# Landesman-Lazer conditions for systems involving twist and positively
homogeneous Hamiltonian systems

Natnael Gezahegn Mamo and Wahid Ullah

###### Abstract

We present multiplicity results for the periodic and Neumann-type boundary
value problems associated with coupled Hamiltonian systems. For the periodic
problem, we couple a system having twist condition with another one whose
nonlinearity lies between the gradients of two positive and positively
$2$-homogeneous Hamiltonain functions. Concerning the Neumann-type problem, we
treat the same system without any twist assumption. We examine the cases of
nonresonance, simple resonance, and double resonance by imposing some kind of
Landesman–Lazer conditions.

##  1 Introduction

The main motivation of this paper is arising from the two open problems given
under a recent paper [16, Section 5, Problems 2 and 4]. To this aim, we first
consider the periodic problem associated with a system of the form

$\begin{cases}\dot{x}=\nabla_{y}\mathcal{H}(t,x,y)+\nabla_{y}P(t,x,y,w)\,,\\\
\dot{y}=-\nabla_{x}\mathcal{H}(t,x,y)-\nabla_{x}P(t,x,y,w)\,,\\\
J\dot{w}=F(t,w)+\,\nabla_{w}P(t,x,y,w)\,,\end{cases}$ (1.1)

where $F(t,w)$ is the gradient of a Hamiltonian function
$K:{\mathbb{R}}\times{\mathbb{R}}^{2}\to{\mathbb{R}}$, i.e.,

$F(t,w)=\nabla_{w}K(t,w)\,,$

and $\mathcal{H}:{\mathbb{R}}\times{\mathbb{R}}^{2M}\to{\mathbb{R}}$. In
addition, all the involved functions in (1.1) are continuous, $T$-periodic in
the variable $t$ and continuously differentiable with respect to $(x,y,w)$. We
denote by $J=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$ the standard symplectic
matrix.

To be more precise, the idea is to consider a Poincaré–Birkhoff situation for
system

$\dot{x}=\nabla_{y}H(t,x,y)\,,\qquad\dot{y}=-\nabla_{x}H(t,x,y)\,,$ (1.2)

while for system

$J\dot{w}=F(t,w)\,,$ (1.3)

we assume that $F(t,w)$ is the combination of gradients of two positive and
positively $2$-homogeneous Hamiltonian functions together with a perturbation
term.

We now explain some basic preliminary concepts about positively homogeneous
Hamiltonian systems. We refer the reader to [8, 9, 18] for the detailed
explanation. If $\mathscr{H}:{\mathbb{R}}^{2}\to{\mathbb{R}}$ is a
$C^{1}$-function satisfying

$0<\mathscr{H}(\lambda w)=\lambda^{2}\mathscr{H}(w),\quad\mbox{ for every
}w\in{\mathbb{R}}^{2}\backslash\\{0\\}\,\mbox{ and }\lambda>0\,,$ (1.4)

then $\mathscr{H}$ is said to be positively homogeneous of degree 2 and
positive. In this setting, the origin is an isochronous center for the
autonomous Hamiltonian system

$J\dot{w}=\nabla\mathscr{H}(w),\quad\mbox{ for every
}\,w\in{\mathbb{R}}^{2}\,,$ (1.5)

which means that aside from the origin, which is an equilibrium, all solutions
of system (1.5) are periodic and have the same minimal period.

Notice that if $\mathscr{H}$ is positively homogeneous of degree $2$, then the
Euler identity holds:

$\big{\langle}\nabla\mathscr{H}(w)\,,w\big{\rangle}=2\mathscr{H}(w),\quad\mbox{
for every }w\in{\mathbb{R}}^{2}\,.$ (1.6)

In [20], a higher dimensional version of the Poincaré–Birkhoff Theorem was
studied for a periodic problem associated with the Hamiltonian system (1.2).
Recently, several authors have discussed about the existence of multiple
solutions for a coupled systems, where the coupling is between system (1.2)
with some other systems involving different types of structures, see for
example [4, 10, 11, 17]. In [18, 19], system (1.2) is coupled with some
isochronous centre by using a nonresonance condition. In [13], the coupling
was with a scalar second order equation, where the nonlinearity has an
asymmetric behaviour combined with some Landesman–Lazer conditions. In that
paper, nonresonance, simple resonance and double resonance situations were
considered by implementing some kind of Landesman–Lazer conditions.

Let us explain what the Landesman–Lazer condition is. This condition has been
introduced by Lazer and Leach [23] for the periodic problem associated with a
scalar second order ordinary differential equations of the form

$\ddot{u}+\sigma u+h(t,u)=0\,,$ (1.7)

where $h$ is continuous, uniformly bounded and $T$-periodic in $t$ with
$\sigma=\big{(}\frac{2\pi n}{T}\big{)}^{2}$ for some positive integer $n$. In
this setting they proved that a sufficient condition for the existence of a
$T$-periodic solution is the following: for every non-zero $\eta$ satisfying
$\ddot{\eta}+\sigma\eta=0$, we have

$\int_{\\{\eta<0\\}}\limsup\limits_{u\to-\infty}h(t,u)\eta(t)dt+\int_{\\{\eta>0\\}}\limsup\limits_{u\to+\infty}h(t,u)\eta(t)dt>0\,.$
(1.8)

Notice that when $h(t,u)$ is increasing in $u$ this condition becomes a
necessary and sufficient condition for the existence of a solution of (1.7).
One year later, Landesman and Lazer [22] introduced a similar condition for a
Dirichlet problem associated with an elliptic operator. Since then, (1.8) is
referred to as Landesman–Lazer condition. This condition is crucial for the
nonlinearity to be kept sufficiently far from resonance. A lot of
generalizations have been made by taking their work as a steeping stone, see
for example [3, 5, 6, 25, 27].

The paper is organized as follows. In Section 2 we state our main results for
the periodic problem concerning nonresonance, simple resonance and double
resonance situations. The proofs of these results are provided in Section 3.
Section 4 is devoted to some variants of our results. The application will be
given through some examples in Section 5. In Section 6 we state and prove an
analogue result concerning a Neumann-type boundary value problem associated
with system (1.1), and finally we state some open problems in Section 7.

##  2 Main results for the periodic problem

We first list all our main assumptions and then the corresponding results
related to system (1.1), where all the involved functions are continuous,
$T$-periodic in the variable $t$ and continuously differentiable with respect
to $(x,y,w)$. The proofs will be given in the next section.

Here are our main assumptions. The first one is concerning the periodicity for
the Hamiltonian function ${\cal H}$. We write

$\displaystyle x=(x_{1},\dots,x_{M})\in{\mathbb{R}}^{M},\quad
y=(y_{1},\dots,y_{M})\in{\mathbb{R}}^{M}.$

$A1.$ The function ${\cal H}(t,x,y)$ is $2\pi$-periodic in each $x_{i}$ for
$i=1,\dots,M$.

The following assumption is on the periodicity and boundedness condition for
the coupling function $P$.

$A2.$ The function $P(t,x,y,w)$ is $2\pi$-periodic in each $x_{i}$ and has a
bounded gradient with respect to $(x,y,w)$. In particular, there exists a
constant $\overline{m}$ such that

$|\nabla_{w}P(t,x,y,w)|\leq\overline{m}\,,\qquad\text{for every
$(t,x,y,w)$}\,.$

In order to introduce a twist assumption, adapted to our setting (cf. [10, 18,
19]), we first consider the case when $\mathcal{D}$ is a rectangle in
$\mathbb{R}^{M}$, i.e.,

$\mathcal{D}=[a_{1},b_{1}]\times\cdots\times[a_{M},b_{M}]\,,$

and we denote by $\mathring{\mathcal{D}}$ its interior.

$A3.$ There exists an $M$-tuple
$\sigma=(\sigma_{1},\dots,\sigma_{M})\in\\{-1,1\\}^{M}$ such that for every
$C^{1}$-function $\mathcal{W}:[0,T]\to{\mathbb{R}}^{2}$, all the solutions
$(x,y)$ of system

$\begin{cases}\dot{x}=\nabla_{y}\mathcal{H}(t,x,y)+\nabla_{y}P(t,x,y,\mathcal{W}(t))\,,\\\
\dot{y}=-\nabla_{x}\mathcal{H}(t,x,y)-\nabla_{x}P(t,x,y,\mathcal{W}(t))\,,\end{cases}$
(2.1)

starting with $y(0)\in{\cal D}$ are defined on $[0,T]$, and for every
$i=1,\dots,M$, we have

$\begin{cases}y_{i}(0)=a_{i}\quad\Rightarrow\quad\sigma_{i}[x_{i}(T)-x_{i}(0)]<0\,,\\\
y_{i}(0)=b_{i}\quad\Rightarrow\quad\sigma_{i}[x_{i}(T)-x_{i}(0)]>0\,.\end{cases}$

We now introduce a structural assumption for the function $F$.

$A4.$ There are two functions
$\gamma:{\mathbb{R}}\times{\mathbb{R}}^{2}\to[0,1]$ and
$Q:{\mathbb{R}}\times{\mathbb{R}}^{2}\to{\mathbb{R}}$ such that

$F(t,w)=(1-\gamma(t,w))\nabla\mathscr{H}_{1}(w)+\gamma(t,w)\nabla\mathscr{H}_{2}(w)+\nabla_{w}Q(t,w)\,,$
(2.2)

where all the functions are continuous, $T$-periodic in $t$, and
$\nabla_{w}Q(t,w)$ is uniformly bounded, i.e., there exists $\widetilde{C}>0$
such that

$|\nabla_{w}Q(t,w)|\leq\widetilde{C}\,,\quad\hbox{ for every
}\,(t,w)\in[0,T]\times{\mathbb{R}}^{2}\,.$ (2.3)

It is our aim to consider the case when $F(t,w)$ in some sense interacts
asymptotically with the gradients of the two positive and positively
$2$-homogeneous Hamiltonian functions $\mathscr{H}_{1}$ and $\mathscr{H}_{2}$
which satisfy (1.4) and

$\mathscr{H}_{1}(w)\leq\mathscr{H}_{2}(w)\,,\quad\mbox{for every
}w\in{\mathbb{R}}^{2}\,.$ (2.4)

We notice that the function $F(t,\cdot)$ has at most linear growth, i.e.,
there exists $C>0$ such that

$|F(t,w)|\leq C(1+|w|)\quad\hbox{ for every }w\in{\mathbb{R}}^{2}\,.$ (2.5)

We fix $\varphi$ and $\psi$ such that

$J\dot{\varphi}=\nabla{\mathscr{H}}_{1}(\varphi),\quad
J\dot{\psi}=\nabla{\mathscr{H}}_{2}(\psi)\,,$

and

$\mathscr{H}_{1}(\varphi(t))=\mathscr{H}_{2}(\psi(t))=\frac{1}{2}\,,\quad\mbox{
for every }t\in[0,T].$ (2.6)

The choice is not restrictive.

Notice that once we fixed $\varphi$ and $\psi$, all the non-zero solutions of

$J\dot{w}=\nabla\mathscr{H}_{1}(w)\quad\mbox{and}\quad
J\dot{\widetilde{w}}=\nabla\mathscr{H}_{2}(\widetilde{w})\,,$

respectively, are of the form $w(t)=R\,\varphi(t+s)$ for all
$s\in[0,\tau_{1}[$ and $\widetilde{w}(t)=\widetilde{R}\,\psi(t+s)$ for all
$s\in[0,\tau_{2}[$ with some positive constants $R$ and $\widetilde{R}$, where
$\tau_{1}$ and $\tau_{2}$ represent their minimal periods, respectively.

Here is our first main result which is associated with a nonresonant
situation.

######  Theorem 2.1.

Let $A1-A4$ hold true, and assume that there exists a positive integer $N$
such that

$\frac{T}{N+1}<\tau_{2}\leq\tau_{1}<\frac{T}{N}\,.$ (2.7)

Then system (1.1) has at least $M+1$ geometrically distinct $T$-periodic
solutions, with $y(0)\in\mathring{\mathcal{D}}$.

Due to the $2\pi$-periodicity assumption of $\mathcal{H}$ in the variable
$x_{i}$, if $(x(t),y(t))$ is a $T$-periodic solution of system (1.2), then we
can find infinitely many others by adding an integer multiple of $2\pi$ to
$x_{i}(t)$. If one is not obtained from the other in this way, then we call
the two $T$-periodic solutions geometrically distinct.

