÞ; bðnÞ
n 2 N0
are asymptotically equivalent as n ! 1. This implies that the sequence
xðnÞ ; bðnÞ
n 2 N0
is asymptotically x0 -periodic. Hence, it is asymptotically LCMðx; x0 Þ periodic as well. From the x-periodicity of the sequence fbðnÞg and the condition (32), it follows that the sequence fbðnÞg is x-periodic. Hence also LCMðx; x0 Þ is a period of fbðnÞg. This implies that
xðn þ LCMðx; x0 ÞÞ bðn þ LCMðx; x0 ÞÞðc þ Aðn þ LCMðx; x0 ÞÞÞ ¼ bðnÞðc þ AðnÞÞ xðnÞ when n ! 1. This gives
xðn þ LCMðx; x0 ÞÞ xðnÞ as n ! 1 or, equivalently,
xðnÞ ¼ xðn þ LCMðx; x0 ÞÞ þ oð1Þ if n ! 1 since xðn þ LCMðx; x0 ÞÞ is bounded and
xðn þ LCMðx; x0 ÞÞð1 þ oð1ÞÞ ¼ xðn þ LCMðx; x0 ÞÞ þ oð1Þ: The proof is completed. h Example 2. Let us assume that, in Eq. (1), 1 faðnÞgn¼0 ¼ f1; 0; 1; 1; 0; 1; . . .g; n o1 nþ1 1 1 1 ¼ ; 2; ; 2; . . . ; fbðnÞgn¼0 ¼ 2ð1Þ n¼0 2 2
and
Kðn; iÞ ¼
1 2nþiþ4
;
n; i 2 N0 :
We show that all the assumptions of the Theorem 1 hold. The sequence bðnÞ is x-periodic with x ¼ 2, the sequence 1 fbðnÞgn¼0 ¼
n o1 n 1 1 1; ; 1; ; . . . ¼ 22b2cn n¼0 2 2
is also 2-periodic, bð2Þ ¼ 1 and
0
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence Josef Diblík a,⇑, Ewa Schmeidel b a b
Brno University of Technology, Brno, Czech Republic ´ University of Technology, Poznan ´ , Poland Poznan
a r t i c l e
i n f o
a b s t r a c t Schauder’s fixed point technique is applied to asymptotical analysis of solutions of a linear Volterra difference equation
Keywords: Linear Volterra difference equation Asymptotic formula Asymptotic equivalence
xðn þ 1Þ ¼ aðnÞ þ bðnÞxðnÞ þ
n P Kðn; iÞxðiÞ i¼0
where n 2 N0 , x : N0 ! R, a : N0 ! R, K : N0 N0 ! R, and b : N0 ! R n f0g is x-periodic. In the paper, sufficient conditions are derived for the validity of a property of solutions that, for every admissible constant c 2 R, there exists a solution x ¼ xðnÞ such that
nP 1 xðnÞ c þ
aðiÞ bðnÞ; i¼0 bði þ 1Þ
Q where bðnÞ ¼ n1 j¼0 bðjÞ, for n ! 1 and inequalities for solutions are derived. Relevant comparisons and illustrative examples are given as well. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction In the paper we study the asymptotic behavior of solutions of a linear Volterra difference equation
xðn þ 1Þ ¼ aðnÞ þ bðnÞxðnÞ þ
n P
Kðn; iÞxðiÞ
ð1Þ
i¼0
for n ! 1. In (1) we assume n 2 N0 :¼ f0; 1; 2; . . .g, a : N0 ! R, K : N0 N0 ! R, and the sequence b : N0 ! R n f0g is x-periodic, x 2 N :¼ f1; 2; . . .g. By a solution of Eq. (1) we mean a sequence x : N0 ! R which satisfies (1) for every n 2 N0 . In the last few years, there has been an interest among many authors to study the asymptotic behavior of solutions of Volterra difference equations. The results were published, e.g., by Appleby et al. [2], Elaydi and Murakami [8], Gil and Medina [9], Györi and Horváth [10], Györi and Reynolds [11], Medina [13–16] and Song and Baker [22]. Appleby et al. [2] is concerned with the asymptotic behavior of solutions of difference equations of general linear Volterra non-convolution system of difference equations. Sufficient conditions are given for the asymptotic constancy of solutions to the initial value problem with a formula for the rate of convergence. More detailed results are obtained in the particular case of linear Volterra convolution system.
