text:
\begin{document}

\title{Robust Topology Identification and Control of LTI Networks}

\author{\IEEEauthorblockN{Mahyar Fazlyab and Victor M. Preciado}
\IEEEauthorblockA{Department of Electrical and Systems Engineering\\
University of Pennsylvania\\
Philadelphia, PA 19104-6228 USA\\
\texttt{\small \{mahyarfa,preciado\}@seas.upenn.edu}\\}
}

\maketitle

\begin{abstract}

This paper reports a robust scheme for topology identification and control of networks running on linear dynamics. In the proposed method, the unknown network is enforced to asymptotically follow a reference dynamics using the combination of Lyapunov based adaptive feedback input and sliding mode control. The adaptive part controls the dynamics by learning the network structure, while the sliding mode part rejects the input uncertainty. Simulation studies are presented in several scenarios (detection of link failure, tracking time varying topology, achieving dynamic synchronization) to give support to theoretical findings.
\end{abstract}

\begin{IEEEkeywords} LTI Networks, Topology Identification, Model Reference Adaptive Control, Sliding Mode Control \end{IEEEkeywords}

\IEEEpeerreviewmaketitle

\section{Introduction}
Complex networks are capable of modeling many real world dynamical processes which can be described by a set of interacting elements (nodes) having some sort of connections or causal relationships (edges) among them.
For example, in molecular biology, complex networks are used to describe regulatory relationships between transcription factors and their target genes \cite{lezon2006using}.  In a communication network, amount of traffic flowing between nodes constitute a dynamical network. \cite{kelly1998rate}

In recent years, there has been an extensive attention to topology identification and control of complex dynamical networks. Topology identification aims at finding the strength of connections between nodes, such as protein-DNA interactions in the regulation of various cellular processes. or detecting failures or anomalies in connections. Also, the ability to control the dynamic variables of nodes by external inputs is another interesting issue which has appreciable applications in, for example, synchronization of coupled oscillators \cite{Yu2009429}, or rate control in communication networks.

Topology identification of networks has been addressed in various research works. In \cite{yu2006estimating} for example, the topology of the network is estimated on-line using Lyapunov based adaptive feedback input. In \cite{5718112}, a node-knockout
procedure is proposed for the complete characterization of the interaction geometry in consensus-type networks. In a different approach addressed in \cite{shahrampour2013reconstruction} and \cite{shahrampour2013topology}, nodes are stimulated by wide-sense stationary input of unknown power spectral density and the topology of the network is identified via measuring cross power spectral densities of the outputs, which encode the direction and weights of the edges.  The control of complex networks has also been studied extensively. In \cite{li2006controlling}, for example, a linear state feedback controller is designed to synchronize the states with a desired orbit.

The purpose of this paper is two fold. We address both the control and topology identification of complex networks running on linear dynamics and in presence of input uncertainty. We assume that the topology of the underlying graph is unknown or uncertain. We design the input of the  network so as to enforce the dynamic variables of the nodes to track those of a predefined reference network, without having the knowledge of the network topology. In this setting, the unknown topology of the network is also estimated using Lypunov theory and adaptive feedback gains. Next, we develop the robust counterpart of the control-identification scheme using sliding mode technique. With this method the additive bounded disturbance to the input of the network is rejected, translating into smooth tracking and estimation. The rest of the paper is organized as follows: In section II the problem is defined and the tracking controller and the identifier are designed. In section III we robustify the algorithm through incorporating additional sliding mode control law. In section IV we provide simulations to verify the effectiveness of the proposed approach. Concluding remarks are drawn is section V.

\section{Simultaneous Identification and Control}
Consider the following linear time invariant dynamics on a directed network:

\begin{equation}
\dot{x}(t)=\mbox{A}x(t)+\mbox{B}u(t)  \label{1}
\end{equation}

where $\mbox{A}=[a_{ij}] \in \mathbb{R}^{N\times N}$ is the unknown weighted adjacency matrix of the directed network with $N$ nodes. $a_{ij}$ denotes the strength of connection from node $j$ to node $i$, and is zero when there is no connection. Positive or negative weights refer to excitatory or inhibitory effect. In consensus dynamics, the adjacency matrix $\mbox{A}$ is replaced by the negative Laplacian of the underlying network. Vector $x(t)\in \mathbb{R}^N$ captures the evolution of the nodes which can be, for instance, the transcription factor in gene regulatory networks or amount of traffic entering each node in a communication network. Here, without loss of generality, we are assuming scalar variables for each node. Extention to multi variable nodes, where each node is represented by an LTI system, is straightforward. Boolean matrix $\mbox{B}=[b_{ij}] \in \mathbb{R}^{n\times m}$, $b_{ij}\in \{0,1\}$ determines which nodes are accessible by external inputs, where $b_{ij}=1$  when input $j$ drives node $i$ and $b_{ij}=0$ otherwise. External input $u(t)\in \mathbb{R}^m$ is responsible for driving the states of the network. The goal is to control the dynamic variables of the nodes and to estimate the weights of connection between nodes. For this goal to be possible, we need structural controllability assumption of the pair $(\mbox{A},\mbox{B})$. In the context of networks, controllability translates into minimum number of driver nodes that can uniquely determine the time evolution of all nodes \cite{liu2011controllability}.

\begin{remark}
If the topology of the network is completely unknown, all nodes must be accessible for input injection, i.e. $B=I_{N}$. However, if the boolean structure is known a priori and the weights of connections are unknown, fewer nodes are required for input injection. In the development of the identification algorithm, we consider the latter case.
\end{remark}

To begin, we define the following reference network:

\begin{equation}
\dot{x}_m(t)=\mbox{A}_m x_m(t)+Br(t) \label{2}
\end{equation}

where $A_m$ is the reference adjacency matrix, and $r(t)$ is the reference control input. If the pair $(A_m,B)$ is controllable, one can generate the desired trajectory $x_m(t)$ using appropriate input $r(t)$.

\begin{assumption}
There exists matrices $K^* \in \mathbb{R}^{m\times n}$ such that the algebraic equations $A+BK^*=A_m$ is satisfied. For the special case of $B=I_{N}$, this assumption is held.
\end{assumption}

Consider the following input for the real network:
\begin{equation}
u(t)=K^*x(t)+L^*e(t)+r(t) \label{3}
\end{equation}

Substituting Eq. \eqref{3} in Eq. \eqref{1} results in the following dynamics for the real network:

\begin{equation}
\dot{x}(t)=(A+BK^*)x(t)+Br(t)+BL^*e(t) \label{b}
\end{equation}

Subtracting \eqref{2} from \eqref{b} yields the error dynamics:

\begin{equation}
\dot{e}(t)=(A+BK^*)x(t)-A_mx_m(t)+BL^*e(t) \label{4}
\end{equation}

Now if $K^*$ and $L^*$ are deigned such that $A+BK^*=A_m$ and $A_m+BL^*$ is Hurwitz, the error dynamics reduces to $\dot{e}(t)=(A_m+BL^*)e(t)$, translating to asymptotic convergence of $e(t)$ to zero.

It is clear that when matrix $A$ is unknown or uncertain, the feedback matrix $K^*$ cannot be designed. By estimating $K^*$ and using the identity $A+BK^*$, one can recover the unknown topology. Denoting this estimate at time $t$ by $K(t)$, the closed loop network admits the following dynamics:
\begin{equation}
\dot{x}(t)=(A+BK(t))x(t)+Br(t)+BL^*e(t) \label{5}
\end{equation}

Similarly, by subtracting \eqref{2} from \eqref{5}, the error dynamics can be written as:
\begin{equation}
\dot{e}(t)=(A_m+BL^*)e(t)+B\tilde{K}(t)x(t) \label{6}
\end{equation}

where $\tilde{K}(t):= K(t)-K^*$ is the estimation residual.\\

\begin{proposition}
If $K(t)$ satisfies the following differential equation, the tracking error will asymptotically converge to zero:
\begin{equation}
\dot{K}(t)=\dot{\tilde{K}}(t)=-W^{-1}B^TPe(t)x(t)^T, \label{7}
\end{equation}
where $P\in \mathbb{S}_{++}^{n\times n}$ and $W \in \mathbb{S}_{++}^{n\times n}$ are positive definite symmetric matrices.
\end{proposition}

\begin{proof}
We define the following Lyapunov function:
\begin{equation}
V=\frac{1}{2}e^TPe+\frac{1}{2}tr(\tilde{K}^TW\tilde{K}), \label{8}
\end{equation}
The first term is the weighted two norm of the tracking error, while the second terms is the weighted Forbenius norm of $\tilde{K}$ which vanishes only when $\tilde{K}$ is identically zero, resulting in a well defined Lyapunov function. Taking time derivative of Eq. \eqref{8} we see:
\begin{align}
\dot{V} =&-\frac{1}{2}e^TQe+x^T\tilde{K}^TB^TPe
+tr(\tilde{K}^TW\dot{\tilde{K}})\nonumber \\
\nonumber\\
=&-\frac{1}{2}e^TQe+tr(\tilde{K}^T(B^TPex^T+W\dot{\tilde{K}})) \label{9}
\end{align}
Here in the first equality, we have used the fact that $(A_m+BL^*)P+P(A_m+BL^*)^T=-Q$ for some positive definite matrix $Q$ for Hurwitz $A_m+BL^*$. In the second equality, we have used the identity $x^T\tilde{K}^TB^TPe=tr(\tilde{K}^TB^TPex^T)$. Substituting Eq. \eqref{7} in \eqref{9} yields:
\begin{equation}
\dot{V} =-\frac{1}{2}e^TQe \label{10}
\end{equation}
Let M be the set of all points for which $\dot{V}=0$, i.e. $M=\{(e,\tilde{K})|e=0\}$. According to Lasalle's invariance principle, all dynamic variables will asymptotically converge to the largest invariant M, where $e=0$, completing the proof.
\end{proof}
The largest invariant set can be stated as:
\begin{equation}
M_{L}=\{( e,\tilde{K})|e=0 , \  \dot{\tilde{K}}=0 , \ B\tilde{K}x_m(t)=0 \}
\end{equation}  \label{M}
According to Eq. \ref{7}, the learning process is stopped. In order to have zero steady state estimation residual ($\tilde{K}$ in \eqref{M}), matrix $B$ must be full rank, and the reference signal $x_m(t)$ must span the N-dimensional space in time, i.e. it must satisfy the persistent excitation condition. Mathematically speaking, the condition of persistent excitation translates into the condition $\int\limits_{t}^{t+T} x_m(\tau)x_m^T(\tau)d\tau \geq \alpha I_N$ for some $\alpha , T>0$ and all $t \geq 0$ \cite{ioannou2012robust}. It can be readily verified that if $x_m(t)=[sin(\omega_0 t) \ sin(2\omega_0 t) ... sin(N\omega_0 t)]^T$ \textit{PE} condition is satisfied with $T=\frac{2\pi}{\omega_0}$ and $\alpha=\frac{\pi}{\omega_0}$.\\

\begin{remark}
In dynamic synchronization, i.e. when $x_1(t)=x_2(t)=...=x_N(t)$, the condition of persistent excitation is failed and the network topology becomes unidentifiable \cite{4806038}.\\
\end{remark}

