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Homoclinic Solutions of Singular Nonautonomous Second-Order Differential Equations
Boundary Value Problems volume 2009, Article number: 959636 (2009)
Abstract
This paper investigates the singular differential equation , having a singularity at
. The existence of a strictly increasing solution (a homoclinic solution) satisfying
,
is proved provided that
has two zeros and a linear behaviour near
.
1. Introduction
Having a positive parameter we consider the problem


under the following basic assumptions for and






Then problem (1.1), (1.2) generalizes some models arising in hydrodynamics or in the nonlinear field theory (see [1–5]). However (1.1) is singular at because
.
Definition 1.1.
If , thena solution of (1.1) on
is a function
satisfying (1.1) on
. If
is a solution of (1.1) on
for each
, then
is a solution of (1.1) on
.
Definition 1.2.
Let be a solution of (1.1) on
. If
moreover fulfils conditions (1.2), it is called a solution of problem(1.1), (1.2).
Clearly, the constant function is a solution of problem (1.1), (1.2). An important question is the existence of a strictly increasing solution of (1.1), (1.2) because if such a solution exists, many important physical properties of corresponding models can be obtained. Note that if we extend the function
in (1.1) from the half–line onto
(as an even function), then any solution of (1.1), (1.2) has the same limit
as
and
. Therefore we will use the following definition.
Definition 1.3.
A strictly increasing solution of problem (1.1), (1.2) is called a homoclinic solution.
Numerical investigation of problem (1.1), (1.2), where and
,
, can be found in [1, 4–6]. Problem (1.1), (1.2) can be also transformed onto a problem about the existence of a positive solution on the half-line. For
,
and for
,
, such transformed problem was solved by variational methods in [7, 8], respectively. Some additional assumptions imposed on
were needed there. Related problems were solved, for example, in [9, 10].
Here, we deal directly with problem (1.1), (1.2) and continue our earlier considerations of papers [11, 12], where we looked for additional conditions which together with (1.3)–(1.8) would guarantee the existence of a homoclinic solution.
Let us characterize some results reached in [11, 12] in more details. Both these papers assume (1.3)–(1.8). In [11] we study the case that has at least three zeros
. More precisely, the conditions,

are moreover assumed. Then there exist ,
and a solution
of (1.1) on
such that


We call such solution an escape solution. The main result of [11] is that (under (1.3)–(1.8), (1.9)) the set of solutions of (1.1), (1.10) for consists of escape solutions and of oscillatory solutions (having values in
) and of at least one homoclinic solution. In [12] we omit assumptions (1.9) and prove that assumptions (1.3)–(1.8) are sufficient for the existence of an escape solution and also for the existence of a homoclinic solution provided the
fulfils

If (1.12) is not valid, then the existence of both an escape solution and a homoclinic solution is proved in [12], provided that satisfies moreover


Assumption (1.13) characterizes the case that has just two zeros
and
in the interval
. Further, we see that if (1.14) holds, then
is either bounded on
or
is unbounded earlier and has a sublinear behaviour near
.
This paper also deals with the case that satisfies (1.13) and is unbounded above on
. In contrast to [12], here we prove the existence of a homoclinic solution for
having a linear behaviour near
. The proof is based on a full description of the set of all solutions of problem (1.1), (1.10) for
and on the existence of an escape solutions in this set.
Finally, we want to mention the paper [13], where the problem

is investigated under the assumptions that is continuous, it has three distinct zeros and satisfies the sign conditions similar to those in [11, (3.4)]. In [13], an approach quite different from [11, 12] is used. In particular, by means of properties of the associated vector field
together with the Kneser's property of the cross sections of the solutions' funnel, the authors provide conditions which guarantee the existence of a strictly increasing solution of (1.15). The authors apply this general result to problem

and get a strictly increasing solution of (1.16) for a sufficiently small . This corresponds to the results of [11], where
may be arbitrary.
2. Initial Value Problem
In this section, under the assumptions (1.3)–(1.8) and (1.13) we prove some basic properties of solutions of the initial value problem (1.1), (1.10), where .
Lemma 2.1.
For each there exists a maximal
such that problem (1.1), (1.10) has a unique solution
on
and

