\documentclass[twoside]{article} \usepackage{amsmath, amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil A second order ODE \hfil EJDE--2001/75} {EJDE--2001/75\hfil Pablo Amster \& Mar\'\i a Cristina Mariani \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2001}(2001), No. 75, pp. 1--9. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A second order ODE with a nonlinear final condition % \thanks{ {\em Mathematics Subject Classifications:} 34B15, 34C37. \hfil\break\indent {\em Key words:} Nonlinear boundary-value problems, fixed point methods. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Submitted: October 15, 2000. Published December 10, 2001.} } \date{} % \author{Pablo Amster \& Mar\'\i a Cristina Mariani} \maketitle \begin{abstract} We study a semilinear second-order ordinary differential equation with initial condition $u(0)=u_0$. We prove the existence of solutions satisfying a nonlinear final condition $u(T)=h'(u(T))$, under a certain growth condition. Also we state conditions ensuring that any solution with Cauchy data $u(0) = u_0$, $u'(0)=v_0$ is defined on the whole interval $[0,T]$. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \section{Introduction} We study the differential equation $$u''(t)+r(t) u'(t) + g(t,u(t)) = f(t) \label{*}$$ with initial condition $u(0) = u_0$. In the first section, we state the basic assumptions and results concerning the Dirichlet problem associated with (\ref{*}). In the second section, we define a fixed point setting for solving a problem with final value $u(T)$ depending on the velocity at time $T$. We prove that if $g$ satisfies a growth condition that holds for example when $g$ is {\sl sublinear}, then there exist a class of functions $h$ such that (\ref{*}) admits at least one solution $u$ with $u(0)=u_0$, $u(T)=h(u'(T))$. A physical example of this equation is the forced pendulum equation, for which existence results under Dirichlet and periodic conditions are known, see \cite{C,H,M1} and their references. For nonexistence results, see e.g. \cite{A,OST}. Finally, in the third section we prove the existence of a continuous real function $\psi = \psi_{u_0}$ such that a solution of (\ref{*}) with initial value $u_0$ is defined over $[0,T]$ if and only if the equation $\psi(s)=u'(0)$ is solvable. Furthermore, if $g$ is locally Lipschitz on $u$ the union over $u_0$ of the sets $\{ u_0\} \times \mathop{\rm Range}(\psi_{u_0})$ is a simply connected open subset of $\mathbb{R}^2$. \section {Basic assumptions and unique solvability of the Dirichlet problem} Let $S:H^2(0,T)\to L^2(0,T)$ be the semi-linear operator $Su= u''+ru' + g(t,u)$. We assume throughout this paper that $g$ is continuous and satisfies the condition $$\frac {g(t,u) - g(t,v)}{u-v}\le c < \big(\frac\pi T\big)^2 \quad \text {for all } t\in [0,T], u,v \in \mathbb{R}, u\ne v \label{G1}$$ Moreover, we shall assume that the friction coefficient $r\in H^1(0,T)$ is non-decreasing. Concerning the Dirichlet problem for (\ref{*}), we recall the following