\section{Robust Topology identification}
In practical settings, there usually exists uncertainty or disturbance to the input of the real network. In this case, the perturbation adversely affects tracking and learning. In order to reject this disturbance, we might add an additional control input to robustify the adaptation process. To see this, consider the following dynamics for the real network:
\begin{equation}
\dot{x}(t)=Ax(t)+B(u(t)+d(t))  \label{11}
\end{equation}
where $d(t)$ is the unknown additive disturbance with known bound. We resort to sliding mode technique which is a robust control scheme in order to reject this disturbance \cite{khalil2002nonlinear}. To begin, we first define the following sliding surface:
\begin{equation}
s(t)\triangleq\Gamma\left(e(t)-\int\limits_{0}^{t}(A_m+BL^*)e(\tau)d\tau\right) \label{12}
\end{equation}
Where $\Gamma\in\mathbb{R}^{m\times N}$ is a full rank matrix to be designed. By this definition, it is clear that once $s(t)=0,  \forall t\geq t_0$ for some $t_0\geq 0$ (or i.e. $s(t) \equiv \dot{s(t)}\equiv 0$), the tracking error would follow a stable manifold toward the origin. Hence, tracking becomes equivalent to regulation of $s(t)$ to zero. Consider the following input:
\begin{equation}
u(t)=K(t)x(t)+L^*e(t)+r(t)-\rho M\frac{s(t)}{\|s(t)\|} \label{13}
\end{equation}
Here $\|.\|$ refers to two norm. Substituting Eq. \eqref{13} in Eq. \eqref{11} results in the following dynamics for the real network:
\begin{align}
\dot{x}(t)=&\left(A+BK(t)\right)x(t)+B (r(t)+L^*e(t) \nonumber \\
&-\rho M\frac{s(t)}{\|s(t)\|}+ d(t)) \label{14}
\end{align}
Subtracting Eq.\eqref{2} from Eq. \eqref{14} results in the following error dynamics:
\begin{equation}
\dot{e}(t)=(A_m+BL^*)e(t)+B\left(\tilde{K}(t)x(t)-\rho M\frac{s(t)}{\|s(t)\|}+ d(t)\right) \label{15}
\end{equation}
\begin{proposition}
Consider the control input \eqref{13}. If $K(t)$ satisfies the differential equation \eqref{16}, and $\rho=\|P\Gamma B\|\|d(t)\|+\epsilon$ with $\epsilon >0$, the tracking error, as described by Eq. \ref{15} will asymptotically converge to zero.

\begin{equation}
\dot{K}(t)=\dot{\tilde{K}}(t)=-W^{-1}B^T\Gamma^TPs(t)x(t)^T \label{16}
\end{equation}
\end{proposition}

\begin{proof}
Define the following Lypunov function candidate:
\begin{equation}
V=\frac{1}{2}s^TPs+\frac{1}{2}tr(\tilde{K}^TW\tilde{K}) \label{17}
\end{equation}
Differentiating Eq. \eqref{17} w.r.t. time yields:
\begin{align}
\dot{V}=& s^TP\Gamma\left[\dot{e}-(A_m+BL^*)e\right] \nonumber \\
=& s^TP\Gamma B(d-\rho M \frac{s}{\|s\|})
+s^TP\Gamma B\tilde{K}x + tr(\tilde{K}^TW\dot{\tilde{K}}) \nonumber \\
=& s^TP\Gamma B(d-\rho M \frac{s}{\|s\|})
+ tr(\tilde{K}^T(B^T\Gamma^TPsx^T+W\dot{\tilde{K}})) \nonumber \\
\label{18}
\end{align}
The differential equation \eqref{16} crosses out the second term of $\dot{V} $. Now by choosing $M=(P\Gamma B)^{-1}$ we get:
\begin{align}
\dot{V}=& s^TP\Gamma Bd -\rho\|s\| \nonumber \\
\leq& \|s\|(\|P\Gamma B\|\|d\|-\rho) \nonumber \\
\leq& -\epsilon \|s\| \label{19}
\end{align}
where the last inequality follows from the assumption $\rho=\|P\Gamma B\|\|d\|+\epsilon$. We maintain that inequality \eqref{19} ensures asymptotic convergence of the tracking error to zero as follows: Integrating the inequality \ref{19} implies $\epsilon \int\limits_{0}^{t}{\|s\|d\tau}\leq (V(0)-V(t))$. Since $\dot{V}$ is negative semi-definite and $V(t)$ is lower bounded, $\lim\limits_{t\to \infty}V(t)$ exists and hence by Schwartz inequality $\int\limits_{0}^{\infty}{\|s\|^2d\tau}\leq \left(\int\limits_{0}^{\infty}{\|s\|d\tau}\right)^2$, the function $g(t)\triangleq\frac{1}{2} \int\limits_{0}^{t}{\|s\|^2d\tau}$ has finite limit at $t=\infty$. Also $\ddot{g}(t)=s^T\dot{s}$ is bounded due to the boundedness of $s$ and $\dot{s}$. By Barbalat's lemma \cite{khalil2002nonlinear} one can conclude that $\lim\limits_{t\to\infty}s(t)=0$; and by the definition of $s(t)$, Eq. \eqref{12} convergence of $e(t)$ to zero is guaranteed.
\end{proof}

\section{NUMERICAL SIMULATIONS}
In this section, we present a simulation study to validate the effectiveness of the proposed approach. We consider a weighted undirected network whose weights are realizations of a uniform distribution in the unit interval, i.e. $a_{ij} \sim \mathcal{U}(0,1)$ for $i\neq j$ and $a_{ii}=0$.

Each node has its own input, i.e $B=I_{5\times 5}$. For the reference network, we assume $A_m=-I_{5\times5}$ and $B_m=I_{5\times 5}$. We also set the weights of the  Lyapunov function as $P=I_{5\times 5}$ and $W=10I_{5\times 5}$. The initial condition for the network is chosen as $x(0) \sim N(0,1)$. The initial condition for the reference network is zero, $x_m(0)=0$. The initial condition for the estimator $K(t)$ is also set to zero, $K(0)=0_{5\times 5}$. To fulfill the persistence excitation condition, the reference input is chosen as : $r_i(t)=r_{0i}sin(w_{i}t)$ where $r_{0i} \sim \mathcal{U}(1,2)$ and $w_{i} \sim \mathcal{U}(1,2)$
Figure.\ref{Fig1} illustrates the estimation of all elements of the adjacency matrix $A$.

\begin{figure}[h!]
  \centering
  \includegraphics[trim=1cm 0.5cm 1cm 1cm, clip=false, width=0.3\textwidth]{Fig_1}
  \caption{Estimation of weights}
  \label{Fig1}
\end{figure}

The proposed topology estimator can also be used for detecting link failures, i.e. sudden changes in the weights of the links as well as tracking slow temporal variations in the weights. Figure.\ref{Fig3} depicts the scenario when there is an abrupt failure in the link $a_{12}$. Figure.\ref{Fig4} shows the estimation behavior when the link $a_{12}$ has temporal variation $a_{12}(t)=0.56cos^2(2\pi t/800)$.
\begin{figure}[h!]
  \centering
    \includegraphics[trim=1cm 1cm 1cm 1cm, clip=false, width=0.3\textwidth]{Fig_2}
  \caption{Detection of link failure between node 1 and node 2}
    \label{Fig3}
\end{figure}

\begin{figure}
  \centering
    \includegraphics[trim=1cm 1cm 1cm 1cm, clip=false,width=0.3\textwidth]{Fig_3}
  \caption{Tracking time varying connection strength between node 1 and node 2}
    \label{Fig4}
\end{figure}

In order to show the robustness of the identifier when sliding mode control is added, we consider a Gaussian disturbance added to all inputs, $d(t)\sim \mathcal{N}(0,1)$. Figure.\ref{Fig5} shows the sensitivity of learning in presence of noise. Figure.\ref{Fig6} verifies that the estimator perfectly smooths out the disturbance with the aid of sliding mode control. In figure.\ref{Fig7} we plot the state of node 1 along with the reference input and in presence of input disturbance. Note that for tracking purposes, the reference input may not be sufficiently rich, and hence the learning process will become useless.\\
\textit{Application in Dynamic Synchronization}: In order to show the effectiveness of the control algorithm on achieving dynamic synchronization, we choose the reference network as a dynamic consensus filter \cite{spanos2005dynamic} : $\dot{x}_m(t)=-Lx_m(t)+{r}(t)$ where $L$ is the Laplacian matrix of a connected undirected graph. Figure. \ref{Fig8} illustrates dynamic synchronization where $r(t)$ is the vector of unit step function and all nodes will asymptotically agree on unit ramp function.

\begin{figure}[h!]
  \centering
    \includegraphics[trim=1cm 1cm 1cm 1cm, clip=false,width=0.3\textwidth]{Fig_4}
  \caption{Estimation of link $a_{12}$ without disturbance rejection}
    \label{Fig5}
\end{figure}

\begin{figure}[h!]
  \centering
    \includegraphics[trim=1cm 1cm 1cm 1cm, clip=false,width=0.3\textwidth]{Fig_5}
  \caption{Estimation of link $a_{12}$ with disturbance rejection}
    \label{Fig6}
\end{figure}

\begin{figure}[h!]
  \centering
    \includegraphics[trim=1cm 1cm 1cm 1cm, clip=false,width=0.3\textwidth]{Fig_6}
  \caption{Tracking the reference signal in node 1 with disturbance rejection}
    \label{Fig7}
\end{figure}

\begin{figure}[h!]
  \centering
    \includegraphics[trim=1cm 1cm 1cm 1cm, clip=false,width=0.3\textwidth]{Fig_7}
  \caption{Dynamic Synchronization in the unknown network}
    \label{Fig8}
\end{figure}

\section{CONCLUSIONS}
This papers deals with topology identification and control of complex networks with linear dynamics. We used model reference adaptive scheme to design the proper input so as to drive the states of the nodes to desired trajectories. The proposed method is capable of controlling the nodes evolution without any information about the topology of the graph. The topology estimator is shown, via simulation, to be capable of tracking temporal variations in the weights of the connections. By combining the learning controller with sliding mode control, we could also robustify the algorithm to input uncertainty.