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (J. Diblík), [email protected] (E. Schmeidel). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.03.010
J. Diblík, E. Schmeidel / Applied Mathematics and Computation 218 (2012) 9310–9320
9311
In [8] the existence of bounded solutions of nonhomogeneous linear Volterra difference equations is studied. For linear Volterra difference equations of a nonconvolution type, the authors characterize a uniform asymptotic stability of the zero solution by the summability of the resolvent matrix. Gil and Medina [9] deals with explicit stability conditions for a class of vector nonlinear discrete-time Volterra equations in the space ‘p by deriving estimates of the norms of solutions. The asymptotic behaviour of solutions of linear Volterra difference equations is analyzed in [10]. Sufficient conditions under which the solutions to a general linear equation converge to limits, given by a limit formula are presented. This result is then used to obtain the exact asymptotic representation of the solutions of a class of convolution scalar difference equations, which have real characteristic roots. In [11], it is shown that a class of linear nonconvolution discrete Volterra equations has asymptotically periodic solutions. Boundedness and stability properties of some classes of equations is studied in [13,14], sufficient conditions for the asymptotic equivalence of some linear and nonlinear equations are derived in [15], and [16] deals with the behavior of a fundamental system of solutions. The theory of admissibility for linear discrete Volterra operators is developed in [22]. Here, several necessary and sufficient conditions for admissibility in various sequence spaces are obtained. Also the existence of solutions (such as bounded, exponential or convergent solutions), of linear or nonlinear discrete Volterra summation equations is studied. For the reader’s convenience, we note that the background for discrete Volterra equations can be found, e.g., in the wellknown books by Agarwal [1], Elaydi [7], and Kocic´ and Ladas [12]. For some further results on Volterra equations and periodic solutions of difference equations see, e.g., [17–19], [21,23,24] and the related references therein. For example, in [17] the convergence of solutions for two delays Volterra integral equations in the critical case is studied. The authors of present paper studied a Volterra difference equation of the form (1) in [3] and prove that Eq. (1) has asymptotically periodic solutions within the meaning of the following definition. Definition 1. Let x be a positive integer. The sequence y : N0 ! R is called x-periodic if yðn þ xÞ ¼ yðnÞ for all n 2 N0 . The sequence y is called asymptotically x-periodic if there exist two sequences u; v : N0 ! R such that u is x-periodic, limn!1 v ðnÞ ¼ 0 and
yðnÞ ¼ uðnÞ þ v ðnÞ for all n 2 N0 . The existence of asymptotically periodic solutions of systems of Volterra equations is studied in recent papers [4,5]. In [6] the authors consider Eq. (1) under the assumption
1 aðiÞ P 0, Eq. (22) has a solution that satisfies inequality (9), i.e.
jyðnÞ ðc0 AðnÞÞbðnÞj 6
ðc0 þ AÞM jbðnÞj 1M
ð23Þ
for every n 2 N0 . In other words, Eq. (1) has a solution x ¼ xðnÞ satisfying inequality
jxðnÞ ðc þ AðnÞÞbðnÞj 6
ðj c j þAÞM jbðnÞj 1M
which is derived from (23) if yðnÞ ¼ xðnÞ. Moreover, from formula (11), it is clear that
yðnÞ ðc0 AðnÞÞbðnÞ
ð24Þ
for n ! 1 where c0 > 0. Substituting xðnÞ ¼ yðnÞ and c ¼ c0 < 0 in (24) we get
xðnÞ ðc þ AðnÞÞbðnÞ for n ! 1 where, due to (10), c þ AðnÞ – 0.
h
Remark 1. For A we assume A 2 ð0; 1Þ in Theorem 1 (see inequality (7)). If A ¼ 0, then aðnÞ ¼ 0 for every n 2 N0 and (1) has a trivial solution xðnÞ ¼ 0; n 2 N0 . Nevertheless, carefully tracing the proof, we state that Theorem 1 remains valid even in this case. That is if A ¼ 0 holds instead of inequality (7) and the rest of the assumptions of Theorem 1 is valid then, for every c – 0, there exists a nontrivial solution of (1), satisfying (9) and (11), i.e.