results whose proofs can be found in \cite{AM}. For related results and a general overview of this problem, we refer the reader to \cite{D,M2}. \begin{lemma} \label{lm1} Let $u, v\in H^2(0,T)$ with $u-v \in H_0^1(0,T)$. Then $$\| Su-Sv\|_2 \ge \big((\frac \pi T)^2-c\big)\| u-v\|_2$$ and $$\| Su-Sv\|_2 \ge \frac {(\pi/T)^2-c}{\pi/T} \| u'-v'\|_2$$ \end{lemma} \begin{theorem} \label{thm2} The Dirichlet problem \begin{gather*} Su=f(t) \quad\text {in } (0,T) \cr u(0)=u_0, \quad u(T)=u_T \end{gather*} is uniquely solvable in $H^2(0,T)$ for any $f\in L^2(0,T)$, $u_0, u_T \in \mathbb{R}$. \end{theorem} \begin{theorem} \label{thm3} Let $f\in L^2(0,T)$ and $\mathcal{S} = S^{-1}(f)$ with the topology induced by the $H^2$-norm. Then the trace function, $\mathop{\rm Tr}: \mathcal{S}\to \mathbb{R}^2$, given by $\mathop{\rm Tr}(u)= (u(0),u(T))$ is an homeomorphism. \end{theorem} \section{Nonlinearities at the endpoint} In this section we study the problem $$\begin{gathered} u''+ru'+g(t,u) = f \quad \text { in } (0,T) \\ u(0)=u_0,\quad u(T)=h(u'(T)) \end{gathered} \label{e1}$$ for $f\in L^2(0,T)$ and $h$ continuous. First we transform the problem in a one-dimensional fixed point problem: Indeed, for $s\in \mathbb{R}$, we define $u_s$ as the unique solution of the problem \begin{gather*} u''+ru'+g(t,u) = f \quad \text {in } (0,T) \\ u(0)=u_0,\quad u(T)=h(s) \end{gather*} Hence, when $\varphi_s(t)= \frac {h(s)-u_0}T t + u_0$, we have $$u_s(t)- \varphi_s(t) = \int_0^T (f -ru_s'-g(\theta, u_s')) G(t,\theta)d\theta$$ where $G$ is the Green function associated with the second order differential operator. Namely, $$G(t,\theta) = \begin{cases} \frac{t(\theta-T)}T & \text {if } \theta \ge t \\ \frac{\theta(t-T)}T &\text {if } \theta \le t \end{cases}$$ By simple computation we obtain $$u_s'(T) = \frac {h(s)-u_0}T + \int_0^T (f -ru_s'-g(\theta, u_s))\frac\theta T d\theta$$ and from Theorem \ref{thm2} we have \begin{theorem} \label{thm4} Let $\xi:\mathbb{R} \to \mathbb{R}$ with $$\xi (s) = \frac {h(s)-u_0}T + \int_0^T (f -ru_s'-g(\theta, u_s))\frac \theta T d\theta\,.$$ Then $\xi$ is a continuous fixed point operator for (\ref{e1}), i.e. $u$ is a solution of (\ref{e1}) if and only if $u=u_s$ for some $s \in \mathbb{R}$ such that $\xi(s) = s$. \end{theorem} \paragraph{Proof} Continuity of $\xi$ follows immediately from the continuity of $\mathop{\rm Tr}^{-1}: \mathbb{R}^2 \to S^{-1}(f)$. Moreover, if $\xi(s)=s$, then $u_s(T)=h(u_s'(T))$, proving that $u_s$ is a solution of (\ref{e1}). Conversely, if $u$ is a solution of (\ref{e1}), then $u=u_s$ for $s=u'(T)$. \quad\hfill$\Box$ We establish an existence result for (\ref{e1}) assuming that the graph of $h$ crosses the constant $u_0$. \begin{theorem} \label{thm5} Assume that (\ref{G1}) holds and that $h - u_0$ has nonconstant sign on $\mathbb{R}$. Then (\ref{e1}) admits a solution for $T$ small enough. \end{theorem} \paragraph{Proof} First we give a slightly different formulation of the equality $\xi(s)=s$. Integrating by parts, we see that $$\int_0^Tr(\theta)u_s'(\theta) \theta d\theta= r(T)Th(s) - \int_0^T[r(\theta) + \theta r'(\theta)] u_s(\theta) d\theta$$ and then $$\xi (s) = (\frac 1T - r(T)) h(s) + \frac 1T \left[\int_0^T \theta f(\theta)d\theta - u_0\right] +\frac 1T \int_0^T (r+\theta r')u_s -\theta g(\theta, u_s)d\theta$$ Hence, $s$ is a fixed point of $\xi$ if and only if $$sT= (1-r(T)T)h(s) - u_0 + \int_0^T (r+\theta r')u_s -\theta g(\theta, u_s)d\theta + \int_0^T \theta f(\theta)d\theta \label{e2}$$ >From Lemma \ref{lm1}, $$\| u_s-\varphi_s\|_2\le \frac {T^2}{\pi^2 - cT^2} \| Su_s - S\varphi_s\|_2 = \frac {T^2}{\pi^2 - cT^2} \| f - r\varphi_s' - g(\cdot ,\varphi_s)\|_2$$ and $$\| u_s-\varphi_s\|_\infty \le \frac {\pi T^{3/2}}{\pi^2 - cT^2} \| f - r\varphi_s' - g(\cdot ,\varphi_s)\|_2$$ Moreover, $$\|\varphi_s\|_2 = \sqrt{\frac T3 (h(s)^2+h(s)u_0+u_0^2)} := c(s) \sqrt{T}$$ and as $$\|\varphi_s\|_\infty = \max \{ | u_0|, | h(s)|\}, \quad \quad \varphi_s' =\frac {h(s)-u_0}T$$ then letting $T\to 0$ for fixed $s$ we have that $\| u_s\|_2 \to 0$ and $\| u_s\|_\infty$ is bounded. Hence, we conclude that the right-hand side of (\ref{e2}) converges to $h(s) - u_0$. Setting $s_\pm \in \mathbb{R}$ such that $h(s_+) < u_0 < h(s_-)$, it follows, for small $T$, that $$T\xi (s_+) \le h(s_+) - u_0 + B(s_+)$$ and $$T\xi (s_-) \ge h(s_-) - u_0 + B(s_-)$$ for some $B$ such that $B(s_\pm)\to 0$. Hence it suffices to take $T$ such that $$h(s_+) - u_0+ B(s_+) \le Ts_+, \quad h(s_-) - u_0+ B(s_-) \ge Ts_-$$ \hfill$\Box$ For the next existence result, we assume that $g$ grows at most linearly, i.e. $$|g(t,x)| \le \alpha |x| + \beta \label{G2}$$ for some positive constants $\alpha$, $\beta$. We remark that (\ref{G1}) and (\ref{G2}) are independent: for example, $g(x) = -x^3$ satisfies (\ref{G1}) but not (\ref{G2}). Conversely, $g(x) = \sin (Kx)$ does not satisfy (\ref{G1}) for $K \ge \left(\frac {\pi}{T}\right)^2$. For simplicity we define the constants $$c_T= \sqrt{\frac T3} + \frac {T^2}{\pi^2 - cT^2} \Big(\alpha \sqrt{\frac T3} + \frac {\| r\|_2}T\Big), \quad M = \Big(\| r+\theta r'\|_2 + \sqrt{\frac {T^3}3} \alpha\Big) c_T$$ and the functions $$C_\pm(s) = \Big( (1-r(T)T) \mathop{\rm sgn} \big(\frac {h(s)}s\big) \pm M\Big) \Big| \frac {h(s)}s\Big|\,.$$ \begin{theorem} \label{thm6} Assume that (\ref{G1}) and (\ref{G2}) hold. Then (\ref{e1}) admits at least one solution $u\in H^2(0,T)$ in each of the following cases \noindent Case A: $M < |1-r(T)T|$, with $$T < \limsup_{s\to +\infty} C_-(s) \quad \text {or} \quad T > \liminf_{s\to -\infty} C_+(s) \label{A1}$$ and $$\quad T < \limsup_{s\to -\infty} C_-(s) \quad \text {or} \quad T > \liminf_{s\to +\infty} C_+(s)\label{A2}$$ \noindent Case B: $M > |1-r(T)T|$, with $T > \liminf_{s\to \pm\infty} C_+(s)$ \noindent Case C: $M = |1-r(T)T|$, and there exist sequences $s^-_j \to -\infty$, $s^+_j \to +\infty$ such that $T > C_+(s^\pm_j)$ for every $j$, each one of them satisfying one of the following conditions: $$\mathop{\rm sgn} \big(\frac {h(s_j)}{s_j}\big) = \mathop{\rm sgn}(1-r(T)T) \quad\text{for