\addtolength{\textheight}{-12cm}

\bibliographystyle{unsrt}
\bibliography{Refs}

\end{document}


TITLE: 2D Taylor expansion of F(x,y) where F(x,y) is harmonic (a solution of Laplace equation)
QUESTION [0 upvotes]: I would like to know if, given F(x,y) a real function of 2 variables that obeys  $\nabla^{2} \left(F \left( x,y \right)\right) = 0$ , is it true that F(x,y) always equals its Taylor expansion within the radius of convergence of the Taylor series? Or are there more conditions that are required?

REPLY [0 votes]: Harmonic functions are analytic, so the answer is yes.


TITLE: What does it mean for a matrix to induce a norm?
QUESTION [1 upvotes]: I'm given the following math problem:
Consider the scalar product   $\langle x,y\rangle = x^TAy$  given by the matrix:
$$
     A=  \begin{pmatrix}
        1 & 0 & 2 \\
        0 & 2 & 0 \\
        2 & 0 & 8 \\
        \end{pmatrix}
$$
Determine the length of $(2, 1, 0)^T$ with respect to the norm induced by the scalar product.
What does it mean for a norm to be induced by something? In the answers they say that the norm is induced by the matrix $A$. Does it mean that this vector is not in a normal coordinate system but somehow inside this matrix?
Thanks for the help!

REPLY [2 votes]: A matrix $A$ which is symmetric and positive definite defines (as your posts states) an inner product:
$$
\langle x, y\rangle \stackrel{\rm def}{=} x^T A y, \tag{$\forall x,y$}
$$
Now, an inner product defines (induces) a norm by setting $\lVert x\rVert \stackrel{\rm def}{=} \sqrt{\langle x, x\rangle}$.
So "a symmetric positive definite matrix $A$ induces a norm."


TITLE: Recreational integral problem: $\int_0^1 (x+\sqrt[3]{x^3-1})^{2018}dx$
QUESTION [5 upvotes]: I have a (recreational) integral problem any interested inhabitants of MSE to enjoy:

Evaluate the following integral:
  $$\int_0^1 \bigg(x+\sqrt[3]{x^3-1}\bigg)^{2018}dx$$

I promise, it's not from an ongoing math competition (I just put the $2018$ in there to be funny). If you don't believe me, you can try solving the problem with $1,000,000$ (or any large even number, really) instead of $2018$.
Cheers!

REPLY [11 votes]: Write
$$I = \int_0^1 \bigg(x+\sqrt[3]{x^3-1}\bigg)^{2018}dx.$$
Making the substitution $u^3 + x^3 = 1$, we get
$$I = \int_0^1 \frac{u^2}{\sqrt[3]{1-u^3}^2}\bigg(u+\sqrt[3]{u^3-1}\bigg)^{2018}\,du.$$
(The fact that $2018$ is even is important here, as it removes a minus sign inside the parentheses).
Hence
$$2I = \int_0^1 \left(\frac{u^2}{\sqrt[3]{1-u^3}^2}+1\right)\bigg(u+\sqrt[3]{u^3-1}\bigg)^{2018}\,du.$$
But, if $v = u + \sqrt[3]{u^3-1}$ then $\frac{dv}{du} = \frac{u^2}{\sqrt[3]{1-u^3}^2}+1$, which is exactly the left factor.
So,
$$2I = \int_{-1}^1 v^{2018}\,dv=\frac{2}{2019}$$
Hence $I = \frac{1}{2019}$.


TITLE: Absolute value of numbers
QUESTION [2 upvotes]: The absolute value of the sum of -5 and twice a number is 19. Find the number. I have a problem with this question because i do not fully understand absolute value and this question is a little trucky for me.

REPLY [3 votes]: If the absolute value of the sum of -5 and twice the number is 19, then that means either:
$-5+2x=19$ or $-5+2x=-19$.
Thus:
$2x=24$ or $2x=-14$.
Therefore:
$x=12$ or $x=-7$.

REPLY [1 votes]: As @Dr.SonnhardGraubner wrote, the equation resulting from the words in your question is
$$|-5+2x|=19$$
Now you just need to solve that. Use one of these key facts, true if we know that $b\ge 0$:

$|a|=b\qquad$ is equivalent to $\qquad a=-b\quad$ or $\quad a=b$.
$|a|<b\qquad$ is equivalent to $\qquad -b<a<b$.
$|a|>b\qquad$ is equivalent to $\qquad a<-b\quad$ or $\quad b<a$.

I hope this helps you to better understand absolute value. Let us know if you cannot finish.


text:
\begin{document}
\renewcommand{\arraystretch}{1.5}

\title[Averaging for continuous differential equations]
{Higher order analysis on the existence of\\ periodic solutions in continuous differential equations via coincidence degree}

\author[D.D. Novaes and F. B. G. Silva]
{Douglas D. Novaes and Francisco B. G. Silva}

\address{Departamento de Matem\'{a}tica, Universidade
Estadual de Campinas, Rua S\'{e}rgio Buarque de Holanda, 651, Cidade
Universit\'{a}ria Zeferino Vaz, 13083--859, Campinas, SP, Brazil}
\email{ddnovaes@unicamp.br, fbruno@ime.unicamp.br}

\subjclass[2010]{34C29, 34C25, 47H11}

\keywords{continuous differential equations, periodic orbits, averaging method, Brouwer degree, coincidence degree}

\begin{abstract}
Recently, the higher order averaging method for studying periodic solutions of both Lipschitz differential equations and discontinuous piecewise smooth differential equations was developed in terms of Brouwer degree theory. Between the Lipschitz and the discontinuous piecewise smooth differential equations, there is a huge class of differential equations lacking in a higher order analysis on the existence of periodic solutions, namely the class of continuous non-Lipschitz differential equations. In this paper, based on the coincidence degree theory for nonlinear operator equations, we perform a higher order analysis of continuous (non-Lipschitz) perturbed differential equations and derive sufficient conditions for the existence of periodic solutions for such systems. We apply our results to study continuous (non-Lipschitz) higher order perturbations of a harmonic oscillator.
\end{abstract}

\maketitle

\section{Introduction and statements of the main results}
 The Averaging Method is a classical tool which is concerned in providing asymptotic estimates for the solutions of non-autonomous differential equations given in the following standard form \begin{equation}\label{eq:e1}
	x^\prime = \e F(t,x,\e),
	\end{equation}
	where $F\colon\R\times D\times[0,\e_0]\to \R^n$ is a continuous function $T-$periodic in the variable $t,$ with $D$ an open subset of $\R^n$ and $\e_0 > 0.$ Such asymptotic estimativies are given in terms of solutions of an ``averaged equation''.

The averaging method dates back to the works of Lagrange and Laplace, who provided an intuitive justification of the process and applied it to the problem of perturbations in the solar system \cite{sanders2007averaging}. The first formalization of this procedure was given by Fatou in 1928 \cite{Fa}.  Important contributions to the theory were made by Krylov and Bogoliubov \cite{BK} in the 1934 and Bogoliubov \cite{Bo} in 1945. From then on, it has been an efective tool in studying qualitative properties of ordinary differential equations.

In particular, the averaging method has been proved to be very useful in detecting periodic solutions. In \cite{Hale,sanders2007averaging,Ver06}, it was used to provide sufficient conditions for the existence of periodic solutions for sufficiently smooth differential equations. These studies performed a first order analysis, which means that the obtained sufficient conditions depend only on the function $(t,x)\mapsto F(t,x,0),$ neglecting all the information that  the derivatives with respect to $\e$ of $F(t,x,\e)$ can give. Later, \cite{buicua2004averaging} extended these results for studying periodic solutions of Lipschitz-continuous differential equations and continuous differential equations via Brouwer degree theory.  More recently, \cite{LliNovTei2014} performed a higher order analysis for studying periodic solutions of Lipschitz-continuous differential equations in terms of Brouwer degree theory. The averaging method has also been extended for non-smooth differential equations. In this context, the works \cite{ItiLliNov17, LliNovRod17, LlilNovTei15, LliMerNov15} generalize the averaging method at any order for studying periodic solutions of discontinuous piecewise smooth differential equations.

Between the Lipschitz-continuous and the discontinuous piecewise smooth differential equations, there is a huge class of differential equations \eqref{eq:e1} lacking in a higher order analysis on the existence of periodic solutions, namely the class of continuous (non-Lipschitz) differential equations.  Continuous differential equations with non-Lipschitz nonlinearities appears naturally in applications. We may quote, for instance, neural networks \cite{Bian12,Forti06,Huaiqin11}, weather and climate models \cite{Hanke20,Porz18}, incompressible fluid dynamics \cite{Frisch95,Kol91}, and biological models of competition \cite{Zuowei13}. Under the Lipschitz assumption, the analysis strongly rely on the uniqueness property enjoyed by the solutions of differential equations. However, in general, this property is lost for continuous differential equations. Here, motivated by the analysis performed in \cite{buicua2004averaging}, we take advantage of the coincidence degree theory for nonlinear operator equations to overcome this difficult and perform a higher order analysis on the existence of periodic solutions for continuous differential equation in the standard form \eqref{eq:e1}.

