Content-Type: multipart/mixed; boundary="-------------0003311244569" This is a multi-part message in MIME format. ---------------0003311244569 Content-Type: text/plain; name="00-139.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-139.keywords" periodic solutions,pendulum equation,variational methods ---------------0003311244569 Content-Type: application/x-tex; name="pendu.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="pendu.tex" \input vanilla.sty \pagewidth{13cm} \pageheight{19cm} \normalbaselineskip=15pt \normalbaselines \parindent= 15pt \parskip=3pt \mathsurround=1.7pt \overfullrule=0pt \scaledocument{\magstep1} \def\tit#1{\bigskip \bf\noindent #1 \medskip\rm} \def\lema#1{\medskip\smc\noindent #1\quad\sl} \def\demost#1{\smallskip\noindent\underbar{\it #1}\quad\rm} \def\note#1{\medskip \smc\noindent #1\quad\rm} \def\lra{\longrightarrow} \def\noi{\noindent} \def\li#1#2{\smash{\mathop{#1}\limits\sb{#2}}} \def\lie#1#2#3{\smash{\mathop{#1}\limits\sb{{\scriptstyle #2}\atop{\scriptstyle#3}}}} \def\cor{\allowmathbreak} \def\inb{\int\sb B } \def\dde{\sober{$d$}/{$d\va$}} \def\R{\text{{\rm I}\!{\rm R}}} \def\N{\text{{\rm I}\!{\rm N}}} \def \ee{\varepsilon} \def \a{\alpha} \def \sin {\text {\rm sin }} \def \cos {\text {\rm cos }} \title{PERIODIC SOLUTIONS OF THE FORCED} \endtitle \bigskip \centerline{\bf PENDULUM EQUATION WITH FRICTION} \bigskip \centerline{\bf Pablo Amster and Mar\'\i a Cristina Mariani} \medskip \centerline{Universidad de Buenos Aires} \medskip \medskip \lema {Abstract: } This paper is devoted to the study of the general forced pendulum equation in the presence of friction, $$u'' + a(t)u' + b(t) \sin u = f(t)$$ with $a,b\in C([0,T])$ and $f\in L^2(0,T)$. We'll show that $T$-periodic solutions may be obtained as zeroes of a $2\pi$-periodic continuous real function. Furthermore, the existence of infinitely many solutions is proved under appropiate conditions on $a,b$ and $f$. \rm \tit {Introduction} The periodic problem for the forced pendulum equation has been studied by different authors. In 1922 Hamel [H] found that the equation $$u'' + b \sin u = \beta \sin (t)$$ with $b,\beta$ constants admits a $2\pi$-periodic solution that can be obtained as a minimum in $C_{per}[0,2\pi]$ of the functional $$J(u)=\int_0^{2\pi} {u'^2\over 2} + b \cos u + u\beta \sin t$$ and the same argument can be extended for $T$-periodic solutions of the equation $u'' + b \sin u = f (t)$ where $f$ is $T$-periodic and orthogonal to constants (see [M1-2]). If we allow the presence of friction, namely the equation $$u'' + au' + b \sin u = f (t)\tag1$$ (where $a$ is a positive constant) then variational methods are not appliable to the periodic problem. In fact, examples of $T$-periodic functions $f$ such that $f\perp 1$ and (1) is not solvable are given in [A] and [O], under some conditions on the parameters $a,b$ and $T$. In a recent work a rather general class of counterexamples is constructed for arbitrary $a,b,T$ [OST]. Periodic type conditions for a general system are studied