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Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics
Advances in Difference Equations volume 2010, Article number: 714891 (2010)
Abstract
The paper provides conditions sufficient for the existence of strictly increasing solutions of the secondorder nonautonomous difference equation , , where is a parameter and is Lipschitz continuous and has three real zeros . In particular we prove that for each sufficiently small there exists a solution such that is increasing, and . The problem is motivated by some models arising in hydrodynamics.
1. Formulation of Problem
We will investigate the following secondorder nonautonomous difference equation
where is supposed to fulfil
Let us note that means that for each there exists such that for all . A simple example of a function satisfying (1.2)–(1.4) is , where is a positive constant.
A sequence which satisfies (1.1) is called a solution of (1.1). For each values there exists a unique solution of (1.1) satisfying the initial conditions
Then is called a solution of problem (1.1), (1.5).
In [1] we have shown that (1.1) is a discretization of differential equations which generalize some models arising in hydrodynamics or in the nonlinear field theory; see [2–6]. Increasing solutions of (1.1), (1.5) with has a fundamental role in these models. Therefore, in [1], we have described the set of all solutions of problem (1.1), (1.6), where
In this paper, using [1], we will prove that for each sufficiently small there exists at least one such that the corresponding solution of problem (1.1), (1.6) fulfils
Note that an autonomous case of (1.1) was studied in [7]. We would like to point out that recently there has been a huge interest in studying the existence of monotonous and nontrivial solutions of nonlinear difference equations. For papers during last three years see, for example, [8–22]. A lot of other interesting references can be found therein.
2. Four Types of Solutions
Here we present some results of [1] which we need in next sections. In particular, we will use the following definitions and lemmas.
Definition 2.1.
Let be a solution of problem (1.1), (1.6) such that
Then is called a damped solution.
Definition 2.2.
Let be a solution of problem (1.1), (1.6) which fulfils
Then is called a homoclinic solution.
Definition 2.3.
Let be a solution of problem (1.1), (1.6). Assume that there exists , such that is increasing and
Then is called an escape solution.
Definition 2.4.
Let be a solution of problem (1.1), (1.6). Assume that there exists , , such that is increasing and
Then is called a nonmonotonous solution.
Lemma 2.5 (see [1] (on four types of solutions)).
Let be a solution of problem (1.1), (1.6). Then is just one of the following four types:

(I)
is an escape solution;

(II)
is a homoclinic solution;

(III)
is a damped solution;

(IV)
is a nonmonotonous solution.
Lemma 2.6 (see [1] (estimates of solutions)).
Let be a solution of problem (1.1), (1.6). Then there exists a maximal satisfying
Further, if , then moreover
for if , and for if , where
In [1] we have proved that the set consisting of damped and nonmonotonous solutions of problem (1.1), (1.6) is nonempty for each sufficiently small . This is contained in the next lemma.
Lemma 2.7 (see [1] (on the existence of nonmonotonous or damped solutions)).
Let , where is defined by (1.4). There exists such that if , then the corresponding solution of problem (1.1), (1.6) is nonmonotonous or damped.
In Section 4 of this paper we prove that also the set of escape solutions of problem (1.1), (1.6) is nonempty for each sufficiently small . Note that in our next paper [23] we prove this assertion for the set of homoclinic solutions.
3. Properties of Solutions
Now, we provide other properties of solutions important in the investigation of escape solutions.
Lemma 3.1.
Let be an escape solution of problem (1.1), (1.6). Then is increasing.
Proof.
Due to (1.1), fulfils
According to Definition 2.3 there exists , such that is increasing and (2.3) holds. By (1.3) we get . Consequently, by (3.1) and (2.3), and . Similarly and
This yields that is increasing.
Lemma 3.2.
Assume that for . Choose an arbitrary . Let and let and be a solution of problem (1.1), (1.6) with and , respectively. Let be the Lipschitz constant for on . Then
where , .
Proof.
By (3.1) we have
Summing it for , we get by (1.6)
Summing it again for , we get
and similarly
From this and by using summation by parts we easily obtain
By the discrete analogue of the GronwallBellman inequality (see, e.g., [24, Lemma ]), we get
which yields (3.3).
By (3.6) and (3.3) we have for , ,
4. Existence of Escape Solutions
Lemma 4.1.
Assume that and . Let be a solution of problem (1.1), (1.6) with , . For choose a maximal such that for if is finite, and for if , and is increasing if . Then there exists such that for any there exists a unique , , such that
Moreover, if the sequence is unbounded, then there exists such that the solution of problem (1.1), (1.6) with is an escape solution.
Proof.
Choose such that
For denote by a solution of problem (1.1), (1.6) with . The existence of is guaranteed by Lemma 2.6. By Lemma 2.5, is just one of the types (I)–(IV), and if , then the monotonicity of yields a unique , , satisfying (4.1).
For , consider the sequence and assume that it is unbounded. Then we have
(otherwise we take a subsequence.) Assume on the contrary that for any , is not an escape solution. Choose . If is damped, then by Definition 2.1, we have and
If is homoclinic, then by Definition 2.2, we have and
If is nonmonotonous, then by Definition 2.4, we have and
To summarize if is not an escape solution, then by (4.4), (4.5), and (4.6), we have
Since , there exists satisfying
Consider (3.5) with . By dividing it by , multiplying such obtained equality by and summing in from 1 to we get
Denote
Then we get
Let us put and to (4.11) and subtract. By (4.7) and (4.8) we get
Let us put and to (4.10) and subtract. We get
Choose and such that
Let . Then (4.6) holds. Since , and , (3.1) yields
and hence
Clearly, if , then by (4.4) and (4.5), inequality (4.16) holds, as well. By (1.2), is integrable on . So, having in mind (4.1), we can find such that if
then
Therefore, due to (1.3) and (4.7),
Let be such that
If , then (2.7) implies (4.17) and hence (4.19) holds.
Now, let us put and choose . Then, (4.2), (4.14), (4.20), and (4.13)–(4.19) yield
Finally, (4.12) and (4.21) imply
Letting , we obtain, by (4.3), that , contrary to (4.17). Therefore an escape solution of problem (1.1), (1.6) with must exist.
Now, we are in a position to prove the next main result.
Theorem 4.2 (On the existence of escape solutions).
There exists such that for any the initial value problem (1.1), (1.6) has an escape solution for some .
Proof.
We have the following steps.
Step 1.
Let us define
and consider an auxiliary equation
Let be the constant of Lemma 4.1 for problem (4.24), (1.6). Choose , and let be the Lipschitz constant for on . Consider a sequence such that . Then, for each there exists such that
Let for . Then the sequence is the unique solution of problem (4.24), (1.6) with . Let be a solution of problem (4.24), (1.6) with , , and let be the sequence corresponding to by Lemma 4.1. We prove that is unbounded. According to Lemma 3.2, for each ,
Consequently, (4.25) and (4.26) give
and hence
Therefore
which yields that is unbounded. By Lemma 4.1, the auxiliary initial value problem (4.24), (1.6) has an escape solution for some . Denote this solution by .
Step 2.
By Definition 2.3, there exists such that
Now, consider the solution of our original problem (1.1), (1.6) with . Due to (4.23), for . Using (4.30) and Definition 2.3, we get that is an escape solution of problem (1.1), (1.6).
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Acknowledgments
The paper was supported by the Council of Czech Government MSM 6198959214. The authors thank the referees for valuable comments.
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Rachůnek, L., Rachůnková, I. Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics. Adv Differ Equ 2010, 714891 (2010). https://doi.org/10.1155/2010/714891
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DOI: https://doi.org/10.1155/2010/714891