As a fundamental hypothesis on differential equation \eqref{eq:e1}, we shall assume that for a given open bounded subset $V\subset \R^n,$ with $\ov V\subset D,$
\begin{itemize}
\item[$({\bf H})$] there exists $\e_1\in (0,\e_0]$ such that, for each $\la\in(0,1]$ and $\e\in(0,\e_1],$ any $T$-periodic solution of the differential equation
\[
x'=\e\la F(t,x,\e),\, x\in\ov V,
\] is entirely contained in $V.$
\end{itemize}

\begin{remark}\label{rmk:hyp-h}
In applications, hypothesis $({\bf H})$ can be checked by obtaining a contradiction when its negation is assumed. The negation of hypothesis $({\bf H})$ provides numerical convergent sequences $(\e_m)_{m\in\mathbb{N}}\subset(0,\e_0]$ and $(\la_m)_{m\in\mathbb{N}}\subset(0,1],$ such that $\e_m\to 0$ as $m\to \infty,$ and a sequence of $T$-periodic solutions $x_m(t)\in\ov V$ of $x'=\e_m\la_m F(t,x,\e_m)$ for which there exists $t_m\in[0,T]$ such that $x_m(t_m)\in\partial V$ for each $m\in\mathbb{N}.$ In particular,
\[
x_m(t)=x_m(0)+\e_m\la_m\int_0^t F(s,x_m(s),\e_m)ds\, \text{ and }\, \int_0^T F(t,x_m(t),\e_m)dt=0,
\]
for each $m\in\mathbb{N}.$ Furthermore, as an application of  Arzel\'{a}-Ascoli's Theorem, the sequence of functions $(x_m)_{m\in\mathbb{N}}$ can be taken uniformly convergent to a constant function in $\partial V.$
\end{remark}

Define the {\it full averaged function}
$f:D\times [0,\e_0]\rightarrow\R^n$ as the average of $F(t,x,\e),$ that is,
	\[\label{eq:avgd-functions}
	f(z,\e) = \dfrac{1}{T} \int_{0}^{T} \e F(s, z,\e)\d s.
	\]
Our first main result relates the existence of periodic solutions of the differential equation \eqref{eq:e1} with the Brouwer degree of the full averaged function.

\begin{mtheorem}\label{thm:main1}
Consider the continuous $T$-periodic non-autonomous differential equation \eqref{eq:e1}. Assume that for a given open bounded subset $V\subset \R^n,$ with $\ov V\subset D,$ hypothesis $({\bf H})$ holds,
	\begin{equation}\label{Ha}
	 f(z,\e)\neq0 \text{ for all } z\in\partial V \text{ and } \e\in(0,\e_1],
	\end{equation}
and $d_B(f(\cdot,\e),V,0)\neq 0,$ for every $\e\in (0,\e_1].$
	Then, for each $\e\in (0,\e_1],$ there exists a $T-$periodic solution $\f(t,\e)$ of the differential equation \eqref{eq:e1} such that $\f(t,\e)\in V$ for every $t\in[0,T].$
\end{mtheorem}

In many situations, the first derivatives of $F$ with respect to $\e$ are known.
In these cases, the differential equation \eqref{eq:e1} writes
	\[\label{eq:order-k-ode}
	x^\prime = \sum_{i=1}^k\e^i F_i(t,x)  + \e^{k+1} R(t, x, \e),
	\]
	where $F_j\colon\R\times  D \to \R^n,$ for $j\in\{1, \ldots, k\},$ and $R\colon\R\times D \times [0,\e_0]$ are continuous functions $T-$periodic in the variable $t.$ Accordingly, define $f_0=0$ and, for each $j\in\{1, \ldots, k\},$ denote by $f_j\colon D \to \R^n$ the average of $F_j,$ that is,
		\[
	f_j(z) = \dfrac{1}{T} \int_{0}^{T} F_j(s, z)\d s.
		\]
	Also, define the {\it $k$-truncated averaged function} $\CF_k\colon  D \times [0,\e_0]\to \R^n$ by
	\begin{equation}
	\CF_k(z,\e) = \sum\limits_{j = 1}^{k}\e^j f_j(z).
	\end{equation}

Our second main result relates the existence of periodic solutions of the differential equation \eqref{eq:order-k-ode} with the Brouwer degree of the $k$-truncated averaged function. This is a continuous (non-Lipschitz) version of the higher order averaging theorem given in \cite{LliNovTei2014}.

\begin{mtheorem}\label{thm:main2}
Consider the continuous $T$-periodic non-autonomous differential equation \eqref{eq:e1}. Assume that for a given open bounded subset $V\subset \R^n,$ with $\ov V\subset D,$ hypothesis $({\bf H})$ holds,
	\begin{equation}\label{eq:lemma-hyp}
	\lim_{\e\to 0}\inf_{z\in\p V}\left|\dfrac{\CF_{k}(z,\e)}{\e^{k+1}}\right|>\max\{|r(z,\e)|: (z,\e)\in\ov V\times[0,\e_1]\},
	\end{equation}
and $d_B(\CF_{k}(\cdot,\e),V,0)\neq 0,$ for every $\e\in(0,\e_1].$ Then, for each $\e > 0$ sufficiently small, there exists a $T-$periodic solution $\f(t,\e)$ of \eqref{eq:order-k-ode} such that $\f(t,\e)\in V$ for every $t\in[0,T].$
\end{mtheorem}

As a consequence of Theorem \ref{thm:main2} we get our third main result

\begin{mtheorem}\label{thm:main3}
Consider the continuous $T$-periodic non-autonomous differential equation \eqref{eq:e1}. Suppose that for some $\ell\in\{1,2,\ldots,k\},$ $f_0=f_1=f_2=\ldots=f_{\ell-1}=0,$ and $f_{\ell}\neq0,$ and let $z^*\in D$ be an isolated zero of $f_{\ell}.$ Assume that there exists a bounded neighbourhood  $V\subset \R^n$ of $z^*,$ with $\ov V\subset D$ and $f_{\ell}(z)\neq0$ for every $z\in \ov{V}\setminus\{z^*\},$ such that hypothesis $({\bf H})$ holds and $d_B(f_{\ell},V,0)\neq 0.$ Then, for $\e > 0$ sufficiently small, there exists a $T-$periodic solution $\f(t,\e)$ of \eqref{eq:order-k-ode} such that $\f(t,\e)\to z^*$ uniformly as $\e\to 0.$
\end{mtheorem}

It is worth mentioning that, for $\ell=1,$ the existence of an isolated zero of $f_1$ ensures that $({\bf H})$ holds. Indeed, let $V\subset \R^n$ be a bounded neighbourhood of $z^*,$ with $\ov V\subset D,$ such that $f_{1}(z)\neq0$ for every $z\in \ov{V}\setminus\{z^*\}.$ If $({\bf H})$ does not hold on $V$, from Remark \ref{rmk:hyp-h}, we get a sequence of $T$-periodic functions $(x_m)_{m\in\N}$ uniformly converging to a constant function in $\p V,$ let us say $z_0,$ such that
\[
\int_0^T F(t,x_m(t),\e_m)dt=0, \quad m\in\N.
\]
Taking the limit in the integral above, we get that $f_1(z_0)=0,$ which contradicts the fact that $f_{1}(z)\neq0$ for every $z\in \ov{V}\setminus\{z^*\}.$ Therefore, \cite[Theorem 1.2]{buicua2004averaging} follows as a corollary of Theorem \ref{thm:main3}, namely:

\begin{corollary}
Consider the continuous $T$-periodic non-autonomous differential equation \eqref{eq:e1}. Let $z^*\in D$ be an isolated zero of $f_{1}$ and assume that there exists a bounded neighbourhood $V\subset \R^n$ of $z^*,$ with $\ov V\subset D$ and $f_{1}(z)\neq0$ for every $z\in \ov{V}\setminus\{z^*\},$ such that $d_B(f_{1},V,0)\neq 0.$ Then, for $\e > 0$ sufficiently small, there exists a $T-$periodic solution $\f(t,\e)$ of \eqref{eq:order-k-ode} such that $\f(t,\e)\to z^*$ uniformly as $\e\to 0.$
\end{corollary}

\smallskip

This paper is structured as follows. Section \ref{sec:prel} contains some basic notions and definitions on degree theory as well as some preliminary results. More specifically, in Section \ref{sec:BD}, we introduce the Brouwer degree for studying zeros of functions defined in finite dimensional spaces; in Section \ref{sec:LSD}, we introduce the Leray-Schauder degree, which is an extension of the Brouwer degree for functions defined in infinite dimensional spaces; and, in Section \ref{sec:CD}, we introduce the coincidence degree for studying fixed points of nonlinear operator equations. Section \ref{sec:proof} is completely devoted to the proof of our main results. Finally, in Section \ref{sec:applic}, we analyze the following continuous higher order perturbation of a harmonic oscillator
\begin{equation}\label{ap1}
\ddot x=-x +\e \big(x^2+\dot x^2)+\e^k \, x  \sqrt[3]{x^2+\dot x^2-1} +\e^{k+1}E(x,\dot x,\e), \vspace{0.2cm}\\
\end{equation}
where $k$ is a positive integer and $E$ is a continuous function on $\R^3.$ Clearly, the differential equation \eqref{ap1} is not Lipschitz in any neighborhood of $\s^1=\{(x,\dot x)\in\R^2:\,x^2+\dot x^2=1\}.$ As an application of our main results, we get  the existence of a periodic solution $x_{\e}(t)$ of \eqref{ap1} satisfying $(x_{\e}(t),\dot x_{\e}(t))\to\s^1$  uniformly  as $\e\to 0$ (see Proposition \ref{prop:ap}). Notice that any previous version of the averaging method could be applied to detect such a periodic solution.

\section{Degree theory and preliminary results}\label{sec:prel}

This section is devoted to provide the basic notions and definitions on degree theory as well as some preliminary results.

\subsection{Brouwer degree}\label{sec:BD}
The Brouwer degree is defined as an integer-valued function that assigns to every triple $ (f,V,y_0),$ where $ V\subset\R^n $ is an open bounded subset of $ \R^n, $ $ f:\ov{V}\to \R^n $ is a continuous function, and $y_0\notin f(\p V),$ the number $ d_B(f,V,y_0)$ whose defining properties are:
\begin{enumerate}[label={\bf B.\arabic*},ref=B.\arabic*]
	\item\label{brouwer:existence} (Existence) If $ d_B(f,V,y_0)\neq 0, $ then $ y_0\in f(V). $ Furthermore, if $ \iota:\ov{V}\to\R^n $ is the identity function and $ y_0\in V, $ then $ d_B(\iota, V, y_0) = 1. $
	\item\label{brouwer:additivity} (Additivity) If $V_1, V_2\subset V$ are disjoint open subsets of $V$ such
	that $y_0\notin f(\ov{V}\backslash (V_1\cup V_2)),$ then $$d_B(f, V, y_0) =
	d_B(f_{\vert {V_1}}, V_1, y_0) + d_B(f_{\vert {V_2}}, V_2, y_0).$$
	\item\label{brouwer:homotopy} (Invariance under homotopy) If $ \{f_t:\ov{V}\to Y\, |\, t\in [0,1] \} $ is a continuous homotopy and $ \{ y_t\, |\, t\in [0,1] \} $ is a continuous curve such that $ y_t\notin f_t(\p V),\,\forall t\in [0, 1] $ then $ d(f_t,V,y_t) $ is constant in $ t. $
\end{enumerate}

The next result was proved in  \cite{candido-libre-novaes-2017} and concerns the invariance of the Brouwer degree under small perturbations. This will be necessary for proving Theorem \ref{thm:main2}.