Further, for each there exists
such that

Proof.
Let be a solution of problem (1.1), (1.10) on
. By (1.1), we have

and multiplying by and integrating between
and
, we get

Let for some
. Then (2.4) yields
, which is not possible, because
is decreasing on
. Therefore
for
.
Let . Consider the Banach space
(with the maximum norm) and an operator
defined by

A function is a solution of problem (1.1), (1.2) on
if and only if it is a fixed point of the operator
. Using the Lipschitz property of
we can prove that the operator is contractive for each sufficiently small
and from the Banach Fixed Point Theorem we conclude that there exists exactly one solution of problem (1.1), (1.2) on
. This solution
has the form

for . Hence,
can be extended onto each interval
where
is bounded. So, we can put
.
Let . Then there exists
such that
for
. So, (2.6) yields

Put

Then

and, by "per partes" integration we derive . Multiplying (2.7) by
and integrating it over
, we get

Estimates (2.2) follow from (2.7)–(2.10) for

Remark 2.2.
The proof of Lemma 2.1 yields that if , then
.
Let us put

and consider an auxiliary equation

Similarly as in the proof of Lemma 2.1 we deduce that problem (2.13), (1.10) has a unique solution on . Moreover the following lemma is true.
Lemma 2.3 ([12]).
For each ,
and each
, there exists
such that for any
,

Here is a solution of problem (2.13), (1.10) with
,
.
Proof.
Choose ,
,
. Let
be the Lipschitz constant for
on
. By (2.6) for
,
,
,
,

From the Gronwall inequality, we get

Similarly, by (2.6), (2.9), and (2.16),

If we choose such that

we get (2.14).
Remark 2.4.
Choose and
, and consider the initial conditions

Arguing as in the proof of Lemma 2.1, we get that problem (2.13), (2.19) has a unique solution on . In particular, for
and
, the unique solution of problem (2.13), (2.19) (and also of problem (1.1), (2.19)) is
and
, respectively.
Lemma 2.5.
Let be a solution of problem (1.1), (1.10). Assume that there exists
such that

Then for
and

Proof.
By (1.13) and (2.20), on
and thus
and
are positive on
. Consequently, there exists
. Further, by (1.1),

and, by multiplication and integration over ,

Therefore,

and hence exists. Since
is bounded on
, we get

By (1.3), (1.8), and (2.22), exists and, since
is bounded on
, we get
. Hence, letting
in (2.22), we obtain
. Therefore,
and (2.21) is proved.
Lemma 2.6.
Let be a solution of problem (1.1), (1.10). Assume that there exist
and
such that

Then for all
and (2.21) holds.
Proof.
Since fulfils (2.26), we can find a maximal
such that
for
and consequently
for
. By (4.23) and (2.26),
on
and thus
and
are negative on
. So,
is positive and decreasing on
which yields
(otherwise, we get
, contrary to (2.26)). Consequently there exists
. By multiplication and integration (2.22) over
, we obtain

By similar argument as in the proof of Lemma 2.5 we get that and
. Therefore (2.21) is proved.
3. Damped Solutions
In this section, under assumptions (1.3)–(1.8) and (1.13) we describe a set of all damped solutions which are defined in the following way.
Definition 3.1.
A solution of problem (1.1), (1.10) (or of problem (2.13), (1.10)) on is calleddamped if

Remark 3.2.
We see, by (2.12), that is a damped solution of problem (1.1), (1.10) if and only if
is a damped solution of problem (2.13), (1.10). Therefore, we can borrow the arguments of [12] in the proofs of this section.
Theorem 3.3.
If is a damped solution of problem (1.1), (1.10), then
has a finite number of isolated zeros and satisfies (2.21); or
is oscillatory (it has an unbounded set of isolated zeros).
Proof.
Let be a damped solution of problem (1.1), (1.10). By Remark 2.2, we have
in Lemma 2.1 and hence

