# Weighted Asymptotically Periodic Solutions of Linear Volterra Difference Equations

Authors: Josef Diblík; Miroslava Růžičková; Ewa Schmeidel; Małgorzata Zbąszyniak
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

A linear Volterra difference equation of the formx(n+1)=a(n)+b(n)x(n)+∑i=0nK(n,i)x(i), where x:ℕ0→ℝ, a:ℕ0→ℝ, K:ℕ0×ℕ0→ℝ and b:ℕ0→ℝ∖{0} is ω-periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on ∏j=0ω-1b(j) is assumed. The results generalize some of the recent results.


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## Body

## 1. Introduction

In the paper, we study a linear Volterra difference equation(1.1)x(n+1)=a(n)+b(n)x(n)+∑i=0nK(n,i)x(i),
where n∈ℕ0:={0,1,2,…}, a:ℕ0→ℝ, K:ℕ0×ℕ0→ℝ, and b:ℕ0→ℝ∖{0} is ω-periodic, ω∈ℕ:={1,2,…}. We will also adopt the customary notations (1.2)∑i=k+skO(i)=0,∏i=k+skO(i)=1,
where k is an integer, s is a positive integer, and  “𝒪” denotes the function considered independently of whether it is defined for the arguments indicated or not.In [1], the authors considered (1.1) under the assumption(1.3)∏j=0ω-1b(j)=1,
and gave sufficient conditions for the existence of asymptotically ω-periodic solutions of (1.1) where the notion for an asymptotically ω-periodic function has been given by the following definition.Definition 1.1.
Letω be a positive integer. The sequence y:ℕ0→ℝ is called ω-periodic if y(n+ω)=y(n) for all n∈ℕ0. The sequence y is called asymptotically ω-periodic if there exist two sequences u,v:ℕ0→ℝ such that u is ω-periodic, limn→∞v(n)=0, and
(1.4)y(n)=u(n)+v(n)
for all n∈ℕ0.In this paper, in general, we do not assume that (1.3) holds. Then, we are able to derive sufficient conditions for the existence of a weighted asymptotically ω-periodic solution of (1.1). We give a definition of a weighted asymptotically ω-periodic function.Definition 1.2.
Letω be a positive integer. The sequence y:ℕ0→ℝ is called weighted asymptotically ω-periodic if there exist two sequences u,v:ℕ0→ℝ such that u is ω-periodic and limn→∞v(n)=0, and, moreover, if there exists a sequence w:ℕ0→ℝ∖{0} such that
(1.5)y(n)w(n)=u(n)+v(n),
for all n∈ℕ0.Apart from this, when we assume(1.6)∏k=0ω-1b(k)=-1,
then, as a consequence of our main result (Theorem 2.2), the existence of an asymptotically 2ω-periodic solution of (1.1) is obtained.For the reader's convenience, we note that the background for discrete Volterra equations can be found, for example, in the well-known monograph by Agarwal [2], as well as by Elaydi [3] or Kocić and Ladas [4]. Volterra difference equations were studied by many others, for example, by Appleby et al. [5], by Elaydi and Murakami [6], by Győri and Horváth [7], by Győri and Reynolds [8], and by Song and Baker [9]. For some results on periodic solutions of difference equations, see, for example, [2–4, 10–13] and the related references therein.