every j} \label{C1}$$ or $$\lim_{j\to \infty} \frac {h(s_j)}{s_j^2} = 0 \label{C2}$$ \end{theorem} \paragraph{Remarks:} i) The left-hand-side in condition \ref{A1} (resp. \ref{A2}) implies $$\limsup_{s\to +\infty} \frac {h(s)}{s }\mathop{\rm sgn}(1-r(T)T) > \frac T{|1-r(T)T| - M} \quad \text { (resp. s\to -\infty) }$$ ii) The following assumptions are sufficient for the right-hand-side in condition \ref{A1} (resp. \ref{A2}) to be satisfied. $$%\text {2')} \liminf_{s\to -\infty} \Big| \frac {h(s)}s\Big| < \frac T{M + |1-r(T)T|} \quad \text { (resp. s\to +\infty)}$$ or $$%\text {2'')} \mathop{\rm sgn}\big(\frac {h(s_j)}{s_j}\big) = -\mathop{\rm sgn}(1-r(T)T)$$ for a sequence $s_j \to -\infty$ (resp. $s_j \to +\infty$). \smallskip \noindent iii) Conditions in case B are not fulfilled when $$|h(s)|\ge a|s| + b, \quad \text{with}\quad a \ge \frac T{M-|1-r(T)T|}$$ In the same way, conditions in case C imply $$\liminf_{|s|\to \infty} \Big| \frac {h(s)}s\Big| < \frac T{2M}$$ \paragraph{Proof of Theorem \ref{thm6}} As in the previous theorem, $$\| u_s\|_2 \le \sqrt{T} c(s) + \frac {T^2}{\pi^2 - cT^2} \big(\alpha \sqrt{T} c(s) + |h(s)-u_0| \frac {\| r\|_2}T +\| f\|_2 + \beta\big):=A(s)$$ and then $$\| u_s\|_2 \le c_T|h(s)| + \gamma |h(s)|^{1/2}+ \delta$$ for some constants $\gamma, \delta \in \mathbb{R}$. Moreover, $$\Big|\int_0^T (r+\theta r')u_s -\theta g(\theta, u_s)d\theta\Big| \le \Big(\| r+\theta r'\|_2 + \sqrt{\frac {T^3}3} \alpha\Big) c_T|h(s)|+ R(s)$$ with $R(s) \le C_1|h(s)|^{1/2} + C_2$ for some constants $C_1,C_2$. We remark that $\frac {R(s)}s \to 0$ for $|s|\to \infty$ if $h$ is {\sl subquadratic} (i.e. $\frac {h(s)}{s^2}\to 0$ for $|s|\to \infty$). Hence, \begin{align*} [(1-r(T)T) &-M \mathop{\rm sgn}(h(s))] h(s) - R(s)\\ \le& T\xi(s) \\ \le& [(1-r(T)T) + M \mathop{\rm sgn}(h(s))] h(s) + R(s) \end{align*} and it suffices to find $s_\pm$ satisfying: \begin{gather} s_- T\le [(1-r(T)T) -M \mathop{\rm sgn}(h(s_-))] h(s_-) - R(s_-)\label{e-} \\ s_+ T\ge [(1-r(T)T) + M \mathop{\rm sgn}(h(s_+))] h(s_+) + R(s_+) \label{e+} \end{gather} Assuming that $s_- > 0$ then (\ref{e-}) is equivalent to $$T\le \Big[ \mathop{\rm sgn} \big(\frac{h(s_-)}{s_-}\big) (1-r(T)T) -M\Big] \Big|\frac{h(s_-)}{s_-}\Big| - \frac {R(s_-)}{s_-}$$ Hence, if $M < |1-r(T)T|$ then left-hand-side of (\ref{A1}) is a sufficient condition for (\ref{e-}): indeed, if $T < k \Big|\frac{h(s_j)}{s_j}\Big|$ for $s_j \to +\infty$ and some $k>0$, then $$k\Big|\frac{h(s_j)}{s_j}\Big| - \frac {R(s_j)}{s_j} = \Big|\frac{h(s_j)}{s_j}\Big| \Big( k - \frac {R(s_j)}{|h(s_j)|}\Big)$$ As $|h(s_j)| \to \infty$, we have that $R(s_j)/|h(s_j)| \to 0$ and the result follows. In the same way, if we assume that $s_- < 0$, then (\ref{e-}) is equivalent to $$T\ge \Big[ \mathop{\rm sgn} \Big(\frac{h(s_-)}{s_-}\Big) (1-r(T)T) +M\Big] \Big|\frac{h(s_-)}{s_-}\Big| - \frac {R(s_-)}{s_-}$$ and right-hand-side of (\ref{A1}) is sufficient, as well as conditions in cases B and C. The same conclusions can be obtained for (\ref{e+}), which completes