We now consider the situation where resonance can occur from below and for
this we state the following assumption.

$A5.$ For every $\theta\in[0,T]$, one has

$\int_{0}^{T}\liminf\limits_{(\lambda,\,s)\to(+\infty,\,\theta)}\big{[}\big{\langle}F\big{(}t,\lambda\varphi(t+s)\big{)},\,\varphi(t+s)\big{\rangle}-2\lambda\mathscr{H}_{1}\big{(}\varphi(t)\big{)}\big{]}\,dt\\\
>\overline{m}\int_{0}^{T}|\varphi(t)|dt\,.$ (2.8)

######  Theorem 2.2.

Let $A1-A5$ hold true, and assume that there exists a positive integer $N$
such that

$\frac{T}{N+1}<\tau_{2}\leq\tau_{1}=\frac{T}{N}\,.$ (2.9)

Then the same conclusion of Theorem 2.1 holds.

Symmetrically, we can discuss the situation of simple resonance from above,
and for this we state the following assumption.

$A6.$ For every $\theta\in[0,T]$, one has

$\int_{0}^{T}\liminf\limits_{(\lambda,\,s)\to(+\infty,\,\theta)}\big{[}2\lambda\mathscr{H}_{2}\big{(}\psi(t)\big{)}-\big{\langle}F\big{(}t,\lambda\psi(t+s)\big{)},\,\psi(t+s)\big{\rangle}\big{]}\,dt\\\
>\overline{m}\int_{0}^{T}|\psi(t)|dt\,.$ (2.10)

######  Theorem 2.3.

Let $A1-A4$, $A6$ hold true, and assume that there exists a positive integer
$N$ such that

$\frac{T}{N+1}=\tau_{2}\leq\tau_{1}<\frac{T}{N}\,.$ (2.11)

Then the same conclusion of Theorem 2.1 holds.

Concerning the double resonance case, we have the following result.

######  Theorem 2.4.

Let $A1-A6$ hold true, and ${\mathscr{H}}_{1}(w)<{\mathscr{H}}_{2}(w)$ for all
$w\in{\mathbb{R}}^{2}\setminus\\{0\\}$. Assume moreover that there exists a
positive integer $N$ such that

$\tau_{2}=\frac{T}{N+1}\quad\hbox{ and }\quad\tau_{1}=\frac{T}{N}\,.$ (2.12)

Then the same conclusion of Theorem 2.1 holds.

##  3 Proofs of the main results

We first modify the original system, adapted to our setting (cf. [16]) as
follows. From $A4$, and the fact that $F(t,w)$ is the gradient of a
Hamiltonian function $K$, we can find a function
$\varPhi:{\mathbb{R}}\times{\mathbb{R}}^{2}\to{\mathbb{R}}$ such that
$F(t,w)=\nabla_{w}\varPhi(t,w)+\nabla_{w}Q(t,w)$, and

$\nabla_{w}\varPhi(t,w)=(1-\gamma(t,w))\nabla\mathscr{H}_{1}(w)+\gamma(t,w)\nabla\mathscr{H}_{2}(w).$
(3.1)

Writing

$\varPhi(t,w)=\varPhi(t,0)+\int_{0}^{1}\langle\nabla_{w}\varPhi(t,sw),w\rangle\,ds\,,$

we can assume without loss of generality that $\varPhi(t,0)=0$. Thus, using
(1.6), (2.4) and the fact that for $i=1,2$, $\nabla\mathscr{H}_{i}$ is
positively homogeneous of order one, we have

$\mathscr{H}_{1}(w)\leq\varPhi(t,w)\leq\mathscr{H}_{2}(w)\,.$ (3.2)

For every $\rho>1$, let $\eta_{\rho}:{\mathbb{R}}\to[0,1]$ be a
$C^{\infty}$-function such that

$\eta_{\rho}(\xi)=\begin{cases}1\quad\mbox{if }\xi\leq\rho\,,\\\
0\quad\mbox{if }\xi\geq\rho^{3}\,,\end{cases}$

and

$-\frac{1}{\xi\ln\xi}\leq\eta_{\rho}^{\prime}(\xi)\leq 0\,,\quad\mbox{for
every }\xi\geq\rho\,.$ (3.3)

The existence of such a function is guaranteed by the fact that

$\int_{\rho}^{\rho^{3}}\frac{d\xi}{\xi\ln\xi}>1\,.$

We now define the function
$\varPhi_{\rho}:{\mathbb{R}}\times{\mathbb{R}}^{2}\to{\mathbb{R}}$ as

$\varPhi_{\rho}(t,w)=\begin{cases}\varPhi(t,w)\quad&\mbox{ if
}|w|\leq\rho\,,\\\
\eta_{\rho}(|w|)\varPhi(t,w)+(1-\eta_{\rho}(|w|)\frac{1}{2}[\mathscr{H}_{1}(w)+\mathscr{H}_{2}(w)]\quad&\mbox{
if }\rho\leq|w|\leq\rho^{3}\,,\\\
\frac{1}{2}[\mathscr{H}_{1}(w)+\mathscr{H}_{2}(w)]&\mbox{ if
}|w|\geq\rho^{3}\,,\end{cases}$

and consider the following modified system

$\begin{cases}\dot{x}=\nabla_{y}{\cal H}(t,x,y)+\nabla_{y}P(t,x,y,w)\,,\\\
\dot{y}=-\nabla_{x}{\cal H}(t,x,y)-\nabla_{x}P(t,x,y,w)\,,\\\
J\dot{w}=F_{\rho}(t,w)+\,\nabla_{w}P(t,x,y,w)\,,\end{cases}$ (3.4)

where

$F_{\rho}(t,w)=\nabla_{w}\varPhi_{\rho}(t,w)+\nabla_{w}Q(t,w)\,.$ (3.5)

Notice that $\nabla_{w}\varPhi_{\rho}(t,w)$ can be decomposed as

$\nabla_{w}\varPhi_{\rho}(t,w)=(1-\gamma_{\rho}(t,w))\nabla
H_{1}(w)+\gamma_{\rho}(t,w)\nabla H_{2}(w)+v_{\rho}(t,w)\,,$

where $\gamma_{\rho}:{\mathbb{R}}\times{\mathbb{R}}^{2}\to[0,1]$ is defined by

$\gamma_{\rho}(t,w)=\begin{cases}\gamma(t,w)&\quad\mbox{ if }|w|\leq\rho\,,\\\
\eta_{\rho}(|w|)\gamma(t,w)+\frac{1}{2}(1-\eta_{\rho}(|w|))&\quad\mbox{ if
}\rho\leq|w|\leq\rho^{3}\,,\\\ \frac{1}{2}&\quad\mbox{ if
}|w|\geq\rho^{3}\,,\end{cases}$

and $v_{\rho}:{\mathbb{R}}\times{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$ by

$v_{\rho}(t,w)=\begin{cases}0&\quad\mbox{ if }|w|\leq\rho\,,\\\
\eta_{\rho}^{\prime}(|w|)|w|^{-1}[\varPhi(t,w)-\frac{1}{2}(\mathscr{H}_{1}(w)+\mathscr{H}_{2}(w))]w&\quad\mbox{
if }\rho\leq|w|\leq\rho^{3}\,,\\\ 0&\quad\mbox{ if
}|w|\geq\rho^{3}\,.\end{cases}$

By (1.4), for $i=1,2$ there exist $C_{i}>0$ such that
$0<{\mathscr{H}}_{i}(w)\leq C_{i}|w|^{2}$. So, if $\rho\leq|w|\leq\rho^{3}$,
then we have

$\displaystyle|v_{\rho}(t,w)|$
$\displaystyle\leq\frac{1}{|w|\ln(|w|)}|\mathscr{H}_{2}(w)-\mathscr{H}_{1}(w)|$
$\displaystyle\leq\frac{C_{1}+C_{2}}{\ln(|w|)}|w|\leq\frac{C_{3}}{2\ln\rho}|w|\,,$
(3.6)

where $C_{3}=C_{1}+C_{2}$.

###  3.1 Proof of Theorem 2.1

We first provide some a priori bounds so to ensure that the $T$-periodic
solutions of the modified system are indeed solutions of the original one.

######  Proposition 3.1.

There exists $\overline{\rho}>1$ such that, for any $\rho>\overline{\rho}$,
every $T$-periodic solution of (3.4) satisfies
$\|w\|_{\infty}\leq\overline{\rho}$.

###### Proof.

Assume by contradiction that there is a sequence $(\rho_{n})_{n}$ in
$]1,+\infty[$ and a sequence of $T$-periodic solutions $(x_{n},y_{n},w_{n})$
of (3.4), with $\rho=\rho_{n}$, such that $\rho_{n}\to+\infty$ and
$\|w_{n}\|_{\infty}>n$. Let $z_{n}=\frac{w_{n}}{\|w_{n}\|_{\infty}}$. Then,
$z_{n}$ is $T$-periodic and satisfies

$\begin{cases}J\dot{z}_{n}=(1-\Gamma_{n}(t))\nabla\mathscr{H}_{1}(z_{n})+\Gamma_{n}(t)\nabla\mathscr{H}_{2}(z_{n})+\kappa_{n}(t)\,,\\\
z_{n}(0)=z_{n}(T)\,,\par\end{cases}$ (3.7)

where

$\Gamma_{n}(t)=\gamma_{\rho_{n}}(t,\|w_{n}\|_{\infty}z_{n}(t))\,,$

and

$\kappa_{n}(t)=\frac{1}{\|w_{n}\|_{\infty}}\big{[}v_{\rho_{n}}(t,\|w_{n}\|_{\infty}z_{n}(t))+\nabla_{w}Q(t,\|w_{n}\|_{\infty}z_{n}(t))+\\\
\nabla_{w}P(t,x_{n}(t),y_{n}(t),\|w_{n}\|_{\infty}z_{n}(t))\big{]}\,.$ (3.8)

Notice that $\Gamma_{n}(t)\in[0,1]$ for every $t\in[0,T]$. Moreover, using
(3), and by the boundedness of $\nabla_{w}Q$ and $\nabla_{w}P$, we see that
$\kappa_{n}\to 0$ uniformly in $[0,T]$. The sequence $(z_{n})_{n}$ is thus
bounded in $C^{1}([0,T],{\mathbb{R}}^{2})$; therefore there exists a function
$z\in C([0,T],{\mathbb{R}}^{2})$ such that, up to a subsequence, $z_{n}\to z$
uniformly and weakly in $H^{1}$. Since the sequence $(\Gamma_{n})_{n}$ is
bounded, we can suppose that, up to a subsequence, it converges to some
$\Gamma$ weakly in $L^{2}$ with $\Gamma(t)\in[0,1]$ for almost every
$t\in[0,T]$. So, $\|z\|_{\infty}=1$, and passing to the weak limit in (3.7),
$z$ solves

$\begin{cases}J\dot{z}=(1-\Gamma(t))\nabla\mathscr{H}_{1}(z)+\Gamma(t)\nabla\mathscr{H}_{2}(z)\,,\\\
z(0)=z(T)\,.\end{cases}$

By [9, Lemma 2.3], either $z$ is a solution of
$J\dot{z}=\nabla\mathscr{H}_{1}(z)$, or it is a solution of
$J\dot{z}=\nabla\mathscr{H}_{2}(z)$. But this is a contradiction to the fact
that $J\dot{z}=\nabla\mathscr{H}_{1}(z)$ and
$J\dot{z}=\nabla\mathscr{H}_{2}(z)$ have only a trivial solution by (2.7).
Hence our supposition was wrong, and this completes the proof of the
proposition. ∎

To conclude the proof of the theorem, we fix $\rho\geq\bar{\rho}$, and rewrite
the Hamiltonian function of system (3.4) as

$K_{\rho}(t,x,y,w)={\cal
H}(t,x,y)+\frac{1}{2}\big{[}\mathscr{H}_{1}(w)+\mathscr{H}_{2}(w)\big{]}+\widetilde{P}_{\rho}(t,x,y,w)\,,$
(3.9)

with

$\widetilde{P}_{\rho}(t,x,y,w)=P(t,x,y,w)+Q(t,w)+\Phi_{\rho}(t,w)-\frac{1}{2}\big{[}\mathscr{H}_{1}(w)+\mathscr{H}_{2}(w)\big{]}\,.$

Proposition 3.1 provides an a priori bound in $C([0,T],{\mathbb{R}}^{2})$ for
the $w$ component of the solutions of system (3.4). Using the second system in
(3.4) we see that the a priori bound can be extended to the derivative of $w$.
This implies that we have an a priori bound in
$C^{1}([0,T],{\mathbb{R}}^{2})$. Hence, by the Ascoli–Arzelà Theorem that $w$
belongs to a compact set $W\subseteq C([0,T],{\mathbb{R}}^{2})$. Now since
$\varPhi_{\rho}(t,w)-\frac{1}{2}\big{[}\mathscr{H}_{1}(w)+\mathscr{H}_{2}(w)\big{]}=0$
for $|w|\geq\rho^{3}$, we have that $\widetilde{P}_{\rho}(t,x,y,w)$ has a
bounded gradient with respect to $(x,y,w)$. Thus by using [18, Theorem 1.1]
system (3.4) has at least $M+1$ geometrically distinct $T$-periodic solutions
such that $y(0)\in\,\mathring{\mathcal{D}}\,$. By Proposition 3.1, these
solutions are indeed solutions of the original system (1.1). ∎

###  3.2 Proof of Theorem 2.4

Similar to the previous section, we first state and prove the following
proposition which is crucial for the proof of Theorem 2.4.