\begin{lemma}[\cite{candido-libre-novaes-2017}]\label{NNL}
Let $V$ be an open bounded subset of $\R^m.$ Consider the continuous
functions $f_i:\ov{V}\to \mathbb{R}^n,$ $i=0,1,\cdots,\kappa,$ and
$f,g,r:\ov{V}\times [0,\e_0] \rightarrow \mathbb{R}^n$ given by
\[
g(z,\e)=f_0(z)+\e f_1(z)+\cdots+\e^\kappa f_\kappa(z) \mbox{ and }
f(z,\e)=g(z,\e)+\e^{\kappa+1}r(z,\e).
\]
Let $V_\e \subset V,$ $R=\max\{\vert r(z,\e) \vert :
(z,\e)\in\ov{V}\times [0,\e_0]\}$ and assume that $\vert g(z,\e)
\vert> R\vert \e\vert^{\kappa+1}$ for all $ z \in \p V_\e $ and
$\e\in(0,\e_0].$ Then, for each
$\e\in(0,\e_0]$ we have $d_B\left(f(\cdot,
\e),V_\e,0 \right)=d_B\left(g(\cdot, \e),V_\e,0  \right).$
\end{lemma}

\subsection{Leray-Schauder degree}\label{sec:LSD}
The Leray-Schauder degree was introduced by Leray and Schauder in \cite{LerayShcauder1934} and defined in the context of compact perturbations of the identity on normed linear spaces.

\begin{definition}[\cite{deimling1985}]\label{def:lsd}
Let $ X $ be a normed linear space, $ \O\subset X $ be an open subset of $X,$ and $ M:\ov{\O}\to X $ be a compact map. If $ y_0\notin (\Id - M)(\p\O), $ then the Leray-Schauder degree is an integer-valued function defined by
\begin{align*}
d_{LS}(\Id - M, \O, y_0) = d_B((\Id - M_1)_{\vert \O\cap X_1}, \O\cap X_1, 0),
\end{align*}
where $ M_1 $ is any compact mapping with finite dimensional range satisfying  \[\sup_{x\in\ov{\O}}|M_1 x - Mx| < \dist (y_0,(\Id - M)(\p\O)),\]  and $ X_1 $ is any finite dimensional space such that $ y_0\in X_1 $ and $ M_1(\ov{\O})\subset X_1. $
\end{definition}

Considering the setting in definition \ref{def:lsd}, then one can prove that the Leray-Schauder degree has the following properties (see, for instance \cite{deimling1985}):

\begin{enumerate}[label={\bf LS.\arabic*},ref=LS.\arabic*]
	\item\label{leray-schauder:existence} (Existence) If $ d_{LS}(\Id - M, \O, y_0)\neq 0, $ then there exists $ x\in\O $ such that $ Mx - x = y_0. $
	\item\label{leray-schauder:additivity} (Additivity) If $ \O_1, \O_2\subset \O $ are open disjoint subsets of $ \O $ such that $ 0\notin (\Id - M)(\ov{\O}\backslash (\O_1\cup\O_2), $ then
	\begin{multline*}
		d_{LS}(\Id - M, \O, y_0) = d_{LS}((\Id - M)_{\vert \O_1}, \O_1, y_0) + d_{LS}((\Id - M)_{\vert \O_2}, \O_2, y_0).
	\end{multline*}
	\item\label{leray-schauder:homotopy} (Invariance under homotopy) Let $ H:\ov{\O}\times[0,1]\to X $ be a compact map and $ \{ y_t\in X\, |\, t\in [0,1] \} $ be a continuous curve such that $ H(x,t) - x\neq y_t, $ for all $ (x,t)\in\p\O\times [0,1]. $  Then, $d_{LS}(\Id- H(\cdot, t), \O, y_t)$ is constant in $ t. $
\end{enumerate}

One can readily see the similarity between these properties and the properties \ref{brouwer:existence}-\ref{brouwer:homotopy}. Indeed, as we can see from Definition \ref{def:lsd}, the Leray-Schauder degree is obtained from the Brouwer degree by approximating the infinite dimensional space $ X $ by finite dimensional ones. In particular, in this scenario, if $ X $ is finite dimensional, then $d_{LS}(I - M,\O,0) = d_B(I - M, \O, 0).$ Such fact is key for the proof of Lemma \ref{thm:consequence}.

\subsection{Coincidence degree}\label{sec:CD}
Finally, consider two Banach spaces $ X $ and $ Z. $ Let $ L:\dom L\subset X\to Z $ be a linear map, $\O$ an open subset of $ X,$ and $ N:\ov{\O}\subset X\to Z $ any map. The coincidence degree concerns the existence of solutions of the operator equation  $Lx = Nx$ under some suitable assumptions on $L,$ $N,$ and $\O.$

We say that $ L $ is a Fredholm operator of index 0 if $ \Im L $ is a closed subset of $ Z ,$ $ \ker L, \coker L = Z/\Im L $ are finite dimensional, and $ \dim\ker L = \dim\coker L. $ Let  $ P:X\to X $ and $ Q:Z\to Z $ be continuous projectors such that the sequence
\begin{equation}\label{eq:exact-seq}
\begin{tikzcd}
X \arrow{r}{P} & \dom L \arrow{r}{L} & Z \arrow{r}{Q} & Z
\end{tikzcd}
\end{equation}
is exact, i.e. $\Im P = \ker L$ and $\Im L = \ker Q.$ It is easily seen that $ L_P = L_{\vert \ker P~\cap~\dom L} $ is an isomorphism. Therefore, we can take its inverse, denoted by $ K_P, $ and define the generalized inverse of $ L $ by $K_{P,Q} = K_P (\Id- Q). $ Denote by $ \Pi:Z\to \coker L $ the canonical projection that sends any $ y\in Z $ onto its equivalence class in $ \coker L. $

Recall that
an operator $ T:\ov{\O}\to Z $ is said to be completely continuous whenever it maps bounded subsets of $ \ov{\O} $ onto relatively compact subsets of $ Z. $

The following is an important definition for the development of the theory of the coincidence degree.

\begin{definition}[\cite{gaines1977coincidence}]\label{def:Lcomp}
Let $ L:\dom L\subset X\to Z $ be a continuous linear Fredholm operator of index 0 and $ N:\ov{\O}\subset X\to Z $ be a continuous operator. The operator $ N $ is said to be $ L- $compact if $(i)$ $\Pi N $ is continuous and bounded and $(ii)$ $ K_{P,Q} N $ is completely continuous.
\end{definition}

In this paper we work with the theory of coincidence degree for $ L- $compact perturbations of Fredholm maps of index zero.

\begin{proposition}[{\cite[Proposition III.2]{gaines1977coincidence}}]\label{Prop:fixedpoint}
Let $ L:\dom L\subset X\to Z $ be a continuous linear Fredholm operator of index 0 and $ N:\ov{\O}\subset X\to Z $ be a continuous operator. If there exists an isomorphism $ \L:\coker L\to\ker L, $ then $ Lx = Nx $ if, and only if,
\[ (\Id - P)x = (\Lambda\Pi + K_{P, Q})Nx. \]
\end{proposition}

This proposition says that, as long as $ \ker L $ and $ \coker L $ are isomorphic vector spaces, the set of solutions of $ Lx = Nx $ is equal to the set of fixed points of the operator $ M = P + (\Lambda\Pi + K_{P, Q})N,$ which can be proved to be a compact operator. If, in addition, one has $ 0 \notin (L - N)(\p \O\cap \dom L), $ then the Leray-Schauder degree of $ \Id - M $ with respect to $ \O $ and 0, $ d_{LS}(\Id - M,\O,0), $ is well-defined. This motivates the following definition.

\begin{definition}\label{def:cd}
	Let $ L:\dom L\subset X\to Z $ be a continuous linear Fredholm operator of index 0 and $ N:\ov{\O}\subset X\to Z $ be a continuous operator.  The coincidence degree of $ L \mbox{ and } N $ with respect to $ \O $ is defined as $ d((L,N),\O) := d_{LS}(\Id - M,\O,0). $
\end{definition}

\begin{remark}
 In the above definition, it is worthwhile to point out that  $ |d((L,N),\O)| $ depends only on $ L, N $ and $ \O. $ The sign of $ d((L,N),\O) $ depends on wether or not $ \Lambda $ is an orientation preserving isomorphism.
\end{remark}

	In the above setting, the properties \ref{leray-schauder:existence}-\ref{leray-schauder:homotopy} of the Leray-Schauder degree induce the following properties on the coincidence degree.
\begin{enumerate}[label={\bf C.\arabic*},ref={\bf C.\arabic*}]
	\item\label{coincidence:existence} If $ d((L,N),\O)\neq 0, $ then there exists $ x\in\O $ such that $ Lx = Nx. $
	\item\label{coincidence:additivity} If $ \O_1,\O_2\subset\O $ are open disjoint subsets of $ \O, $ then
	\[d((L,N),\O) = d((L,N),\O_1) + d((L,N),\O_2). \]
	\item\label{coincidence:homotopy} Let $ N:\ov{\O}\times [0, 1]\to Z $ be $ L- $compact in $ \ov{\O}\times [0,1] $ such that $0 \notin (L - N(\cdot,t))(\dom L\cap \p \O)$ for each $ t \in [0, 1],$ then $ d((L,N(\cdot,t)),\O) $ is constant in $ t. $
\end{enumerate}

\begin{proposition}[{\cite[Lemma IV.1]{gaines1977coincidence}}]\label{lemma:basic-lemma-02}
Let $ L:\dom L\subset X\to Z $ be a continuous linear Fredholm operator of index 0 and, for each $\e\in(0,\e_1],$ let $N_{\e}$ be a $ L- $compact operator on $ \ov{\O}.$ For $ y\in\Im L, $ consider the problem
\begin{equation}\label{eq:iv.1}
Lx = \e\, N_{\e}(x) + y.
\end{equation}
Then, for each $\e\in (0, \e_1],$ the set of solutions of \eqref{eq:iv.1} is equal to the
set of solutions of the equation
\[
Lx = QN_{\e}(x) + \e (\Id - Q)N_{\e}(x) + y.
\]
\end{proposition}
Now, we prove the following result.