in [AM] by fixed point methods. The purpose of this work is to give conditions for the existence of $T$-periodic solutions of the general equation $$u'' + a(t)u' + b(t) \sin u = f(t)\tag2$$ In the first section we'll show that the Dirichlet problem for (2) is solvable for any boundary data, and uniquely solvable under an appropiate condition on $b$. In the second section we prove that the $T$-periodic solutions of (2) are the zeroes of a $2\pi$-periodic continuous real function. Finally, in the third section we give explicit conditions on $a,b$ and $f$ in order to obtain $T$-periodic solutions of (2). \tit{1. Unique solvability of the Dirichlet problem} The existence of solutions of (2) for arbitrary Dirichlet boundary data $\varphi$ is easily obtained from classical results available in the literature (see e.g. [M3]). We sketch a simple proof here which makes use of Schauder fixed point theorem. For any $\overline u\in L^2(0,T)$ let $K \overline u=u $ be the unique solution of the linear problem $$\cases u''+a(t)u' = f - b(t)\sin(\overline u) &\\ u|_{\partial I} = \varphi & \endcases$$ As $\sin \overline u \in L^2(0,T)$ then $K: L^2\to L^2$ is well defined and compact. Moreover, using a standard apriori bound for the operator $Lu=u''+a(t)u'$, we obtain: $$\| K\overline u - K0\|_2 \le c \| L(K \overline u - K0)\|_2 = c\| b\sin \overline u\|_2\le c\| b\|_2$$ Hence, taking $M = c\| b\|_2$ we obtain that $R(K) \subset B_M(K0)$ and the result follows. A sufficient condition for the uniqueness is: $\| b\|_\infty < \frac 1c$. However, for further applications we'll state a slightly more precise uniqueness result providing also an apriori bound for the nonlinear operator $Qu:= u'' + au' + b\sin u$. \lema{Lemma 1} Let $p$ be a positive solution of the equation $p' = a(t) p - k(t)$ for $k\in H^1(0,T)$ with $k'\ge 0$ a.e. and $\lambda_p$ the first eigenvalue of the problem $-(pu')' = \lambda u$. Then, if $\| bp\|_\infty <\lambda_p$, $$\| u- v\|_2 \le \frac 1{\lambda_p -\| pb\|_\infty}\| p(Qu-Qv)\|_2$$ and $$\left(\int_0^T p[(u- v)']^2\right)^{1/2} \le \frac {\sqrt{\lambda_p}}{\lambda_p -\| pb\|_\infty} \| p(Qu-Qv)\|_2$$ for any $u,v\in H^2(0,T)$ such that $u=v$ on $\partial I$. In particular, Dirichlet problem is uniquely solvable for any boundary data $\varphi$. \demost{Proof} Let $w=u-v$, then $$\|p(Qu-Qv)\|_2\| w\|_2\ge -\int_0^T p(Qu-Qv).w = \int p(w')^2 - kw'w - pb(\sin u - \sin v)w$$ As $-\int kw'w = \int k'\frac {w^2}2 \ge 0$, and $|\int pb(\sin u - \sin v)w| \le \|pb\|_\infty \|w\|_2^2$, it follows that $$\|p(Qu-Qv)\|_2\| w\|_2\ge \int p(w')^2 -\|pb\|_\infty \|w\|_2^2$$ and the result holds since $\|w\|_2^2 \le \frac 1{\lambda_p} \int p(w')^2$. \lema{Remark:} If $a\in H^1(0,T)$ with $a'\ge 0$ a.e., we may take $p \equiv 1$, and the result holds for $\| b\|_\infty < (\frac \pi T)^2$. \rm \tit{2. An embedded curve in $H^1(0,T)$} In this section we'll assume that $b$ satisfies the assumptions of lemma 1 for some $\widetilde p>0$, and prove that all the possible $T$-periodic solutions