######  Proposition 3.2.

There exists $\overline{\rho}>1$ such that, for any $\rho>\overline{\rho}$,
every $T$-periodic solution $(x(t),y(t),w(t))$ of (3.4) satisfies
$\|w\|_{\infty}\leq\overline{\rho}$.

###### Proof.

Assume by contradiction that there is a sequence $(\rho_{n})_{n}$ in
$]1,+\infty[$ and a sequence of $T$-periodic solutions $(x_{n},y_{n},w_{n})$
of (3.4), with $\rho=\rho_{n}$, such that $\rho_{n}\to+\infty$ and
$\|w_{n}\|_{\infty}>n$. Let $z_{n}=\frac{w_{n}}{\|w_{n}\|_{\infty}}$. Then by
similar arguments in the proof of Proposition 3.1, we can find a function
$z\in C([0,T],{\mathbb{R}}^{2})$ such that $\|z\|_{\infty}=1$, and it solves

$\begin{cases}J\dot{z}=(1-\Gamma(t))\nabla\mathscr{H}_{1}(z)+\Gamma(t)\nabla\mathscr{H}_{2}(z)\,,\\\
z(0)=z(T)\,.\end{cases}$

By [9, Lemma 2.3], either $z$ is a solution of
$J\dot{z}=\nabla\mathscr{H}_{1}(z)$, or it is a solution of
$J\dot{z}=\nabla\mathscr{H}_{2}(z)$. Let us assume that
$J\dot{z}=\nabla\mathscr{H}_{1}(z)$, and so $\Gamma(t)\equiv 0$. The other
case can be treated similarly.

For suitable $R_{0}>0$, $\theta_{0}\in[0,\tau_{1}[$ , it will be
$z(t)=R_{0}\varphi(t+\theta_{0})$. By $A2$ and (2.5), there exists $C_{4}>0$
such that

$|\dot{w}_{n}(t)|\leq C_{4}(1+|w_{n}(t)|)\,.$

So by Gronwall Lemma we deduce that

$|w_{n}(t_{0})|\leq|w_{n}(t_{1})|e^{C_{4}(t_{1}-t_{0})}\,,\quad\hbox{ for
every }t_{0}\leq t_{1}\,,$

and as a consequence we can say that if $\|w_{n}\|_{\infty}\to+\infty$, then
$|w_{n}(t)|\to+\infty$ uniformly in $t\in[0,T]$. This shows that $w_{n}(t)\neq
0$ for every $t\in[0,T]$. Now writing, in generalized polar coordinates,
$w_{n}(t)=r_{n}(t)\varphi(t+\theta_{n}(t))$, with
$\theta_{n}(0)\in[0,\tau_{1}[$ for every $n$, we see that $r_{n}(t)\to\infty$
uniformly in $t\in[0,T]$. The second system in (3.4) with $\rho=\rho_{n}$
gives

$\displaystyle\dot{\theta}_{n}(t)$ $\displaystyle=\frac{\langle
J\dot{w}_{n}(t)\,,w_{n}(t)\rangle}{r_{n}^{2}(t)}-1$
$\displaystyle=\frac{\Big{\langle}F_{\rho_{n}}\big{(}t,r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\Big{\rangle}-r_{n}(t)}{r_{n}(t)}$
$\displaystyle\quad+\frac{\Big{\langle}\nabla_{w}P\big{(}t,x_{n}(t),y_{n}(t),r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\Big{\rangle}}{r_{n}(t)}\,.$

Since $z$ solves $J\dot{z}=\nabla\mathscr{H}_{1}(z)$, by (2.12) it performs
$N$ rotations around the origin when $t$ varies from $0$ to $T$. Moreover,
since $z_{n}\to z$ uniformly, for $n$ sufficiently large every $z_{n}$
performs $N$ rotations around the origin, and also every $w_{n}$, since
$w_{n}=\|w_{n}\|_{\infty}z_{n}$. As a consequence, for such $n$ it is
$\theta_{n}(0)=\theta_{n}(T)$, thus integrating in the above computation , we
get

$\displaystyle 0$ $\displaystyle=\int_{0}^{T}\Bigg{[}\frac{\langle
F_{\rho_{n}}\big{(}t,r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\rangle-
r_{n}(t)}{r_{n}(t)}$
$\displaystyle\quad+\frac{\Big{\langle}\nabla_{w}P\big{(}t,x_{n}(t),y_{n}(t),r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\Big{\rangle}}{r_{n}(t)}\Bigg{]}\,dt\,.$

So, for $n$ large,

$0=\int_{0}^{T}\Bigg{[}\frac{\langle
F_{\rho_{n}}\big{(}t,r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\rangle-
r_{n}(t)}{r^{v}_{n}(t)}\\\
+\frac{\Big{\langle}\nabla_{w}P\big{(}t,x_{n}(t),y_{n}(t),r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\Big{\rangle}}{r^{v}_{n}(t)}\Bigg{]}\,dt\,,$
(3.10)

where $r^{v}_{n}(t)=\frac{r_{n}(t)}{\|w_{n}\|_{\infty}}$. We now prove that
the expressions in the above computation are bounded below in order to apply
Fatou’s Lemma. Clearly, the second term in (3.10) is bounded below by $A2$.
For the first one, we discuss the following cases.

Case 1. If $|r_{n}(t)\varphi(t+\theta_{n}(t))|\leq\rho_{n}$, then
$F_{\rho_{n}}=F$ and so by (1.6), (2.2), (2.3), (2.4), and (2.6), we have

$\big{\langle}F\big{(}t,r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\big{\rangle}-r_{n}(t)\geq-\widetilde{C}\|\varphi\|_{\infty}\,.$

Case 2. If $|r_{n}(t)\varphi(t+\theta_{n}(t))|\geq\rho_{n}^{3}$, then

$F_{\rho_{n}}(t,r_{n}(t)\varphi(t+\theta_{n}(t)))=\frac{r_{n}(t)}{2}\big{[}\nabla\mathscr{H}_{1}(\varphi(t+\theta_{n}(t)))+\nabla\mathscr{H}_{2}(\varphi(t+\theta_{n}(t)))\big{]}\\\
+\nabla_{w}Q(t,r_{n}(t)\varphi(t+\theta_{n}(t)))\,.$

Using the fact that ${\mathscr{H}}_{1}(w)<{\mathscr{H}}_{2}(w)$, there exists
$\varepsilon>0$ such that

${\mathscr{H}}_{2}(\varphi(s))-{\mathscr{H}}_{1}(\varphi(s))\geq\varepsilon\,,\quad\hbox{
for every }s\in[0,T]\,,$

and thus by (1.6), (2.3) and (2.6), we have

$\displaystyle\big{\langle}F_{\rho_{n}}\big{(}t,r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\big{\rangle}-r_{n}(t)$
$\displaystyle\geq\frac{r_{n}(t)}{2}\bigg{(}1+(1+2\varepsilon)\bigg{)}-r_{n}(t)+\big{\langle}\nabla_{w}Q(t,r_{n}(t)\varphi(t+\theta_{n}(t))),\varphi(t+\theta_{n}(t))\big{\rangle}$
$\displaystyle\geq\varepsilon
r_{n}(t)-\widetilde{C}\|\varphi\|_{\infty}\to+\infty\,,$ (3.11)

when $n\to+\infty$, since $r_{n}(t)\to+\infty$.

Case 3. Finally, if
$\rho_{n}\leq|r_{n}(t)\varphi(t+\theta_{n}(t))|\leq\rho_{n}^{3}$, we just
interpolate the previous two cases so to get the desired bound.

Thus we can use Fatou’s Lemma to (3.10) and get

$0\geq\int_{0}^{T}\liminf_{n}\Bigg{[}\frac{\langle
F_{\rho_{n}}\big{(}t,r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\rangle-
r_{n}(t)}{r^{v}_{n}(t)}\\\
+\frac{\Big{\langle}\nabla_{w}P\big{(}t,x_{n}(t),y_{n}(t),r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\Big{\rangle}}{r^{v}_{n}(t)}\Bigg{]}\,dt\,.$

Now using standard properties of the inferior limit, taking into account that,
since $z_{n}\to z$ uniformly, $r^{v}_{n}\to R_{0}$ uniformly and by $A2$, we
have

$\overline{m}\int_{0}^{T}|\varphi(t)|\,dt\\\
\geq\int_{0}^{T}\liminf_{n}\big{[}\langle
F_{\rho_{n}}\big{(}t,r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\rangle-
r_{n}(t)\big{]}\,dt\,.$ (3.12)

We fix $t\in[0,T]$ and discuss the possible three cases to verify that

$\big{\langle}F_{\rho_{n}}\big{(}t,r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\big{\rangle}\\\
\geq\big{\langle}F\big{(}t,r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\big{\rangle}\,.$
(3.13)

Case 1. If $|r_{n}(t)\varphi(t+\theta_{n}(t))|\leq\rho_{n}$, then
$F_{\rho_{n}}=F$.

Case 2. If $|r_{n}(t)\varphi(t+\theta_{n}(t))|\geq\rho_{n}^{3}$, then by (3.2)
for $n$ large enough, (3.13) follows.

Case 3. Finally, if
$\rho_{n}\leq|r_{n}(t)\varphi(t+\theta_{n}(t))|\leq\rho_{n}^{3}$, we just
interpolate the previous two cases so to get (3.13), for $n$ large enough.

Thus (3.12) implies that

$\overline{m}\int_{0}^{T}|\varphi(t)|\,dt\geq\int_{0}^{T}\liminf_{n}\big{[}\langle
F\big{(}t,r_{n}(t)\varphi(t+\theta_{n}(t))\big{)}\,,\varphi(t+\theta_{n}(t))\rangle-
r_{n}(t)\big{]}\,dt\,.$

By using the fact that $z_{n}\to z$ uniformly, we can assume without loss of
generality that, up to a subsequence, $\theta_{n}(t)\to\theta_{0}\in[0,T]$
uniformly. Recalling (2.6), for every $t\in[0,T]$ we are computing the
inferior limit which appears in (2.8) along the particular subsequence
$(r_{n}(t)\,,\theta_{n}(t))$ for which, $\theta_{n}(t)\to\theta_{0}$ and
$r_{n}(t)=||w_{n}||_{\infty}r^{v}_{n}(t)\to+\infty$. We obtain

$\overline{m}\int_{0}^{T}|\varphi(t)|dt\geq\int_{0}^{T}\liminf\limits_{(\lambda,\,s)\to(+\infty,\,\theta_{0})}\big{[}\big{\langle}F\big{(}t,\lambda\varphi(t+s)\big{)},\,\varphi(t+s)\big{\rangle}-\lambda\big{)}\big{]}\,dt\,,$

which contradicts (2.8), thus completing the proof of the proposition. ∎

To complete the proof of Theorem 2.4, we can write exactly the same lines as
for Theorem 2.1 after the proof of Proposition 3.1. ∎

###  3.3 Proof of Theorems 2.2 and 2.3

We first prove Theorem 2.2, and for this we assume without loss of generality
that ${\mathscr{H}}_{1}(w)<{\mathscr{H}}_{2}(w)$ for all
$w\in{\mathbb{R}}^{2}\setminus\\{0\\}$. Because if this is not true, then we
can define a function $\widetilde{{\mathscr{H}}}_{2}$ by

$\widetilde{{\mathscr{H}}}_{2}(w)={\mathscr{H}}_{2}(w)+\epsilon|w|^{2}$

for some $\epsilon>0$ small enough so that

${\mathscr{H}}_{1}(w)<\widetilde{{\mathscr{H}}}_{2}(w).$

Denote by $\widetilde{\tau}_{\psi}$, the minimal period corresponding to
system

$J\dot{w}=\nabla\widetilde{{\mathscr{H}}}_{2}(w)\,.$

We see that for $\epsilon$ small enough, it still satisfies

$\frac{T}{N+1}<\widetilde{\tau}_{\psi}<\tau_{2}\leq\tau_{1}=\frac{T}{N}\,.$

The proof of Theorem 2.2 now follows similarly as the proof of Theorem 2.4.