## 2. Weighted Asymptotically Periodic Solutions

In this section, sufficient conditions for the existence of weighted asymptoticallyω-periodic solutions of (1.1) will be derived. The following version of Schauder's fixed point theorem given in [14] will serve as a tool used in the proof.Lemma 2.1.
LetΩ be a Banach space and S its nonempty, closed, and convex subset and let T be a continuous mapping such that T(S) is contained in S and the closure T(S)¯ is compact. Then, T has a fixed point in S.We set(2.1)β(n):=∏j=0n-1b(j),n∈N0,(2.2)B:=β(ω).
Moreover, we define (2.3)n*:=n-1-ω⌊n-1ω⌋,
where ⌊·⌋ is the floor function (the greatest-integer function) and n* is the “remainder” of dividing n-1 by ω. Obviously, {β(n*)}, n∈ℕ is an ω-periodic sequence.Now, we derive sufficient conditions for the existence of a weighted asymptoticallyω-periodic solution of (1.1).Theorem 2.2 (Main result).
Letω be a positive integer, b:ℕ0→ℝ∖{0} be ω-periodic, a:ℕ0→ℝ, and K:ℕ0×ℕ0→ℝ. Assume that
(2.4)∑i=0∞|a(i)β(i+1)|<∞,∑j=0∞∑i=0j|K(j,i)β(i)β(j+1)|<1,
and that at least one of the real numbers in the left-hand sides of inequalities (2.4) is positive.
Then, for any nonzero constantc, there exists a weighted asymptotically ω-periodic solution x:ℕ0→ℝof (1.1) with u,v:ℕ0→ℝ and w:ℕ0→ℝ∖{0} in representation (1.5) such that
(2.5)w(n)=B⌊(n-1)/ω⌋,u(n):=cβ(n*+1),lim⁡n→∞⁡v(n)=0,
that is,
(2.6)x(n)B⌊(n-1)/ω⌋=cβ(n*+1)+v(n),n∈N0.Proof.
We will use a notation(2.7)M:=∑j=0∞∑i=0j|K(j,i)β(i)β(j+1)|,
whenever this is useful. Case 1.
First assumec>0. We will define an auxiliary sequence of positive numbers {α(n)}, n∈ℕ0. We set
(2.8)α(0):=∑i=0∞|a(i)/(β(i+1))|+c∑j=0∞∑i=0j|(K(j,i)β(i))/(β(j+1))|1-∑j=0∞∑i=0j|(K(j,i)β(i))/(β(j+1))|,
where the expression on the right-hand side is well defined due to (2.4). Moreover, we define
(2.9)α(n):=∑i=n∞|a(i)β(i+1)|+(c+α(0))∑j=n∞∑i=0j|K(j,i)β(i)β(j+1)|,
for n≥1. It is easy to see that
(2.10)lim⁡n→∞α(n)=0.
We show, moreover, that
(2.11)α(n)≤α(0),
for any n∈ℕ. Let us first remark that
(2.12)α(0)=∑i=0∞|a(i)β(i+1)|+(c+α(0))∑j=0∞∑i=0j|K(j,i)β(i)β(j+1)|.
Then, due to the convergence of both series (see (2.4)), the inequality
(2.13)α(0)=∑i=0∞|a(i)β(i+1)|+(c+α(0))∑j=0∞∑i=0j|K(j,i)β(i)β(j+1)|≥∑i=n∞|a(i)β(i+1)|+(c+α(0))∑j=n∞∑i=0j|K(j,i)β(i)β(j+1)|=α(n)
obviously holds for every n∈ℕ and (2.11) is proved.
LetB be the Banach space of all real bounded sequences z:ℕ0→ℝ equipped with the usual supremum norm ∥z∥=supn∈ℕ0|z(n)| for z∈B. We define a subset S⊂B as
(2.14)S:={z∈B:c-α(0)≤z(n)≤c+α(0),n∈N0}.
It is not difficult to prove that S is a nonempty, bounded, convex, and closed subset of B.
Let us define a mappingT:S→B as follows:
(2.15)(Tz)(n)=c-∑i=n∞a(i)β(i+1)-∑j=n∞∑i=0jK(j,i)β(i)β(j+1)z(i),
for any n∈ℕ0.
We will prove that the mappingT has a fixed point in S.
We first show thatT(S)⊂S. Indeed, if z∈S, then |z(n)-c|≤α(0) for n∈ℕ0 and, by (2.11) and (2.15), we have
(2.16)|(Tz)(n)-c|≤∑i=n∞|a(i)β(i+1)|+(c+α(0))∑j=n∞∑i=0j|K(j,i)β(i)β(j+1)|=α(n)≤α(0).