Step 1.
If has no zero in
, then
for
and, by Lemma 2.5,
fulfils (2.21).
Step 2.
Assume that is the first zero of
on
. Then, due to Remark 2.4,
. Let
for
. By virtue of (1.4),
for
and thus
is decreasing. Let
be positive on
. Then
is also decreasing,
is increasing and
, due to (3.1). Consequently,
. Letting
in (2.22), we get
, which is impossible because
is bounded below. Therefore there are
and
satisfying (2.26) and, by Lemma 2.6, either
fulfils (2.21) or
has the second zero
with
. So
is positive on
and has just one local maximum
in
. Moreover, putting
and
in (2.23), we have

and hence

Step 3.
Let have no other zeros. Then
for
. Assume that
is negative on
. Then, due to (2.1),
. Putting
in (2.23) and letting
, we obtain

Therefore, exists and, since
is bounded, we deduce that

Letting in (2.22), we get
, which contradicts the fact that
is bounded above. Therefore,
cannot be negative on the whole interval
and there exists
such that
. Moreover, according to (3.2),
.
Then, Lemma 2.5 yields that fulfils (2.21). Since
is positive on
,
has just one minimum
on
. Moreover, putting
and
in (2.23), we have

which together with (3.4) yields

Step 4.
Assume that has its third zero
. Then we prove as in Step 2 that
has just one negative minimum
in
and (3.8) is valid. Further, as in Step 2, we deduce that either
fulfils (2.21) or
has the fourth zero
,
is positive on
with just one local maximum
on
, and
. This together with (3.8) yields

If has no other zeros, we deduce as in Step 3 that
has just one negative minimum
in
,
and
fulfils (2.21).
Step 5.
If has other zeros, we use the previous arguments and get that either
has a finite number of zeros and then fulfils (2.21) or
is oscillatory.
Remark 3.4.
According to the proof of Theorem 3.3, we see that if is oscillatory, it has just one positive local maximum between the first and the second zero, then just one negative local minimum between the second and the third zero, and so on. By (3.8), (3.9), (1.4)–(1.6) and (1.13), these maxima are decreasing (minima are increasing) for
increasing.
Lemma 3.5.
A solution of problem (1.1), (1.10) fulfils the condition

if and only if fulfils the condition

Proof.
Assume that fulfils (3.10). Then there exists
such that
,
for
. Otherwise
, due to Lemma 2.5. Let
be such that
on
,
. By Remark 2.4 and (3.10),
. Integrating (1.1) over
, we get

Due to (1.4), we see that is strictly decreasing for
as long as
. Thus, there are two possibilities. If
for all
, then from Lemma 2.6 we get (2.21), which contradicts (3.10). If there exists
such that
, then in view Remark 2.4 we have
. Using the arguments of Steps 3–5 of the proof of Theorem 3.3, we get that
is damped, contrary to (3.10). Therefore, such
cannot exist and
on
. Consequently,
. So,
fulfils (3.11). The inverse implication is evident.
Remark 3.6.
According to Definition 1.3 and Lemma 3.5, is a homoclinic solution of problem (1.1), (1.10) if and only if
is a homoclinic solution of problem (2.13), (1.10).
Theorem 3.7 (on damped solutions).
Let satisfy (1.5) and (1.6). Assume that
is a solution of problem (1.1), (1.10) with
. Then
is damped.
Proof.
Let be a solution of (1.1), (1.10) with
. Then, by (1.4)–(1.6),

