1. Introduction
In the theory of differential equations, there are several aspects and interesting features to study or explore further. Among these is the study of impulsive differential equations or systems and the qualitative analysis of solutions in the real line. In this context, this article aims to study the existence of solutions of the heteroclinic type in nonlinear second order differential equations with generalized, infinite impulse effects, which to the best of our knowledge is a rare occurrence.
Impulses incorporated in differential equations are intended to describe and represent the effects of small and sudden changes in a given system over certain periods of time. In the literature, there are several areas of study associated with impulses such as biotechnology, medicine, population dynamics, logging, etc (see, [
12,
29]) and references therein. Various theoretical approaches, as well as numerous applications of second-order nonlinear differential equations featuring impulses can be found in ([
2,
11,
19,
21,
22,
23,
24,
27]).
On the other hand, in the analysis of the qualitative aspects of differential equations, the investigation into the existence of heteroclinic or homoclinic solutions is useful and necessary. When a system of ordinary differential equations has equilibria (that is, constant solutions), studying the connections between them through the trajectories of the system’s solutions, known as homoclinic or heteroclinic solutions, becomes an essential task. It is common for homoclinic and heteroclinic solutions to emerge in mathematical models dealing with dynamical systems, bifurcations, mechanics, chemistry and biology [
4,
13,
14]. The existence of heteroclinic orbits is also crucial for analyzing the spatio-temporal chaotic patterns of nonlinear evolution equations [
16].
Over the years, some studies have been carried out on the topic of heteroclinic solutions. In [
26], the author investigates heteroclinic solutions pertaining to a second-order equation that is asymptotically autonomous
. In [
9], Coti Zelati along with Rabinowitz investigated heteroclinic orbits for a non-autonomous differential equation that connects stationary points with distinct energy levels. For a fourth-degree ordinary differential equation, heteroclinic solutions linking nonconsecutive equilibria of a triple-well potential were found [
5]. Cabada et al., in [
6] examine the existence of heteroclinic type solutions in semi-linear second-order difference equations pertaining to the Fisher-Kolmogorov’s equation. Monotonicity and continuity arguments form the basis of the proof for these results. Hale and Rybakowski also demonstrated the existence of heteroclinic solutions for retarded functional differential equations. [
15]. Furthermore, works involving heteroclinics and impulses can be found in [
3,
7,
8,
17,
20,
25,
28] and references therein.
Using findings concerning the existence of non-principal solutions, in [
1], the authors study Leighton and Wong theorems of oscillation regarding a class of second-order impulsive equations having the form
and
in which
,
q,
f are left continuous piece-wise functions in
, and
,
and
are real number sequences with
. The set
, of impulse points constitutes a strictly increasing, unbounded sequence of positive real numbers.
In [
10], Cupini, Marcelli and Papalini present the strongly nonlinear boundary-value problem
In this work, the authors consider nonlinear mixed differential operators depending both on x and . Where , and with being a general increasing homeomorphism, , a being a positive continuous function and f a nonlinear Caratheódory function.
In a recent paper [
11], Sousa and Minhós consider the following coupled system
where
and
are increasing homeomorphisms satisfying adequate relations on their inverses, with
being continuous functions, and
,
-Carathéodory functions, along with the following asymptotic conditions
for
Motivated by these works, in our paper we consider a similar problem but with the inclusion of infinite impulsive conditions, more precisely, we study the following real nonlinear second-order differential equation
with
being an increasing homeomorphism satisfying adequate relations on its inverse,
a continuous function, and
an
-Carathéodory function, considering the following asymptotic conditions
for
, together with the generalized and infinite impulse conditions
where, for
(
is the set of all integers),
,
, and
,
are the right and left limits for
, respectively,
and
has a similar meaning for
.
are Carathéodory sequences and
are moments such that
, and
Our results are based on [
11,
20], extending the results for a
-Laplacian operator with infinite impulses of more or less intensity. This can be very interesting for modeling phenomena with minor changes and with different intensities, which occur very quickly and for long periods of time, opening new fields for investigation on the subject.
The outline of the present paper is given as follows:
Section 2 comprises of the functional backgrounds. In
Section 3, we present a result of existence. Lastly, an example application illustrates the main result.
3. Main Theorem
Here we present the main result of this work, that is, the Theorem that guarantees the existence of a solution to the problem (
1)-(
3), for
Theorem 3.
Let be an increasing homeomorphism and a continuous function satisfying (H1) and (H2). Assume that is an Carathéodory function, are Carathéodory sequences and there are , and non-negative constants such that
when
Then, for , satisfying condition (2) and such that (6) is satisfied, the problem (1)-(3) has, at least heteroclinic solutions .
Proof. Let us define the operator
with
with
, satisfying condition (
2) and
such that (
6) is satisfied.
To apply the Theorem 2, we will prove that T is compact and that it has a fixed point, that is, the proof follows five steps.
Step 1.Tis well defined and it is continuous inX.
Allow
and let us take
such that
. Being
f an
Carathéodory function and
,
are Carathéodory sequence, there exists a positive function
, and non-negative constants
such that
Thus,
, as
and
Furthermore, by (
2), (
7), (
8) and (H2),
and
Therefore, .
Step 2.is uniformly bounded on, for some bounded K.
Take
K to be a bounded set of
X, with the definition
for some
.
By (
7), (
8), (H1) and (H2), we have
and
So, , that is, is uniformly bounded on X.
Step 3.is equicontinuous, on each interval,for
Consider
and let us suppose, without losing generality, that
. So, for
and by (
7), (
8), and (H1), follow
uniformly for
, as
,
uniformly for
, as
. Then,
is equicontinuous on each intervalfor
Step 4.is equiconvergent at each impulse point, and at, that is, is equiconvergent at, () and at infinity.
First, let us prove that
is equiconvergent at
, for
. Let
. So, by (
7), (
8), and (H1), it follows
uniformly in
, as
, for
and
uniformly in
, as
, for
. Therefore,
is equiconvergent at each point
, for
.
Identically, we will prove that
is equiconvergent at . In this way we have,
uniformly in
, as
.
In turn,
uniformly in
, as
.
It follows for the derivative that,
uniformly in
, as
, and
uniformly in
, as
.
Therefore, is equiconvergent at and by Theorem 1, is relatively compact.
Step 5.has a fixed point.
To be able to apply Schauder’s fixed point Theorem for the operator , we have to prove that , for some, bounded, closed and convex .
Let us consider
with
such that
with
given by (
9).
Following arguments similar to step 2, we have that for
and
. Then, the operator
, by Theorem 2, has a fixed point
. Using standard arguments, we can demonstrate that this fixed point determines a pair of heteroclinic or homoclinic solutions for the problem (
1)-(
3). □
5. Discussion
This work presents the existence of heteroclinic solutions of strongly nonlinear second order equations. While the existence of the solutions is guaranteed by Schauder’s fixed point theorem, which is very well known, our work extends the results in the literature, using a -Laplacian operator in the equations with conditions of infinite impulses of greater or lesser intensity.
This can be very interesting for modeling phenomena with small changes and different intensities, which occur very quickly and for long periods of time, opening new fields of investigation on the topic. As illustrated in the example and application section, an example of this is that this type of models can incorporate at the same time, the dynamics of continuous growth governed by a differential equation, as well as the discrete changes represented by infinite impulse conditions. Therefore, they provide a broad vision and field to investigate the complex dynamics of bird population growth in a natural reserve, integrating biological, environmental and behavioral factors.