Next, we prove that T is continuous. Let z(p) be a sequence in S such that z(p)→z as p→∞. Because S is closed, z∈S. Now, utilizing (2.15), we get
(2.17)|(Tz(p))(n)-(Tz)(n)|=|∑j=n∞∑i=0jK(j,i)β(i)β(j+1)(z(p)(i)-z(i))|≤Msup⁡i≥0|z(p)(i)-z(i)|=M‖z(p)-z‖,n∈N0.
Therefore,
(2.18)‖Tz(p)-Tz‖≤M‖z(p)-z‖,lim⁡p→∞‖Tz(p)-Tz‖=0.
This means that T is continuous.
Now, we show thatT(S)¯ is compact. As is generally known, it is enough to verify that every ɛ-open covering of T(S)¯ contains a finite ɛ-subcover of T(S)¯, that is, finitely many of these open sets already cover T(S)¯ ([15], page 756 (12)). Thus, to prove that T(S)¯ is compact, we take an arbitrary ɛ>0 and assume that an open ɛ-cover 𝒞ɛ of T(S)¯ is given. Then, from (2.10), we conclude that there exists an nɛ∈ℕ such that α(n)<ɛ/4 for n≥nɛ.
Suppose thatxT1∈T(S)¯ is one of the elements generating the ɛ-cover 𝒞ɛ of T(S)¯. Then (as follows from (2.16)), for an arbitrary xT∈T(S)¯,
(2.19)|xT1(n)-xT(n)|<ɛ
if n≥nɛ. In other words, the ɛ-neighborhood of xT1-c*:
(2.20)‖xT1-c*‖<ɛ,
where c*={c,c,…}∈S covers the set T(S)¯ on an infinite interval n≥nɛ. It remains to cover the rest of T(S)¯ on a finite interval for n∈{0,1…,nɛ-1} by a finite number of ɛ-neighborhoods of elements generating ɛ-cover 𝒞ɛ. Supposing that xT1 itself is not able to generate such cover, we fix n∈{0,1,…,nɛ-1} and split the interval
(2.21)[c-α(n),c+α(n)]
into a finite number h(ɛ,n) of closed subintervals
(2.22)I1(n),I2(n),…,Ih(ɛ,n)(n)
each with a length not greater then ɛ/2 such that
(2.23)⋃i=1h(ɛ,n)Ii(n)=[c-α(n),c+α(n)],int⁡Ii(n)∩int⁡Ij(n)=∅,i,j=1,2,…,h(ɛ,n),i≠j.
Finally, the set
(2.24)⋃n=0nɛ-1[c-α(n),c+α(n)]
equals
(2.25)⋃n=0nɛ-1⋃i=1h(ɛ,n)Ii(n)
and can be divided into a finite number
(2.26)Mɛ:=∑n=0nɛ-1h(ɛ,n)
of different subintervals (2.22). This means that, at most, Mɛ of elements generating the cover 𝒞ɛ are sufficient to generate a finite ɛ-subcover of T(S)¯ for n∈{0,1,…,nɛ-1}. We remark that each of such elements simultaneously plays the same role as xT1(n) for n≥nɛ. Since ɛ>0 can be chosen as arbitrarily small, T(S)¯ is compact.
By Schauder's fixed point theorem, there exists az∈S such that z(n)=(Tz)(n) for n∈ℕ0. Thus,
(2.27)z(n)=c-∑i=n∞a(i)β(i+1)-∑j=n∞∑i=0jβ(i)β(j+1)K(j,i)z(i),
for any n∈ℕ0.
Due to (2.10) and (2.16), for fixed point z∈S of T, we have
(2.28)lim⁡n→∞|z(n)-c|=lim⁡n→∞|(Tz)(n)-c|≤lim⁡n→∞α(n)=0,
or, equivalently,
(2.29)lim⁡n→∞z(n)=c.
Finally, we will show that there exists a connection between the fixed point z∈S and the existence of a solution of (1.1) which divided by ℬ⌊(n-1)/ω⌋ provides an asymptotically ω-periodic sequence. Considering (2.27) for z(n+1) and z(n), we get
(2.30)Δz(n)=a(n)β(n+1)+∑i=0nβ(i)β(n+1)K(n,i)z(i),
where n∈ℕ0. Hence, we have
(2.31)z(n+1)-z(n)=a(n)β(n+1)+1β(n+1)∑i=0nβ(i)K(n,i)z(i),n∈N0.
Putting
(2.32)z(n)=x(n)β(n),n∈N0
in (2.31), we get (1.1) since
(2.33)x(n+1)β(n+1)-x(n)β(n)=a(n)β(n+1)+1β(n+1)∑i=0nK(n,i)x(i),n∈N0
yields
(2.34)x(n+1)=a(n)+b(n)x(n)+∑i=0nK(n,i)x(i),n∈N0.
Consequently, x defined by (2.32) is a solution of (1.1). From (2.29) and (2.32), we obtain
(2.35)x(n)β(n)=z(n)=c+o(1),
