1. Introduction
We denote by
the set of positive integers and by
the set of real numbers. Assume
is an integer,
,
and consider difference equations of the form
By a solution of (
1) we mean a sequence
satisfying (1) for all large
n. We say that Equation (1) is of monotone type if one of the following conditions is satisfied:
- (a)
f is non-decreasing with respect to the second variable and for all ;
- (b)
f is non-increasing with respect to the second variable and for all .
By studying the hereditary influences in population growth models Vito Volterra obtained an equation of the form
which was termed the Volterra integro-differential equation. The non-linear Volterra integro-differential equation of the form
appears also in many problems. Volterra equations are frequently used to describe many real world phenomena concerning biology, chemistry, physics, mechanics, economy, medicine, population dynamics, and others. For more information on the theory and applications of linear and non-linear Volterra integro-differential equations we refer readers to the books by Burton [
1] and Wazwaz [
2] and, for example, the papers [
3,
4,
5,
6].
In the last four decades many authors have studied the qualitative properties of solutions of discrete Volterra equations. In particular, asymptotic properties of solutions of first order Volterra difference equations were considered, e.g., in [
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20] or [
21]. For example, in [
19] the necessary and sufficient condition for boundedness of all solutions of the linear Volterra equation
are obtained. In [
8], the authors established conditions under which every solution of the system of linear Volterra equations
is convergent. Some population models described by Volterra difference equations can be found in the recent monograph by Raffoul [
22]. However, there are relatively few papers devoted to the higher order discrete Volterra equations, see [
23,
24,
25].
In this paper we investigate asymptotic behavior of solutions to Equation (1) which is a discrete analog of Equation (
2). We mainly deal with problems of two types. The first is the problem of the existence of solutions with prescribed asymptotic behavior. The second problem is the approximation of a given solution of Equation (1). Studies on solutions with prescribed asymptotic behavior are usually based on the application of the Schauder or Darboux type theorems. In this case, conditions of the continuity type are superimposed on the function
f. We use the Knaster-Tarski theorem. Using the Knaster-Tarski theorem, we replace the conditions of the continuity type with the conditions of the monotonicity type. This allows us to apply our results to, e.g., floor function, ceiling function, or other locally constant functions. To our knowledge, the asymptotic properties of solutions to the Volterra equations of the monotonic type have not been studied. We believe that the case of monotonic type equations, e.g., with a locally constant function
f, is important in the application of numerical methods.
We use techniques from [
26] based on the use of the iterated remainder operator. This allows us to control the degree of approximation of solutions. In this paper, we choose a positive non-increasing sequence
u and use
as a measure of approximation. Two particularly important approximation cases can be obtained when
u is a power sequence or a geometric sequence. More precisely, if
for some fixed
, then we have the so-called harmonic approximation. If
, where
is fixed, then we have the geometric approximation. It is worth noting that even in the case of
, i.e., in the case when
is a degree of approximation, our results are new.
The organization of the paper is as follows. In
Section 2, we introduce some notations and terminology. Moreover, we present two basic lemmas. In
Section 3, we present and prove two theorems. They are the main results of the paper. In
Section 4, we present a number of different consequences of Theorems 1 and 2.
Section 5 provides examples, remarks and additional results. Some conclusions are given in
Section 6.
2. Preliminaries
We denote by
the set of all integers and
is the space of all sequences
. We will use the convention
whenever
and
. Let
. We will use the following notations
For any
we define the sequence
by
Then
and
for any
and any
. Moreover
for any
and any
. It is easy to see that if
and
, then
. For more information about the operator
see [
26]. We will use the following consequence of the Knaster-Tarski fixed point theorem.
Lemma 1. ([
27], Lemma 4.9)
. Let and let S denote the setwith natural order defined by: if for any . Then every non-decreasing map has a fixed point. We will also need the following lemma.
Lemma 2. ([
28], Lemma 2.3)
. Assume u is a positive and non-decreasing sequence,Then there exists a sequence such that and .
For
we use the factorial notation
Moreover, we will use the ceiling function
defined by
3. Main Results
We present two theorems in this section. In Theorem 1 we deal with the problem of the existence of solutions with prescribed asymptotic behavior. More precisely, for a given solution y of the equation and a given positive and non-increasing sequence u we present the sufficient conditions for the existence of a solution x to Equation (1) such that . The proof of this theorem is based on the Knaster-Tarski fixed point theorem. To use the Knaster-Tarski theorem, it is necessary to assume that Equation (1) is of the monotone type.
Theorem 2 is devoted to the problem of approximating the solutions of (1). For a given solution x of Equation (1) and a given positive and non-increasing sequence u, we establish the sufficient conditions for the existence of a solution y to equation such that . In Theorem 2, we do not need to assume that Equation (1) is of monotone type.