jxðnÞ cbðnÞj 6
jcjM jbðnÞj 1M
for every n 2 N0 and
xðnÞ cbðnÞ for n ! 1. Remark 2. For M we assume M 2 ð0; 1Þ in Theorem 1 (see inequality (8)). If M ¼ 0, then (1) reduces to a linear difference equation
xðn þ 1Þ ¼ aðnÞ þ bðnÞxðnÞ
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with a general solution
xðnÞ ¼ ðc þ AðnÞÞbðnÞ
ð25Þ
where c is an arbitrary constant. In this case, Theorem 1 remains valid as well (in the proof aðnÞ ¼ 0 for all n 2 N0 and the set S reduces to one sequence zðnÞ ¼ c þ AðnÞ only) and inequality (9) gives equivalent with (25) inequality
jxðnÞ ðc þ AðnÞÞbðnÞj 6 0; n 2 N0 : Example 1. Let us assume that, in Eq. (1),
aðnÞ ¼ 2nþ2 ; bðnÞ ¼ ð1Þn 2; n 2 N0 and
Kð4k; iÞ ¼ ð1Þiði1Þ=2 22k2i2 ; Kð4k þ 1; iÞ ¼ 0; Kð4k þ 2; iÞ ¼ ð1Þiði1Þ=2 22k2i ; Kð4k þ 3; iÞ ¼ 0 where k; i 2 N0 . We show that all assumptions of the Theorem 1 hold. The sequence bðnÞ is x-periodic with x ¼ 2 and
bðnÞ ¼
n1 Q
bðjÞ ¼ ð1Þnðn1Þ=2 2n ; n 2 N0 ;
j¼0
bð4kÞ ¼
4k1 Q
bðjÞ ¼ ð1Þ4kð4k1Þ=2 24k ¼ 24k ; k 2 N0 ;
j¼0
bð4k þ 1Þ ¼ ð1Þð4kþ1Þ4k=2 24kþ1 ¼ 24kþ1 ; k 2 N0 ; bð4k þ 2Þ ¼ ð1Þð4kþ2Þð4kþ1Þ=2 24kþ2 ¼ 24kþ2 ; k 2 N0 ; bð4k þ 3Þ ¼ ð1Þð4kþ3Þð4kþ2Þ=2 24kþ3 ¼ 24kþ3 ; k 2 N0 : Therefore, we get
aðnÞ ¼ ð1Þðnþ1Þn=2 2; n 2 N0 ; bðn þ 1Þ að4kÞ að4k þ 1Þ ¼ 2; ¼ 2; k 2 N0 ; bð4k þ 1Þ bð4k þ 2Þ að4k þ 2Þ að4k þ 3Þ ¼ 2; ¼ 2; k 2 N0 : bð4k þ 3Þ bð4k þ 4Þ This implies that
Að4kÞ ¼
4k1 P i¼0
Að4k þ 1Þ ¼
Að4k þ 2Þ ¼
4k1 P aðiÞ ¼ ð1Þiðiþ1Þ=2 2 ¼ 0; k 2 N0 ; bði þ 1Þ i¼0 4k P
4k P aðiÞ ¼ ð1Þiðiþ1Þ=2 2 ¼ 2; k 2 N0 ; bði þ 1Þ i¼0 i¼0 4kþ1 P i¼0
Að4k þ 3Þ ¼
4kþ2 P i¼0
4kþ1 P aðiÞ ¼ ð1Þiðiþ1Þ=2 2 ¼ 0; k 2 N0 ; bði þ 1Þ i¼0 4kþ2 P aðiÞ ¼ ð1Þiðiþ1Þ=2 2 ¼ 2; k 2 N0 : bði þ 1Þ i¼0
From the above,
A ¼ supfjAðnÞjg ¼ 2 < 1: n2N0
Now, we calculate that
Kð4k; iÞbðiÞ 1 Kð4k þ 1; iÞbðiÞ ¼ iþ2kþ3 ; ¼ 0; bð4k þ 1Þ bð4k þ 2Þ 2 Kð4k þ 2; iÞbðiÞ 1 Kð4k þ 3; iÞbðiÞ ¼ iþ2kþ3 ; ¼ 0: bð4k þ 3Þ bð4k þ 4Þ 2
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Hence
0<M¼ ¼
2 1 1 ! j Kðj; iÞbðiÞ 1 P 1 1 1 1 1 1 P P1 1 1 6 P PKðj; iÞbðiÞ ¼ P PKð2j; iÞbðiÞ ¼ PP 1 ¼ 1 P 1 ¼ iþjþ3 j 8 i¼0 2i 8 1 1=2 j¼0 i¼0 bðj þ 1Þ j¼0 i¼0 bðj þ 1Þ j¼0 i¼0 bð2j þ 1Þ j¼0i¼0 2 j¼0 2
1 1 4 ¼ < 1; 8 2
i.e., 0 < M 6 1=2 and M=ð1 MÞ 6 1. By virtue of Theorem 1 (inequality (9)), for any constant c, there exists a solution x : N0 ! R of (1) such that