To prove Theorem 2.3, we can define a function $\widetilde{{\mathscr{H}}}_{1}$
by

$\widetilde{{\mathscr{H}}}_{1}(w)={\mathscr{H}}_{1}(w)-\epsilon|w|^{2}$

for some $\epsilon>0$ small enough so that

${\mathscr{H}}_{2}(w)>\widetilde{{\mathscr{H}}}_{1}(w).$

Denote by $\widetilde{\tau}_{\varphi}$, the minimal period corresponding to
system

$J\dot{w}=\nabla\widetilde{{\mathscr{H}}}_{1}(w)\,.$

It is clear that for $\epsilon$ small enough, it satisfies

$\frac{T}{N+1}=\tau_{2}\leq\tau_{1}<\widetilde{\tau}_{\varphi}<\frac{T}{N}\,,$

and the proof follows similarly as the proof of Theorem 2.4.

##  4 Some variants of the main results

We first discuss some variants of our results by changing assumption $A4$. Let
us denote respectively the four quadrants of the plane ${\mathbb{R}}^{2}$ by
$E_{1},E_{2},E_{3},$ and $E_{4}$. Instead of $A4$, we assume the following
assumption.

$A4^{\prime}.$ There are five functions $\gamma_{j}:{\mathbb{R}}\times
E_{j}\to[0,1]$ and $Q:{\mathbb{R}}\times{\mathbb{R}}^{2}\to{\mathbb{R}}$ such
that

$F(t,w)=(1-\gamma_{j}(t,w))\nabla\mathscr{H}_{1}(w)+\gamma_{j}(t,w)\nabla\mathscr{H}_{2}(w)+\nabla_{w}Q(t,w)\,,$
(4.1)

for all $w\in E_{j}$, where all the functions are continuous, $T$-periodic in
$t$, and $\nabla_{w}Q(t,w)$ is uniformly bounded, i.e., there exists
$\widetilde{C}>0$ such that

$|\nabla_{w}Q(t,w)|\leq\widetilde{C}\,,\quad\hbox{ for every
}\,(t,w)\in[0,T]\times{\mathbb{R}}^{2}\,.$ (4.2)

We now state the variant of our main results.

######  Theorem 4.1.

If in the statement of Theorem 2.1 we replace assumption $A4$ by
$A4^{\prime}$, then the same conclusion holds.

###### Proof.

Similar to Section 3, we can find a function
$\varPhi:{\mathbb{R}}\times{\mathbb{R}}^{2}\to{\mathbb{R}}$ such that
$F(t,w)=\nabla_{w}\varPhi(t,w)+\nabla_{w}Q(t,w)$, and

$\nabla_{w}\varPhi(t,w)=(1-\gamma_{j}(t,w))\nabla\mathscr{H}_{1}(w)+\gamma_{j}(t,w)\nabla\mathscr{H}_{2}(w)\,,\quad\hbox{
for every }\,w\in E_{j}\,.$ (4.3)

Now we modify system (1.1) by the similar way we modified in Section 3 with
the only change in defining $\gamma_{\rho}$, i.e., we define
$\gamma_{j_{\rho}}:{\mathbb{R}}\times E_{j}\to[0,1]$ instead of
$\gamma_{\rho}:{\mathbb{R}}\times{\mathbb{R}}^{2}\to[0,1]$ as

$\gamma_{j_{\rho}}(t,w)=\begin{cases}\gamma_{j}(t,w)&\quad\mbox{ if
}|w|\leq\rho\,,\\\
\eta_{\rho}(|w|)\gamma_{j}(t,w)+\frac{1}{2}(1-\eta_{\rho}(|w|))&\quad\mbox{ if
}\rho\leq|w|\leq\rho^{3}\,,\\\ \frac{1}{2}&\quad\mbox{ if
}|w|\geq\rho^{3}\,.\end{cases}$

We can now prove the a priori bound for the modified system (3.4) in this case
following the same lines as in the proof of Proposition 3.1 with the only
difference in the $\Gamma_{n}(t)$, which we can define in this case as
follows:

$\Gamma_{n}(t)=\gamma_{j_{\rho_{n}}}(t,\|w_{n}\|_{\infty}z_{n}(t))\,,\quad\hbox{
when }z_{n}\in\mathring{E}_{j}\,.$

We notice that $\Gamma_{n}(t)$ is not defined if $z_{n}(t)$ belongs to an
axis. Anyway, the set of those $t^{\prime}s$ has zero measure, since if $n$ is
large enough, one has

$\langle\dot{z}_{n}(t),-Jz_{n}(t)\rangle>0\,,\quad\hbox{ for every
}t\in[0,T]\,,$

hence $z_{n}(t)$ crosses the axes transversally.

After the a priori bound, we rewrite the Hamiltonian function of system (3.4)
as in (3.9) and by a similar way, we obtain $M+1$ geometrically distinct
$T$-periodic solutions $(x(t),y(t),w(t))$ of system (1.1) such that
$y(0)\in\,\mathring{\mathcal{D}}\,$. The proof of the theorem is thus
completed. ∎

We now state the case of double resonance. We skip the proof for briefness.

######  Theorem 4.2.

If in the statement of Theorem 2.4 we replace assumption $A4$ by
$A4^{\prime}$, then the same conclusion holds.

The cases of simple resonance can be treated similarly.

######  Remark 4.3.

It is worth noticing that we could discuss a more general case in assumption
$A4^{\prime}$, i.e., instead of using the quadrants $E_{j}$, consider
$\alpha_{1}<\alpha_{2}<\dots<\alpha_{n}=\alpha_{1}+2\pi$, and define the
planar sectors

$\widehat{E}_{j}=\\{re^{i\theta}\;:\;\alpha_{j}\leq\theta\leq\alpha_{j+1}\,,\;r\geq
0\\}\,.$

We now consider some variants of our main results by modifying $A3$, and to
this aim, we first recall some definitions. A closed convex bounded subset
$\mathcal{D}$ of $\mathbb{R}^{M}$ having nonempty interior
$\mathring{\mathcal{D}}$ is said to be a convex body of $\mathbb{R}^{M}$. If
we assume that $\mathcal{D}$ has a smooth boundary, then we denote the unit
outward normal at $\xi\in\partial\mathcal{D}$ by $\nu_{\mathcal{D}}(\xi)$.
Moreover, we say that $\mathcal{D}$ is strongly convex if for any
$k\in\partial\mathcal{D}$, the map $\mathcal{F}:\mathcal{D}\to{\mathbb{R}}$
defined by $\mathcal{F}(\xi)=\langle\xi-k,\nu_{\mathcal{D}}(k)\rangle$ has a
unique maximum point at $\xi=k$.

First we state the following “avoiding rays” assumption.

$A3^{\prime}$. There exists a convex body $\mathcal{D}$ of $\mathbb{R}^{M}$,
having a smooth boundary, such that for $\sigma\in\\{-1,1\\}$ and for every
$C^{1}$-function $\mathcal{W}:[0,T]\to{\mathbb{R}}^{2L}$, all the solutions
$(x,y)$ of system (2.1) starting with $y(0)\in{\cal D}$ are defined on
$[0,T]$, and

$y(0)\in\partial{\mathcal{D}}\quad\Rightarrow\quad
x(T)-x(0)\notin\\{\sigma\lambda\,\nu_{\mathcal{D}}(y(0)):\lambda\geq 0\\}\,.$

######  Theorem 4.4.

If in the statements of Theorem 2.1, 2.2, 2.3, 2.4, we replace assumption $A3$
by $A3^{\prime}$, the same conclusion holds.

###### Proof.

The arguments are the same as in Section 3, with the only difference that
instead of applying [18, Theorem 1.1], we apply [18, Theorem 3.1]. ∎

Now we state the following “indefinite twist” assumption.

$A3^{\prime\prime}$. There are a strongly convex body ${\cal D}$ of
${\mathbb{R}}^{M}$ having a smooth boundary and a symmetric regular $M\times
M$ matrix $\mathbb{A}$ such that for every $C^{1}$-function
$\mathcal{W}:[0,T]\to{\mathbb{R}}^{2L}$, all the solutions $(x,y)$ of system
(2.1) starting with $y(0)\in{\cal D}$ are defined on $[0,T]$, and

$y(0)\in\partial{\mathcal{D}}\quad\Rightarrow\quad\langle
x(T)-x(0)\,,\,\mathbb{A}\nu_{\mathcal{D}}(y(0))\rangle>0\,.$

######  Theorem 4.5.

If in the statements of Theorem 2.1, 2.2, 2.3, 2.4, we replace assumption $A3$
by $A3^{\prime\prime}$, the same conclusion holds.

###### Proof.

Apply [18, Theorem 3.2] instead of [18, Theorem 1.1] in Section 3. ∎

##  5 Applications to scalar second order equations

As a direct consequence of Theorem 2.4, we discuss the main results in [13,
Theorem 5, Theorem 14]. We consider the following system

$\begin{cases}\dot{x}=\nabla_{y}\mathcal{H}(t,x,y)+\nabla_{y}P(t,x,y,u)\,,\\\
\dot{y}=-\nabla_{x}\mathcal{H}(t,x,y)-\nabla_{x}P(t,x,y,u)\,,\\\
\ddot{u}+g(t,u)=\partial_{u}P(t,x,u)\,,\end{cases}$ (5.1)

where all the involved functions are continuous and $T$ periodic in the
variable $t$.

$B4.$ There exist some positive constants
$\mu_{1},\mu_{2},\nu_{1},\nu_{2},C_{6},C_{7}$ such that

$\begin{cases}g(t,u)\leq\nu_{1}u+C_{6}\quad\hbox{if }u\leq 0\,,\\\
g(t,u)\geq\mu_{1}u-C_{6}\quad\hbox{if }u\geq 0\,,\end{cases}$ (5.2)
$\begin{cases}g(t,u)\geq\nu_{2}u-C_{7}\quad\hbox{if }u\leq 0\,,\\\
g(t,u)\leq\mu_{2}u+C_{7}\quad\hbox{if }u\geq 0\,.\end{cases}$ (5.3)

Moreover, we consider the following Landesman–Lazer conditions.

$B5.$ For a nontrivial $T$-periodic solution $\varphi$ of
$\ddot{\varphi}+\mu_{1}\varphi^{+}-\nu_{1}\varphi^{-}=0$, one has

$\int_{\\{\varphi<0\\}}\liminf\limits_{u\to-\infty}\big{[}\nu_{1}u-g(t,u)\big{]}|\varphi(t)|dt\\\
+\int_{\\{\varphi>0\\}}\liminf\limits_{u\to+\infty}\big{[}g(t,u)-\mu_{1}u\big{]}\varphi(t)dt>\overline{m}\int_{0}^{T}|\varphi(t)|dt\,.$
(5.4)

$B6.$ For a nontrivial $T$-periodic solution $\psi$ of
$\ddot{\psi}+\mu_{2}\psi^{+}-\nu_{2}\psi^{-}=0$, one has

$\int_{\\{\psi<0\\}}\liminf\limits_{u\to-\infty}\big{[}g(t,u)-\nu_{2}u\big{]}|\psi(t)|dt\\\
+\int_{\\{\psi>0\\}}\liminf\limits_{u\to+\infty}\big{[}\mu_{2}u-g(t,u)\big{]}\psi(t)dt>\overline{m}\int_{0}^{T}|\psi(t)|dt\,.$
(5.5)

######  Theorem 5.1.

Let $A1-A3$ and $B4-B6$ hold true, and assume that there exists a positive
integer $N$ such that

$\frac{T}{N+1}=\frac{\pi}{\sqrt{\mu_{2}}}+\frac{\pi}{\sqrt{\nu_{2}}}\leq\frac{\pi}{\sqrt{\mu_{1}}}+\frac{\pi}{\sqrt{\nu_{1}}}=\frac{T}{N}\,.$
(5.6)

Then system (5.1) has at least $M+1$ geometrically distinct $T$-periodic
solutions, with $y(0)\in\mathring{\mathcal{D}}$.

###### Proof.

It can be shown (see e.g. [7, Lemma 1]) that using conditions (5.2) and (5.3),
there exists a bounded function $h(t,u)$ such that

$g(t,u)=\zeta_{1}(t,u)u^{+}-\zeta_{2}(t,u)u^{-}+h(t,u)\,,$ (5.7)

where $u^{+}=\max\\{u,0\\}$, $u^{-}=\max\\{-u,0\\}$, and

$\mu_{1}\leq\zeta_{1}(t,u)\leq\mu_{2}\,,\qquad\nu_{1}\leq\zeta_{2}(t,u)\leq\nu_{2}\,,$
(5.8)

for almost every $t\in[0,T]$, and every $u\in{\mathbb{R}}$. System (5.1) can
be written as

$\begin{cases}\dot{x}=\nabla_{y}\mathcal{H}(t,x,y)+\nabla_{y}P(t,x,y,u)\,,\\\
\dot{y}=-\nabla_{x}\mathcal{H}(t,x,y)-\nabla_{x}P(t,x,y,u)\,,\\\
\dot{u}=v\,,\quad-\dot{v}=g(t,u)-\partial{u}P(t,x,u)\,.\end{cases}$ (5.9)

Now for $w=(u,v)$ with the aim of applying Theorem 4.2, define

${\mathscr{H}}_{1}(w)=\frac{1}{2}\bigg{[}\mu_{1}(u^{+})^{2}+\nu_{1}(u^{-})^{2}+v^{2}\bigg{]}\,,\quad{\mathscr{H}}_{2}(w)=\frac{1}{2}\bigg{[}\mu_{2}(u^{+})^{2}+\nu_{2}(u^{-})^{2}+v^{2}\bigg{]}$
$Q(t,w)=\int_{0}^{u}h(t,s)ds,\quad F(t,w)=\Bigg{(}\begin{matrix}g(t,w)\\\
v\end{matrix}\Bigg{)}\,.$

For $u>0,$ $w\in E_{1}\cup E_{4}$ and $u^{-}=0$. Thus by choosing

$\gamma_{1}(t,w)=\gamma_{4}(t,w)=\frac{\zeta_{1}(t,u)-\mu_{1}}{\mu_{2}-\mu_{1}}\,,$

it is clear by (5.8) that

$0\leq\gamma_{1}(t,w)=\gamma_{4}(t,w)\leq 1\,.$

Similarly, for $u<0$, $u^{+}=0$ and $w\in E_{2}\cup E_{3}$. Thus by defining

$\gamma_{2}(t,w)=\gamma_{3}(t,w)=\frac{\zeta_{2}(t,u)-\nu_{1}}{\nu_{2}-\nu_{1}}\,,$

we see by (5.8) that

$0\leq\gamma_{1}(t,w)=\gamma_{4}(t,w)\leq 1\,.$

Hence we can use Theorem 4.2, and then system (5.1) has at least $M+1$
geometrically distinct $T$-periodic solutions $(x,y,u)$ such that
$y(0)\in\mathring{\mathcal{D}}.$ ∎

We remark here that the main result for the nonresonance case [13, Theorem 2]
can also be obtained by a similar way using Theorem 4.1.

As an example, for $M=1$, we consider the coupling of a pendulum-like equation
with an asymmetric oscillator, i.e.,

$\begin{cases}\ddot{q}+A\sin q=e(t)+\partial_{q}P(t,x,u)\,,\\\
\ddot{u}+f(u)+h(t,u)=\partial_{u}P(t,x,u)\,,\end{cases}$ (5.10)

where $h$ is bounded,

$f(u)=\Bigg{(}\frac{\mu_{1}+\mu_{2}}{2}+\frac{\mu_{2}-\mu_{1}}{2}\cos
u\Bigg{)}u^{+}-\Bigg{(}\frac{\nu_{1}+\nu_{2}}{2}+\frac{\nu_{2}-\nu_{1}}{2}\sin(u^{3})\Bigg{)}u^{-}\,$

and the constants $A,\mu_{1},\mu_{2},\nu_{1},\nu_{2}$ are all positive with
$\mu_{1}\leq\mu_{2}$ and $\nu_{1}\leq\nu_{2}$. Assume that $P(t,x,u)$ is
$T$-periodic in $t$ and $2\pi$-periodic in $x$, and that it has a bounded
gradient with respect to $(x,u)$. Setting $E(t)=\int_{0}^{t}e(s)\,ds$, system
(5.10) is equivalent to

$\begin{cases}\dot{q}=p+E(t)\,,\quad\dot{p}=-A\sin
q+\partial_{q}P(t,x,u)\,,\\\
\dot{u}=v\,,\quad\dot{v}=-f(u)-h(t,u)+\partial_{u}P(t,x,u)\,.\end{cases}$

Assuming $e(t)$ to be $T$-periodic with

$\int_{0}^{T}e(t)\,dt=0\,,$

the function $E(t)$ is $T$-periodic, as well.

It has been proved in [18, Section 3], that the twist condition $A3$ holds
true.

On the other hand, if we define $\mathscr{H}_{1}$, ${\mathscr{H}}_{2}$ by

${\mathscr{H}}_{1}(w)=\frac{1}{2}\bigg{[}\mu_{1}(u^{+})^{2}+\nu_{1}(u^{-})^{2}+v^{2}\bigg{]}\,,\quad{\mathscr{H}}_{2}(w)=\frac{1}{2}\bigg{[}\mu_{2}(u^{+})^{2}+\nu_{2}(u^{-})^{2}+v^{2}\bigg{]}\,,$

with $w=(u,v)$, then $\mathscr{H}_{1}$ and ${\mathscr{H}}_{2}$ are positive,
positively homogeneous of degree $2$,
${\mathscr{H}}_{1}(w)<{\mathscr{H}}_{2}(w)$ for all $w\neq 0$. Thus all the
solutions of system $J\dot{w}=\nabla\mathscr{H}_{1}(w)$ and
$J\dot{w}=\nabla{\mathscr{H}}_{2}(w)$ are periodic with fixed period,
respectively

$\tau_{1}=\frac{\pi}{\sqrt{\mu_{1}}}+\frac{\pi}{\sqrt{\nu_{1}}}\,,\quad\tau_{2}=\frac{\pi}{\sqrt{\mu_{2}}}+\frac{\pi}{\sqrt{\nu_{2}}}\,.$

Moreover, we assume the following Landesman–Lazer conditions.

$B5^{\prime}.$ For a nontrivial $T$-periodic solution $\varphi$ of
$\ddot{\varphi}+\mu_{1}\varphi^{+}-\nu_{1}\varphi^{-}=0$, one has

$\int_{\\{\varphi<0\\}}\liminf\limits_{u\to-\infty}\big{[}-h(t,u)\big{]}|\varphi(t)|dt\\\
+\int_{\\{\varphi>0\\}}\liminf\limits_{u\to+\infty}h(t,u)\varphi(t)dt>\overline{m}\int_{0}^{T}|\varphi(t)|dt\,.$
(5.11)

$B6^{\prime}.$ For a nontrivial $T$-periodic solution $\psi$ of
$\ddot{\psi}+\mu_{2}\psi^{+}-\nu_{2}\psi^{-}=0$, one has

$\int_{\\{\psi<0\\}}\liminf\limits_{u\to-\infty}h(t,u)|\psi(t)|dt\\\
+\int_{\\{\psi>0\\}}\liminf\limits_{u\to+\infty}\big{[}-h(t,u)\big{]}\psi(t)dt>\overline{m}\int_{0}^{T}|\psi(t)|dt\,.$
(5.12)

All the assumptions of Theorem 5.1 are satisfied, and we can thus state the
following.

######  Corollary 5.2.

In the above setting, assume that $B5^{\prime},B6^{\prime}$ hold, and

$\frac{T}{N+1}=\frac{\pi}{\sqrt{\mu_{2}}}+\frac{\pi}{\sqrt{\nu_{2}}}\leq\frac{\pi}{\sqrt{\mu_{1}}}+\frac{\pi}{\sqrt{\nu_{1}}}=\frac{T}{N}\,.$

Then system (5.10) has at least two geometrically distinct $T$-periodic
solutions.

The above result generalizes a classical result by Mawhin and Willem [26] on
the multiplicity of periodic solutions for the pendulum equation.

##  6 Neumann-type boundary value problem

Consider system (1.1) with Neumann-type boundary conditions

$\begin{cases}y(a)=0=y(b)\,,\\\ v(a)=0=v(b)\,,\end{cases}$ (6.1)

with $w=(u,v)$. Notice that in this case, we omit the periodicity in the
variable $t$ of all the functions in system (1.1). We only consider all the
involved functions to be continuous and continuously differentiable with
respect to $(x,y,w)$. For more details in this direction, we refer the reader
to some recent advancements [1, 12, 15, 24, 27].

For the Neumann-type boundary value problem, instead of assumptions $A3$,
$A4$, $A5$ and $A6$, we state the following assumptions.

$A3^{\prime\prime\prime}$. All the solutions of system (1.1) satisfying
$y(a)=v(a)=0$ are defined on $[a,b]$.

$A4^{\prime\prime\prime}.$ There are two functions
$\gamma:[a,b]\times{\mathbb{R}}^{2}\to[0,1]$ and
$Q:[a,b]\times{\mathbb{R}}^{2}\to{\mathbb{R}}$ such that

$F(t,w)=(1-\gamma(t,w))\nabla\mathscr{H}_{1}(w)+\gamma(t,w)\nabla\mathscr{H}_{2}(w)+\nabla_{w}Q(t,w)\,,$
(6.2)

where all the functions are continuous and $\nabla_{w}Q(t,w)$ is uniformly
bounded, i.e., there exists $\widetilde{C}>0$ such that

$|\nabla_{w}Q(t,w)|\leq\widetilde{C}\,,\quad\hbox{ for every
}\,(t,w)\in[a,b]\times{\mathbb{R}}^{2}\,.$ (6.3)

$A5^{\prime\prime\prime}.$ For every $\theta\in[a,b]$, one has

$\int_{a}^{b}\liminf\limits_{(\lambda,\,s)\to(+\infty,\,\theta)}\big{[}\big{\langle}F\big{(}t,\lambda\varphi(t+s)\big{)},\,\varphi(t+s)\big{\rangle}-2\lambda\mathscr{H}_{1}\big{(}\varphi(t)\big{)}\big{]}\,dt>\overline{m}\int_{a}^{b}|\varphi(t)|dt\,.$
(6.4)

$A6^{\prime\prime\prime}.$ For every $\theta\in[a,b]$, one has

$\int_{a}^{b}\liminf\limits_{(\lambda,\,s)\to(+\infty,\,\theta)}\big{[}2\lambda\mathscr{H}_{2}\big{(}\psi(t)\big{)}-\big{\langle}F\big{(}t,\lambda\psi(t+s)\big{)},\,\psi(t+s)\big{\rangle}\big{]}\,dt>\overline{m}\int_{a}^{b}|\psi(t)|dt\,.$
(6.5)

Considering problem (1.1)–(6.1), if $u_{0}<0$, for all solutions $\zeta$ of

$\dot{u}=\nabla_{v}\mathscr{H}_{j}(u,v)\,,\qquad\dot{v}=-\nabla_{u}\mathscr{H}_{j}(u,v)\,,$
(6.6)

starting with $\zeta(0)=(u_{0},0)$, there is a first time $\tau_{j_{+}}>0$ for
which $v(\tau_{j_{+}})=0$, while $v(t)>0$ for all $t\in\,]0,\tau_{j_{+}}[$ ,
and this time $\tau_{j_{+}}$ is independent of $u_{0}<0$. Similarly, if
$u_{0}>0$, there is a first time $\tau_{j_{-}}>0$ for which
$v(\tau_{j_{-}})=0$, while $v(t)<0$ for all $t\in\,]0,\tau_{j_{-}}[$ , and
this time $\tau_{j_{-}}$ is independent of $u_{0}>0$. Clearly enough,
$\tau_{j}=\tau_{j_{+}}+\tau_{j_{-}}$ for $j=1,2$.