the proof. \hfill$\Box$ \paragraph{Example} We consider the forced pendulum equation $$u''(t) + \sin u = f(t) \label{eP}$$ for which it is clear that (\ref{G2}) holds, and (\ref{G1}) holds when $T <\pi$. In this case $c_T= \sqrt{\frac T3}$, $M=0$, and $C_-(s) = C_+(s) =\frac {h(s)}s$. If we assume, further, that $$\lim_{s\to \pm \infty}\frac {h(s)}s = L_\pm$$ then (\ref{e1}) is solvable, unless $$L_- \le T \le L_+ \quad \text {or} \quad L_+ \le T \le L_-$$ In particular, (\ref{e1}) is solvable when $h$ is sublinear or superlinear (and obviously when $h$ is linear, $h(s) = as + b$, for $T\neq a$). It is well known that (\ref{eP}) admits $T$-periodic solutions when $f$ is $T$-periodic and $\int_0^T f=0$. Furthermore, in \cite{C} it has been proved that for any $2\pi$-periodic $f_0 \in L^2(0,2\pi)$ such that $\int_0^{2\pi} f_0 =0$ there exist two numbers $d(f_0) \le 0 \le D(f_0)$ such that (P) admits $2\pi$-periodic solutions for $f(t) = f_0(t) + f_1$ if and only if $$d(f_0) \le f_1\le D(f_0)$$ \paragraph{Remark} Assuming (\ref{G1}) and (\ref{G2}) we may define the functions $\xi^\pm:\mathbb{R}\to \mathbb{R}$ as \begin{align*} \xi^\pm(s)=& \frac 1T \Big( (1-r(T)T)h(s)\pm \Big[ \| r+\theta r'\|_2 A(s) + \sqrt{\frac {T^3}3} (\alpha A(s) + \beta) \Big] \\ &+ \int_0^T\theta f(\theta)d\theta - u_0\Big) \end{align*} with $A(s)$ as in the previous proof. Then a sufficient condition for the solvability of (\ref{e1}) is the existence of $s_\pm \in \mathbb{R}$ such that $s_- \le \xi^-(s_-)$ and $\xi^+(s_+) \le s_+$. Indeed, from the previous computations we have $$|\int_0^T (r+\theta r')u_s -\theta g(\theta, u_s)d\theta| \le \| r+\theta r'\|_2 A(s) + \sqrt{\frac {T^3}3} (\alpha A(s) + \beta)$$ Then $\xi^- \le \xi\le \xi^+$ and the result the result follows from Theorem \ref{thm4}. \hfill$\Box$ \section{Blow-up results} In this section we study the behavior of the solutions of the Cauchy problem $$\begin{gathered} u''+ru'+g(t,u) = f \quad \text { in } (0,T) \\ u(0)=u_0, \quad u'(0)= v_0 \end{gathered}\label{e3}$$ As a simple remark, under condition (\ref{G1}) we see that if $g$ is locally Lipschitz on $u$, then there exists an interval $I(u_0)$ such that $v_0\in I(u_0)$ if and only if $u$ is defined over $[0,T]$. Indeed, it suffices to show that the set $$I:= \{ v_0: \text{ the local solution of (\ref{e3}) does not blow up on [0,T]}\}$$ is connected. Let $v_0, v_2 \in I$ and $v_1 \notin I$ such that $v_0 |v_0|$ for $t>0$, we have that $|u(\frac 12)| > \frac {v_0}2$. Moreover, $|u'| > u^2$, and hence $$\frac 1 {|u(\frac 12)|} - \frac 1 {|u(1)|} > \frac 12$$ Thus, $$\frac 2{|v_0|} -\frac 12 > \frac 1{|u(1)|}$$ proving that $|v_0| < 4$. This shows that $I(0) \subset (-4,4)$. \medskip The following theorem shows that the Lipschitz condition is not necessary in order to prove the existence of $I(u_0)$. Further, we give an explicit expression for $I(u_0)$ as the range of a continuous function. \begin{theorem} \label{thm7} Assume that (\ref{G1}) holds. Then there exists an interval $I(u_0)$ such that the following two conditions are equivalent:\\ i) $v_0\in I(u_0)$ \\ ii) At least one local solution of (\ref{e3}) is defined over $[0,T]$. \\ Moreover, if $h(s) = u_0+ sT$ and $\psi:\mathbb{R}\to \mathbb{R}$ given by $$\psi(s) = s + \int_0^T (f -ru_s'-g(\theta, u_s))\frac{\theta-T}{T}d\theta,$$ then $I(u_0)= \mathop{\rm Range} (\psi)$. \end{theorem} \paragraph{Proof} As in Section 3, we have $$u_s(t)- \varphi_s(t) = \int_0^T (f -ru_s'-g(\theta, u_s))G(t,\theta)d\theta$$ with $\varphi_s(t)= st + u_0$. By simple computation, $u_s'(0) = \psi(s)$, and the proof is complete. \hfill$\Box$ \paragraph{Remark} In particular, if $g$ is locally Lipschitz on $u$ then $\psi$ is injective and hence $I(u_0)$ is open. \begin{theorem} \label{thm8} Assume (\ref{G1}) and that $g$ is locally Lipschitz on $u$. Then the set $$\bigcup_{u_0\in \mathbb{R}} \{ u_0\}\times I(u_0)$$ is open and simply connected in $\mathbb{R}^2$. \end{theorem} \paragraph{Proof} Let $\mathcal{S} = S^{-1}(f)$ and consider the continuous mapping $\rho:\mathcal{S} \to \mathbb{R}^2$, $\rho(u) = (u(0),u'(0))$. Then $v_0 \in I(u_0)$ if and only if $(u_0,v_0)\in \mathop{\rm Range} (\rho)$. As $g$ is locally Lipschitz, $\rho$ is injective, and hence $\mathop{\rm Range} (\rho) = \rho\circ \mathop{\rm Tr}^{-1}(\mathbb{R}^2)$ is open and simply connected. \hfill$\Box$ \paragraph{Acknowledgement} The authors want to thank Professor Alfonso Castro for the careful reading of the manuscript and his fruitful suggestions and remarks. \begin{thebibliography}{0} \frenchspacing \bibitem{A} Alonso, J.: Nonexistence of periodic solutions for a damped pendulum equation. Diff. and Integral Equations, 10 (1997), 1141-8. \bibitem{AM} Amster, P., Mariani, M.C.: Nonlinear two-point boundary value problems and a Duffing equation. Submitted. \bibitem{C} Castro, A: Periodic solutions of the forced pendulum equation. Diff. Equations 1980, 149-60. \bibitem{D} Dolph, C.L.: Nonlinear integral equations of the Hammerstein type, Trans. Amer. Math. Soc. 66 (1949), 289-307. \bibitem{H} Hamel, G.: \"Uber erzwungene Schwingungen bei endlichen Amplituden. Math. Ann., 86 (1922), 1-13. \bibitem{M1} Mawhin, J.: The forced pendulum: A paradigm for nonlinear analysis and dynamical systems. Expo. Math., 6 (1988), 271-87. \bibitem{M2} Mawhin, J.: Boudary value problems for nonlinear ordinary differential equations: from successive approximations to topology. Recherches de math\'ematique (1998), Inst. de Math Pure et Apliqu\'ee, Univ.Cath. de Louvain. Prepublication \bibitem{OST} Ortega, R., Serra, E., Tarallo, M.: Non-continuation of the periodic oscillations of a forced pendulum in the presence of friction. To appear. \end{thebibliography} \noindent\textsc{Pablo Amster} (e-mail: pamster@dm.uba.ar)\\ \textsc{Maria Cristina Mariani} (e-mail: mcmarian@dm.uba.ar)\\[3pt] Departamento de Matem\'atica, \\ Facultad de Ciencias Exactas y Naturales, \\ Universidad de Buenos Aires - CONICET, \\ Pab. I, Ciudad Universitaria, (1428) Buenos Aires, Argentina \end{document}