jxð4kÞ c 24k j 6 ð2 þ jcjÞ 24k ; jxð4k þ 1Þ ðc þ 2Þ 24kþ1 j 6 ð2 þ jcjÞ 24kþ1 ; jxð4k þ 2Þ þ c 24kþ2 j 6 ð2 þ jcjÞ 24kþ2 ; jxð4k þ 3Þ þ ðc 2Þ 24kþ3 j 6 ð2 þ jcjÞ 24kþ3 where k 2 N0 . Moreover, for any constant c such that c þ AðnÞ – 0, if n ! 1, i.e., for c–0 and c– 2, there exists a solution x : N0 ! R of (1) such that (11) holds, i.e.,
xð4kÞ c 24k ; xð4k þ 1Þ ðc þ 2Þ 24kþ1 ; xð4k þ 2Þ c 24kþ2 ; xð4k þ 3Þ ðc 2Þ 24kþ3 for k ! 1. 3. Corollaries We set
B :¼ bðxÞ:
ð26Þ
Moreover, we define
n1 n :¼ n 1 x
ð27Þ
x
where b c is the floor function (the greatest-integer function) and n is the ‘‘remainder’’ of dividing n 1 by x. Obviously, fbðn Þg, n 2 N0 is an x-periodic sequence. We denote
A :¼ lim AðnÞ ¼ n!1
1 P
aðiÞ þ 1Þ
i¼0 bði
provided that the series 1 P
aðiÞ bði þ 1Þ i¼0
ð28Þ
is convergent. The following corollary is a consequence of Theorem 1. Corollary 1. Let x be a positive integer, b : N0 ! R n f0g be x-periodic, a : N0 ! R and K : N0 N0 ! R, series (28) be convergent and inequality (8) hold. Then, for any constant c – A there exists a weighted asymptotically x-periodic solution x ¼ xðnÞ; x : N0 ! R of (1) with u; v : N0 ! R and w : N0 ! R n f0g in representation (4) such that n1
wðnÞ ¼ Bb x c ; uðnÞ :¼ ðc þ A Þbðn þ 1Þ; lim v ðnÞ ¼ 0; n!1
i.e,
xðnÞ n1
Bb x c
¼ ðc þ A Þbðn þ 1Þ þ v ðnÞ; n 2 N0 :
Proof. By virtue of Theorem 1, for any constant c satisfying (10), there exists a solution x : N0 ! R of (1) that fulfills (11). Since c – A , (10) is satisfied for n ! 1. Then
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lim
xðnÞ
n!1 bðnÞ
¼ c þ A
or
xðnÞ ¼ c þ A þ oð1Þ bðnÞ for n ! 1. Hence
xðnÞ ¼ bðnÞðc þ A þ oð1ÞÞ; n ! 1:
ð29Þ
From (3), (26) and (27) follows
bðnÞ ¼
n1 Q
n1
bðjÞ ¼ Bb x c bðn þ 1Þ
j¼0
for n 2 N0 Then, utilizing (29), n1
xðnÞ ¼ Bb x c bðn þ 1Þðc þ A þ oð1ÞÞ; n ! 1 or
xðnÞ n1
Bb x c
¼ ðc þ A Þbðn þ 1Þ þ bðn þ 1Þoð1Þ; n ! 1:
We will complete the proof. The sequence fbðn þ 1Þg is x-periodic, hence bounded and, due to the properties of Landau order symbols, we have
bðn þ 1Þoð1Þ ¼ oð1Þ as n ! 1. It is easy to see that the choice n1
uðnÞ :¼ ðc þ A Þbðn þ 1Þ; wðnÞ :¼ Bb x c ; and an appropriate function
n 2 N0
v : N0 ! R such that
lim v ðnÞ ¼ 0
n!1
finishes the proof.