\begin{lemma}\label{thm:consequence}
Let $  L:\dom L\subset X\to Z $  be a continuous linear Fredholm operator of index 0 and, for each $\e\in(0,\e_1],$ let $N_{\e}$ be a $ L- $compact operator on $ \ov{\O}.$ Consider the operators $P,Q,K_{P,Q},\Lambda,$ and $\Pi$ as defined above and assume that
\begin{equation}\label{eq:hypothesis-on-the-boundary2}
\|x-Px-(\L\Pi+K_{P,Q})\wt{N}_\e(x,\la)\|\neq 0,
\end{equation}
for every $ x\in\p\O,\,\la\in [0,1] $ and $ \e\in (0,\e_1], $ where
\begin{equation}\label{homotopy}
 \wt{N}_\e(x,\la) = QN_{\e}(x) + \la\e (\Id - Q) N_{\e}(x).
 \end{equation}
Then,
 \[d((L,QN_{\e} + \e (\Id - Q) N_{\e}),\O)= d_B({JQN_{\e}}_{\,|\, \O\cap\ker L},\O\cap\ker L, 0)\] for every  $ \e\in(0,\e_1], $ where $ J:\Im Q\to\ker L$ is the isomorphism $J=-\Lambda \Pi_Q,$ with $\Pi_Q=\Pi\big|_{\Im Q}.$
\end{lemma}
\begin{proof}
For each $ \e\in(0,\e_1] $ fixed,
\[\wt{N}_{\e}(x,\la) = QN_{\e}(x) + \la\e (\Id - Q) N_{\e}(x),\quad \la\in[0,1],\]
is a continuous homotopy between $QN_{\e}$ and $\widehat{N}_{\e}(x) := QN_{\e}(x) + \e (\Id - Q) N_{\e}(x).$ Now, in order to use property \ref{coincidence:homotopy} of the coincidence degree, we have to show that $ \wt{N}_\e $ is $ L- $compact in $ \ov{\O}\times [0,1] $ (see Definition \ref{def:Lcomp}) and that
\[0 \notin (L - \widetilde N_{\e}(\cdot,\la))(\dom L\cap \p \O_{\e})\]
for each $\la \in [0, 1].$

\begin{claim}\label{claim:homotopy-bounded}
For each $\e\in(0,\e_1],$ the operator $ \wt{N}_\e $ is $ L- $compact in $ \ov{\O}\times [0,1] .$
\end{claim}

First, notice that $\Pi\wt{N}_\e$ is continuous. Now, since $Q$ is a continuous projection and $\ker Q=\Im L,$ we get that $\Pi (\Id - Q)=0$  and, consequently, $ \Pi Q = \Pi.$ Therefore, $ \Pi\wt{N}_\e(x,\la) = \Pi N_{\e}(x).$ Hence, $ \Pi\wt{N}_\e(\ov{\O}\times [0,1]) $ is bounded in $ \coker L.$

It remains to check that $ K_{P,Q}\wt{N}_\e$ is completely continuous. The continuity of $ K_{P,Q}\wt{N}_\e $ follows from the continuity of $ K_{P,Q} $ and $ \wt{N}_\e. $ Now, from the definition of the operator $K_{P,Q},$ we have
\begin{align*}
K_{P,Q}\wt{N}_\e(x,\la) & = K_{P,Q} QN_{\e}(x) + \la\e K_{P,Q} (\Id - Q) N_{\e}(x) = \la\e K_{P,Q} N_{\e}(x).
\end{align*}
Now, define $ \CF_\e : \ov{\O}_{\e}\times [0,1]\to Z\times [0,1] $ as
\[ \CF_\e(x,\la) = \left(\e K_{P,Q}N_{\e}(x),\la\right). \]
Since $N$ is $ L- $compact, we conclude that $ \CF_\e $ is a continuous compact operator. Indeed, the $ L- $compactness of $ N$ implies that $\ov{\e K_{P,Q}N(\ov{\O_{\e}}\times\{\e\})} $ is compact and, then,
\[\ov{\CF_\e(\ov{\O_{\e}}\times [0,1])} = \ov{\e K_{P,Q}N(\ov{\O_{\e}}\times\{\e\})\times [0,1]} = \ov{\e K_{P,Q}N(\ov{\O_{\e}}\times\{\e\})}\times [0,1]\]  is a compact set. The continuity of $\CF_\e$ is clear. Hence, defining the continuous operator $ \CM:Z\times\R\to Z $ as $ \CM(z,\la) = \la z, $ we get $ K_{P,Q}\wt{N}_\e = \CM\circ\CF_\e. $
Finally, composition of a compact map with a continuous one is also compact, which implies that $ K_{P,Q}\wt{N}_\e $ is compact. This concludes the proof of the claim.

\begin{claim}\label{claim:hyp-equivalent}
For each $\e\in(0,\e_1]$ and $\la \in [0, 1],$ $0 \notin (L - \widetilde N_{\e}(\cdot,\la))(\dom L\cap \p \O).$
\end{claim}

From Proposition \ref{Prop:fixedpoint}, this claim is equivalent to proving that the operator $P + (\L\Pi + K_{P,Q})\wt{N}_\e(\cdot,\la)$ has no fixed points in $\dom L\cap \p \O$  for $ \e\in (0,\e_1] $ and $ \la\in [0,1]. $
Since, from hypothesis \eqref{eq:hypothesis-on-the-boundary2},
\[
	\| x - Px - (\L\Pi + K_{P,Q})\wt{N}_\e(x,\la) \|\neq 0
\]
for every $\e\in(0,\e_1]$ and $ \la\in [0,1],$
this claim follows.

\smallskip

Now, property \ref{coincidence:homotopy} implies that
\begin{equation}\label{lemma:eq1}
d((L,Q N_{\e} + \e (\Id - Q) N_{\e}),\O)= d((L,QN_{\e}),\O),
\end{equation}
for every $\e\in(0,\e_1].$ By definition of the Coincidence Degree (see Definition \ref{def:cd}), we have
\begin{align*}
d((L,QN_{\e}),\O) & = d_{LS}(\Id - P - (\L\Pi + K_{P,Q})QN_{\e},\O,0)\\
 & = d_{LS}(\Id - P - \L\Pi QN_{\e},\O,0).
\end{align*}
Define $ M_\e = P + \L\Pi QN_{\e} $ and note that $ M_\e(\ov{\O})\subset \ker L,$ for every $\e\in(0,\e_1].$ Therefore, using the definition of the Leray-Schauder Degree (see Definition \ref{def:lsd}) with $ X_1 = \ker L $ and $ M_1 = {M_\e}_{\vert\ker L}, $ and the fact that $ (\Id - P)_{\vert\ker L} = 0, $ we obtain from the identity above that\begin{equation}\label{lemma:eq2}
\begin{array}{rl}
d((L,Q N_{\e}),\O)  & \!\!\! = d_B((\Id - P - \L\Pi Q N_{\e})_{\vert \ker L},\O\cap\ker L,0)\\
& \!\!\! = d_B(-\L\Pi Q {N_{\e}}_{\vert \ker L},\O\cap\ker L,0),\\
&\!\!\! = d_B((-\L\Pi_Q)Q {N_{\e}}_{\vert\ker L},\O\cap\ker L, 0)
\end{array}
\end{equation}
for every $\e\in(0,\e_1],$ where $ \Pi_Q := \Pi_{\vert \Im Q} $ is an isomorphism.
We conclude this proof by defining $ J = -\L\Pi_Q$ and taking \eqref{lemma:eq1} and \eqref{lemma:eq2} into account.
\end{proof}

\section{Proof of the main results}\label{sec:proof}

We denote by $C[0, T]$ the space of all continuous functions defined in $[0,
T]$ with values in $ \R^n $ and define the function spaces \[ C_0 = \{ x\in C[0, T]~\colon~ x(0) = 0\}\quad\mbox{ and }\quad C_T = \{ x\in C[0, T]~\colon~ x(0) =
x(T)\}\] both endowed with the sup-norm making them into real Banach spaces. We also define $ \O = \{ x\in C_T\colon x(t)\in V,\,\forall\, t\in [0, T] \}, $ which is an open bounded subset of $ C_T. $ Now, we define the linear operator $ L\colon C_T\to C_0 $ by putting for every $ t\in [0,T],\, Lx(t) = x(t)- x(0). $ Lastly, for each $\e\in (0,\e_0],$ we define $ N_{\e}:\ov{\O}\to C_0 $ by $$ N_{\e}(x)(t) = \int_0^t  F(s,x(s),\e)\d s. $$

Now, we know that $ x\colon [0,T]\to\R^n $ is a solution of \eqref{eq:e1} if, and only if,
 \[\label{eq:functional-eq}
Lx =\e N_{\e}(x),
\]
which, from Proposition \ref{lemma:basic-lemma-02}, is equivalent to
\begin{equation}\label{eq:functional-eq-2}
Lx =  \widehat N_{\e}(x):= QN_{\e}(x) + \e (\Id - Q) N_{\e}(x).
\end{equation}
Therefore, in order to prove
Theorem \ref{thm:main1}, we must guarantee the existence of solutions of \eqref{eq:functional-eq-2} lying in $ \O.$ From Property \ref{coincidence:existence}, the existence of such solutions is guaranteed provided that \[ d((L,\widehat  N_{\e}),\O)\neq 0. \]

\begin{proof}[{\bf Proof of Theorem \ref{thm:main1}}]
	In the above setting, consider the operators

  $ P\colon C_T\to C_T,$ and $Q\colon C_0\to C_0 $ defined by
 \[
 Px(t) = x(0)\,\, \text{ and }\,\, Qy(t) = \dfrac{t~y(T)}{T}, \,\, \text{ for }\,\, t\in [0,T],
 \]
 respectively.
 By definition of $L$, we have that
 \[
 \ker L = \big\{ x\in C_T:x(t) = x(0),\,\forall\, t\in [0,T]\big\},
 \]
that is, the space of all constant function in $ \R^n,$ which can be identified with $ \R^n.$ In addition,  $\coker L =\big \{[y]:=y+\Im L:y\in C_0\big\}$. One can readily see that

 $[y_1] = [y_2] $ if, and only if,

 $ y_1(T) = y_2(T),$ which means that $ \coker L $ can be identified with $ \R^n. $ Accordingly,
 let $ \Pi:C_0\to\coker L$ and  $\L:\coker L\to\ker L $ be defined by
 \[ \Pi y= [y(T)] \,\, \text{ and }\,\,  \L[z](t) = -z, \,\, \text{ for }\,\,   t\in [0,T], \]
 respectively.

