1. Introduction
In this paper, we study the second order non-autonomous half-linear equation on the whole real line,
with
an increasing homeomorphism,
and
,
such that
for
and
a
-Carathéodory function, together with the asymptotic conditions:
with
such that
Moreover, an application to singular
-Laplacian equations will be shown.
This problem (
1) and (
2) was studied in [
1,
2]. This last paper contained several results and criteria. For example, Theorem 2.1 in [
2] guarantees the existence of heteroclinic solutions under, in short, the following main assumptions:
grows at most linearly at infinity;
for
there exist constants
, a continuous function
and a function
, with
, such that:
for every
, there exist functions
, null in
and positive in
, and
such that:
Motivated by these works, we prove, in this paper, the existence of heteroclinic solutions for (
1) assuming a Nagumo-type condition on the real line and without asymptotic assumptions on the nonlinearities
and
. The method follows arguments suggested in [
3,
4,
5], applying the technique of [
3] to a more general function
, with an adequate functional problem and to classical and singular
-Laplacian equations. The most common application for
is the so-called
p-Laplacian, i.e.,
, and even in this particular case, verifying (
4), the new assumption on
. Moreover, this type of equation includes, for example, the mean curvature operator. On the other hand, to the best of our knowledge, the main result is even new when
that is for equation:
The study of differential equations and boundary value problems on the half-line or in the whole real line and the existence of homoclinic or heteroclinic solutions have received increasing interest in the last few years, due to the applications to non-Newtonian fluids theory, the diffusion of flows in porous media, and nonlinear elasticity (see, for instance, [
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16] and the references therein). In particular, heteroclinic connections are related to processes in which the variable transits from an unstable equilibrium to a stable one (see, for example, [
17,
18,
19,
20,
21,
22,
23,
24]); that is why heteroclinic solutions are often called transitional solutions.
The paper is organized in this way:
Section 2 contains some notations and auxiliary results. In
Section 3, we prove the existence of heteroclinic connections for a functional problem, which is used to obtain an existence and location theorem for heteroclinic solutions for the initial problem.
Section 4 contains an example, to show the applicability of the main theorem. The last section applies the above theory to singular
-Laplacian differential equations.
2. Notations and Auxiliary Results
Throughout this paper, we consider the set of the bounded functions, equipped with the norm , where .
By standard procedures, it can be shown that is a Banach space.
As a solution of the problem (
1) and (
2), we mean a function
such that
and satisfying (
1) and (
2).
The -Carathéodory functions will play a key role throughout the work:
Definition 1. A function is -Carathéodory if it verifies:
- (i)
for each , is measurable on ;
- (ii)
for almost every is continuous in ;
- (iii)
for each , there exists a positive function such that, for ,
The following hypothesis will be assumed:
-
is an increasing homeomorphism with
and
such that:
-
is a continuous and positive function with as
To overcome the lack of compactness of the domain, we apply the following criterion, suggested in [
25]:
Lemma 1. A set is compact if the following conditions hold:
- 1.
M is uniformly bounded in X;
- 2.
the functions belonging to M are equicontinuous on any compact interval of ;
- 3.
the functions from M are equiconvergent at , that is, given , there exists such that:for all and .
3. Existence Results
The first existence result for heteroclinic connections will be obtained for an auxiliary functional problem without the usual asymptotic or growth assumptions on or on the nonlinearity f.
Consider two continuous operators
, with
,
, and
,
, the functional problem composed of:
and the boundary conditions (
2).
Define, for each bounded set
and for the above operators, assume that:
-
For each , there is , with such that whenever
-
as
and:
Theorem 1. Assume that conditions , and hold and there is such that: Then, there exists such that , verifying (5) and (2), given by:where is the unique solution of: Moreover, for such that with:and: Proof. For every
, define the operator
by
where
is the unique solution of:
To show that
is the unique solution of (
10), consider the strictly-increasing function in
:
and remark that:
and:
Moreover, for
given by (
12) and
given by (
13),
and
have opposite signs, as:
As
G is strictly increasing in
, by (
14), there is
such that
and
Therefore, the equation
has a unique solution
, and by Bolzano’s theorem,
when
, for some
.
It is clear that if
T has a fixed point
, then
u is a solution of the problem (
5) and (
2).
To prove the existence of such a fixed point, we consider several steps:
Step 1. is well defined
By the positivity of
A and the continuity of
A and
F, then
and:
are continuous on
, that is
Moreover, by (
),
, and (
10),
and
are bounded. Therefore,
.
Step 2. T is compact.
Let
be a bounded subset,
and
such that
Consider
given by (
6) with
Claim:TB is uniformly bounded in X.
