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Oscillatory Solutions of Singular Equations Arising in Hydrodynamics
Advances in Difference Equations volume 2010, Article number: 872160 (2010)
Abstract
We investigate the singular differential equation on the half-line [
), where
satisfies the local Lipschitz condition on
and has at least two simple zeros. The function
is continuous on [
) and has a positive continuous derivative on (
) and
. We bring additional conditions for
and
under which the equation has oscillatory solutions with decreasing amplitudes.
1. Introduction
We study the equation

on the half-line , where


Equation (1.1) is singular at because
. If
in (1.1) fulfils moreover assumptions



then (1.1) generalizes equations which appear in hydrodynamics or in the nonlinear field theory [1–5].
Definition 1.1.
A function which has continuous second derivative on
and satisfies (1.1) for all
is called a solution of (1.1).
Consider and the initial conditions

The initial value problem (1.1), (1.7) has been investigated, for example, in [6–12]. In particular in [10] it was proved that for each negative there exists a unique solution of problem (1.1), (1.7) under the assumptions (1.2)–(1.6). Consider such solution
and denote

Definition 1.2.
If (
or
), then
is called a damped solution (a homoclinic solution or an escape solution) of problem (1.1), (1.7).
In [10, 12] these three types of solutions of problem (1.1), (1.7) have been studied, and the existence of each type has been proved for sublinear or linear asymptotic behaviour of near
. In [11],
has been supposed to have a zero
. Here we generalize and extend the results of [10–12] concerning damped solutions. We prove their existence under weaker assumptions than in the above papers. Moreover, we bring conditions under which each damped solution is oscillatory; that is, it has an unbounded set of isolated zeros.
We replace assumptions (1.4)–(1.6) by the following ones.
There exist ,
,
such that


( is possible).
2. Damped Solutions
Theorem 2.1 (Existence and uniqueness).
Assume that (1.2), (1.3), (1.9), and (1.10) hold and let . Then problem (1.1), (1.7) has a unique solution
, and moreover the solution
satisfies

Proof.
Step 1.
Put

We will study the auxiliary differential equation:

By virtue of (1.2) we find the Lipschitz constant for
on
, and due to (1.2), (1.10), and (2.2), we find
such that

Put for
. Having in mind (1.3), we see that
is increasing and so

Consequently we can choose such that

Consider the Banach space (with the maximum norm) and define an operator
by

Using (2.4) and (2.6), we have

that is maps the ball
to itself. Due to (2.2) and the choice of
, we have for
,

Hence is a contraction on
, and the Banach fixed point theorem yields a unique fixed point
of
.
Step 2.
The fixed point of Step 1 fulfils

Hence satisfies (2.3) on
. Finally, (2.4) and (2.5) yield

Consequently fulfils (1.7). Choose an arbitrary
. Then, by (2.5) and (2.10),

Having in mind that ,
can be (uniquely) extended as a function satisfying (2.3) onto
. Since
is arbitrary,
can be extended onto
as a solution of (2.3). We have proved that problem (2.3), (1.7) has a unique solution.
Step 3.
According to Step 2 we have

Multiplying (2.13) by and integrating between
and
, we get

Put

So,(2.14) has the form

Let for some
. Then (2.16) yields
which is not possible because
is decreasing on
by (1.9) and (2.2). Therefore
for
. Consequently, due to (2.2),
is a solution of (1.1).
Step 4.
Assume that there exists another solution of problem (1.1), (1.7). Then we can prove similarly as in Step 3 that
for
. This implies that
is also a solution of problem (2.3), (1.7) and by Step 2,
. We have proved that problem (1.1), (1.7) has a unique solution.
Lemma 2.2.
Let and let
be a solution of (1.1). Assume that there exists
such that

Then for all
.
Proof.
We see that the constant function is a solution of (1.1). Let
be a solution of (1.1) satisfying (2.17) and let
for some
. Then the regular initial problem (1.1), (2.17) has two different solutions
and
, which contradicts (1.2).
Remark 2.3.
Let us put

Due to (1.2) and (1.9) we see that is continuous on
, decreasing and positive on
, increasing and positive on
. Therefore we can define
by

(.
Theorem 2.4 (Existence of damped solutions).
Assume that (1.2), (1.3), (1.9), and (1.10) hold. Let be given by (2.19), and assume that
is a solution of problem (1.1), (1.7) with
. Then
is a damped solution.
Proof.
Since , we can find
such that

Assume on the contrary that is not damped, that is,

Then, according to Lemma 2.2, there exists such that

By (1.1), (1.3), and (1.9) we have on
. So,
is increasing and positive on
and hence
on
. Assumption (2.21) implies that there exists
such that

Since fulfils (1.1), we have

Multiplying (2.24) by and integrating between
and
we get

This contradicts (2.20).
3. Oscillatory Solutions
In this section we assume that, in addition to our basic assumptions (1.2), (1.3), (1.9), and (1.10), the following conditions are fulfilled:


Then the next lemmas can be proved.
Lemma 3.1.
Let be a solution of problem (1.1), (1.7) with
. Then there exists
such that

Proof.
Step 1.
Assume that such does not exist. Then

Hence (1.1), (1.7), and (1.9) yield and
on
. Therefore
is increasing on
and

Multiplying (2.24) by and integrating between
and
, we get due to (2.18)

Letting , we get

Since the function is positive and increasing, it follows that there exists
. If
, then
contrary to (3.5). Consequently,

