Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics
© L. Rachůnek and I. Rachůnková. 2010
Received: 19 December 2009
Accepted: 10 March 2010
Published: 16 March 2010
The paper provides conditions sufficient for the existence of strictly increasing solutions of the second-order 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 second-order non-autonomous 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  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 , we have described the set of all solutions of problem (1.1), (1.6), where
In this paper, using , 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 . 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  which we need in next sections. In particular, we will use the following definitions and lemmas.
Then is called a damped solution.
Then is called a homoclinic solution.
Then is called an escape solution.
Then is called a non-monotonous solution.
Lemma 2.5 (see  (on four types of solutions)).
is an escape solution;
is a homoclinic solution;
is a damped solution;
is a non-monotonous solution.
Lemma 2.6 (see  (estimates of solutions)).
In  we have proved that the set consisting of damped and non-monotonous solutions of problem (1.1), (1.6) is nonempty for each sufficiently small . This is contained in the next lemma.
Lemma 2.7 (see  (on the existence of non-monotonous 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 non-monotonous 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  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.
Let be an escape solution of problem (1.1), (1.6). Then is increasing.
This yields that is increasing.
where , .
which yields (3.3).
4. Existence of Escape Solutions
Moreover, if the sequence is unbounded, then there exists such that the solution of problem (1.1), (1.6) with is an escape solution.
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).
If , then (2.7) implies (4.17) and hence (4.19) holds.
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 .
We have the following steps.
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 .
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).
The paper was supported by the Council of Czech Government MSM 6198959214. The authors thank the referees for valuable comments.
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