By (
4), (
11), and (
), we have:
and:
Therefore, is uniformly bounded in .
Claim:TB is equicontinuous on X.
For , consider and without loss of generality, .
Then, by (
4), (
11) and (
),
and:
Therefore, is equicontinuous on X.
Claim:TB is equiconvergent at ±∞.
Let
As in the claims above:
and:
Moreover, by
,
and:
Therefore, is equiconvergent at and by Lemma 1, T is compact.
Step 3. Letbe a closed and bounded set. Then,.
Consider
defined as:
with
such that:
with:
and:
Let
Following similar arguments as in the previous claims, with
given by (
6) and
and:
Therefore,
By Schauder’s fixed point theorem,
has a fixed point in
That is, there is a heteroclinic solution of the problem (
5) and (
2). □
To make the relation between the functional problem and the initial one, we apply the lower and upper solution method, according to the following definition:
Definition 2. A function is a lower solution of the problem (1) and (2) if and: An upper solution of the problem (1) and (2) satisfies and the reversed inequalities. To have some control on the first derivative, we apply a Nagumo-type condition:
Definition 3. A -Carathéodory function satisfies a Nagumo-type growth condition relative to with if there are positive and continuous functions such that:and: Lemma 2. Let be a -Carathéodory function satisfying a Nagumo-type growth condition relative to with . Then, there exists (not depending on u) such that for every solution u of (1) and (2) with:we have: Proof. Let
u be a solution of (
1) and (
2) verifying (
19). Take
such that:
If , the proof would be complete by taking
Suppose there is such that
In the case
, by (
17), we can take
such that:
with
which is finite by (
17).
By (
2), there are
such that
,
, and
. Therefore, the following contradiction with (
22) holds, by the change of variable
and (
17):
Therefore, .
By similar arguments, it can be shown that . Therefore, □
The next lemma, in [
26], provides a technical tool to use going forward:
Lemma 3. For such that , for every , define: Then, for each , the next two properties hold:
- (a)
exists for a.e..
- (b)
If and in , then:
The main result will be given by the next theorem:
Theorem 2. Suppose that is a -Carathéodory function verifying a Nagumo-type condition and hypotheses , and (8). If there are lower and upper solutions of the problem (1) and (2), α and β, respectively, such that:then there is a function with the solution of the problem (1) and (2) and: Proof. Define the truncation operator
given by:
Consider the modified equation:
for
which is well defined by Lemma 3.
Claim 1:Every solutionof the problem (23) and (2) verifies: Let
u be a solution of the problem (
23) and (
2), and suppose, by contradiction, that there is
such that
Remark that, by (
16),
as
Therefore, there is an interval
such that
for a.e.
and by (
15), this contradiction is achieved:
Therefore, Following similar arguments, it can be proven that
Claim 2:The problem (23) and (2) has a solution. Let
and
be the operators given by
and:
As, for:
with
N given by (
20),
then
verifies
Moreover, from:
we have that
A satisfies
with
Therefore, by Schauder’s fixed point theorem, the problem (
23) and (
2) has a solution, which, by Claim 1, is a solution of the problem (
1) and (
2). □
4. Example
Consider the boundary value problem, defined on the whole real line, composed by the differential equation:
coupled with the boundary conditions:
Remark that the null function is not solution of the problem (
24) and (
25), which is a particular case of (
1) and (
2), with:
All hypotheses of Theorem 2 are satisfied. In fact:
Therefore, by Theorem 2, there is a heteroclinic connection
u between two equilibrium points
and one of the problem (
24) and (
25), such that:
5. Singular ϕ-Laplacian Equations
The previous theory can be easily adapted to singular
-Laplacian equations, that is for equations:
where
verifies:
, for some
is an increasing homeomorphism with
and
such that:
In this case, a heteroclinic solution of (
1s), that is a solution for the problem (
1s) and (
2), is a function
such that
for
and
satisfying (
1s) and (
2).
The theory for singular -Laplacian equations is analogous to Theorems 1 and 2, replacing the assumption by
As an example, we can consider the problem, for
and
Clearly, Problem (26) is a particular case of (
1) and (
2), with:
which models mechanical oscillations under relativistic effects,
Moreover, the nonlinearity
f given by (28) is a
-Carathéodory function with:
The conditions of Theorem 2 are satisfied with replaced by as:
the function defined by (27), verifies ;
the constant functions and are lower and upper solutions of Problem (26), respectively.
verifies (
8) for
and satisfies a Nagumo-type condition for
with:
Therefore, there is a heteroclinic connection
u between two equilibrium points
and one, for the singular
-Laplacian problem (26), such that: