\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 247, pp. 1--17.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/247\hfil Antiperiodic solutions] {Antiperiodic solutions to van der Pol equations with state-dependent impulses} \author[I. Rach\accent23unkov\'a, J. Tome\v{c}ek \hfil EJDE-2017/247\hfilneg] {Irena Rach\accent23unkov\'a, Jan Tome\v{c}ek} \address{Irena Rach\accent23unkov\'a \newline Department of Mathematical Analysis, and Applications of Mathematics, Faculty of Science, Palack\'y University, 17. Listopadu 12, 771 46 Olomouc, Czechia} \email{irena.rachunkova@upol.cz} \address{Jan Tome\v{c}ek \newline Department of Mathematical Analysis, and Applications of Mathematics, Faculty of Science, Palack\'y University, 17. Listopadu 12, 771 46 Olomouc, Czechia} \email{jan.tomecek@upol.cz} \dedicatory{Communicated by Pavel Drabek} \thanks{Submitted June 2, 2017. Published October 6, 2017.} \subjclass[2010]{34A37, 34B37} \keywords{van der Pol equation; state-dependent impulses; existence; \hfill\break\indent distributional equation; periodic distributions; antiperiodic solution} \begin{abstract} In this article we give sufficient conditions for the existence of an antiperiodic solution to the van der Pol equation \begin{equation*} x'(t) = y(t), \quad y'(t) = \mu \Big(x(t) - \frac{x^3(t)}{3}\Big)' - x(t) + f(t) \text{for a. e. }t \in \mathbb{R}, \end{equation*} subject to a finite number of state-dependent impulses \begin{equation*} \Delta y(\tau_i(x)) = \mathcal{J}_i(x), \quad i = 1,\ldots,m\,. \end{equation*} Our approach is based on the reformulation of the problem as a distributional differential equation and on the Schauder fixed point theorem. The functionals $\tau_i$ and $\mathcal{J}_i$ need not be Lipschitz continuous nor bounded. As a direct consequence, we obtain an existence result for problem with fixed-time impulses. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \newcommand {\dist}[2] {\langle #1,#2\rangle} \section{Introduction} \label{introduction} The study of anti-periodic solutions is closely related to the study of periodic solutions and their existence plays an important role in characterizing the behaviour of nonlinear differential equations. On the other hand impulsive problems are characterized by the occurrence of abrupt changes of their solutions which implies that such solution does not preserve the basic properties which are associated with non-impulsive problems. In real world problems, the impulses often do not occur at fixed times, but moments of their appearance depend on the state and situation of a differential model. Then the corresponding impulse conditions are called \emph{state-dependent} in contrast to \emph{fixed-time} impulse conditions where the moments of discontinuity are prescribed. First order differential systems with fixed-time impulses can be found for example in \cite{lsn05cma,an08na}. They mostly appear as models of neural networks and their anti-periodic solutions are investigated in many papers \cite{aj15n,cg14aaa,sd10amc,w13ade,wflsd14n,x16ejde, xw15ade,xz14npl}. For state-dependent impulses in such models see \cite{sy16aor}, where Lipschitz nonlinearities are assumed. Second order differential equations can serve as physical models, for example: Rayleigh equation (acoustics), Duffing, Li\'enard or van der Pol equations (oscillation theory). Anti-periodic solutions of these equations without impulses are discussed in \cite{clz08jamc,lh09narwa,lyl10cnsns,xl14amp} and of Rayleigh equation with fixed-time impulses in \cite{lz10cnsns}. The first result about the existence and uniqueness of anti-periodic solutions of the distributional Li\'enard equation with state-dependent impulses has been reached by Belley and Bondo \cite{bb16jde} under the assumption that functionals describing moments and values of impulses are globally Lipschitz continuous and bounded. Close results for periodic problems can be found in \cite{bv05jmaa,bv06natma}. Here, we focus our considerations on anti-periodic solutions of the van der Pol equation with state-dependent impulses both in ``classical'' and distributional formulations. Namely, we investigate the existence of solutions to the van der Pol differential equation \begin{equation} \label{1} x'(t) = y(t), \quad y'(t) = \mu \Big(x(t) - \frac{x^3(t)}{3}\Big)' - x(t) + f(t) \quad \text{for a. e. }t \in \mathbb{R}, \end{equation} with a parameter $\mu\in (0,\infty)$ and a function $f$ which is Lebesgue integrable on $[0,T]$ and satisfies \begin{equation}\label{f} f(t + T) = -f(t) \quad \text{for a. e. }t\in \mathbb{R}. \end{equation} We are interested in the existence of a solution fulfilling the antiperiodic conditions \begin{equation} \label{3} x(0) = -x(T), \quad y(0) = -y(T). \end{equation} It is natural to search for a solution $(x,y)$ such that \begin{equation}\label{x} x(t + T) = -x(t), \quad t\in \mathbb{R}. \end{equation} In addition, \eqref{1} is subject to the state-dependent impulse conditions \begin{equation} \label{2} \Delta y(\tau_i(x)) = \mathcal{J}_i(x), \quad i=1,\ldots,m, \end{equation} where $\tau_i, \mathcal{J}_i$ $i=1,\ldots,m$, are real-valued functionals, $\tau_i$ have values in $(0,T)$ and $\Delta y(\tau)=y(\tau+)-y(\tau-)$ for $\tau\in \mathbb{R}$. Then \eqref{1}, \eqref{x} and \eqref{2} lead to \begin{equation}\label{y} y(t+T)=-y(t), \quad t\in \mathbb{R},\; t\not=\tau_i(x),\; i=1,\dots, m. \end{equation} Since $x$ satisfying \eqref{x} is $2T$-periodic and has zero mean value, i.e., \begin{equation*} \bar x = \frac{1}{2T}\int_0^{2T} x(t)\,dt =0, \end{equation*} the functionals $\tau_i$ and $\mathcal{J}_i$ are defined on the set of $2T$-periodic functions of bounded variation with zero mean value. We will consider such solutions $(x,y)$ for which $y$ is piecewise absolutely continuous with the only instants of discontinuity at $t = \tau_i(x)$, $i=1,\ldots,m$. Then the assumption that $\tau_i$, $i=1,\dots,m$, have values in $(0,T)$ guarantees the continuity of $y$ at the points $nT$, $n\in \mathbb{Z}$, and consequently the second equality in \eqref{3}. Our main result is contained in the next theorem, which is a direct consequence of Theorem \ref{existence} from Section \ref{main}. \begin{theorem} \label{theo_main} Assume that $T \in (0,\sqrt{3})$, $\tau_1,\ldots,\tau_m$ are continuous with values in $(0,T)$ and if $i\ne j$, then $\tau_i(x) \ne \tau_j(x)$ for each $2T$-periodic absolutely continuous function $x$ with zero mean value. Further assume that $\mathcal{J}_1,\ldots,\mathcal{J}_m$ are continous and bounded. Then there exists $\mu_0 > 0$ such that for each $\mu \in (0,\mu_0]$ the problem \eqref{1}, \eqref{3}, \eqref{2} has a solution. \end{theorem} The novelty of this paper is the following: \begin{itemize} \item Our existence result for problem \eqref{1}, \eqref{3}, \eqref{2} is the first in the literature. \item We need not the Lipschitz continuity of functionals $\tau_i$ and $\mathcal{J}_i$ in problem \eqref{1}, \eqref{3}, \eqref{2} as well as in \eqref{vanderpol} in contrast to \cite{bb16jde}. \item We also get the solvability provided these functionals are unbounded. \item Our solvability conditions can be very easily checked, which we illustrate on two nontrivial examples. \end{itemize} \section{Preliminaries} Motivated by the paper \cite{bb16jde} we construct a distributional differential equation equivalent to the problem \eqref{1}, \eqref{3}, \eqref{2}. This enables to work in more advantageous space $\widetilde{{\rm NBV}}$ and to use properties of Fourier series of distributions. To this aim, by ${\mathcal{P}_{2T}}$ we denote the complex vector space of all complex-valued $2T$-periodic functions of one real variable having continuous derivatives of all orders on $\mathbb{R}$. The elements of ${\mathcal{P}_{2T}}$ are called \emph{test functions} and ${\mathcal{P}_{2T}}$ is equipped with a locally convex topological space structure (see \cite{e82sv}). Its topological dual is denoted by $({\mathcal{P}_{2T}})'$. The elements of $({\mathcal{P}_{2T}})'$ are called \emph{$2T$-periodic distributions} or only \emph{distributions}, i.e., these elements are complex-valued continuous linear functionals on ${\mathcal{P}_{2T}}$. For a distribution $u \in ({\mathcal{P}_{2T}})'$ and a test function $\varphi \in {\mathcal{P}_{2T}}$, the symbol $\dist{u}{\varphi}$ stands for a value of the distribution $u$ at $\varphi$. The distributional derivative $Du$ of a distribution $u$ is a distribution which is defined by \begin{equation*} \dist{Du}{\varphi}=-\dist{u}{\varphi'} \quad\text{for each } \varphi \in {\mathcal{P}_{2T}}. \end{equation*} Let us take $n\in \mathbb{Z}$ and introduce a complex-valued function $e_n \in {\mathcal{P}_{2T}}$ by \begin{equation*}\label{en} e_n(t) := \mathrm{e}^{\mathrm{i} n\omega t}, \quad t\in \mathbb{R}, \end{equation*} where $\omega = \pi /T$. Then every distribution $u$ can be expressed uniquely by the \emph{Fourier series} \begin{equation}\label{fouru} u=\sum_{n\in \mathbb{Z}}\hat