Assume on the contrary that is not damped. Then
is defined on the interval
and
or there exists
such that
,
and
for
. If the latter possibility occurs, (2.22) and (3.13) give by integration

a contradiction. If , then, by Lemma 3.5,
fulfils (3.11). So
has a unique zero
. Integrating (2.22) over
, we get

and so

Integrating (2.22) over , we obtain for

Therefore, on
, and letting
, we get
. This together with (3.16) contradicts (3.13). We have proved that
is damped.
Theorem 3.8.
Let be the set of all
such that corresponding solutions of problem (1.1), (1.10) are damped. Then
is open in
.
Proof.
Let and
be a solution of (1.1), (1.10) with
. So,
is damped and
is also a solution of (2.13).
-
(a)
Let
be oscillatory. Then its first local maximum belongs to
. Lemma 2.3 guarantees that if
is sufficiently close to
, the corresponding solution
of (2.13), (1.10) has also its first local maximum in
. This means that there exist
and
such that
satisfies (2.26). Now, we can continue as in the proof of Theorem 3.3 using the arguments of Steps 2–5 and Remark 3.2 to get that
is damped.
-
(b)
Let
have at most a finite number of zeros. Then, by Theorem 3.3,
fulfils (2.21). Choose
. Since
fulfils (2.22), we get by integration over
(3.18)
For we get, by (2.21),

Therefore, we can find such that

Let be the constant of Lemma 2.1. Choose
. Assume that
and
is a corresponding solution of problem (2.13), (1.10). Using Lemma 2.1, Lemma 2.3 and the continuity of
, we can find
such that if
, then

moreover for
and

Therefore, we have

Consequently, integrating (2.13) over and using (3.19)–(3.23), we get for

We get for
. Therefore,
for
and, due to (1.4)–(1.6),

Assume that there is such that
,
. Then, since
if
and
, we get
and
for
, contrary to (3.25). Hence we get that
fulfils (3.1).
4. Escape Solutions
During the whole section, we assume (1.3)–(1.8) and (1.13). We prove that problem (1.1), (1.10) has at least one escape solution. According to Section 1 and Remark 2.2, we work with the following definitions.
Definition 4.1.
Let . A solution of problem (1.1), (1.10) on
is called an escape solution if

Definition 4.2.
A solution of problem (2.13), (1.10) is called an escape solution, if there exists
such that

Remark 4.3.
If is an escape solution of problem (2.13), (1.10), then
is an escape solution of problem (1.1), (1.10) on some interval
.
Theorem 4.4 (on three types of solutions.).
Let be a solution of problem (1.1), (1.10). Then
is just one of the following three types
(I) is damped;
(II) is homoclinic;
(III) is escape.
Proof.
By Definition 3.1, is damped if and only if (3.1) holds. By Lemma 3.5 and Definition 1.3,
is homoclinic if and only if (3.10) holds. Let
be neither damped nor homoclinic. Then there exists
such that
is bounded on
,
,
. So,
has its first zero
and
on
. Assume that there exist
such that
and
. Then, by Lemma 2.6, either
fulfils (2.21) or
has its second zero and, arguing as in Steps 2–5 of the proof of Theorem 3.3, we deduce that
is a damped solution. This contradiction implies that
on
. Therefore, by Definition 4.1,
is an escape solution.
Theorem 4.5.
Let be the set of all
such that the corresponding solutions of (1.1), (1.10) are escape solutions. The set
is open in
.
Proof.
Let and
be a solution of problem (1.1), (1.10) with
. So,
fulfils (4.1) for some
. Let
be a solution of problem (2.13), (1.10) with
. Then
on
and
is increasing on
. There exists
and
such that
. Let
be a solution of problem (2.13), (1.10) for some
. Lemma 2.3 yields
such that if
, then
. Therefore,
is an escape solution of problem (2.13), (1.10). By Remark 4.3,
is also an escape solution of problem (1.1), (1.10) on some interval
.
To prove that the set of Theorem 4.5 is nonempty we will need the following two lemmas.
Lemma 4.6.
Let . Assume that
is a solution of problem (1.1), (1.10) on
and
is a maximal interval where
is increasing and
for
. Then