Here is our first main result which is associated with a nonresonant
situation.

######  Theorem 6.1.

Let $A1-A2$ and $A3^{\prime\prime\prime}-A4^{\prime\prime\prime}$ hold true
and for $j=1,2$, $\tau_{j_{+}}=\tau_{j_{-}}$. Assume moreover that there
exists a positive integer $N$ such that

$\frac{b-a}{N+1}<\tau_{2_{+}}\leq\tau_{1_{+}}<\frac{b-a}{N}\,.$ (6.7)

Then problem (1.1)– (6.1) has at least $M+1$ geometrically distinct solutions.

Concerning the double resonance case, we have the following result.

######  Theorem 6.2.

Let $A1-A2$ and $A3^{\prime\prime\prime}-A6^{\prime\prime\prime}$ hold true,
for $j=1,2$, $\tau_{j_{+}}=\tau_{j_{-}}$ and
${\mathscr{H}}_{1}(w)<{\mathscr{H}}_{2}(w)$ for all
$w\in{\mathbb{R}}^{2}\setminus\\{0\\}$. Assume moreover that there exists a
positive integer $N$ such that

$\tau_{2_{+}}=\frac{b-a}{N+1}\quad\hbox{ and
}\quad\tau_{1_{+}}=\frac{b-a}{N}\,.$ (6.8)

Then the same conclusion of Theorem 6.1 holds.

Let us remark here that a sufficient condition for having satisfied the
assumption $\tau_{j_{+}}=\tau_{j_{-}}$ is that the function $\mathscr{H}_{j}$
is even in $v$. This is a frequent case in the applications, where, e.g.,
$\mathscr{H}_{j}$ is quadratic in $v$.

We only prove the double resonance result. In order to prove this theorem, we
require the following result for the Neumann-type problem which is similar to
[9, Lemma 2.3] (for the periodic problem). We omit the proof of nonresonance
case and statements of simple resonance cases for briefness.

######  Lemma 6.3.

Assume that (6.8) is satisfied. Then, if $v\in H^{1}(a,b)$ solves

$\begin{cases}J\dot{v}=\alpha(t)\nabla H_{1}(v)+(1-\alpha(t))\nabla
H_{2}(v)\,,\\\ v(a)=0=v(b)\,,\end{cases}$ (6.9)

being $\alpha\in L^{2}(a,b)$, with $0\leq\alpha(t)\leq 1$ for almost every
$t\in[0,T]$, then $v$ solves either

$J\dot{v}=\nabla H_{1}(v)\,,$

or

$J\dot{v}=\nabla H_{2}(v)\,.$

###### Proof.

By a remark after [9, Lemma 2.2], we observe that a nontrivial solution of
(6.9) never reaches the origin. Indeed, if $v(t)$ solves (6.9) then also
$s_{0}v(t)$ does, for every $s_{0}>0$, thanks to the homogeneity of the right-
hand side; moreover, since the right-hand side grows at most linearly in $v$,
[9, Lemma 2.2] applies and also the remark after that lemma. It follows that,
$v(\bar{t})\neq 0$ for some $\bar{t}\in[a,b]$, then $v(t)\neq 0$ for every
$t\in[a,b]$.

Consequently, the usual system in polar coordinates $(\rho,\theta)$ is well
defined for system (6.9). Writing $v(t)=\rho(t)(\cos\theta(t),\sin\theta(t))$,
recalling Euler’s formula we get

$-\dot{\theta}(t)=2\alpha(t)H_{1}\big{(}\cos\theta(t),\sin\theta(t)\big{)}+2(1-\alpha(t))H_{2}\big{(}\cos\theta(t),\sin\theta(t)\big{)}\,.$

Since $0\leq\alpha(t)\leq 1$, it follows that

$-\frac{\dot{\theta}(t)}{2H_{2}(\cos\theta(t),\sin\theta(t))}\leq
1\leq-\frac{\dot{\theta}(t)}{2H_{1}(\cos\theta(t),\sin\theta(t))}\,,$ (6.10)

for almost every $t\in[a,b]$. Since $v(a)=0=v(b)$, $v$ performs an integer
number of half rotations around the origin, say $m$ half rotations. Recalling
that

$\int_{0}^{\pi}\frac{d\theta}{2H_{1}(\cos\theta,\sin\theta)}=\tau_{1_{+}}\,,\quad\int_{0}^{\pi}\frac{d\theta}{2H_{2}(\cos\theta,\sin\theta)}=\tau_{2_{+}}\,,$

integrating in (6.10) from $a$ to $b$, we get

$m\tau_{2_{+}}\leq b-a\leq m\tau_{1_{+}}\,,$

from which, using (6.8),

$N=\frac{b-a}{\tau_{1_{+}}}\leq m\leq\frac{b-a}{\tau_{2_{+}}}=N+1\,.$

Since $m$ is integer, this gives a contradiction unless $m=N$ and
$\tau_{1_{+}}=\frac{b-a}{N}$ or $m=N+1$ and $\tau_{2_{+}}=\frac{b-a}{N+1}$.

We only consider the first case, the other can be treated in a similar way.
Introducing the generalized polar coordinates in (6.9) by writing
$v(t)=r(t)\psi(t+\omega(t))$, we get

$\dot{r}(t)=-r(t)\alpha(t)\big{\langle}\nabla
H_{1}\big{(}\psi(t+\omega(t))\big{)},\dot{\psi}(t+\omega(t))\big{\rangle}\,,$
(6.11)

and

$\dot{\omega}(t)=\alpha(t)(2H_{1}(\psi(t+\omega(t))-1).$ (6.12)

Since $\omega(a)=0=\omega(b)$, integrating in (6.12) from $a$ to $b$, we have

$0=\int_{a}^{b}\alpha(t)(2H_{1}(\psi(t+\omega(t))-1)dt\,,$

and from the fact that $-\alpha(t)(2H_{1}(\psi(t+\omega(t))-1)\geq 0$ for
almost every $t\in[a,b]$, this expression satisfies

$\alpha(t)(2H_{1}(\psi(t+\omega(t))-1)=0$ (6.13)

almost everywhere, that is to say, $\dot{\omega}(t)=0$ for almost every
$t\in[a,b]$. Thus, since $\omega(t)$ is absolutely continuous, there exists
$\omega_{0}\in{\mathbb{R}}$ such that $\omega(t)=\omega_{0}$ for every
$t\in[a,b]$. Concerning (6.11), we have

$\dot{r}(t)=-r(t)\alpha(t)\big{\langle}\nabla
H_{1}\big{(}\psi(t+\omega(t))\big{)},\dot{\psi}(t+\omega(t))\big{\rangle}\,.$

We now need to prove that $\dot{r}(t)=0$ for almost every $t\in[a,b]$, Indeed,
if $t\in[a,b]$, (6.13) implies that either $\alpha(t)=0$, or
$H_{1}\big{(}\psi(t+\omega_{0}\big{)}=\frac{1}{2}$. If $\alpha(\bar{t})=0$,
then $\dot{r}(\bar{t})=0$; on the other hand, if $\alpha(\bar{t})>0$, then
$\bar{t}$ is a zero of the function $t\mapsto
H_{1}\big{(}\psi(t+\omega_{0}\big{)}-H_{2}\big{(}\psi(t+\omega_{0}\big{)}$,
which is of class $C^{1}$ and nonnegative. Necessarily $\bar{t}$ is then a
minimum of this function, and so

$\frac{d}{dt}H_{1}\big{(}\psi(t+\omega_{0}\big{)}_{|_{t=\bar{t}}}=\frac{d}{dt}H_{2}\big{(}\psi(t+\omega_{0}\big{)}_{|_{t=\bar{t}}}=0\,,$

as $H_{2}$ is preserved along $\psi$. It follows that $\big{\langle}\nabla
H_{1}\big{(}\psi(\bar{t}+\omega_{0}\big{)},\dot{\psi}(\bar{t}+\omega_{0}\big{\rangle}=0$,
so that $\dot{r}(\bar{t})=0$. Summing up, $\dot{r}(t)=0$ for almost every
$t\in[a,b]$, and, since $r(t)$ is absolutely continuous, this implies that
$r(t)$ is constant; being $v(t)=R_{0}\varphi(t+\omega_{0})$ for some
nonnegative constant $R_{0}$, it follows that $v$ is a solution of
$J\dot{v}=\nabla H_{2}(v)$ . This concludes the proof. ∎

To conclude the proof of Theorem 6.2, one can follow the same lines of the
proof of Theorem 2.4. We need to use Lemma 6.3 instead of [9, Lemma 2.3] in
Proposition 3.2 and at the end, we have to use [19, Theorem 1.1] (for $p=q=2$)
instead of [18, Theorem 1.1].∎

We can discuss variants of the above results as like Theorem 4.1 and Theorem
4.2. We are not entering into detail in order to avoid repetition of
arguments. Furthermore, we can adapt an example of application for this
situation as in Section 5.

##  7 Open problems

We conclude the paper by suggesting some open problems.

1. In [4], the coupling of twist dynamics with a resonant equation involving the Ahmad–Lazer–Paul condition has been addressed. However, under such a condition, the scenario of double resonance remains unexplored, even when dealing only with a scalar second order equation. Notice that the Ahmad– Lazer–Paul condition does not guarantee an a priori bound as the one proved in Proposition 3.1, as shown in [2]. A similar result for Neumann-type boundary condition is still open for consideration.

2. In [19], the authors studied about a Hamiltonian system coupled with positive and positively $(p,q)$-homogeneous Hamiltonian system. They have discussed both periodic and Neumann-type problems for the nonresonance case. It would be interesting to try the nonresonance/resonance cases by taking positive and positively $(p,q)$-homogeneous Hamiltonian functions in (2.2).

3. In [14], they investigated a system ruled by a positively-$(p,q)$-homogeneous Hamiltonian function with a friction term and used a nonresonance condition developed by Frederickson–Lazer [21]. We wonder whether in our setting, Frederickson–Lazer-condition could be applied.

Acknowledgement. We are deeply grateful to Prof. Alessandro Fonda for
generously allowing us to explore this research problem combined with his
insightful guidance and valuable discussions. We extend our sincere gratitude
to Prof. Andrea Sfecci for his valuable comments.

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Authors’ addresses:

Natnael Gezahegn Mamo and Wahid Ullah
---
Dipartimento di Matematica, Informatica e Geoscienze
Università degli Studi di Trieste
P.le Europa 1, 34127 Trieste, Italy
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

Mathematics Subject Classification: 34B15; 34C25

Keywords: Hamiltonian systems; periodic solutions; positively 2-homogeneous
systems; Landesman-Lazer condition; Poincaré–Birkhoff Theorem; Resonance;
Neumann-type boundary conditions.