u(n) e_n, \end{equation} where $\hat u(n)\in \mathbb{C}$ are \emph{Fourier coefficients} of $u$, \begin{equation*} \hat u(n) = \dist{u}{e_{-n}}, \quad n\in \mathbb{Z}. \end{equation*} For a distribution $u$ we define the \emph{mean value} $\bar u$ as \begin{equation}\label{mean} \bar u := \hat u(0)=\dist{u}{e_0}=\dist{u}{1}, \end{equation} and, for simplicity of notation, we write $\widetilde u := u - \bar u$. In general, the Fourier series in \eqref{fouru} need not be pointwise convergent and the equality in \eqref{fouru} is understood in the sense of distributions written as \begin{equation*}\label{fouru1} \lim_{N\to \infty}\dist{s_N}{\varphi}= \dist{u}{\varphi}\in \mathbb{C} \quad \text{for each }\varphi\in {\mathcal{P}_{2T}}, \quad \text{where } s_N=\sum_{|n|\le N}\hat u(n)e_n. \end{equation*} In particular, the Dirac $2T$-periodic distribution $\delta$ is defined by \begin{equation*} \dist{\delta}{\varphi}=\varphi(0) \quad\text{for each } \varphi\in {\mathcal{P}_{2T}}, \end{equation*} and it has the Fourier series \begin{equation} \label{Dirac} \delta = \sum_{n\in\mathbb{Z}} e_n. \end{equation} The convolution $u*v$ of two distributions has the Fourier series \begin{equation}\label{fouruv} u*v=\sum_{n\in \mathbb{Z}}\hat u(n)\hat v(n) e_n, \end{equation} and the Fourier series for \emph{distributional derivatives} $Du$ and $D^2 u$ reads \begin{equation}\label{fourDu} Du = \sum_{n\in \mathbb{Z}, n\not=0}\mathrm{i} n\omega\, \hat u(n)e_n \quad \text{and} \quad D^2 u= \sum_{n\in \mathbb{Z}, n\not=0} (\mathrm{i} n\omega)^2 \hat u(n)e_n. \end{equation} This immediately implies that \begin{equation} \label{Dumean} u*\delta=u,\quad \overline{Du} = \overline{D^2 u} = 0, \quad D \widetilde u = D u, \quad D^2 \widetilde u = D^2 u. \end{equation} Let us introduce distributions $E_1$ and $E_2$ by \begin{equation}\label{fourE1} E_{1}: = \sum_{n\in \mathbb{Z}, n\not=0}\frac{1}{\mathrm{i} n\omega}e_n, \quad E_{2}: = E_1* E_1 = \sum_{n\in \mathbb{Z}, n\not=0}\frac{1}{(\mathrm{i} n\omega)^2}e_n, \end{equation} and define linear operators $I$ and $I^2$ by \begin{equation} \label{I} \begin{gathered} Iu := E_1 * u = \sum_{n\in\mathbb{Z}, n \ne 0} \frac 1 {\mathrm{i} n\omega}\hat u(n)e_n, \\ I^2u := I(Iu) = E_1*(E_1*u) = \sum_{n\in\mathbb{Z}, n \ne 0} \frac 1 {(\mathrm{i} n\omega)^2}\hat u(n)e_n = E_2 * u. \end{gathered} \end{equation} Using \eqref{fouruv} and \eqref{fourDu}, for every distribution $u$, we obtain \begin{equation} \label{ID} \begin{gathered} D(Iu)=I(Du)=\widetilde u,\quad D^2(I^2u) = I^2(D^2 u) = \widetilde u,\\ I^2(Du) = Iu = I\widetilde u, \quad D^2(Iu) = Du = D\widetilde u. \end{gathered} \end{equation} From these identities we see that $I$ is an inverse to $D$ on the set of all distributions with zero mean value and therefore we call $I$ an \emph{antiderivative operator}. Consider $\tau\in \mathbb{R}$. Let us remind the translation operator $\mathbb{T}_\tau$ on test functions and distributions. For a function $\varphi \in {\mathcal{P}_{2T}}$ we define $\mathbb{T}_\tau \varphi \in {\mathcal{P}_{2T}}$ by \begin{equation*} (\mathbb{T}_\tau \varphi)(t) := \varphi(t - \tau), \quad t \in \mathbb{R}, \end{equation*} and for a distribution $u\in ({\mathcal{P}_{2T}})' $ we define $\mathbb{T}_\tau u\in ({\mathcal{P}_{2T}})' $ by \begin{equation*} \dist{\mathbb{T}_\tau u}{\varphi} := \dist{u}{\mathbb{T}_{-\tau}\varphi}, \quad \varphi \in {\mathcal{P}_{2T}}. \end{equation*} Since \begin{equation}\label{fourte} \widehat{(\mathbb{T}_\tau u)}(n) = \dist{\mathbb{T}_\tau u}{e_{-n}} = \dist{u}{\mathbb{T}_{-\tau} e_{-n}} = \mathrm{e}^{-\mathrm{i} n\omega\tau} \dist{u}{e_{-n}} = \mathrm{e}^{-\mathrm{i} n\omega\tau} \hat u(n), \end{equation} for $n \in \mathbb{Z}$, the Fourier series of $\mathbb{T}_\tau u$ reads \begin{equation}\label{transtau} \mathbb{T}_{ \tau} u=\sum_{n\in\mathbb{Z}} \mathrm{e}^{-\mathrm{i} n\omega\tau}\hat u(n) e_n, \quad u\in ({\mathcal{P}_{2T}})', \end{equation} in particular, for $\tau=T$ \begin{equation}\label{transT} \mathbb{T}_{ T} u=\sum_{n\in\mathbb{Z}} (-1)^n \hat u(n) e_n, \quad u\in ({\mathcal{P}_{2T}})'. \end{equation} Further, by \eqref{Dirac} and \eqref{transtau}, the \emph{ Dirac $2T$-periodic distribution $\delta_\tau$ at the point } $\tau\in \mathbb{R}$ which is defined as $$ \delta_\tau := \mathbb{T}_\tau \delta, $$ and satisfies \begin{gather} \label{fourdirac} \delta_\tau = \sum_{n\in\mathbb{Z}} \mathrm{e}^{-\mathrm{i} n\omega\tau} e_n,\quad \overline{\delta_\tau} = 1, \\ u * \delta_\tau = \mathbb{T}_\tau u, \quad u\in ({\mathcal{P}_{2T}})'. \nonumber \end{gather} Hence \begin{equation} \label{E2ac} I\delta_\tau = E_1 * \delta_\tau = \mathbb{T}_\tau E_1, \quad I^2\delta_{\tau} = E_2 * \delta_\tau = \mathbb{T}_\tau E_2. \end{equation} We are interested in solutions of \eqref{1} satisfying the antiperiodic conditions \eqref{3} and so we work here with \emph{antiperiodic distributions}. Exactly, we say that a distribution $u\in ({\mathcal{P}_{2T}})'$ is called \emph{antiperiodic} provided $u$ satisfies \begin{equation} \label{antiperiodic} \mathbb{T}_T u = -u. \end{equation} By \eqref{transT} we see that $u\in ({\mathcal{P}_{2T}})' $ is antiperiodic if and only if $\hat u(n)=0$ for each even $n\in \mathbb{Z}$. Consequently, if $u\in ({\mathcal{P}_{2T}})' $ is antiperiodic, then $\hat u(0)=\overline{u}=0$ and $Du$, $Iu$ are antiperiodic, as well. On the other hand, \eqref{fourdirac} yields that the Dirac $2T$-periodic distribution $\delta_\tau$, which could characterize impulses from \eqref{2}, is not antiperiodic. Therefore, motivated by \cite{bb16jde}, we introduce the distribution \begin{equation} \label{velkedelta} \varepsilon_{\tau} := \delta_{\tau} - \mathbb{T}_T\delta_{\tau} \end{equation} which is antiperiodic for any $\tau\in \mathbb{R}$. Now, we turn our attention to real-valued functions and distributions which we use in next sections. To this aim the functional spaces defined below consist of \emph{real-valued $2T$-periodic functions}. Clearly it suffices to prescribe their values on a semiclosed interval with the length $2T$: \begin{itemize} \item ${{L^1}}$ is the Banach space of Lebesgue integrable functions equipped with the norm $\|x\|_{{L^1}} := \frac{1}{2T}\int_0^{2T} |x(t)|\,dt$, \item ${\rm BV}$ is the space of functions of bounded variation; the total variation of $x\in {\rm BV}$ is denoted by $\operatorname{var}(x)$; for $x\in {\rm BV}$ we also define $\|x\|_\infty := \sup\{|x(t)|: t\in [0,2T]\}$, \item ${\rm NBV}$ is the space of functions from ${\rm BV}$ normalized in the sense that $x(t)=\frac{1}{2}(x(t+)+x(t-))$, \item $\widetilde{{\rm NBV}}$ represents the Banach space of functions from ${\rm NBV}$ having zero mean value ($\bar x : = \frac 1 {2T} \int_0^{2T} x(t)\,dt =0$), which is equipped with the norm equal to the total variation $\operatorname{var}(x)$, \item for an interval $J\subset[0, 2T]$ we denote by ${\rm AC}(J)$ the set of absolutely continuous functions on $J$, and if $J=[0,2T]$ we simply write ${\rm AC}$, \item ${C^\infty}\subset {\mathcal{P}_{2T}}$ is the classical Fr\'echet space of functions having derivative of an arbitrary order, \item for finite $\Sigma \subset [0,2T)$ we denote by ${\rm PAC}_\Sigma$ the set of all functions $x \in {\rm NBV}$ such that $x \in {\rm AC}(J)$ for each interval $J \subset [0,2T]$ for which $\Sigma \cap J = \emptyset$. For $\tau \in [0,2T)$, we write ${\rm PAC}_\tau := {\rm PAC}_{\{\tau\}}$, \item $\widetilde{{\rm AC}} = {\rm AC} \cap \widetilde{{\rm NBV}}$; for finite $\Sigma \subset [0,2T)$ we denote $\widetilde{{\rm PAC}}_\Sigma = {\rm PAC}_\Sigma \cap \widetilde{{\rm NBV}}$, \item $\Delta y(\tau) : = y(\tau+)-y(\tau-)$ for $y\in \widetilde{{\rm NBV}}$, $\tau\in \mathbb{R}$. \end{itemize} Further, ${\rm Car}$ designates the set of real functions $f(t,x)$ such that $f(\cdot,x) \in {{L^1}}$ for each $x \in \mathbb{R}$ and satisfy the Carath\'eodory conditions on $[0,2T]\times \mathbb{R}$. We say that $u \in ({\mathcal{P}_{2T}})'$ is a \emph{real-valued distribution} if \begin{equation*} \dist{u}{\varphi} \in \mathbb{R}\quad\text{for each } \varphi \in {C^\infty}. \end{equation*} A real-valued distribution $u$ is characterized by the fact that its Fourier coefficients $\hat u(n)$ and $\hat u(-n)$ are complex conjugate for each $n \in \mathbb{Z}$. Obviously, if $\tau\in \mathbb{R}$ and $u$ and $v$ are real-valued distributions, then $u* v$, $\widetilde{u}$, $Du$, $D^2u$, $Iu$, $I^2u$, $\mathbb{T}_\tau u$, $\delta_\tau$ and $\varepsilon_\tau$ are real-valued distributions, as well. We say that $u\in ({\mathcal{P}_{2T}})'$ is a \emph{regular distribution} if $u$ is a real-valued distribution and there exists $y \in {{L^1}}$ such that \begin{equation}\label{regular} \dist{u}{\varphi} = \frac 1 {2T} \int_0^{2T} y(s)\varphi(s)\,ds \quad\text{for each } \varphi \in {C^\infty}. \end{equation} Then we say that $u=y$ in the sense of distributions and write $y$ in place of $u$ in \eqref{regular}. Hence all functions from ${{L^1}}$ can be