Theorem 1. Assume , , g is locally bounded,w is bounded, , and (1) is of monotone type. Then, for any solution y of the equation such that, there exists a solution x of (1) with the property . Proof. Assume
,
, and
. Let
By (7), there exists a constant
K such that if
and
, then
Since
g is locally bounded, there exists a positive constant
M such that
. Therefore, using (7) we have
for
and
. Let
be defined by
The sequence
u is positive and non-increasing. Hence, using (8), we have
. So there exists an index
p such that
for
. Let
,
Define an operator
by
Hence
for any
. Let
Then
. Define an operator
by
Hence
. Now we assume that the condition (a) of the definition of monotonicity of (1) is fulfilled. The proof in the case (b) is analogous. Let
,
. If
, then
Hence
. Since the operator
is non-decreasing, we get
By Lemma 1, there exists a sequence
such that
. Then, for
, we have
Hence
for
. Therefore
x is a solution of (1). Now we will show that
Define sequences
by
Then
. Hence
and using (5) we get
By (3),
. Hence
. Analogously,
. Thus
Now, using (10), we obtain . The proof is complete. □
Theorem 2. Assume , , g is locally bounded,w is bounded, and . Then for any solution x of (1) such that there exists a solution y of the equation such that, . Proof. Assume
x is a solution of (1) such that
. There exists a positive constant
K such that
for any
n. Since
g is locally bounded, there exists a positive constant
M such that
. By (11) we have
for any
. Define a sequence
by
Since
x is a solution of (1), we have
for large
n. Hence there exists a constant
such that
for any
. Using (12) we get
By Lemma 2, there exists a sequence
z such that
Let
. Then
for any
. Moreover
. □
We say that a sequence
is standard if
For example, if , then the sequence is standard. If , then the sequence is standard. It is easy to see that a sum of two standard sequences is standard. In particular any polynomial sequence is standard. The sequence is not standard.
Remark 1. Assume is a positive standard sequence. Then the sequence is also standard. In this case, condition in Theorem 1 can be replaced by condition . Similarly, condition in Theorem 2 can be replaced by condition .
5. Examples, Remarks, and Additional Results
We start with an example illustrating Theorem 1.
Example 1. Then, Equation (1) takes the form So, (1) is of monotone type. It is easy to check that is a solution of equation Set , . Then . It is easy to check that Thus, by Theorem 1, there exists a solution x of (15) such that . For example the sequenceis such a solution. Condition
may be difficult to verify. The following lemma may facilitate the verification of this condition.
Lemma 3. Assume , , , ,and at least one of the following conditions is satisfied Then, the condition (17) is satisfied.
Proof. Using ([
27], Lemma 4.4, Lemma 4.5) and ([
29], Lemma 6.4) we get
Since for any n, we obtain (17). □
Example 2. Let , . Define a kernel K and a sequence u by If is defined by (18), then and Hence, by Lemma 3 we get (17).
The following lemma can be the basis for the theory of ‘geometric approximation’ of the solutions of Equation (1).
Lemma 4. Assume , , is defined by (18), and Proof. Define a sequence
and a number
by
It is clear that, in Lemma 4, condition (19) can be replaced by condition:
Example 3. Therefore, by Corollary 3, for any solution y of the equation such that , there exists a solution x of the equationsuch that . Now we turn to the problem of asymptotically periodic solutions to Equation (1). Let
. We say that a sequence
is
q-balanced if it is
q-periodic and
Example 4. If , then the sequenceis 6-balanced. More generally, we say that a sequence is q-symmetric iffor any . It is easy to see that any q-symmetric sequence γ is -balanced. Lemma 5. ([
27], Lemma 7.7)
. Assume and is q-balanced. Then there exists a q-periodic sequence such that . Corollary 12. Assume the assumptions of Theorem 1 are satisfied, , and the sequence b is q-balanced. Then there exists a q-periodic solution y of the equation such that for any there exists an asymptotically q-periodic solution x of (1) such that .
Proof. By Lemma 5 there exists a q-periodic solution y of the equation . Let . Then the sequence is bounded and . By Corollary 2 there exists a solution x of (1) such that .□
Remark 2. If the assumptions of Theorem 1 are satisfied, , a sequence is q-symmetric and , then, by Corollary 2, there exists an asymptotically symmetric solution x of (1), such that .
Below we establish conditions under which any bounded solution of (1) is asymptotically periodic.
Corollary 13. Assume the assumptions of Theorem 2 are satisfied, , and the sequence b is q-balanced. Then, for any bounded solution x of (1) there exists a q-periodic sequence y such that .
Proof. Let
x be a bounded solution of (1). By Corollary 8 there exists a solution
y of the equation
such that
. By Lemma 5 there exists a
q-periodic sequence
such that
. Let
. Then
. Hence
is a polynomial sequence. Moreover,
Hence is bounded. Therefore, the sequence is constant and is q-periodic. □
We say that a sequence is -bounded if the sequence is bounded. For -bounded solutions of Equation (1) we have the following simple version of Theorem 2.
Theorem 3. Then, for any -bounded solution x of (1) there exists a solution y of the equation such that, .
Proof. Let
x be an
-bounded solution of (1) and let
M be a positive constant such that
for any
. Now, repeating the second part of the proof of Theorem 2 we get the result. □
Corollary 14. Then, for any -bounded solution x of (1) there exists a polynomial φ such that and .
Proof. Assume x is an -bounded solution of (1). By Theorem 3 there exists a sequence y such that and . By Lemma 2 there exists a sequence z such that and . Let . Then is a polynomial sequence, , and . □
Below we present conditions under which any solution of (1) is asymptotically polynomial.
Corollary 15. Assume (22), (23), and f is bounded. Then for any solution x of (1) there exists a polynomial φ with the property and .
Proof. If f is bounded, then any sequence is -bounded. Hence the assertion follows from Corollary 14. □
Finally, we present a version of Theorem 1 relating to the case of an ordinary difference equation. In this case, our result is also new.
Theorem 4. Assume , , g is locally bounded,w is bounded, , and one of the following conditions is satisfied: - (a)
f is non-decreasing with respect to the second variable and
for all n,
- (b)
f is non-increasing with respect to the second variable and
for all n.
Then, for any solution y of the equation such that, there exists a solution x of the equationsuch that . Proof. Let us define a map
by
Then, the assumptions of Theorem 1 are satisfied and Equation (1) takes the form (26). Hence, using Theorem 1, we obtain the result. □