for n→∞ (where o(1) is the Landau order symbol). Hence,
(2.36)x(n)=β(n)(c+o(1)),n⟶∞.
It is easy to show that the function β defined by (2.1) can be expressed in the form
(2.37)β(n)=∏j=0n-1b(j)=B⌊(n-1)/ω⌋⋅β(n*+1),
for n∈ℕ0. Then, as follows from (2.36),
(2.38)x(n)=B⌊(n-1)/ω⌋⋅β(n*+1)(c+o(1)),n⟶∞,
or
(2.39)x(n)B⌊(n-1)/ω⌋=cβ(n*+1)+β(n*+1)o(1),n⟶∞.
The proof is completed since the sequence{β(n*+1)} is ω-periodic, hence bounded and, due to the properties of Landau order symbols, we have
(2.40)β(n*+1)o(1)=o(1),n⟶∞,
and it is easy to see that the choice
(2.41)u(n):=cβ(n*+1),w(n):=B⌊(n-1)/ω⌋,n∈N0,
and an appropriate function v:ℕ0→ℝ such that
(2.42)lim⁡n→∞v(n)=0
finishes this part of the proof. Although for n=0, there is no correspondence between formula (2.36) and the definitions of functions u and w, we assume that function v makes up for this.Case 2.
Ifc<0, we can proceed as follows. It is easy to see that arbitrary solution y=y(n) of the equation
(2.43)y(n+1)=-a(n)+b(n)y(n)+∑i=0nK(n,i)y(i)
defines a solution x=x(n) of (1.1) since a substitution y(n)=-x(n) in (2.43) turns (2.43) into (1.1). If the assumptions of Theorem 2.2 hold for (1.1), then, obviously, Theorem 2.2 holds for (2.43) as well. So, for an arbitrary c>0, (2.43) has a solution that can be represented by formula (2.6), that is,
(2.44)y(n)B⌊(n-1)/ω⌋=cβ(n*+1)+v(n),n∈N0.
Or, in other words, (1.1) has a solution that can be represented by formula (2.44) as
(2.45)x(n)B⌊(n-1)/ω⌋=c0β(n*+1)+v*(n),n∈N0,
with c0=-c and v*(n)=-v(n). In (2.45), c0<0 and the function v*(n) has the same properties as the function v(n). Therefore, formula (2.6) is valid for an arbitrary negative c as well.Now, we give an example which illustrates the case where there exists a solution to equation of the type (1.1) which is weighted asymptotically periodic, but is not asymptotically periodic.Example 2.3.
We consider (1.1) with
(2.46)a(n)=(-1)n+1(1-13n+1),b(n)=3(-1)n,K(n,i)=(-1)n+(i(i-1))/2132i,
that is, the equation
(2.47)x(n+1)=(-1)n+1(1-13n+1)+3(-1)nx(n)+∑i=0n(-1)n+(i(i-1))/2132ix(i).
The sequence b(n) is 2-periodic and
(2.48)β(n)=∏j=0n-1b(j)=(-1)n(n-1)/23n,B=β(ω)=β(2)=-9,β(n*+1)=-3+6(-1)n+1,a(n)β(n+1)=(-1)(-n2+n+2)/2(13n+1-132(n+1)),∑i=0∞|a(i)β(i+1)|<∞,∑j=0∞∑i=0j|K(j,i)β(i)β(j+1)|<∑j=0∞∑i=0∞|K(j,i)β(i)β(j+1)|=∑j=0∞∑i=0∞13i+j+1=13(∑j=0∞13j)(∑i=0∞13i)=13⋅11-1/3⋅11-1/3=13⋅32⋅32=34<1.
By virtue of Theorem 2.2, for any nonzero constant c, there exists a solution x:ℕ0→ℝ of (1.1) which is weighed asymptotically 2-periodic. Let, for example, c=2/3. Then,
(2.49)w(n)=(-9)⌊(n-1)/2⌋,u(n)=cβ(n*+1)=23(-3+6(-1)n+1)=-2+4(-1)n+1,
and the sequence x(n) given by
(2.50)x(n)(-9)⌊(n-1)/2⌋=-2+4(-1)n+1+v(n),n∈N0,
or, equivalently,
(2.51)x(n)=(-9)⌊(n-1)/2⌋(-2+4(-1)n+1)+v(n),n∈N0
is such a solution. We remark that such solution is not asymptotically 2-periodic in the meaning of Definition 1.1.
It is easy to verify that the sequencex*(n) obtained from (2.51) if v(n)=0, n∈ℕ0, that is,
(2.52)x*(n)=(-9)⌊(n-1)/2⌋(-2+4(-1)n+1)=23⋅(-1)n(n-1)/2⋅3n,n∈N0
is a true solution of (2.47).