[1]Pejman Habibiroudkenar

1,2,3]Mechanical Engineering, Aalto University, , Espoo, 02150, Uusimaa,
Finland

# DynaHull: Density-centric Dynamic Point Filtering in Point Clouds

<EMAIL_ADDRESS>Risto Ojala<EMAIL_ADDRESS>Kari Tammi
<EMAIL_ADDRESS>[

###### Abstract

In the field of indoor robotics, accurately navigating and mapping in dynamic
environments using point clouds can be a challenging task due to the presence
of dynamic points. These dynamic points are often represented by people in
indoor environments, but in industrial settings with moving machinery, there
can be various types of dynamic points. This study introduces DynaHull, a
novel technique designed to enhance indoor mapping accuracy by effectively
removing dynamic points from point clouds. DynaHull works by leveraging the
observation that, over multiple scans, stationary points have a higher density
compared to dynamic ones. Furthermore, DynaHull addresses mapping challenges
related to unevenly distributed points by clustering the map into smaller
sections. In each section, the density factor of each point is determined by
dividing the number of neighbors by the volume these neighboring points occupy
using a convex hull method. The algorithm removes the dynamic points using an
adaptive threshold based on the point count of each cluster, thus reducing the
false positives. The performance of DynaHull was compared to state-of-the-art
techniques, such as ERASOR, Removert, OctoMap, and a baseline statistical
outlier removal from Open3D, by comparing each method to the ground truth map
created during a low activity period in which only a few dynamic points were
present. The results indicated that DynaHull outperformed these techniques in
various metrics, noticeably in the Earth Mover’s Distance. This research
contributes to indoor robotics by providing efficient methods for dynamic
point removal, essential for accurate mapping and localization in dynamic
environments.

###### keywords:

Dynamic Points Removal, Point Cloud, SLAM, ConvexHull

##  1 Introduction

Autonomous robotics have become increasingly popular in recent years in
research in which indoor positioning and navigation systems have been of
particular interest [1]. Mapping stands out as a crucial element in the indoor
robotics ecosystem, being essential for several tasks, such as localization,
robot control, route-planning, perception, and overall system management.
Localization in outdoor settings is often supported by GPS systems, but this
traditional method lacks accuracy for indoor environments due to the
limitations of the line-of-sight (LoS) path from the satellite [2]. Moreover,
indoor mapping primarily relies on RGB cameras [3] and 2D LiDAR [4], with
fewer studies exploring the utilization of 3D LiDAR for indoor mapping as 3D
LiDAR has been mainly used in outdoor environments [5]. To establish an
accurate map, the robot must achieve precise localization; conversely,
effective localization is contingent upon a clean, stable map. This
interdependence is addressed through the implementation of a technique known
as Simultaneous Localization and Mapping (SLAM), a concept that has been set
in motion over the past three decades [6].

Mapping in highly dynamic environments, such as in a warehouse teeming with
moving people and operational automation robots, presents a unique challenge.
The mapping process in such settings is prone to capturing ’dynamic points’,
also known as ghost or phantom points, which can, for example, compromise the
accuracy of a robot’s localization in subsequent operations. Removing the
dynamic points affected by the ghosting effect, not only significantly
enhances localization accuracy and reduces drift [7], but can also enable
creation of a clean and accurate map capable of being used for various
reconstruction purposes. For instance, it can facilitate the generation of a
digital representation of the environment; this reconstructed map would
provide valuable insights and can be leveraged in multiple applications. In
the context of industrial settings, the reconstructed map can be used to
optimize material flow and analyze the efficiency of construction operations.

The proposed dynamic point removal method in this study relies on the
distinctive difference in density between the stationary and dynamic points.
After several scans, the stationary points are close to each other while the
dynamic points are sparse due to their movement. The density is calculated at
the point level using our algorithm. To alleviate the undistributed point
distribution in mapping processes, the filtering algorithm proposed in this
study also clusters the map into smaller sections with different densities.

###  1.1 Contributions

This study introduces a novel density-based dynamic point removal using a
convex hull, DynaHull, for point clouds, that is flexible with maps containing
non-uniformly distributed points. Furthermore, this research tailors a dynamic
points removal process for point clouds in indoor environments, deviating from
the conventional dynamic point elimination methods prevalent in outdoor
mapping. Our method performed better than state-art-dynamic-point-removal
methods in a highly dynamic indoor environment.

##  2 Related Work

Dynamic points in point clouds resulting from dynamic objects undermine
navigational reliability. In response, researchers have developed several
strategic approaches, each one addressing dynamic point issue from a different
technological and methodological standpoint. These solutions include ray-
tracing methods that consider space occupancy, visibility-based methods
focusing on logical geometric consistencies, classical and machine-learning
segmentation methods using distinctive features, and advanced data
partitioning of dynamic points.

###  2.1 Voxel Ray Casting Methods

Voxel ray-casting methods, commonly known as ’ray-tracing,’ effectively manage
the removal of dynamic objects by adjusting the likelihood of space being
occupied along the path of the ray as referenced in various studies [8, 9, 10,
11, 12]. Areas previously occupied by moving objects become increasingly
likely to be considered free as more rays intersect with that space. The early
work in this domain, grounded in Dempster-Shafer Theory (DST), also known as
the theory of belief, utilizes probability equations to determine the
occupation status of a voxel. Xiao et al. [8] enhanced the reliability of
spatial representations by fusing occupancy information and integrating data
from adjacent rays using the Weighted Dempster-Shafer theory. Building on
this, Georgi Postica et al. [9] further refined dynamic points detection by
discretizing the occupancy representation based on its distance from the
sample scan origin in DST, thus significantly reducing false positives and
increasing both speed and accuracy. Schauer et al. [10] improved this approach
by traversing a voxel occupancy grid along the lines of sight between the
sensor to find differences in volumetric occupancy between the scans,
resulting in an increase of the number of scans used for calculations and
reducing the required parameter tuning to just one: voxel size. To address the
computational cost challenges associated with voxel-based methods, the OctoMap
[11] method utilized an efficient oct-tree data structure. Building on this,
Kin-Zhang et al. [12] further optimized the approach by minimizing the impact
of noise and abnormal points in the OctoMap using the Statistical Outlier
Removal (SOR) technique for filtering, and Sample Consensus for performing
ground segmentation. Despite their precision and utility, ray-tracing methods
face the challenge of incidence angle ambiguity. As the range of measurement
extends, the ambiguity in the incidence angle of a point increases, leading to
the potential misidentification of adjacent ground points as dynamic objects.
This limitation has spurred the development of visibility-based techniques.

###  2.2 Visibility

Visibility-based methods [13, 14, 15] operate under the assumption that if a
query point is discerned behind a point previously integrated into the map, it
indicates that the initial point is dynamic. These approaches require finding
an association between a single point and a map point. Consequently, [13, 14]
have focused on iteratively updating the dynamic and stationary states by
repeatedly fusing multiple measurements and checking the visibility between a
query scan and a map. However, pose-estimate errors accumulated in LiDAR
motion can render this association inaccurate, thus raising the possibility of
erroneously deleting stationary points. To address this, Kim Removert proposed
a multiresolution range image-based false prediction-reverting algorithm, also
known as Removert. Initially, this method retains certain static points, then
progressively reinstates more ambiguous static points by enlarging the query-
to-map association window. This approach compensates for errors in LiDAR
motion or registration, hence the name ”remove, then revert.” However,
visibility-based methods struggle with occlusion and the elimination of
dynamic points at which static points were not detectable behind these dynamic
elements commonly known as Irremovable dynamic point, as demonstrated in Fig.
1. This challenge impedes accurately differentiating between static and
dynamic points, which is crucial for creating reliable and detailed maps. To
address these challenges and reduce false positives, segmentation methods have
gained popularity.

Figure 1: Irremovable dynamic points are presented in the color red because
there are no stationary objects present behind those points. Meanwhile, the
green areas are removed because there is a stationary box behind the observed
point. Table 1: Dynamic point removal methods and their limitations

Author Paper Method Type Designed for indoors Removal time Xiao et al. RChange
Detection in 3D Point Clouds Acquired by a Mobile Mapping System [8] Ray
tracing $\times$ Post-mapping Shishir Pagad et al. Robust Method for Removing
Dynamic Objects from Point Clouds [9] Ray tracing $\times$ Post-mapping
Johannes Schauer et al. The Peopleremover Removing Dynamic Objects From 3-D
Point Cloud Data by Traversing a Voxel Occupancy Grid [10] Ray tracing
$\times$ Post-mapping Francois Pomerlea et al. Long-term 3D map maintenance in
dynamic environments [13] Visibility based $\times$ Post-mapping R. Ambruş et
al. Meta-rooms: Building and maintaining long term spatial models in a dynamic
world [14] Visibility based $\times$ Post mapping Giseop Kim et al. Remove,
then Revert: Static Point cloud Map Construction using Multiresolution Range
Images [15] Visibility based $\times$ Post-mapping Hyungtae Lim et al. ERASOR:
Egocentric Ratio of Pseudo Occupancy-based Dynamic Object Removal for Static
3D Point Cloud Map Building [16] Classical Segmentation $\times$ Post-mapping
Andres Milioto et al. RangeNet++: Fast and Accurate LiDAR Semantic
Segmentation [17] Machine learning Segmentation $\times$ Real-time Yin Zhou et
al. VoxelNet: End-to-End Learning for Point Cloud Based 3D Object Detection
[18] Machine learning Segmentation $\times$ Real-time Xieyuanli Chen et al.
Moving Object Segmentation in 3D LiDAR Data: A Learning-based Approach
Exploiting Sequential Data [19] Machine learning Segmentation $\times$ Real-
time Xixun Wang et al. Navigation of Mobile Robots in Dynamic Environments
Using a Point Cloud Map [20] Classical Segmentation $\checkmark$ Post-mapping

###  2.3 Segmentation

Segmentation strategies for dynamic point removal in LiDAR data are broadly
categorized into classical and machine learning-based methods. Classical
approaches, exemplified by Hyungtae Lim et al.’s ERASOR [16], leverage
distinctive features of dynamic points. ERASOR, focusing on urban settings, is
predicated on the idea that dynamic objects typically touch the ground. This
method, by comparing height differences between the original map and the
current frame, identifies areas likely containing dynamic elements, and then
applies the Region-wise Ground Plane Fitting technique to separate static from
dynamic points in these areas.

Machine learning-based methods, as referenced in [17, 21, 18, 22, 19], involve
segmenting environmental data into distinct entities. This segmentation helps
the system recognize entities producing ’ghost tails’. These entities, such as
people, cars, and bicycles, are identified and removed from the point cloud.
Notably, Ayush Dewan and colleagues [21] devised a method using motion cues
for detecting and tracking dynamic objects in urban environments, bypassing
the need for prior maps. They employed Bayesian methods for segmentation and
RANSAC for motion model estimation. Complementarily, Zhou et al.’s VoxelNet
[18] bypassed traditional hand-crafted feature representations, using an end-
to-end trainable network for directly processing sparse 3D points. This has
significantly improved LiDAR-based detection of cars, pedestrians, and
cyclists. Furthermore, [22] introduced PointRCNN, a two-stage framework for 3D
object detection from point clouds, which has advanced proposal generation and
refinement techniques. Despite these advancements, segmentation-based
strategies have limitations. A notable challenge is their struggle with
unknown classes [23] and incomplete detection in cases of occlusion. This
highlights the ongoing need for developing more robust, versatile, and
accurate dynamic point segmentation methods in LiDAR data-processing.

###  2.4 Research Gap

Ray-tracing techniques, while powerful, require significant computational
resources and often struggle with incidence angle ambiguity. Conversely,
methods based on visibility face difficulties with irremovable dynamic points.
Segmentation approaches also pose their own challenges, particularly in
processing unknown classes and segmenting under occlusion. Table 1 provides a
comprehensive comparison of the aforementioned methods. Notably, the most
dynamic object removal techniques have been applied in outdoor settings, to
best of our knowledge, with the exception of [20] which has been used indoors
to create a 2D-occupancy map rather than a dynamic-free point cloud. This
study focuses on indoor environments characterized by slower movement dynamics
compared to outdoor settings. It aims to detect dynamic points without needing
pre-trained data and while addressing incidence angle ambiguity in ray tracing
and irremovable dynamic point challenges present in visibility-based methods.
The proposed approach, DynaHull, which centers on density calculation, is not
limited to the challenges explained before as it does not require a prior map
and is capable of addressing incidence angle ambiguity and irremovable dynamic
point by evaluating each point dynamic feature based on its surrounding
points. Furthermore, occlusion also becomes less severe since the density of
the point is not related to the viewing angle. By capitalizing on these
strengths, the proposed method holds promise for robust and efficient dynamic
point removal in indoor environments.