of (2) belong to a continuous curve $\Cal C\subset H^1(0,T)$. More precisely, if the functional $I: H^1(0,T)\to \R$ is given by $I(u)= \int_0^T a(t)u'+ b(t)\sin u$, then the set of $T$-periodic solutions of (2) is $\Cal C \cap I^{-1} (\int_0^T f)$. Indeed, we may prove directly the following \newpage \lema{Theorem 2} Let $\psi:\R\to \R$ given by $\psi (s) = I(u_s) - \int_0^T f$, where $u_s$ is the unique solution of the problem $$\cases u''+a(t)u' + b(t)\sin(u)= f &\\ u|_{\partial I} = s & \endcases$$ Then $\psi$ is continuous and $2\pi$-periodic. Moreover, the set of $T$-periodic solutions of (2) can be characterized as $$\{ u_s : \psi (s) =0\}$$ \demost{Proof} Let $s\to s_0$, and define $w_s=u_s-u_{s_0}$. In order to prove that $I(u_s)\to I(u_{s_0})$, by dominated convergence it will suffice to show that $w_s, w_s'\to 0$ a.e. As $$0 = Q u_s - Qu_{s_0} = w_s'' + a w_s'+ b(\sin u_s - \sin u_{s_0})= w_s'' + a w_s'+ b\cos \xi w_s$$ it follows that $\| w_s\|_{1,2}\le c|s-s_0|$ for some constant $c$. Hence, $\psi$ is continuous. Moreover, by uniqueness $u_{s+2\pi}=u_s + 2\pi$, and by definition of $I$ we conclude that $\psi$ is $2\pi$-periodic. Finally, $u$ is a $T$-periodic solution of (2) if and only if $u\in \Cal C= \{ u_s : s\in \R\}$, and $0 = u'(T)-u'(0) = \int_0^T f - (au'+b\sin u)$. Hence, the proof is complete. \lema{Remarks:} i) With the notation of the preceeding theorem, it's immediate that $\Cal C$ is an embedded curve of $H^1(0,T)$. ii) A more general version of Theorem 2 may be obtained if we define $p>0$ verifying: $$p' = a(t)p - k(t), \qquad p(0) = p(T)\tag{P}$$ and $$\psi_p (s) = \int_0^T p(t) (b(t)\sin u_s - f) + k(t)u_s' := I_p(u_s) - \int_0^T pf $$ iii) By periodicity, extremals $s_{min}$ and $s_{max}$ of $\psi_p$ are achieved in $[0,2\pi]$. Using dominated convergence, it can be shown that if $w = \frac {\partial}{\partial s} u_s$ then satisfies the linear equation $$\cases (pw')'+kw' + b(t)\cos(u_s)w= 0 &\\ w|_{\partial I} = 1 & \endcases$$ For a solution of the equation $\psi_p '(s)=0$, if $k$ is constant then $w$ is periodic. \rm \medskip As an immediate consequence of theorem 2 we have: \lema{Corollary 3} Let $p>0$ verify (P) and assume that $I_p(u_{s_-})\le \int_0^T pf \le I_p(u_{s_+})$ for some $s_-\le s_+\in \R$. Then there exist $s_1\in [s_-,s_+]$, $s_2\in [s_+,s_- + 2\pi]$ such that $u_{s_i} + 2n\pi$ is a $T$-periodic solution of (2) for any integer $n$ and $i=1,2$. \rm \tit{3. Sufficient conditions for the existence of $T$-periodic solutions of (2) } In this section we'll obtain $T$-periodic solutions of (2) under rather explicit conditions on $a,b$ and $f$. For a straightforward application of corollary 3 we first note that a positive $p$ satisfying (P) for some constant $k$ may be constructed in a unique way up to a constant factor. Indeed, from the equation $p'= ap - k$ we obtain that $$p(t) = \left(c_0 - k\int_0^t e^{-\int_0^sa}ds\right) e^{\int_0^ta}$$ Without loss of generality we may assume that $c_0 = 1$, and as $p(0)=p(T)$ we deduce that $$k=\frac { e^{\int_0^Ta}-1}{\int_0^T e^{\int_s^Ta}ds}$$ Now, if $k\le 0$ it's immediate that $p>0$. On the other hand, if $k > 0$ and $p$ vanishes in $(0,T)$, since $p(0)=p(T)$ there exists $t_0\in (0,T)$ such that $p(t_0)=0$ and $p'(t_0)\ge 0$. Then $ k = -p'(t_0) \le 0$, a contradiction. \lema {Remark:} in particular, if $a\perp 1$ then $p(t)= e^{\int_0^ta}$, and if $a$ is a constant then $p\equiv 1$. \rm \medskip In the following $p$ will be considered as in the previous construction, and we'll assume that $\| \widetilde p b\|_\infty < \lambda_{\widetilde p}$ for some $\widetilde p$ such that lemma 1 holds. As $Qu_s - Qs = f - b\sin s$, we may denote by $\delta$ the best possible constant such that $$\| u_s -s\|_\infty \le \delta \| p(f-b\sin s)\|_2$$ A simple computation shows that if $\| pb\|_\infty < \lambda_p$, then $$\delta \le \left( \frac {T\lambda_p}{\inf_t p}\right) ^{1/2} \frac 1{\lambda_p - \| pb\|_\infty}$$ and if $\| pb\|_2 \le \| pb\|_\infty$ (for example, when $T\le 1$) then $\| pb\|_\infty$ may be replaced in the formula by $\| pb\|_2$. We obtain the following \lema {Theorem 4} Let us assume that $b$ does not vanish in $[0,T]$, and \noi i) $\| \widetilde p b\|_\infty < \lambda_{\widetilde p}$ for some $ \widetilde p$ satisfying the hypothesis of lemma 1 \noi ii) $\| p(f \pm b) \|_2 \le \frac c\delta $ for some constant $c<\frac {\pi}2$. \noi iii) $|\int_0^T pf| \le \| pb\|_1 \cos (c)$ Then (2) has infinitely many $T$-periodic solutions. More precisely, there exist $s_1\in [\frac {-\pi}2, \frac {\pi}2]$, $s_2\in [\frac {\pi}2, \frac {3\pi}2]$ such that $\{ u_{s_i} + 2n\pi : n\in Z\}$ is a family of $T$-periodic solutions of (2). \demost{Proof} With the previous notations, for $s_n = \frac {(2n-1)\pi}2$ we obtain that $\| u_{s_n} - s_n\|_\infty\le c $. Furthermore, as $$I_p(u_{s_{n}})= \int_0^T k u_{s_n}' + pb\sin u_{s_n} = (-1)^n \int_0^T pb \cos (u_{s_n}-s_n),$$ taking $n$ such that $(-1)^nb > 0$ we obtain: $$I_p(u_{s_{n}})\ge \| pb\|_1 \cos (c)$$ and in the same way $$I(u_{s_{n \pm 1}})\le -\| pb\|_1 \cos (c)$$ Hence, by iii) $$I_p(u_{s_{n \pm 1}}) \le \int_0^T pf \le I_p(u_{s_n})$$ and from corollary 3 the result holds. \medskip For constant $a$, conditions $i)-iii)$ can be written in a more explicit way: \lema{Corollary 5} Let us assume that $a$ is a constant, and $b$ does not vanish in $[0,T]$. Then, if \noi i) $\| b\|_\infty< (\frac {\pi}{T})^2$ \noi ii) $\| f \pm b \|_2 \le c\frac {\pi^2- \|b\|_\infty T^2}{T^{3/2}}$ for some constant $c<\frac {1}2$. \noi iii) $|\int_0^\a f|\le \| b\|_1 \cos (c\pi)$ \noi there exist $s_1\in [\frac {-\pi}2, \frac {\pi}2]$, $s_2\in [\frac {\pi}2, \frac {3\pi}2]$ such that $\{ u_{s_i} + 2n\pi : n\in Z\}$ is a family of $T$-periodic solutions of (2). \lema{Remarks:} 1) Condition ii) in Theorem 4 may be easily generalized in the following way: \noi ii') There exist $s_0, s_1$ such that $\delta \| p(f - b\sin s_i) \|_2 + |s_i - (-1)^i \frac {\pi}2| \le c$ for some constant $c<\frac {\pi}2$. Replacing ii) by ii') a family $\{ u_{s} + 2n\pi\}$ of $T$-periodic solutions of (2) is obtained for some $s \in [s_1,s_0]$ and also for some $s \in [s_0,s_1+2\pi]$. Furthermore, the existence of one of the numbers $s_i$ may be avoided if the following extra condition on $f$ holds: $$\| pb\|_2\| pf\|_2\le \frac 1\delta_2 |\int pf|\tag{F} $$ where $\delta_2$ is the best constant such that $\| u_s - s\|_2 \le \delta_2 \| pf\|_2$. Indeed, for $s = 0,\pi$ $$|\int_0^T pb \sin u_s| = |\int_0^T pb \sin (u_s-s)| \le \delta_2 \| pb \|_2\| pf \|_2\le |\int_0^T pf|,$$ and one of the inequalities required in corollary 3 is proved. For example, (F) holds for $sg(f)$ constant, if $0\le |f| \le \frac 1p$ and $|\int pf| \ge \delta_2 \| pb\|_2$. 2) As a particular case, condition iii) in Theorem 4 (Corollary 5) holds for $f \perp p$ (respectively: $\int_0^T f = 0$). 3) The results of this paper may be simplified and constants can be computed in a sharper way by the use of a weighted Hilbert space $L_p^2(0,T)$. For example, if $\|\cdot \|$ denotes the $p$-weighted norm, and $\mu_p$ is the first eigenvalue of the problem $(-pu')' = \mu pu$, then the inequalities of lemma 1 can be transformed in: $$\| u- v\| \le \frac 1{\mu_p -\|b\|_\infty}\|(Qu-Qv)\|$$ and $$\|(u- v)'\| \le \frac {\sqrt{\mu_p}}{\mu_p -\| b\|_\infty} \| (Qu-Qv)\| $$ \newpage \tit {References} [A] Alonso, J.: Nonexistence of periodic solutions for a damped pendulum equation. Diff. and Integral Equations, 10 (1997), 1141-8. [AM] Amster, P., Mariani, M.C.: Resolution of Semilinear Equations by Fixed Point Methods. To appear in the Bulletin of the Belgian Math. Society, Simon Stevin. [H] Hamel, G.: \"Uber erzwungene Schwingungen bei endlichen Amplituden. Math. Ann., 86 (1922), 1-13. [M1] Mawhin, J.: Periodic oscillations of forced pendulum-like equations. Lecture Notes in Math., Springer, 964 (1982), 458-76. [M2] Mawhin, J.: The forced pendulum: A paradigm for nonlinear analysis and dynamical systems. Expo. Math., 6 (1988), 271-87. [M3] Mawhin, J.: Boudary value problems for nonlinear ordinary differential equations: from successive approximations to topology. [O] Ortega, R.: A counterexample for the damped pendulum equation. Bull. Classe des Sciences, Ac.Roy. Belgique, LXXIII (1987), 405-9. [OST] Ortega, R., Serra, E., Tarallo, M.: Non-continuation of the periodic oscillations of a forced pendulum in the presence of friction. To appear. \bigskip {\bf P.Amster and M. C. Mariani} Dpto. de Matem\'atica Fac. de Cs. Exactas y Naturales, UBA Pab. I, Ciudad Universitaria (1428), Buenos Aires, Argentina CONICET \bigskip {\bf Address for correspondence:} \noi P.Amster and M. C. Mariani, \noi Dpto. de Matem\'atica, Fac. de Cs. Exactas y Naturales, UBA \noi Pab. I, Ciudad Universitaria \noi (1428) Buenos Aires, Argentina \medskip {\bf E-mail:} \quad pamster\@dm.uba.ar - mcmarian\@dm.uba.ar \end [ Part 2, "TeX DVI file" Application/X-DVI 34KB. ] [ Unable to print this part. ] [ Part 3, "ascii text" Text/PLAIN 548 lines. ] [ Unable to print this part. ] ---------------0003311244569--