understood as regular distributions. For $x \in {\rm BV}$, we write $x'$ as a classical derivative, which is defined a.e. on $\mathbb{R}$ and which is an element of ${{L^1}}$ and consequently a regular distribution. If $x \in {\rm AC}$, then $x' = Dx$ in the sense of distributions. Since the first series in \eqref{fourE1} converges pointwise to the $2T$-periodic function \begin{equation*} E_1(t)= \begin{cases} T-t & \text{for } t\in(0,2T), \\ 0 & \text{for } t=0, \end{cases} \end{equation*} we see that $E_1$ is a regular distribution and it can be considered as a function from $\widetilde{{\rm PAC}}_0$. The second series in \eqref{fourE1} uniformly converges to the $2T$-periodic function \begin{equation*} E_2(t)= \frac {t(2T - t)} 2 - \frac{T^2}{3}, \quad t\in [0, 2T], \end{equation*} and so $E_2$ is a regular distribution which can be considered as a function from $\widetilde{{\rm AC}}$. Similarly for $\tau \in \mathbb{R}$, \begin{equation}\label{TE2} \mathbb{T}_\tau E_1\in \widetilde{{\rm PAC}}_\tau, \quad \mathbb{T}_{\tau}E_2\in \widetilde{{\rm AC}}. \end{equation} Obviously, $E'_2 = E_1$, $E'_1 = -1$ a.e. on $[0,2T)$ and \begin{equation} \label{normy} \operatorname{var}(E_1) = 4T,\quad \|E_1\|_\infty = T,\quad \operatorname{var}(E_2) = T^2,\quad \|E_2\|_\infty = \frac{T^2}{3}. \end{equation} Since \begin{equation*} (x* y)(t) := \frac 1 {2T} \int_0^{2T} x(t-s)y(s)\,ds, \quad t \in[0,2T] \text{ for } x,y \in {{L^1}}, \end{equation*} for $h \in {{L^1}}$, we have \begin{equation*} (E_1 * h)(t) = \frac 1 {2T}\int_0^{2T} (s-t)h(s)\,ds + \frac 1 2 \Big(\int_0^th(s)\,ds - \int_t^{2T} h(s)\,ds\Big), \end{equation*} for $t \in [0,2T]$. Therefore $Ih$ is a regular distribution which is equal to the function $E_1 * h \in {\rm AC}$, and we conclude by \eqref{I}, \begin{equation} \label{AC} h \in {{L^1}} \; \Longrightarrow \; I h, \ I^2 h \in \widetilde{{\rm AC}}, \quad (I h)'(t) = h(t) - \bar h \text{ a.e. } t\in [0,2T]\,. \end{equation} Further, for $x \in {\rm BV}$ and $t\in \mathbb{R}$ we have $(\mathbb{T}_\tau x)(t) = x(t-\tau)$ which implies \begin{equation} \label{trans} \operatorname{var} \left(\mathbb{T}_\tau x\right) = \operatorname{var}{x} \quad \text{and}\quad \|\mathbb{T}_\tau x\|_\infty = \|x\|_\infty, \quad x \in {\rm BV}. \end{equation} Let us recall the following inequalities \begin{gather} \label{var-infty} \operatorname{var}(x * y) \leq \operatorname{var}(x) \|y\|_\infty, \quad x,y \in {\rm NBV}\,,\\ \label{var-elj} \operatorname{var}(x * f) \leq \operatorname{var}(x)\|f\|_{{L^1}}, \quad x \in {\rm NBV}, \; f \in {{L^1}}\,,\\ \label{inf-var} \|x\|_{{L^1}}\leq\|x\|_\infty \leq \operatorname{var}(x), \quad x \in \widetilde{{\rm NBV}}\,. \end{gather} \begin{remark} \label{rema0} \rm Let $x$ be antiperiodic. If $x \in {\rm NBV}$, then $x(t+T) = -x(t)$ for $t \in \mathbb{R}$, \begin{equation*} \|x\|_\infty = \sup_{t\in[0,T]} |x(t)|, \end{equation*} and $\operatorname{var}(x)$ is double the total variation of $x$ over the interval $[0,T]$ (or any semiclosed interval of the length $T$). If $x \in {{L^1}}$, then $x(t+T) = -x(t)$ for a.e. $t \in \mathbb{R}$ and \begin{equation*} \|x\|_{{L^1}} = \frac 1 T \int_0^T |x(t)|\,dt. \end{equation*} Therefore it is sufficient to define an antiperiodic function on any interval of the length $T$. \end{remark} By \eqref{E2ac} and \eqref{velkedelta} and \eqref{TE2}, it holds for $\tau\in \mathbb{R}$, \begin{gather}\label{IDeltatau} I\varepsilon_{\tau} = I\delta_\tau-I\mathbb{T}_T\delta_\tau =T_\tau E_1-T_{\tau+T} E_1\in \widetilde{{\rm PAC}}_{\{\tau, \tau+T\}}, \\ \label{I2Deltatau} I^2\varepsilon_{\tau} = I^2\delta_\tau-I^2\mathbb{T}_T\delta_\tau =T_\tau E_2-T_{\tau+T} E_2\in \widetilde{{\rm AC}}. \end{gather} So, for $\tau=0$, we have \begin{equation*} I\varepsilon_0=E_1-\mathbb{T}_T E_1, \quad I^2 \varepsilon_0 =E_2- \mathbb{T}_T E_2, \end{equation*} and in detail \begin{equation} \label{Idelta} I\varepsilon_{0}(t) = \begin{cases} 0 & t = 0,\\ T & t \in (0,T), \end{cases} \quad I^2\varepsilon_0(t) = \frac{T(2t - T)}{2}, \quad t \in [0,T], \end{equation} Since $I\varepsilon_{\tau} = \mathbb{T}_{\tau} I\varepsilon_0$, by \eqref{trans} and \eqref{Idelta} and according to Remark \ref{rema0}, we obtain \begin{equation} \label{normy-delta} \operatorname{var}(I\varepsilon_\tau) = 4T, \quad \|I\varepsilon_\tau\|_\infty = T, \quad \operatorname{var}(I^2\varepsilon_\tau) = 2T^2,\quad \|I^2\varepsilon_\tau\|_\infty = \frac {T^2} 2, \end{equation} for $\tau\in \mathbb{R}$. Choosing $\tau_1, \tau_2\in \mathbb{R}$, where $|\tau_1-\tau_2|0$ such that \begin{equation}\label{kapa} \operatorname{var}(z_n)\le \kappa,\ n\in \mathbb{N}. \end{equation} By \eqref{tau-spoj} and \eqref{je-spoj} we have \begin{equation*} \tau_i(z_n) \in [a,b], \quad |\mathcal{J}_i(z_n)|\le a_i,\quad i = 1,\ldots,m, \; n\in \mathbb{N}, \end{equation*} and we can choose a subsequence $\{z_{n_k}\}_{k=1}^\infty$ such that \begin{equation}\label{limrJ} \lim_{k\to \infty} \tau_i(z_{n_k})=\tau_{0,i}, \quad \lim_{k\to \infty} \mathcal{J}_i(z_{n_k})=J_{0,i}, \end{equation} where $\tau_{0,i}\in (0, T)$, $J_{0,i} \in [-a_i,a_i]$ for $i=1,\ldots,m$. By \eqref{kapa} and the Helly's selection theorem (see e.g. \cite[p. 222]{n64fup}) there exists a subsequence $\{z_{n_\ell}\}_{\ell=1}^\infty \subset \{z_{n_k}\}_{k=1}^\infty$ which is pointwise converging to a function $z^*\in {\rm BV}$ and moreover $\overline{z^*} = 0$. Normalizing $z^*$ in the sense of $z(t) = (z^*(t-) + z^*(t+))/2$ we obtain $z \in \widetilde{{\rm NBV}}$ and a subsequence $\{z_{n_\ell}\}_{\ell=1}^\infty$ converging to $z$ a.e. on $[0,2T]$. Using \eqref{inf-var}, \eqref{kapa} and the Lebesgue convergence theorem, we see that $\|z_{n_\ell} - z\|_{{L^1}} \to 0$ as $n_\ell\to\infty$. Denote \begin{equation*} v:=\mu I \big(z - \frac 1 3 z^3\big) + I^2 \Big(-z + f + \frac{1}{2T}\sum_{i=1}^mJ_{0,i}\varepsilon_{\tau_{0,i}}\Big)\,. \end{equation*} In the same way as in step 1 we get \begin{align*} &\operatorname{var}(v_{n_\ell}-v) \\ &\leq \mu \operatorname{var}(E_1) \|z_{n_\ell} - z\|_{{L^1}} + \frac \mu 3 \operatorname{var}(E_1)\|z_{n_\ell}^3 - z^3\|_{{L^1}} + \operatorname{var}(E_2)\|z_{n_\ell} - z\|_{{L^1}} \\ &\quad + \frac 1 {2T}\sum_{i=1}^m \left(|\mathcal{J}_i(z_{n_\ell}) - J_{0,i}|\operatorname{var}(I^2\varepsilon_{\tau_i(z_{n_\ell})}) + |J_{0,i}|\operatorname{var}(I^2\varepsilon_{\tau_i(z_{n_\ell})} - I^2\varepsilon_{\tau_{0,i}})\right), \end{align*} and derive that the sequence $\{v_{n_\ell}\}_{\ell=1}^\infty$ is convergent to $v$ in $\widetilde{{\rm NBV}}$. This yields that $\mathcal{F}(B)$ is relatively compact in $\widetilde{{\rm NBV}}$. \end{proof} We are ready to prove the existence of a fixed point of the operator $\mathcal{F}$ in $\widetilde{{\rm NBV}}$. To do it we denote \begin{equation} \label{4} c_1 := T \|f\|_{{{L^1}}} + \sum_{i=1}^m a_i, \quad T_0 := 1 - \mu T - \frac {T^2} {3}\,, \quad c_2 := \frac 1 2 \sqrt{\frac {T_0} {\mu T}}\,, \end{equation} assume that $\mu$ and $T$ satisfy \begin{equation} \label{7} T c_1 \leq \frac {T_0} 3 \sqrt{\frac{T_0}{\mu T}}, \end{equation} and define the set \begin{equation} \label{8} \Omega := \big\{ z \in \widetilde{{\rm NBV}} \text{such that }\operatorname{var}(z) \leq c_2, \quad z \text{ is antiperiodic}\big\}. \end{equation} \begin{remark} \label{rema1} \rm The construction of the set $\Omega$ is based on these observations: \begin{itemize} \item The parameter $c_2$ is well defined for $T_0\ge 0$ which requires the inequality $1 - T^2 /{3} > 0$. Therefore we have to assume $T \in (0,\sqrt{3})$. Further, $T_0\ge 0$ implies $\mu\le \frac{1}{T}-\frac{T}{3}$. \item If $c_1>0$, then $c_2 > 0$ and $\Omega$ is nonempty, convex, bounded and closed set in $\widetilde{{\rm NBV}}$. \item If $T \in (0,\sqrt{3})$, then \begin{equation*} \sqrt{ \frac {1 - \mu T - \frac {T^2} {3}} {\mu T} } \to \infty \quad \text{as } \mu \to 0+\,, \end{equation*} and therefore \eqref{7} is always valid for each sufficiently small $\mu$. If $c_1>0$ , then the optimal (maximal) value of the parameter $\mu$ is determined by \begin{equation} \label{mi-nula} T c_1 = \frac {T_0} 3 \sqrt{\frac{T_0}{\mu T}}. \end{equation} \item If $c_1=0$, then $f=0$ a. e. on $\mathbb{R}$ and the impulses \eqref{2} disappear. \end{itemize} \end{remark} \begin{theorem} \label{theo1} Consider \eqref{4} and \eqref{8} and assume that $T \in (0,\sqrt{3})$ and $c_1>0$. Let \eqref{tau-spoj} and \eqref{je-spoj} hold. Then there exists a solution $\mu_0 > 0$ of \eqref{mi-nula} such that for each $\mu \in (0,\mu_0]$ the operator $\mathcal{F}$ maps $\Omega$ into $\Omega$. \end{theorem} \begin{proof} By Remark \ref{rema1}, there exists $\mu_0>0$ satisfying \eqref{mi-nula}. Consider $\mu\in(0,\mu_0]$. Clearly $\mu$ fulfils \eqref{7}. As we mentioned in Section 2, if $z\in \widetilde{{\rm NBV}}$ is antiperiodic, that is $z$ fulfils \eqref{x}, then $Iz$ is antiperiodic as well. Since $f$ is supposed to satisfy \eqref{f} and the distribution $\varepsilon_\tau$ is antiperiodic for any $\tau\in \mathbb{R}$, we can conclude that if $z\in \widetilde{{\rm NBV}}$ is antiperiodic, then $\mathcal{F} z \in \widetilde{{\rm NBV}}$ is antiperiodic, as well. Therefore, if we have the set $\Omega$ from \eqref{8}, we only need to prove \begin{equation}\label{estimate} \operatorname{var}(\mathcal{F} z) \leq c_2 \quad \text{for each } z \in \Omega\,. \end{equation} So, let us choose $z \in \Omega$. By \eqref{ef} and \eqref{I}, \begin{align*} \operatorname{var}(\mathcal{F} z) &\leq \mu\operatorname{var}(I z) + \frac \mu 3 \operatorname{var}\left( I( z^3)\right) + \operatorname{var} (I^2 z) + \operatorname{var} ( I^2 f) \\ &\quad +\frac{1}{2T} \sum_{i=1}^m a_i \operatorname{var}(I^2\varepsilon_{\tau_i(z)})\\ & = \mu\operatorname{var}(E_1* z) + \frac \mu 3 \operatorname{var}\left( E_1 * z^3\right) + \operatorname{var} (E_2* z) + \operatorname{var} ( E_2 * f) \\ &\quad +\frac{1}{2T} \sum_{i=1}^m a_i \operatorname{var}(I^2\varepsilon_{\tau_i(z)}). \end{align*} Consequently, using \eqref{var-infty}, \eqref{var-elj} and \eqref{normy-delta}, we derive \begin{align*} \operatorname{var}(\mathcal{F} z) &\leq \mu\|E_1\|_\infty \operatorname{var}(z) + \frac \mu 3 \operatorname{var}(E_1)\|z^3\|_\infty + \|E_2\|_\infty \operatorname{var}(z) \\ &\quad + \operatorname{var}(E_2)\| f\|_{{{L^1}}} + T \sum_{i=1}^m a_i. \end{align*} Therefore, by \eqref{normy}, \eqref{inf-var}, \eqref{4}, we get \begin{align*} \operatorname{var}(\mathcal{F} z) &\leq \mu T\operatorname{var}(z) + \frac{4\mu T} 3\left( \|z\|_\infty\right)^3 + \frac{T^2} {3} \operatorname{var}(z) + T^2\|f\|_{{L^1}} + T \sum_{i=1}^m a_i \\ &\leq \big(\mu T + \frac{T^2} {3} \big) \operatorname{var}(z) + \frac{4\mu T} 3 \left(\operatorname{var}(z)\right)^3 + Tc_1\,. \end{align*} Hence, to derive \eqref{estimate}, it suffices to prove the inequality \begin{equation} \label{9} \big(\mu T + \frac{T^2} {3} \big) c_2 + \frac{4\mu T} 3 c_2^3 + Tc_1 \leq c_2\,. \end{equation} Subtracting the first term on the left-hand side we get \begin{equation*} \frac{4\mu T} 3 c_2^3 + Tc_1 \leq \big(1 - \mu T - \frac{T^2} {3} \big) c_2\,, \end{equation*} and using \eqref{4} we obtain \begin{equation*} \frac{4\mu T} 3\Big(\frac 1 2\sqrt{\frac{T_0}{\mu T}}\Big)^3 + T c_1 \leq \frac {T_0} 2\sqrt{\frac{T_0}{\mu T}}\,, \end{equation*} which is equivalent to \eqref{7}. Therefore \eqref{9} is proved. \end{proof} \section{Main results}\label{main} \begin{theorem} \label{existence} Consider \eqref{4} and assume that $T \in(0,\sqrt{3})$ and $c_1>0$. Let \eqref{tau-spoj} and \eqref{je-spoj} hold. Then there exists a solution $\mu_0>0$ of \eqref{mi-nula} such that for each $\mu \in (0,\mu_0]$ the distributional equation \eqref{vanderpol} has at least one antiperiodic solution $z$ such that $\operatorname{var}(z) \leq c_2$. If in addition \eqref{taui} holds, then problem \eqref{1},\eqref{2} has an antiperiodic solution $(x,y)=(z,Dz)$. \end{theorem} \begin{proof} By Remark \ref{rema1}, there exists $\mu_0>0$ satisfying \eqref{mi-nula}. Let us consider the operator $\mathcal{F} : \widetilde{{\rm NBV}} \to \widetilde{{\rm NBV}}$ defined in \eqref{ef}, and the set $\Omega$ defined in \eqref{8}, where $\mu\in (0,\mu_0]$. According to Theorem \ref{theo1} the operator $\mathcal{F}$ maps $\Omega$ to $\Omega$. Due to Lemma \ref{lema4} the operator $\mathcal{F}$ is completely continuous. Therefore, by the Schauder fixed point theorem $\mathcal{F}$ has a fixed point $z\in \Omega$. Finally, by Lemma \ref{lema3} we see that $z$ is an antiperiodic solution of the distributional equation \eqref{vanderpol} and that under the assumption \eqref{taui} the couple $(z,Dz)$ is an antiperiodic solution of problem \eqref{1},\eqref{2}. \end{proof} \begin{theorem} \label{thm15} %\label{coro1} Let $T \in(0,\sqrt{3})$. Let \eqref{taui}, \eqref{tau-spoj} and \eqref{je-spoj} hold. Then the equation \begin{equation*} \label{1*} x'(t) = y(t), \quad y'(t) = - x(t) + f(t), \quad \text{for a. e. }t \in \mathbb{R}, \end{equation*} subject to the state-dependent impulse conditions \begin{equation*} \label{2*} \Delta y(\tau_i(x)) = \mathcal{J}_i(x), \quad i=1,\ldots,m, \end{equation*} has at least one antiperiodic solution $(x,y)$ such that \begin{equation*} \operatorname{var}(x) \leq \frac {T^2\|f\|_{{L^1}} + T\sum_{i=1}^m a_i} {1 - \frac{T^2} 3}. \end{equation*} \end{theorem} \begin{proof} Let us put \begin{equation} \label{c2} c_1 = T\|f\|_{{L^1}} + \sum_{i=1}^m a_i, \quad c_2 = \frac {Tc_1}{1 - \frac{T^2} 3}. \end{equation} Consider the operator $\mathcal{F}$ from \eqref{ef}, where $\mu = 0$ and the set $\Omega$ from \eqref{8} with $c_2$ defined by \eqref{c2}. Similarly as in the proof of Theorem \ref{theo1} we prove \eqref{estimate}. Since now $\mu = 0$, we derive \begin{equation*} \frac {T^2} 3 c_2 + Tc_1 \leq c_2 \end{equation*} (compare with \eqref{9}). Using \eqref{c2}, we get \begin{equation*} Tc_1 \leq c_2\big( 1 - \frac{T^2} 3\big) = Tc_1. \end{equation*} Hence $\mathcal{F}$ maps $\Omega$ to $\Omega$, and arguing as in the proof of Theorem \ref{existence}, we finish the proof. \end{proof} If $\tau_i$, $i=1,\dots,m$, do not depend on $x\in \widetilde{{\rm NBV}}$, then the state-dependent impulse conditions \eqref{2} have the form of the fixed-time impulse conditions \begin{equation} \label{fixed} \Delta y(\tau_i) = \mathcal{J}_i(x), \quad i=1,\ldots,m, \end{equation} where the points $\tau_i\in (0,T)$, $i=1,\dots,m$, are known and fixed. It is clear that \eqref{tau-spoj} holds and Theorem \ref{existence} yields the following corollary. \begin{corollary} \label{existence-fixed} Consider \eqref{4} and assume that $T \in(0,\sqrt{3})$ and $c_1>0$. Let \eqref{je-spoj} hold. Then there exists a solution $\mu_0>0$ of \eqref{mi-nula} such that for each $\mu \in (0,\mu_0]$ the distributional equation \begin{equation} \label{vanderpol-fixed} D^2 z = \mu D\big(z - \frac{z^3} 3 \big) - z + f + \frac 1 {2T} \sum_{i=1}^m \mathcal{J}_i(z) \varepsilon_{\tau_i} \end{equation} has an antiperiodic solution $z$. If in addition \begin{equation*} \tau_i \ne \tau_j \quad \text{for all } i,j=1,\ldots,m,\ i\ne j, \end{equation*} then problem \eqref{1},\eqref{fixed} has an antiperiodic solution $(x,y)=(z,Dz)$. \end{corollary} \begin{example}\label{ex1}\rm Put $m=1$, $T=1$, choose $00$ and $T \in(0,\sqrt{3})$. Further assume that there exist $00$, $i=1,\dots, m$, such that \begin{gather} \label{tau-omega} \tau_i(\Omega)\subset[a,b] , \quad \tau_i : \widetilde{{\rm NBV}} \to \mathbb{R}, \quad i=1,\dots, m, \quad \text{are continuous}, \\ \label{je-omega} \mathcal{J}_i(\Omega)\subset [-a_i,a_i], \quad \mathcal{J}_i : \widetilde{{\rm NBV}} \to \mathbb{R}, \quad i=1,\dots, m, \quad \text{are continuous}. \end{gather} Then there exists a solution $\mu_0>0$ of \eqref{mi-nula} such that for each $\mu \in (0,\mu_0]$ the distributional equation \eqref{vanderpol} has at least one antiperiodic solution $z\in \Omega$. If in addition \eqref{taui} holds, then problem \eqref{1},\eqref{2} has an antiperiodic solution $(x,y)=(z,Dz)$. \end{theorem} \begin{proof} By Remark \ref{rema1}, there exists $\mu_0 > 0$ satisfying \eqref{mi-nula}. Let $\mu \in (0,\mu_0]$. Put \begin{equation*} \chi(s)= \begin{cases} 1, & s\in [0,c_2],\\ 2-\frac{s}{c_2}, \quad &s\in (c_2,2c_2),\\ 0, & s\ge 2c_2, \end{cases} \end{equation*} and for $z\in \widetilde{{\rm NBV}}$ define \begin{gather*} \tau_i^*(z):= \chi(\operatorname{var}(z)) \tau_i(z),\quad i=1,\dots,m, \\ \mathcal{J}_i^*(z):= \chi(\operatorname{var}(z)) \mathcal{J}_i(z),\quad i=1,\dots,m. \end{gather*} According to Lemma \ref{lema3}, each fixed point $z$ of the operator $\mathcal{F}^*: \widetilde{{\rm NBV}} \to \widetilde{{\rm NBV}}$, \begin{equation*} \mathcal{F}^* z = \mu I\big(z - \frac{z^3}{3}\big) + I^2\Big(-z + f + \frac 1 {2T} \sum_{i=1}^m \mathcal{J}^*_i(z) \varepsilon_{\tau^*_i(z)}\Big) \end{equation*} is a solution of the distributional equation \begin{equation*} D^2 z = \mu D\big(z - \frac{z^3} 3 \big) - z + f + \frac 1 {2T} \sum_{i=1}^m \mathcal{J}^*_i(z) \varepsilon_{\tau^*_i(z)}. \end{equation*} By \eqref{tau-omega} and \eqref{je-omega}, the functionals $\tau^*_i$ and $\mathcal{J}^*_i$ fulfil \eqref{tau-spoj} and \eqref{je-spoj}. Consequently, due to Lemma \ref{lema4}, the operator $\mathcal{F}^*$ is completely continuous. In addition, if $z\in \Omega$, then $\tau^*_i(z)= \tau_i(z)$, $\mathcal{J}^*_i(z)=\mathcal{J}_i(z)$ and hence $\mathcal{F}^*z = \mathcal{F} z$. Therefore, by Theorem \ref{theo1}, the operator $\mathcal{F}^*$ maps $\Omega$ to $\Omega$. So, by the Schauder fixed point theorem, $\mathcal{F}^*$ has a fixed point $z\in \Omega$. Consequently $z$ is a fixed point of $\mathcal{F}$. Now, as in the proof of Theorem \ref{existence}, we use Lemma \ref{lema3} to get that $z$ is an antiperiodic solution of the distributional equation \eqref{vanderpol}. Moreover, under the assumption \eqref{taui} the couple $(z,Dz)$ is an antiperiodic solution of problem \eqref{1},\eqref{2}. \end{proof} \begin{example}\label{ex2}\rm Put $m=1$, $T=1$, choose $0