 Therefore, $ J = -\L\Pi_{|\Im Q} $ becomes
 \[ Jy(t) = y(T),\quad\forall\,t\in [0,T]. \]

 It is straightforward to check that $ \Im P = \ker L $  and $ \Im L = \ker Q, $ which implies that the sequence in \eqref{eq:exact-seq} is exact. Therefore, $ L $ is Fredholm operator of index zero and $ N_{\e} $ is $ L- $compact for each $ \e\in (0,\e_0]. $ Notice that $\ker L $ is constituted by constant functions. Thus, working out $ {JQN_{\e}}_{\vert \O\cap \ker L} $ we have that, for $ x(t)\equiv z\in\ker L,$
	\begin{align}\label{eq:JQN-full-avg}
	JQN_{\e}(x)(t) = J\left(\dfrac{t}{T}\int_0^T \e F(s,z,\e)\d s\right) = T f(z,\e).
	\end{align}

	This proof will follow by showing that
	\begin{equation}\label{teoA:goal}
	d((L,\widehat  N_{\e}),\O)=d_B(JQN(\cdot,\e)_{\vert \O \cap \ker L}, \O \cap \ker L, 0), \text{ for  }\e\in(0,\e_1],
	\end{equation}
	and some $ \e_1\in (0,\e_0]. $
	 Indeed, in this case, taking \eqref{eq:JQN-full-avg} into account and proceeding with the identification $ \O\cap\ker L \equiv V,$ we get $d((L, \widehat  N_{\e}),\O)=d_B(f(\cdot,\e),V,0),$ which, by hypothesis, is distinct from zero for every $\e\in(0,\e_1].$ Thus, for each $\e\in(0,\e_1],$ Property \ref{coincidence:existence} provides the existence of a solution $x_{\e}\in\Omega$ of \eqref{eq:functional-eq}, which corresponds to a periodic solution $\f(t,\e)=x_{\e}(t)$ of the differential equation \eqref{eq:e1} satisfying $\f(t,\e)\in V$ for every $t\in[0,T].$

	In order to use Lemma  \ref{thm:consequence} to get \eqref{teoA:goal},
we must first check that
\begin{itemize}
	\item[$({\bf C})$] there exists $\e_1\in(0,\e_0]$ such that
	\[\label{teoA:hyp}
	 \| x - Px - (\L\Pi + K_{P,Q})\wt{N}_\e(x,\la) \|\neq 0,
	 \]
	for every $ x\in\p\O,$ $\la\in [0,1], $ and $ \e\in (0,\e_1],$ where $\wt{N}_\e(x,\la)$ is the homotopy defined in \eqref{homotopy}.
\end{itemize}
	Note that
	\[
	\begin{array}{ll}
	& \| x - Px - (\L\Pi + K_{P,Q})\wt{N}_\e(x,\la) \| \\
	=&\displaystyle \sup_{t\in [0,T]} \big|\left[x - Px - \L\Pi QN_{\e}(x) - \la\e (\Id-Q) N_{\e}(x)\right](t)\big|\\
	=&\displaystyle \sup_{t\in [0,T]} \left|x(t) - x(0) +\left(\dfrac{\la\e t}{T}+1\right)\int_0^T F(s,x(s),\e)\d s \right.\\
	&\displaystyle \hspace{7cm}\left. - \la\e\int_0^t F(s,x(s),\e)\d s\right|.
	\end{array}
	\]

	Suppose by contradiction that $({\bf C})$ does not hold. Thus, we can find numerical convergent sequences $(\e_m)_{m\in\mathbb{N}}\subset(0,\e_0]$ and $(\la_m)_{m\in\mathbb{N}}\subset(0,1],$ with $\e_m\to 0$ as $m\to \infty,$ and a sequence of functions
	$(x_m)_{m\in\mathbb{N}}\in\p\O$ such that
	\begin{multline*}
		\sup_{t\in [0,T]} \left|x_m(t) - x_m(0) +\left(\dfrac{\la_m\e_m t}{T}+1\right)\int_0^T F(s,x_m(s),\e_m)\d s\right.\\
		\left. - \la_m\e_m\int_0^t F(s,x_m(s),\e_m)\d s\right| = 0,
	\end{multline*}
	for every $ m\in\N.$ Notice that $(x_m)_{m\in\mathbb{N}}\in\p\O$ means that, for each $m\in\N,$  $x_m(t)\in\ov V$ for every $t\in[0,T]$ and there exists $ t_m\in [0,T] $ such that $ x_m(t_m)\in\p V.$ In addition, the equality above implies that
	\begin{multline*}
		x_m(t) = x_m(0) - \left(\dfrac{\la_m\e_m t}{T}+1\right)\int_0^T F(s,x_m(s),\e_m)\d s\\
		 + \la_m\e_m\int_0^t F(s,x_m(s),\e_m)\d s,
	\end{multline*}
	for every $ t\in [0,T]. $ Plugging $ t=0,$ we obtain that
	\begin{equation}\label{eq:homot-diff-eq0}
	\int_0^T F(s,x_m(s),\e_m)\d s = 0,
	\end{equation}
	for every $ m\in\N, $ which gives us
	\begin{align}\label{eq:homot-diff-eq}
	x_m(t) = x_m(0) + \la_m\e_m\int_0^t F(s,x_m(s),\e_m)\d s,
	\end{align}
	that is, $ x_m $ is a solution of the differential equation
	\[ x' = \la_m\e_m F(t,x,\e_m). \]
	If $ (\la_m)_{m\in\N} $ admits a vanishing subsequence, let us say $ (\la_{m_i})_{i\in\N}$ with $\la_{m_i}=0$ for every $i\in\N,$ then, from \eqref{eq:homot-diff-eq}, $x_{m_i}(t)=x_{m_i}(0)=z_i\in\p V$ for every $t\in[0,T].$ Therefore, we get sequences $z_i\in\p V$ and $\e_{m_i}\to 0$ such that, from \eqref{eq:homot-diff-eq0},
	\[ \int_0^T F(s,z_i,\e_{m_i})\d s = 0, \quad i\in \N, \]
	which contradicts \eqref{Ha}. Thus, we can assume that $\la_m\neq0$ for every $m\in\N.$ In this case, for each $m\in\N,$ $x_m$ is a $T$-periodic solution of $x'=\e_m\la_m F(t,x,\e_m),$ $x\in\ov V,$ which is not entirely contained in $V.$
	This contradicts hypothesis $ ({\bf H}).$ Therefore, condition $({\bf C})$ holds and Lemma \ref{thm:consequence} can be applied to get \eqref{teoA:goal}, which concludes this proof.
\end{proof}

\begin{proof}[{\bf Proof of Theorem \ref{thm:main2}}]
For system \eqref{eq:order-k-ode}, we have
\[ f(z,\e) = \dfrac{1}{T}\int_{0}^{T} \sum_{j=1}^{k} \e^j F_j(s,z) + \e^{k+1}R(s,z,\e)\d s = \CF_k(z,\e) + \e^{k+1} r(z,\e). \]

By hypothesis \eqref{eq:lemma-hyp}, there exists $ \e_1 \in(0,\e_0] $ such that $$\left\lvert \CF(z,\e)\right\rvert > \lvert\e^{k+1}\rvert\, \max\{ |r(z, \e)|\colon (z,\e)\in \ov{V}\times [0,\e_0]\},$$ for every $ \e \in (0,\e_1].$
In particular, hypothesis \eqref{Ha} of Theorem \ref{thm:main1} holds. Moreover, by taking $V_{\e}=V$ in Lemma \ref{NNL}, we get that $ d_B(f(\cdot, \e),V, 0) = d_B(\CF(\cdot, \e), V, 0), $ which is non-zero by hypothesis. Therefore $ d_B(f(\cdot, \e), V, 0) \neq 0 $ and the result follows from Theorem \ref{thm:main1}.
\end{proof}

\begin{proof}[{\bf Proof of Theorem \ref{thm:main3}}]

Without loss of generality, we can assume that $\ell=k.$ Take neighbourhoods $ V_\mu = B(z^*,\mu)\subset V $ for $ \mu > 0 $ sufficiently small. Clearly $ V_\mu\to\{ z^*\} $ as $ \mu\to 0. $ Now, $ f_1 = \cdots = f_{k-1} = 0,$ then $\CF_k(\cdot,\e)=\e^k f_k(z).$
 Moreover, since $f_k(z)\neq0,$  for every $z\in\p V_\mu,$ and $ r(z,\e) $ is continuous, consequently, bounded on compact sets, we conclude that
 \[
 \begin{array}{rl}
 \ds\lim_{\e\to 0} \inf_{z\in\p V_{\mu}} \left\lvert \dfrac{\e^k f_k(z)}{\e^{k+1}} \right\rvert =&\ds\!\!\! \lim_{\e\to 0} \inf_{z\in\p V_{\mu}} \left\lvert \dfrac{f_k(z)}{\e} \right\rvert = \infty \\
 >&\ds\!\!\!\max\{|r(z,\e)|: (z,\e)\in\ov V\times[0,\e_0]\}.
 \end{array}
 \]
 Thus, hypothesis \eqref{eq:lemma-hyp} of Theorem \ref{thm:main2} holds. In addition, $ d_B(\CF_k(\cdot,\e),V_\mu,0) = d_B(f_k(z),V_\mu,0),$ which, by hypothesis, is distinct from zero, for $|\e|\neq0$ sufficiently small. Hence, by Theorem \ref{thm:main2}, there exists $ \e_\mu > 0 $ and a $ T- $periodic solution $ \f(\cdot,\e) $ of \eqref{eq:order-k-ode} such that $ \f(t,\e)\in V_\mu,\,\forall\, t\in [0, T] $ and for each $ \e\in (0,\e_\mu]. $ Now, given any $ \xi > 0, $ put $ \mu = \xi $ and $ \de = \e_\mu. $ By the conclusion above, $ |\e| < \de $ implies $ |\f(t,\e) - z^*| < \xi $ for every $ t\in [0,T]. $ That is precisely to say that $ \f(t,\e) \to z^*$  uniformly  as $\e\to 0.$ This completes the proof.
\end{proof}

\section{Non-Lipschitz perturbation of a harmonic oscillator}\label{sec:applic}

Consider the continuous higher order perturbation of a harmonic oscillator \eqref{ap1},
\[
\ddot x=-x +\e \big(x^2+\dot x^2)+\e^k \, x  \sqrt[3]{x^2+\dot x^2-1} +\e^{k+1}E(x,\dot x,\e).
\]
Recall that $k$ is a positive integer and $E$ is a continuous function on $\R^3.$ Clearly, the differential equation \eqref{ap1} is not Lipschitz in any neighborhood of $\s^1=\{(x,\dot x)\in\R^2:\,x^2+\dot x^2=1\}.$ In the next result, Theorem \ref{thm:main3} is applied to show the existence of a periodic solution $x_{\e}(t)$ of \eqref{ap1} satisfying $(x_{\e}(t),\dot x_{\e}(t))\to\s^1$  uniformly  as $\e\to 0.$ Notice that such a periodic solution is not detectable by any Lipschitz version of averaging method.
\begin{proposition}\label{prop:ap}
For any positive integer $k$ and $|\e|\neq0$ sufficiently small, the differential equation \eqref{ap1} admits a periodic solution $x(t;\e)$ satisfying $(x(t;\e),\dot x(t;\e))\to \s^1$  uniformly  as $\e\to 0.$
\end{proposition}
\begin{proof}
Changing to polar coordinates $x=r \cos\T, \dot x=r\sin\T$ and taking $ \theta $ as the new independent variable, the differential equation \eqref{ap1} becomes
\[
\begin{array}{rl}
\dfrac{d r}{d\T}=\e F(\T,r,\e),
\end{array}
\]
where
\[
\begin{array}{rl}
F(\T,r,\e)=&\displaystyle\sum_{i=1}^{k-1}\e^{i-1} r^{i+1}\cos^{i-1}\T\sin\T\vspace{0.2cm}\\
&+\e^{k-1} r\Big(r^k\cos^{k-1}\T+\sqrt[3]{r^2-1}\sin\T\Big)\sin\T+\e^{k}R(\T,r,\e),
\end{array}
\]

which is not Lipschitz in any neighbourhood of $ r = 1.$ Let $V=(1-\al,1+\al)$ for some $0<\al<1.$