## 3. Concluding Remarks and Open Problems

It is easy to prove the following corollary.Corollary 3.1.
Let Theorem2.2 be valid. If, moreover, |ℬ|<1, then every solution x=x(n) of (1.1) described by formula (2.6) satisfies
(3.1)lim⁡n→∞x(n)=0.
If |ℬ|>1, then, for every solution x=x(n) of (1.1) described by formula (2.6), one has
(3.2)lim⁡⁡inf⁡n→∞x(n)=-∞
or/and
(3.3)lim⁡⁡sup⁡n→∞x(n)=∞.
Finally, if ℬ>1, then, for every solution x=x(n) of (1.1) described by formula (2.6), one has
(3.4)lim⁡n→∞x(n)=∞,
and if ℬ<-1, then, for every solution x=x(n) of (1.1) described by formula (2.6), one has
(3.5)lim⁡n→∞x(n)=-∞.Now, let us discuss the case when (1.6) holds, that is, when (3.6)B=∏j=0ω-1b(j)=-1.Corollary 3.2.
Let Theorem2.2 be valid. Assume that ℬ=-1. Then, for any nonzero constant c, there exists an asymptotically 2ω-periodic solution x=x(n), n∈ℕ0 of (1.1) such that
(3.7)x(n)=(-1)⌊(n-1)/ω⌋u(n)+z(n),n∈N0,
with
(3.8)u(n):=cβ(n*+1),lim⁡n→∞z(n)=0.Proof.
Puttingℬ=-1 in Theorem 2.2, we get
(3.9)x(n)=(-1)⌊(n-1)/ω⌋u(n)+(-1)⌊(n-1)/ω⌋v(n),
with
(3.10)u(n):=cβ(n*+1),lim⁡n→∞v(n)=0.
Due to the definition ofn*, we see that the sequence
(3.11){β(n*+1)}={β(ω),β(1),β(2),…,β(ω),β(1),β(2),…,β(ω),…},
is an ω-periodic sequence. Since
(3.12){⌊n-1ω⌋}={-1,0,…,0︸ω,1,…,1︸ω,2,…},
for n∈ℕ0, we have
(3.13){(-1)⌊(n-1)/ω⌋}={-1,1,…,1︸ω,-1,…,-1︸ω,1,…}.
Therefore, the sequence
(3.14){(-1)⌊(n-1)/ω⌋u(n)}=c{-β(ω),β(1),β(2),…,β(ω),-β(1),-β(2),…,-β(ω),…}
is a 2ω-periodic sequence. Set
(3.15)z(n)=(-1)⌊(n-1)/ω⌋v(n).
Then,
(3.16)lim⁡n→∞z(n)=0.
The proof is completed.Remark 3.3.
From the proof, we see that Theorem2.2 remains valid even in the case of c=0. Then, there exists an “asymptotically weighted ω-periodic solution” x=x(n) of (1.1) as well. The formula (2.6) reduces to
(3.17)x(n)=B⌊(n-1)/ω⌋v(n)=o(1),n∈N0,
since u(n)=0. In the light of Definition 1.2, we can treat this case as follows. We set (as a singular case) u≡0 with an arbitrary (possibly other than “ω”) period and with v=o(1), n→∞.Remark 3.4.
The assumptions of Theorem2.2 [1] are substantially different from those of the present Theorem 2.2. However, it is easy to see that Theorem 2.2 [1] is a particular case of the present Theorem 2.2 if (1.3) holds, that is, if ℬ=1. Therefore, our results can be viewed as a generalization of some results in [1].In connection with the above investigations, some open problems arise.Open Problem 1.
The results of [1] are extended to systems of linear Volterra discrete equations in [16, 17]. It is an open question if the results presented can be extended to systems of linear Volterra discrete equations.Open Problem 2.
Unlike the result of Theorem2.2 [1] where a parameter c can be arbitrary, the assumptions of the results in [16, 17] are more restrictive since the related parameters should satisfy certain inequalities as well. Different results on the existence of asymptotically periodic solutions were recently proved in [8]. Using an example, it is shown that the results in [8] can be less restrictive. Therefore, an additional open problem arises if the results in [16, 17] can be improved in such a way that the related parameters can be arbitrary and if the expected extension of the results suggested in Open Problem 1 can be given in such a way that the related parameters can be arbitrary as well.



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