Proof.
Step 1.
We show that the interval is nonempty. Since
and
satisfies (1.3), (1.13), we can find
such that

Integrating (1.1) over we obtain

So, is an increasing solution of problem (1.1), (1.10) on
and
for
. Therefore the nonempty interval
exists.
Step 2.
By multiplication of (1.1) by and integration over
we obtain

Using the "per partes" integration, we get for

This relation together with (4.6) implies (4.3).
Remark 4.7.
Consider a solution of Lemma 4.6. If
is an escape solution, then
. Assume that
is not an escape solution. Then both possibilities
and
can occur. Let
. By Theorem 4.4 and Lemma 2.5,
,
. Let
. We write
,
. Using Lemmas 3.5 and 2.5 and Theorem 4.4, we obtain
and either
or
.
Lemma 4.8.
Let and let
. Then for each
(i)there exists a solution of problem (1.1), (1.10) with
,
(ii)there exists such that
is the maximal interval on which the solution
is increasing and its values in this interval are contained in
,
(iii)there exists satisfying
.
If the sequence is unbounded, then there exists
such that
is an escape solution.
Proof.
Similar arugmets can be found in [12]. By Lemma 2.1, the assertion (i) holds. The arguments in Step 1 of the proof of Lemma 4.6 imply (ii). The strict monotonicity of and Remark 4.7 yields a unique
. Assume that
is unbounded. Then

(otherwise, we take a subsequence). Assume on the contrary that for any ,
is not an escape solution. Choose
. Then, by Remark 4.7,

Due to (4.9), (1.2) and (ii) there exists satisfying

By (i) and (ii), satisfies

Integrating it over we get

Put

Then, by (4.12),

We see that is decreasing. From (1.4) and (1.6) we get that
is increasing on
and consequently by (4.9) and (4.13), we have

Integrating (4.14) over and using (4.10), we obtain

where

Further, by (4.15),


Conditions (1.8) and (4.8) yield , which implies

By (4.13) and (4.18),

and consequently

which contradicts (4.20). Therefore, at least one escape solution of (1.1), (1.10) with must exist.
Theorem 4.9 (on escape solution).
Assume that (1.3)–(1.8) and (1.13) hold and let

Then there exists such that the corresponding solution of problem (1.1), (1.10) is an escape solution.
Proof.
Let and let
,
,
and
be sequences from Lemma 4.8. Moreover, let

By (4.24) we can find such that
for
. We assume that for any
,
is not an escape solution and we construct a contradiction.
Step 1.
We derive some inequality for . By Remark 4.7, we have

and, by Lemma 4.8, the sequence is bounded. Therefore there exists
such that

Choose an arbitrary . According to Lemma 4.6,
satisfies equality (4.3), that is

Since and
is increasing on
, there exists a unique
such that

Having in mind, due to (1.4)–(1.8), that the inequality

holds, we get

By virtue of (1.6) and (1.13), we see that is decreasing on
, which yields

Hence,

Since and
, the monotonicity of
yields
for
, and consequently

Therefore (4.27) and (4.32) give

Step 2.
We prove that the sequence is bounded below by some positive number. Since
is a solution of (1.1) on
, we have

Integrating it, we get

where satisfies
and
. Having in mind (1.8), we see that
is increasing and
for
. Consequently

Integrating (4.36) over , we obtain

and hence

By (4.23) we get

which, due to (4.39), yields

So, by virtue of (4.37), there exists such that
for
.
Step 3.
We construct a contradiction. Putting in (4.34), we have

Due to (4.23), . Therefore,
, and consequently, by (4.24),

In order to get a contradiction, we distinguish two cases.
Case 1.
Let , that is, we can find
,
,
, such that