Notice that $f_i=0,$ for $i=1,2,\ldots,k-1,$ and $f_k(r)=r\sqrt[3]{r^2-1}/2.$
Moreover, $ f_k $ has a unique positive zero $r^* = 1 $ and is homotopic to the map $r\mapsto r-1$ in $ V .$ Therefore, $ d_B(f_k,V,0) \neq 0. $

In what follows we shall assume that $\e>0.$ The result for $\e<0$ can be obtained analogously just by taking considering $-F(\T,r,\e).$ In order to apply Theorem \ref{thm:main3}, it remains to check hypothesis $ ({\bf H}).$ From Remark \ref{rmk:hyp-h}, the negation of hypothesis $({\bf H})$ provides numerical convergent sequences $(\e_m)_{m\in\mathbb{N}}\subset(0,\e_0]$ and $(\la_m)_{m\in\mathbb{N}}\subset(0,1],$ such that $\e_m\to 0$ as $m\to \infty,$ and a sequence $(r_m)_{m\in\N}$ of $T$-periodic solutions of
	\begin{equation}\label{eq:homot-eq}
		r'=\e_m\la_m F(\T,r,\e_m),\quad r\in\ov V,
	\end{equation}
for which there exists $\T_m\in[0,2\pi]$ such that $r_m(\T_m)\in\partial V$ for each $m\in\mathbb{N}.$ As an application of  Arzel\'{a}-Ascoli's Theorem, the sequence of functions $(r_m)_{m\in\mathbb{N}}$ can be taken uniformly convergent to a constant function in $\partial V=\{1\pm\al\}.$ In particular,
\[\label{contra:edo}
\int_0^{2\pi} F(\T,r_m(\T),\e_m)d\T=0,
\]
which implies that

\begin{equation}\label{contra:rela}
\int_0^{2\pi} r_m(\T)\sqrt[3]{r_m(\T)^2-1}\sin^2\T d\T=\dfrac{1}{\e_m^{k-1}}\sum_{i=1}^{k}\e_m^{i-1} G_m^{i+1,i-1} +\CO(\e_m),
\end{equation}

where
\[ G^{i,j}_m = \int_0^{2\pi} r_m(\t)^i\cos^j(\T)\sin(\T)~\d\T. \]
Here, although $ (\e_m)_{m\in\N} $ is a \emph{numerical} sequence, we borrow the Landau's symbol notation $ h_m=\CO(\e_m^p), $ for some $ p\in\N, $ to mean that there exists a positive constant $ C $ such that $ |h_m|\leq C |\e_m^p|, $ for $ m $ sufficiently large.
Note that, by applying integration by parts and using that $ r_m(\T) $ is $ 2\pi- $periodic, we obtain
\begin{align*}
	G_m^{i,j} & = -\dfrac{i}{j+1}\int_0^{2\pi} r_m(\t)^{i-1}\cos^{j+1}(\t)r_m'(\t)~\d\t.
\end{align*}
Since $r_m(\t)$ satisfies \eqref{eq:homot-eq}, we conclude that

\begin{align*}
	G_m^{i,j}& = -\dfrac{i}{j+1}\sum_{l=1}^{k-1}\la_m\e_m^l\int_0^{2\pi}r_m(\t)^{j+l}\cos^{j+l}(\t)\sin\t~\d\t +\CO(\e_m^k)\\
		& = -\dfrac{i}{j+1}\sum_{l=1}^{k-1}\la_m\e_m^l G_{i+l,j+l}(r_m) +\CO(\e_m^k) \\
		& = \CO(\e_m).
\end{align*}
Applying the above procedure recurrently, we conclude that

$G_m^{i,j} = \CO(\e_m^p), $ for any positive integer $ p.$

In particular, for $ i=1,\ldots,k-1, $ we have $G_m^{i+1,i-1}=\CO(\e_m^{k-1}).$ Now, from \eqref{contra:rela}, we get that
\[
\int_0^{2\pi} r_m(\T)\sqrt[3]{r_m(\T)^2-1}\sin^2\T d\T=\CO(\e_m).
\]
Since $r_m$ converges uniformly to a constant function $r_0\in\{1\pm\alpha\},$ we compute the limit of the integral above as
\[
\int_0^{2\pi} r_0\sqrt[3]{r_0^2-1}\sin^2\T d\T=0,
\]
which is an absurd, because
\[
\int_0^{2\pi} r_0\sqrt[3]{r_0^2-1}\sin^2\T d\T=\pi r_0\sqrt[3]{r_0^2-1}\neq0,
\]
for $r_0\neq 1.$

Thus, we conclude that hypothesis $ ({\bf H}) $ holds and Theorem \ref{thm:main3} can be applied in order to get the result.
\end{proof}

\section*{Acknowledgements}
DDN is partially supported by FAPESP grants 2018/16430-8, 2018/ 13481-0, and 2019/10269-3, and by CNPq grant 306649/2018-7 and 438975/ 2018-9. FBGS has been partially supported by FAPESP grant 2018/22689-4.

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\end{document}

config: arxiv


TITLE: Find an equation of a line that is tangent to the graph of $f$ and parallel to the given line.
QUESTION [2 upvotes]: $$f(x) = x^3$$
$$12x − y + 9 = 0$$
I got $12x-22$ as one of my answers but the system says its wrong, please help. How can I do this correctly?

REPLY [2 votes]: The slope of the straight line is of course $12$.
The tangent to the cubic will be parallel when the slopes match, so we need $12 = f'(x) = 3x^2$, thus $x^2=4$, giving two locations on the curve at $(-2,-8)$ and $(2,8)$ The tangents through these two points cut the $y$-axis at $-8+24 = 16$ and $8-24=-16$  respectively, giving
\begin{align}
y&=12x+16 \\
y&=12x-16 \\
\end{align}


TITLE: Mackeys formula
QUESTION [0 upvotes]: I have been recently reading about Mackey's formula for normal subgroups and I have the following question. Let $H$ be a finite index normal subgroup of $G$ and $V$ a representation of $H$ with character $\chi_V$. Then the character of $coInd^G_H V$ is given by
$$\chi_{coInd^G_H V} = \begin{cases} \sum_{gH \in G/H} \chi_V (g h g^{-1}), \quad h \in H \\ 0, \qquad \qquad \qquad \qquad \quad \text{otherwise} \end{cases}$$
As $\chi_V$ is a class function would it not be easier to write it as:
$$\chi_{coInd^G_H V} = \begin{cases} [G:H] \ \chi_V (g h g^{-1}), \quad h \in H \\ 0, \qquad \quad \qquad \qquad \quad \text{otherwise} \end{cases}$$

REPLY [3 votes]: $ghg^{-1}$ is in the same conjugate class as $h$ in $G,$ but not necessarily in the same conjugate class in $H.$ So there might not be an $h_1\in H$ such that $ghg^{-1}=h_1hh_1^{-1}.$
But $\chi_V$ is only constant on classes in $H.$
For example, in $C_n,$ each element is in its own class, since $C_n$ is commutative. But a rotation clockwise by one step, when conjugated by a reflection in $D_{2n},$ becomes a rotation counter-clockwise by one step.


TITLE: $X$ Banach, $u_n \to u$ and $x^*_n \xrightarrow{w^*} x^*$ implie that $\langle x^*_n, u_n\rangle \to \langle x^*,u\rangle$.
QUESTION [0 upvotes]: Exercise :

Let $X$ be a Banach space, $u_n \to u$ and $x^*_n \xrightarrow{w^*} x^*$. Show that $\langle x^*_n, u_n\rangle \to \langle x^*,u\rangle$.

Attempt-Discussion :
I know that a sequence $x_n^* \in X^*$ converges to $x^*$ provided that $x_n^*(u) \to x^*(u)$ for all $u \in X$. Knowing that the $w^*-$topology coincides  the topology of pointwise convergence of linear functionals, this implies that $x_n^* \xrightarrow{w^*} x^*$.
Now, I know that the duality brackets given are essentialy the expression of the linear functional, aka : $u \mapsto \langle x^*, u \rangle := x^*(u)$. So essentially, since we have haev that the pointwise convergence is coinciding, we essentialy need to prove that $x^*_n(u_n) \to x^*(u)$ and I guess since we have the functional convergence, it boils down to $x^*(u_n) \to x^*(u)$ ?
How would one work further to prove the convergence asked ?
Alternatively, we would be interested in showing that $|\langle x^*_n, u_n\rangle - \langle x^*,u\rangle| < \varepsilon$ but I think I am missing something in re-writting the expression by breaking it up to form an arbitrarily small inequality.
Any hints will be greatly appreciated.

REPLY [2 votes]: Note that by the Banach-Steinhaus theorem $\sup_{n}\|x_{n}^{*}\|<\infty$. We find
$$\lim_{n\rightarrow\infty}|\langle x_{n}^{*},u_{n}\rangle-\langle x^{*},u\rangle|=\lim_{n\rightarrow\infty}|\langle x_{n}^{*},u_{n}-u\rangle+\langle x_{n}^{*},u\rangle-\langle x^{*},u\rangle|$$
$$\leq\lim_{n\rightarrow\infty}|\langle x_{n}^{*},u_{n}-u\rangle|+|\langle x_{n}^{*}-x^{*},u\rangle|\leq \lim_{n\rightarrow\infty}\|x_{n}^{*}\|\|u_{n}-u\|+\lim_{n\rightarrow\infty}|\langle x_{n}^{*}-x^{*},u\rangle|=0$$
as $x_{n}^{*}\stackrel{w^{*}}{\rightarrow}x^{*}$, $u_{n}\rightarrow u$ and $(\|x_{n}^{*}\|)$ is bounded.