- Research Article
- Open Access

# Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics

- Lukáš Rachůnek
^{1}and - Irena Rachůnková
^{1}Email author

**2010**:714891

https://doi.org/10.1155/2010/714891

© L. Rachůnek and I. Rachůnková. 2010

**Received:**19 December 2009**Accepted:**10 March 2010**Published:**16 March 2010

## Abstract

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.

## Keywords

- Differential Equation
- Partial Differential Equation
- Unique Solution
- Ordinary Differential Equation
- Functional Analysis

## 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 [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.

Then
is called *a damped solution*.

Definition 2.2.

Then
is called *a homoclinic solution*.

Definition 2.3.

Then
is called *an escape solution*.

Definition 2.4.

Then
is called *a non-monotonous solution*.

Lemma 2.5 (see [1] (on four types of solutions)).

- (I)
is an escape solution;

- (II)
is a homoclinic solution;

- (III)
is a damped solution;

- (IV)
is a non-monotonous solution.

Lemma 2.6 (see [1] (estimates of solutions)).

In [1] 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 [1] (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 [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.

This yields that is increasing.

Lemma 3.2.

where , .

Proof.

which yields (3.3).

## 4. Existence of Escape Solutions

Lemma 4.1.

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.

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 .

Proof.

We have the following steps.

Step 1.

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.

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).

## Declarations

### Acknowledgments

The paper was supported by the Council of Czech Government MSM 6198959214. The authors thank the referees for valuable comments.

## Authors’ Affiliations

## References

- Rachůnek L:
**On four types of solutions.**submitted, http://phoenix.inf.upol.cz/~rachunekl/mathair/rr7.pdf - Berestycki H, Lions P-L, Peletier LA:
**An ODE approach to the existence of positive solutions for semilinear problems in**.*Indiana University Mathematics Journal*1981,**30**(1):141-157. 10.1512/iumj.1981.30.30012MathSciNetView ArticleMATHGoogle Scholar - Derrick GH:
**Comments on nonlinear wave equations as models for elementary particles.***Journal of Mathematical Physics*1964,**5:**1252-1254. 10.1063/1.1704233MathSciNetView ArticleGoogle Scholar - Dell'Isola F, Gouin H, Rotoli G:
**Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations.***European Journal of Mechanics—B/Fluids*1996,**15:**545-568.MATHGoogle Scholar - Gouin H, Rotoli G:
**An analytical approximation of density profile and surface tension of microscopic bubbles for Van der Waals fluids.***Mechanics Research Communications*1997,**24:**255-260. 10.1016/S0093-6413(97)00022-0View ArticleMATHGoogle Scholar - Kitzhofer G, Koch O, Lima P, Weinmüller E:
**Efficient numerical solution of the density profile equation in hydrodynamics.***Journal of Scientific Computing*2007,**32**(3):411-424. 10.1007/s10915-007-9141-0MathSciNetView ArticleMATHGoogle Scholar - Rachůnek L, Rachůnková I: On a homoclinic point of some autonomous second-order difference equation. submitted to Journal of Difference Equations and ApplicationsGoogle Scholar
- Amleh AM, Camouzis E, Ladas G:
**On second-order rational difference equation. I.***Journal of Difference Equations and Applications*2007,**13**(11):969-1004. 10.1080/10236190701388492MathSciNetView ArticleMATHGoogle Scholar - Berenhaut KS, Stević S:
**The difference equation****has solutions converging to zero.***Journal of Mathematical Analysis and Applications*2007,**326**(2):1466-1471. 10.1016/j.jmaa.2006.02.088MathSciNetView ArticleMATHGoogle Scholar - Berg L:
**On the asymptotics of the difference equation**.*Journal of Difference Equations and Applications*2008,**14**(1):105-108. 10.1080/10236190701503041MathSciNetView ArticleMATHGoogle Scholar - Gutnik L, Stević S:
**On the behaviour of the solutions of a second-order difference equation.***Discrete Dynamics in Nature and Society*2007,**2007:**-14.Google Scholar - Hu L-X, Li W-T, Stević S:
**Global asymptotic stability of a second order rational difference equation.***Journal of Difference Equations and Applications*2008,**14**(8):779-797. 10.1080/10236190701827945MathSciNetView ArticleMATHGoogle Scholar - Iričanin B, Stević S:
**Eventually constant solutions of a rational difference equation.***Applied Mathematics and Computation*2009,**215**(2):854-856. 10.1016/j.amc.2009.05.044MathSciNetView ArticleMATHGoogle Scholar - Rachůnková I, Rachůnek L:
**Singular discrete problem arising in the theory of shallow membrane caps.***Journal of Difference Equations and Applications*2008,**14**(7):747-767. 10.1080/10236190701843371MathSciNetView ArticleMATHGoogle Scholar - Rachůnková I, Rachůnek L:
**Singular discrete and continuous mixed boundary value problems.***Mathematical and Computer Modelling*2009,**49**(3-4):413-422. 10.1016/j.mcm.2008.09.004MathSciNetView ArticleMATHGoogle Scholar - Rachůnek L, Rachůnková I:
**Approximation of differential problems with singularities and time discontinuities.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71:**e1448-e1460. 10.1016/j.na.2009.01.183View ArticleMATHGoogle Scholar - Rouhani BD, Khatibzadeh H:
**A note on the asymptotic behavior of solutions to a second order difference equation.***Journal of Difference Equations and Applications*2008,**14**(4):429-432. 10.1080/10236190701825162MathSciNetView ArticleMATHGoogle Scholar - Stević S:
**Asymptotics of some classes of higher-order difference equations.***Discrete Dynamics in Nature and Society*2007,**2007:**-20.Google Scholar - Stević S:
**Asymptotic periodicity of a higher-order difference equation.***Discrete Dynamics in Nature and Society*2007,**2007:**-9.Google Scholar - Stević S:
**Existence of nontrivial solutions of a rational difference equation.***Applied Mathematics Letters*2007,**20**(1):28-31. 10.1016/j.aml.2006.03.002MathSciNetView ArticleMATHGoogle Scholar - Stević S:
**Nontrivial solutions of higher-order rational difference equations.***Matematicheskie Zametki*2008,**84**(5):772-780.MathSciNetView ArticleGoogle Scholar - Sun T, Xi H, Quan W:
**Existence of monotone solutions of a difference equation.***Discrete Dynamics in Nature and Society*2008,**2008:**-8.Google Scholar - Rachůnek L, Rachůnková I: Homoclinic solutions of non-autonomous difference equations arising in hydrodynamics. in preparationGoogle Scholar
- Elaydi SN:
*An Introduction to Difference Equations, Undergraduate Texts in Mathematics*. 2nd edition. Springer, New York, NY, USA; 1999:xviii+427.View ArticleGoogle Scholar

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