##  3 Methodology

This section is divided into three parts:” Equipment and Mapping”, ”DynaHull
Algorithm”, and” Validation”. The first part offers details about the testing
robot and the testing environment. The subsequent sub-section explains the
DynaHull algorithm and its advantages. Lastly,” Validation” compares our
method to other state-of-the-art methods.

###  3.1 Equipment and Mapping

The map containing dynamic points was collected using a differential wheel
robot named DBot. DBot is equipped with tools, notably the Velodyne VLP 16
sensor, which captures 3D spatial points. The sensor has a 100-meter range and
a 30-degree field of view. For optimal coverage, the sensor is mounted at the
top of the DBot, as illustrated in Fig 2. The mapping in this study is
accomplished by using an HDL graph slam package provided by Koide [24] in ROS
(Robotic Operating System) .

(a) Dbot from side

(b) Dbot from top

Figure 2: Dbot- The robot used for collecting the Data

In this study, the map was created in the form of point cloud data (PCD) in an
indoor environment at a university covering an area of approximately 150
square meters during a demo event in which about 100 people and numerous
dynamic objects were present in the robots proximity. The maximum range of the
Velodyne VLP16 was set to 75 meters and default parameters were used for HDL
graph slam parameters.

Figure 3: Estimating convex hull volume between stationary and dynamic points:
The left figure represents the stationary area (wall), while the picture on
the right illustrates the dynamic area (human). Figure 4: Flowchart of
DynaHull method

###  3.2 DynaHull

The DynaHull method capitalizes on the observation that, after several scans,
stationary points have a smaller convex hull volume compared to dynamic ones,
as demonstrated in Fig 3.

On completing the mapping, the DynaHull process removes dynamic points as
follows (the flowchart is illustrated in Fig. 4): (1) DynaHull segments the
ground points by estimating the slope of the ground points plane inspired by
[25] and segments the floor points accordingly. (2) The map, free from ground
points, is divided into N_(clusters) clusters using Kmeans from [26]. This
division helps to manage points that are non-uniformly distributed during
mapping. (3) After clustering, nearest neighbor distances (in this study,
$k=75$ number of nearest neighbors) of each point are calculated using
sklearn’s efficient KDtree [26]. These distances are then used to calculate
the convex hull, which is the smallest convex set containing a given set of
points. DynaHull integrates a divide-and-conquer algorithm known as Quickhull
from Scipy [27]to efficiently calculate the convex hull volumes (described in
detail in the next paragraph). (4) After calculating the convex hull for each
point, the point density factor is determined by dividing $k$ neighbor counts
by the volume of each convex hull point. (5) After assigning a density factor
for each point, a removal percentage is defined, ranging from 5 to 20 percent
depending on the point count of the cluster. In other words, the clusters with
low point counts will have a lower removal percentage, while the removal
percentage increases as the cluster point count increases. This allows the
algorithm to be flexible for areas with sparse stationary points which may be
due to LiDAR expansion problems or mapping limitations, such as obstacles or
inaccessibility. (6) By incrementally increasing a threshold value based on
the variances of the previously calculated density factors from zero, points
with low density factors are removed until the required removal percentage in
each cluster is achieved. These tasks are achieved by utilizing Open3D [28]
and numpy [29]. (7) This process is repeated for all clusters, and on
completing the last cluster, these refined clusters are merged to reconstruct
the map. (8) Finally, the ground floors are added back to the point cloud
completing the DynaHull algorithm.

The Quick hull in Step (4) operates as follows:

  * •

Initialization: The algorithm begins by identifying the extreme points on each
axis (the smallest and largest) which are key components of the convex hull.
Using these points, it forms an initial shape, often a tetrahedron.

  * •

Face Processing: Each face of the initial shape is examined to find the point
farthest from it, known as the ”furthest point.”

  * •

Horizon Edge Search: For each face being examined, the algorithm searches for
the ”horizon edge” at which a line from the furthest point intersects the
shape. Faces visible from the furthest point are marked for removal.

  * •

Constructing New Faces: Marked faces are removed, and new faces are created
using the horizon edges as a base, linking the furthest point to these edges.

  * •

Termination: This recursive process continues until no more points are found
outside the convex hull. The algorithm concludes when no additional points can
be seen from any face of the shape. The volume is returned by summing the
volumes of the tetrahedrons. The volume of each tetrahedron is calculated
using Equation 1, where A, B, C, and D represent the four vertices of the
tetrahedron.

$V_{tetrahedrons}=\frac{1}{6}\left|(\mathbf{A}-\mathbf{D})\cdot((\mathbf{B}-\mathbf{D})\times(\mathbf{C}-\mathbf{D}))\right|$
(1)

###  3.3 Validation

Our methodology is evaluated using the DynamicMap Benchmark [12] by Kin-Zhang
et al. consisting of well-known algorithms, such as ERASOR [16], Removert
[15], and an improved version of OctoMap [11]. Additionally, we incorporate
the Statistical Outlier Removal (SOR) from Open3D [28], referred to as O3D.
For ERASOR, the configuration for a semi-indoor environment from DynamicMap
Benchmark was selected without any modifications. In Removert, the sequence
field of the view parameter was adjusted from 50 to 29, aligning with the
30-degree FOV of the Velodyne VLP-16 used in our study. The OctoMap parameters
remained unchanged. For statistical outlier removal of O3D, the number of
neighbors was set to 100, and the standard deviation ratio was set to 1.

The core evaluation metric compares the map outputs of each method with a
reference ground truth map created in a low-activity environment. Any ghost
trails made by the robot operator were manually removed using the CloudCompare
program. To compare generated maps to the ground truth, the distances of each
point in the maps of each method are calculated to its closest point in the
ground truth map. Afterwards, the mean absolute error (MAE), variance, root
mean square error (RMSE) and 90th percentile error of these distance are
recorded. Further comparisons employed the Chamfer Distance [30] and Earth
Mover’s Distance (EMD) [31] to assess the similarity between the methods and
the ground truth map. The Chamfer Distance is calculated by computing the
average of the squared distances from each point in one point cloud to its
nearest point in the other point cloud, and vice versa, with a lower value
indicating a closer match. EMD, also known as the Wasserstein distance,
measures the minimum ’work’ required to transform one point cloud into
another. In our study, the EMD is calculated using the ’emd2’ function from
the Python Optimal Transport (POT) library, with maps uniformly downsampled to
reduce computational requirements.

Finally, this paper demonstrates the odometry performance of a map filtered by
DynaHull and compares it with the odometry performance of a ground truth map
and a dynamic map obtained by scan-matching live LiDAR data to each global
map. Odometry data is collected using the ’HDL localization’ package provided
by Koide [24]. The live LiDAR data is replayed using the ROS system for all
maps to ensure a fair comparison.

##  4 Results

For each point in the map, the minimum distance to its corresponding point in
the second map (ground truth map) is calculated. The mean and variances of
these distances are depicted in Fig. 5. It is evident that DynaHull has the
lowest mean and variance. Although the O3D map has the second lowest mean, its
variance is higher than that of the Octomap. ERASOR has the highest mean and
variances among all the methods.

Figure 5: Mean and variance of distances between the points in each method compared to the ground truth Figure 6: Comparison between the DynaHull, Removert, O3D, ERASOR, OctoMap, and ground truth. The green points indicate the point at which each method is considered to be stationary whereas the red parts demonstrate the dynamic points.  Table 2: Methods comparison between DynaHull and other methods File Names | MAE | RMSE | 90 Percentile Errors | CD | EMD
---|---|---|---|---|---
DynaHull | 0.102 | 0.312 | 0.184 | 4.784 | 0.766
Removert | $0.2185$ | $0.459$ | $0.4085$ | $5.669$ | $2.131$
O3D | $0.161$ | $0.398$ | $0.311$ | $6.425$ | $2.444$
OctoMap | $0.191$ | $0.433$ | $0.363$ | $6.0418$ | $2.327$
ERASOR | $0.281$ | $0.522$ | $0.602$ | $7.375$ | $2.753$

Table LABEL:tab:mapscomparison provides a comparison between the filtered map
and ground truth in five metrics, namely MAE, RMSE, 90th percentile error, CD,
and EMD. Table LABEL:tab:mapscomparison demonstrates that the similarity of
the DynaHull filtered map with ground truth is higher than that of other
methods, particularly in the Earth Mover’s Distance.

To provide a visual comparison between all methods, Fig. 6 shows dynamic and
stationary points identified by each method in red and green colors,
respectively. Note that for a better visual comparison, the ground and ceiling
points in each map have been removed. On examining the figure, it is evident
that other methods have excessively removed points from the walls which should
be considered stationary. While Removert, O3D, and OctoMap are capable of
removing the humans walking in the map, ERASOR struggles to do so.

To emphasize the importance of filtering and the effectiveness of DynaHull in
localization, we compared the localization performance of a robot using a
dynamic map, a DynaHull-filtered map, and a ground truth map. The
corresponding odometry is presented in Fig. 7. The figure illustrates that the
DynaHull-filtered map excels in localizing similarly to the ground truth map,
while the robot fails to localize itself in a dynamic map. The failure was
particularly evident in turns; although the robot manages to localize itself
during the first two turns in the dynamic map, it completely fails in
subsequent turns.

Figure 7: Odometry comparison between the ground truth, DynaHull, and Dynamic
Map: The robot fails to localize itself after the third turn in dynamic map

##  5 Discussion

The primary objective of this study was to improve the accuracy of indoors
dynamic points filtering through density filtering. To reach this, a novel
density-based method was developed. The proposed algorithm had a few
parameters which affect the performance, mainly, the neighboring points count
k and removal percentage. In terms of the neighboring points, a larger number
might result in denser point clouds with a higher confidence of stationary
points. However, an extensive increase of the k value results in more
computational power. Another factor influencing the performance of our method
is the removal percentage. Increasing the removal percentage value enhances
the confidence of dynamic point elimination, but an excessive value may lead
to the removal of stationary points.

In comparison with Table LABEL:tab:mapscomparison, the DynaHull method
outperformed other techniques in all metrics. In indoor environments, other
methods might face challenges in effectively performing when objects show
limited dynamism. Furthermore, while other methods utilized 32- or 64-channel
LiDARs, our study employed a 16-channel LiDAR. This could explain the reason
for these outdoor environment methods under-performing. This distinction is
noteworthy as a higher point count is advantageous. This is notably more
visible with the ERASOR method as the algorithm was built on the premise of
the dynamic points being in contact with the ground. However, since the
16-channel LiDAR does not provide a sufficient number of ground points with
which the algorithm can detect the dynamic points, ERASOR performs poorly in
this scenario. The other contributing factor is the low height of the Velodyne
installed on the DBot compared to those installed on top of the autonomous
vehicle.

In Figure 6,visually speaking, DynaHull preserved walls better than other
methods. In contrast, the other methods falsely removed stationary points,
especially in the case of O3D. Although outlier removal methods are efficient
in removing dynamic points, they can lead to a noticeable amount of stationary
point removal, further emphasizing the importance of specific dynamic point
removal techniques such as DynaHull.

One main drawback that affected the filtering performance in all maps was the
inaccurate registration of points. Since the mapping was implemented in a
highly dynamic environment, the points were not accurately registered due to
the presence of a noticeable number of outliers (humans and other dynamic
objects). This has caused walls or other stationary features to appear as
artificially thickened due to cumulative mapping errors. Improving
registration techniques or alignments can enhance the DynaHull method, since
thinner walls will decrease the value of the convex hull volume estimates,
thereby increasing the density of the stationary points.

##  6 Summary and Conclusions

This study introduced the DynaHull method for removing dynamic points from
point clouds post-mapping. The primary objective was to address the
limitations of state-of-the-art techniques, including ray tracing, visibility,
and segmentation methods. The results of our study demonstrate the better
performance of our methods compared to other state-of-the-art techniques in
all the presented metrics, namely, MAE, RMSE, 90th percentile error, CD, and
EMD. The suggested future research directions include applying the DynaHull
method across a broader array of scenarios and extending the range of
hyperparameter optimization. This approach should incorporate more
sophisticated algorithm designs, such as wall detection and machine learning-
based segmentation of dynamic objects like humans. Additionally, the
development of improved registration techniques tailored for highly dynamic
environments is recommended to address the challenges in mapping registration.

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