- Research Article
- Open access
- Published:
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 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 [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 non-monotonous 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 non-monotonous 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 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.
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 Gronwall-Bellman 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 non-monotonous, 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).
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.1981.30.30012
Derrick GH: Comments on nonlinear wave equations as models for elementary particles. Journal of Mathematical Physics 1964, 5: 1252-1254. 10.1063/1.1704233
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.
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)00022-0
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/s10915-007-9141-0
Rachůnek L, Rachůnková I: On a homoclinic point of some autonomous second-order difference equation. submitted to Journal of Difference Equations and Applications
Amleh AM, Camouzis E, Ladas G: On second-order rational difference equation. I. Journal of Difference Equations and Applications 2007,13(11):969-1004. 10.1080/10236190701388492
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.2006.02.088
Berg L:On the asymptotics of the difference equation
. Journal of Difference Equations and Applications 2008,14(1):105-108. 10.1080/10236190701503041
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.
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. 10.1080/10236190701827945
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.2009.05.044
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. 10.1080/10236190701843371
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.2008.09.004
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.2009.01.183
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. 10.1080/10236190701825162
Stević S: Asymptotics of some classes of higher-order difference equations. Discrete Dynamics in Nature and Society 2007, 2007:-20.
Stević S: Asymptotic periodicity of a higher-order difference equation. Discrete Dynamics in Nature and Society 2007, 2007:-9.
Stević S: Existence of nontrivial solutions of a rational difference equation. Applied Mathematics Letters 2007,20(1):28-31. 10.1016/j.aml.2006.03.002
Stević S: Nontrivial solutions of higher-order rational difference equations. Matematicheskie Zametki 2008,84(5):772-780.
Sun T, Xi H, Quan W: Existence of monotone solutions of a difference equation. Discrete Dynamics in Nature and Society 2008, 2008:-8.
Rachůnek L, Rachůnková I: Homoclinic solutions of non-autonomous difference equations arising in hydrodynamics. in preparation
Elaydi SN: An Introduction to Difference Equations, Undergraduate Texts in Mathematics. 2nd edition. Springer, New York, NY, USA; 1999:xviii+427.
Acknowledgments
The paper was supported by the Council of Czech Government MSM 6198959214. The authors thank the referees for valuable comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/714891