Letting in (2.24), we get by (1.3), (1.9), and (3.5)

Due to (3.8), we conclude that and hence
. We have proved that if
fulfilling (3.3) does not exist, then

Step 2.
We define a function

By (1.3) and (3.2), we have ,



Due to (1.3), (3.1), (3.10) and (3.14) there exist and
such that

Due to (3.4), (3.11), (3.13), and (3.15), we get

Thus, is increasing on
and has the limit

If , then
, which contradicts (3.4) and (3.11). If
, then
on
and

In view of (3.16) we can see that

We get which contradicts
. The obtained contradictions imply that (3.4) cannot occur and hence
satisfying (3.3) must exist.
Corollary 3.2.
Let be a solution of problem (1.1), (1.7) with
. Further assume that there exist
and
such that

Then there exists such that

Proof.
We can argue as in the proof of Lemma 3.1 working with and
instead of
and
.
Lemma 3.3.
Let be a solution of problem (1.1), (1.7) with
. Further assume that there exist
and
such that

Then there exists such that

Proof.
We argue similarly as in the proof of Lemma 3.1.
Step 1.
Assume that such does not exist. Then

By (1.1), (1.7), and (1.9) we deduce on
and

Multiplying (2.24) by , integrating between
and
, and using (2.18), we obtain

and we derive as in the proof of Lemma 3.1 that (3.10) holds.
Step 2.
We define by (3.11) and get (3.13) for
. As in the proof of Lemma 3.1 we find
and
satisfying (3.15). Due to (3.24), (3.11), (3.13), and (3.15) we get

So, is decreasing on
and
. If
, then
which contradicts (3.24) and (3.11). If
, then
on
and

In view of (3.27) we can see that

We get contrary to
. The obtained contradictions imply that (3.24) cannot occur and that
satisfying (3.23) must exist.
Theorem 3.4.
Assume that (1.2), (1.3), (1.9), (1.10), (3.1), and (3.2) hold. Let be a solution of problem (1.1), (1.7) with
. If
is a damped solution, then
is oscillatory and its amplitudes are decreasing.
Proof.
Let be a damped solution. By (2.1) and Definition 1.2, we can find
such that

Step 1.
Lemma 3.1 yields satisfying (3.3). Hence there exists a maximal interval
such that
on
. Let
. Then, by (3.30), we get
,
on
and

By (1.1), (1.3), and (1.9), we have on
. So
and
are decreasing on
and, due to (3.31),

Letting in (2.24) and using (1.3), (1.9), and (3.31), we get

which contradicts (3.32). Therefore and there exists
such that (3.22) holds. Lemma 3.3 yields
satisfying (3.23). Therefore
has just one positive local maximum
between its first zero
and second zero
.
Step 2.
By (3.23) there exists a maximal interval , where
. Let
. Then, by (3.30), we have
,
on
, and

By (1.1), (1.3), and (1.9), we get on
and so
is increasing on
. Since
, we deduce that
is increasing on
and, by (3.34), we get (3.32). Letting
in (1.1) and using (1.3), (1.9), and (3.34), we get

which contradicts (3.32). Therefore and there exists
such that (3.20) holds. Corollary 3.2 yields
satisfying (3.21). Therefore
has just one negative minimum
between its second zero
and third zero
.
Step 3.
We can continue as in Step 1 and Step 2 and get the sequences and
of local maxima and local minima of
attained at
and
, respectively. Now, put
,
and write (1.1) as a system

Consider of (2.18) and define a Lyapunov function
by

where . By Remark 2.3, we see that
and
on
. By (3.6) and (3.37), we have

Therefore

By (3.30), for
. We see that
is positive and decreasing (for the damped solution
) and hence

So, sequences and
are decreasing:

for and

Further, due to Remark 2.3, the sequence is decreasing and the sequence
is increasing. Consequently,

Remark 3.5.
There are two cases for the number from the proof of Theorem 3.4:
and
. Denote

If , then
and hence
, that is,
.
Let . Consider an arbitrary sequence
such that
. By (3.40) we have
. By (3.30) and (3.6), the sequence
is bounded and so there exists a subsequence

such that , where
is a point of the level curve:

Note that

Theorem 3.6 (Existence of oscillatory solutions).
Assume that (1.2), (1.3), (1.9), (1.10), (3.1), and (3.2) hold. Let be given by (2.19) and let
be a solution of problem (1.1), (1.7) with
. Then
is an oscillatory solution with decreasing amplitudes.
Proof.
The assertion follows from Theorems 2.4 and 3.4.
Remark 3.7.
The assumption (1.10) in Theorem 3.6 can be omitted, because it has no influence on the existence of oscillatory solutions. It follows from the fact that (1.10) imposes conditions on the function values of the function for arguments greater than
; however, the function values of oscillatory solutions are lower than this constant
. This condition (used only in Theorem 2.1) guaranteed the existence of solution of each problem (1.1), (1.7) for each
on the whole half-line, which simplified the investigation of the problem.
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Acknowledgment
This work was supported by the Council of Czech Government MSM 6198959214.
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Rachůnková, I., Tomeček, J. & Stryja, J. Oscillatory Solutions of Singular Equations Arising in Hydrodynamics. Adv Differ Equ 2010, 872160 (2010). https://doi.org/10.1155/2010/872160
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DOI: https://doi.org/10.1155/2010/872160