h
Tracing the proof of Corollary 1, we conclude that, without loss of generality, it can be reformulated as follows. Corollary 2. Let x be a positive integer, b : N0 ! R n f0g be x-periodic, a : N0 ! R and K : N0 N0 ! R, series (28) be convergent and inequality (8) hold. Then, for any constant c – 0 there exists a weighted asymptotically x-periodic solution x ¼ xðnÞ; x : N0 ! R of (1) with u; v : N0 ! R and w : N0 ! R n f0g in representation (4) such that n1
wðnÞ ¼ Bb x c ; uðnÞ :¼ cbðn þ 1Þ; lim v ðnÞ ¼ 0; n!1
i.e,
xðnÞ n1
Bb x c
¼ cbðn þ 1Þ þ v ðnÞ; n 2 N0 :
In [6], the following theorem is proved. Theorem 2 [6, Theorem 2.2]. Let x be a positive integer, b : N0 ! R n f0g be x-periodic, a : N0 ! R and K : N0 N0 ! R. Assume that
1 aðiÞ P < 1; i¼0 bði þ 1Þ
ð30Þ
j Kðj; iÞbðiÞ 1 P P bðj þ 1Þ < 1;
ð31Þ
j¼0 i¼0
and that at least one of the real numbers in the left-hand sides of inequalities (30), (31) is positive. Then, for any nonzero constant c, there exists a weighted asymptotically x-periodic solution x : N0 ! R of (1) with u; v : N0 ! R and w : N0 ! R n f0g in representation (4) such that n1
wðnÞ ¼ Bb x c ; uðnÞ :¼ cbðn þ 1Þ; lim v ðnÞ ¼ 0; n!1
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i.e,
xðnÞ n1
Bb x c
¼ cbðn þ 1Þ þ v ðnÞ; n 2 N0 :
Comparing Theorem 2 with Corollary 2, we see that Corollary 2 significantly generalizes Theorem 2 because (30) implies the convergence of the series (28). In the following corollary, we use the notation LCMðp; qÞ for the lowest common multiple of two positive integers p and q, i.e., for the smallest positive integer that is a multiple of both p and q. Corollary 3. Let the assumptions of Theorem 1 be true,
bðxÞ ¼ 1;
ð32Þ
and the sequence AðnÞ, defined by (6), be x0 -periodic where x0 is a positive integer. Then, there exists a solution x : N0 ! R of (1) which is asymptotically LCMðx; x0 Þ-periodic. Proof. Since AðnÞ is x0 -periodic, we state that the sequence c þ AðnÞ is also x0 -periodic for an arbitrary c 2 R. By virtue of Theorem 1, the sequences
xðnÞ ; c þ AðnÞ; bðnÞ
n 2 N0
are asymptotically equivalent as n ! 1. This implies that the sequence
xðnÞ ; bðnÞ
n 2 N0
is asymptotically x0 -periodic. Hence, it is asymptotically LCMðx; x0 Þ periodic as well. From the x-periodicity of the sequence fbðnÞg and the condition (32), it follows that the sequence fbðnÞg is x-periodic. Hence also LCMðx; x0 Þ is a period of fbðnÞg. This implies that
xðn þ LCMðx; x0 ÞÞ bðn þ LCMðx; x0 ÞÞðc þ Aðn þ LCMðx; x0 ÞÞÞ ¼ bðnÞðc þ AðnÞÞ xðnÞ when n ! 1. This gives
xðn þ LCMðx; x0 ÞÞ xðnÞ as n ! 1 or, equivalently,
xðnÞ ¼ xðn þ LCMðx; x0 ÞÞ þ oð1Þ if n ! 1 since xðn þ LCMðx; x0 ÞÞ is bounded and
xðn þ LCMðx; x0 ÞÞð1 þ oð1ÞÞ ¼ xðn þ LCMðx; x0 ÞÞ þ oð1Þ: The proof is completed. h Example 2. Let us assume that, in Eq. (1), 1 faðnÞgn¼0 ¼ f1; 0; 1; 1; 0; 1; . . .g; n o1 nþ1 1 1 1 ¼ ; 2; ; 2; . . . ; fbðnÞgn¼0 ¼ 2ð1Þ n¼0 2 2
and
Kðn; iÞ ¼
1 2nþiþ4
;
n; i 2 N0 :
We show that all the assumptions of the Theorem 1 hold. The sequence bðnÞ is x-periodic with x ¼ 2, the sequence 1 fbðnÞgn¼0 ¼
n o1 n 1 1 1; ; 1; ; . . . ¼ 22b2cn n¼0 2 2
is also 2-periodic, bð2Þ ¼ 1 and
0