Then, by (4.43), for each sufficiently large , we get

Putting it to (4.42), we have

Therefore , contrary to (4.25).
Case 2.
Let . We may assume
(otherwise we take a subsequence). Then there exists
,
, such that

Due to (4.43), for each sufficiently large , we get

Putting it to (4.42), we have

Therefore, for
. Integrating it over
, we obtain

which yields, by (4.26), and also
, contrary to (4.25). These contradictions obtained in both cases imply that there exists
such that
is an escape solution.
5. Homoclinic Solution
The following theorem provides the existence of a homoclinic solution under the assumption that the function in (1.1) has a linear behaviour near
. According to Definition 1.2, a homoclinic solution is a strictly increasing solution of problem (1.1), (1.2).
Theorem 5.1 (On homoclinic solution.).
Let the assumptions of Theorem 4.9 be satisfied. Then there exists such that the corresponding solution of problem (1.1), (1.10) is a homoclinic solution.
Proof.
For denote by
the corresponding solution of problem (1.1), (1.10). Let
and
be the set of all
such that
is a damped solution and an escape solution, respectively. By Theorems 3.7, 3.8, 4.5, and 4.9, the sets
and
are nonempty and open in
. Therefore, the set
is nonempty. Choose
. Then, by Theorem 4.4,
is a homoclinic solution. Moreover, due to Theorem 3.7,
.
Example 5.2.
The function

where is a negative constant, satisfies the conditions (1.3)–(1.6), (1.13), and (4.23).
References
Dell'Isola F, Gouin H, Rotoli G: Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations. European Journal of Mechanics. B 1996, 15(4):545-568.
Derrick GH: Comments on nonlinear wave equations as models for elementary particles. Journal of Mathematical Physics 1964, 5: 1252-1254. 10.1063/1.1704233
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(3):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
Lima PM, Konyukhova NB, Sukov AI, Chemetov NV: Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems. Journal of Computational and Applied Mathematics 2006, 189(1-2):260-273. 10.1016/j.cam.2005.05.004
Koch O, Kofler P, Weinmüller EB: Initial value problems for systems of ordinary first and second order differential equations with a singularity of the first kind. Analysis 2001, 21(4):373-389.
Bonheure D, Gomes JM, Sanchez L: Positive solutions of a second-order singular ordinary differential equation. Nonlinear Analysis: Theory, Methods & Applications 2005, 61(8):1383-1399. 10.1016/j.na.2005.02.029
Conti M, Merizzi L, Terracini S:Radial solutions of superlinear equations on
. I. A global variational approach. Archive for Rational Mechanics and Analysis 2000, 153(4):291-316. 10.1007/s002050050015
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
Maatoug L: On the existence of positive solutions of a singular nonlinear eigenvalue problem. Journal of Mathematical Analysis and Applications 2001, 261(1):192-204. 10.1006/jmaa.2001.7491
Rachůnková I, Tomeček J: Singular nonlinear problem for ordinary differential equation of the second-order on the half-line. In Mathematical Models in Engineering, Biology and Medicine: International Conference on Boundary Value Problems Edited by: Cabada A, Liz E, Nieto JJ. 2009, 294-303.
Rachůnková I, Tomeček J: Bubble-type solutions of nonlinear singular problem. submitted
Palamides AP, Yannopoulos TG: Terminal value problem for singular ordinary differential equations: theoretical analysis and numerical simulations of ground states. Boundary Value Problems 2006, 2006:-28.
Acknowledgments
The authors thank the referee for valuable comments. This work was supported by the Council of Czech Government MSM 6198959214.
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Rachůnková, I., Tomeček, J. Homoclinic Solutions of Singular Nonautonomous Second-Order Differential Equations. Bound Value Probl 2009, 959636 (2009). https://doi.org/10.1155/2009/959636
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DOI: https://doi.org/10.1155/2009/959636
Keywords
- Differential Equation
- Functional Equation
- Local Maximum
- Point Theorem
- Lipschitz Constant