Content-Type: multipart/mixed; boundary="-------------0004041715246" This is a multi-part message in MIME format. ---------------0004041715246 Content-Type: text/plain; name="00-151.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-151.keywords" Two-point boundary value problems - Duffing equation ---------------0004041715246 Content-Type: application/x-tex; name="duffing.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="duffing.tex" \input vanilla.sty \def\rect{\hskip 0.1truecm\vbox{\hrule\hbox{\vrule \hskip .2truecm\vbox{\vskip .2truecm}\vrule} \hrule}\hskip 0.1truecm} \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 {Nonlinear two-point boundary value problems and a Duffing equation} \endtitle \centerline {P.Amster and M.C. Mariani} \medskip \medskip \lema{Abstract: } In this paper we study a general semilinear second order ODE $$(pu')'+g(t,u,u') = f \tag{*}$$ Under an appropiate growth condition on $g$ we prove that the Dirichlet problem for (*) is uniquely solvable. Moreover, the set of $H^2$-solutions of (*) is homeomorphic to the two-dimensional real space. We also establish conditions for the existence of periodic solutions of (*). \tit {Introduction} The two-point boundary value problem for a general semilinear second order ODE, namely $$u''+g(t,u,u') =0 \qquad, u(0)=u_0, \quad u(\a)=u_\a \tag {**}$$ has been studied by many authors from the pioneering work of Picard [P], who proved the existence of a solution of $(**)$ by an application of the well known method of successive approximations under a Lipschitz condition on $g$ and a smallness condition on $\a$. Sharper results were obtained by Hamel [H] in the special case of a forced pendulum equation (see also [M1], [M2]). The existence of periodic solutions for this case has been first considered by Duffing [Du] in 1918. Variational methods have been also applied when $g=g(t,u)$ by Lichtenstein [L], who considered the functional $$I(u)= \int_0^\a \frac {u'^2}2 - (\int_0^{u(t)} g(t,s)ds)dt$$ Finally, we mention also the topological approach, introduced in 1905 by Severini [S] with a shooting method, which gave a survey of results by the use of Leray-Schauder techniques and degree theory. For a complete overview of the problem and further results we refer the reader to [M3]. \medskip Let $I=(0,\a)$ and $S:H^2(I)\to L^2(I)$ be the semilinear operator given by $$Su= (pu')'+g(t,u,u')$$ where $p\in C^1(\overline I)$, $p>0$ and $g:\overline I\times \R^2\to \R$ is continuous. We'll assume that $$[g(t,u,x)-g(t,v,y)](u-v)\le c_1(u-v)^2 +c_2(t) (x-y)^2\tag{G1}$$ $$\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad + [r(t)(u-v)+\psi(u-v)](x-y) $$ holds, for some $\psi\in L^1_{loc}(\R)$, $r\in H^1(I)$, $r'\ge 0$ a.e., and $c_1\in \R$, $c_2\in L^\infty(I)$ satisfying the condition $$ \sup_{t \in \overline I} {c_2(t)\over p(t)} < 1 - {c_1\over \lambda_1}$$ where $ \lambda_1$ is the first eigenvalue of the problem $$-(pu')'=\lambda u, \qquad\quad u|_{\partial I} = 0.$$ We remark that (G1) implies a growing condition on $u$: $$\frac{g(t,u,x)-g(t,v,x)}{u-v} \le c_1$$ In particular, if $g$ is differentiable with respect to $u$, then $\frac {\partial g}{\partial u}\le c_1$. Moreover, considering $u=v$ we also obtain that $\psi(0)=0$ and $c_2\ge 0$. For simplicity, we'll consider $\varepsilon = 1 - {c_1\over \lambda_1} - \sup_{t \in \overline I} {c_2(t)\over p(t)} $ and $\delta = \lambda_1\varepsilon \sup_t p(t)$. Then we have: \lema {Lemma 1} Let $u, v\in H^2(I)$ with $u=v$ on $\partial I$. Then $$\| Su-Sv\|_2 \ge \delta\| u-v\|_2$$ and $$\| Su-Sv\|_2 \ge \frac {\delta}{\sqrt{\lambda_1}} \left( \int_I p(u'-v')^2\right)^{1/2}$$ \demost{Proof} Let $w=u-v$, then $$\| Su-Sv\|_2\| w\|_2\ge -\int ( Su-Sv).w = \int p(w')^2 - \int [g(t,u,u')-g(t,v,v')](u-v)$$ $$\ge \int (p- c_2)(w')^2 - c_1 \|w\|_2^2 -\int (rw+\psi(w))w'$$ By Poincar\'e's inequality $\| w\|_2^2 \le \frac 1{\lambda_1}\int p(w')^2$. Moreover, if $\widetilde{\psi}(t) = \int_0^t\psi$ then $\int \psi(w)w'= \left[\widetilde{\psi}(w) \right]_0^\a = 0$ and as $-\int rww'= \int r'\frac {w^2}2 \ge 0$, we obtain: $$\| Su-Sv\|_2\| w\|_2\ge \int ((1-\frac {c_1}{\lambda_1})p - c_2)(w')^2\ge {\delta\over \lambda_1}\int p (w')^2 $$ Hence, the result follows. \hfill\rect \lema {Remarks:} i) For simplicity and by the previous lemma, we may denote by $c_0$ the best constant such that $\| u - v\|_{1,2} \le c_0 \| Su - Sv\|_{2}$ for any $u, v\in H^2(I)$ such that $u=v$ on $\partial I$. ii) It's easy to prove that if $g$ satisfies a Lipschitz condition on $(u,u')$ then there exists an apriori inferior bound for $S$ on $H^2(I)$, i.e. a constant $c$ such that $$c\| Su-Sv\|_2 \ge \| u-v\|_{2,2}$$ for any $u, v\in H^2(I)$ with $u=v$ on $\partial I$. \medskip \rm We'll prove an existence and uniqueness result assuming a weaker condition on $g$, namely: $$|g(t,u,x)|\le g_0(u) k |x| \tag{G2}$$ for any $t\in \overline I$, $u,x\in \R$, with $g_0$ such that $g_R:=g_0|_{[-R,R]} < \infty$ for any $R$. \lema {Theorem 2} Let us assume that $g$ satisfies (G1) and (G2). Then the Dirichlet problem $$\text {\rm (1)} \cases Su=f(t) \qquad \qquad\text { in } (0,\a) &\\ u(0)=u_0, \quad u(\a)=u_\a & \endcases $$ is uniquely solvable in $H^2(I)$ for any $f\in L^2(I)$. \demost {Proof} Let $\sigma \le 1$ and $S_\sigma u:= (pu')'+ \sigma g(t,u,u')$, then the result is obviously true for $S_0$. If the result holds for $S_\sigma$, for any $0\le\sigma < \sigma_0$, then for $\ee >0$ and fixed $\overline u\in H^1(I)$ we may define $u=T\overline u $ as the unique solution of the problem $$ \cases S_{\sigma_0-\ee}u=f(t)-\ee g(t,\overline u,\overline u') \qquad \text { in } (0,\a) &\\ u(0)=u_0, \qquad u(\a)=u_\a & \endcases $$ For $\overline u\in H^1(I)$, $g(\cdot , \overline u,\overline u')\in L^2(I)$ with $\|g(\cdot , \overline u,\overline u')\|_2 \le \| g_0 (\overline u) \|_2 + k\| \overline u'\|_2$. This shows that $T:H^1(I)\to H^1(I) $ is well defined. On the other hand, if $c_0(\sigma)$ is the constant of lemma 1 for $S_\sigma$, we may assume that $c_0(\sigma) \le c_0(1)=c_0$ and for $u=T\overline u$, $v=T\overline v$ we have: $$\| u-v\|_{1,2}\le c_0\|S_{\sigma_0-\ee}u-S_{\sigma_0-\ee}v\|_2= \ee c_0\| g(\cdot,\overline u, \overline u')- g(\cdot,\overline v, \overline v')\|_2$$ If $\overline u \to \overline v$ in $H^1(I)\hookrightarrow C(\overline I)$ by dominated convergence it's easy to prove that $g(\cdot ,\overline u, \overline u')\to g(\cdot,\overline v, \overline v')$ for the $L^2$-norm and continuity of $T$ follows. Furthermore, if $\| \overline u\|_{1,2} \le R$ and $\overline v =0$, $$\| u\|_{1,2}\le \| v\|_{1,2}+ \ee c_0 (\| g(\cdot,\overline u, \overline u')\|_2+ \|g(\cdot,0, 0)\|_2)\le \| v\|_{1,2}+ \ee c_0 (g_{k_1R}\a^{1/2}+ \|g(\cdot,0, 0)\|_2) $$ where $k_1$ is the constant of the imbedding $H^1(I)\hookrightarrow L^\infty(I)$. Moreover, as $$\| (u - v)''\|_2\le \frac 1{inf p(t)} ( \| p'(u'-v') + (\sigma_0 - \ee) (g(t,u,u')-g(t,v,v'))\|_2 $$ $$+ \ee \| g(t,\overline u,\overline u') - g(t,0,0)\|_2 )$$ we conclude that $T(B_R(0))$ is bounded for $\| \cdot \|_{2,2}$. By the compactness of the imbedding $H^2(I)\hookrightarrow H^1(I)$, $T(B_R(0))$ is precompact. Fixing $\varphi\in C^1(\overline I)$ such that $\varphi (0) = u_0$, $\varphi (\a) = u_\a $, we have that $$\| v - \varphi\|_{1,2} \le c_0 \| f - \ee g(t,0,0) - S_{\sigma_0 - \ee}\varphi\|_2 \le k_0$$ for some constant $k_0$ independent of $\ee$. Then, for $R > k_0 + \|\varphi\|_{1,2}$ and $\ee < \frac {R-( k_0 + \|\varphi\|_{1,2})}{ c_0 (g_{k_1R}\a^{1/2}+ \|g(\cdot,0, 0)\|_2) }$ we obtain that $T(B_R(0)) \subset B_R(0)$ and by Schauder Theorem $T$ has a fixed point. As uniqueness follows immediately from lemma 1, the result holds for $\sigma_0$. In the same way, if the result holds for $S_\sigma$, with $0\le\sigma < 1$, then for $\ee >0$ and fixed $\overline u\in H^1(I)$ we may define $u=T\overline u $ as the unique solution of the problem $$ \cases S_{\sigma}u=f(t)-\ee g(t,\overline u,\overline u') \qquad \text { in } (0,\a) &\\ u(0)=u_0, \qquad u(\a)=u_\a & \endcases $$ proving that the result holds for $S_{\sigma + \ee}$ if $\ee$ is small enough. Hence, the proof is complete. \hfill\rect \lema {Remark:} In the previous proof one may have assumed that $f=0$, since for $g$ satisfying (G1)-(G2) it suffices to take $\overline g = g(t,u,u')- f(t)$, even for noncontinuous $f \in L^2$. \rm \medskip As a consequence of Theorem 2, we obtain a version of a Green-type operator for the nonlinear case: \lema {Corollary 3} Let $g$ satisfy (G1) and (G2), and consider the operator $G:L^2(I)\to H^1(I)$ given by $Gf=u$, where $u$ is the unique solution of (1). Then $G$ is compact. \demost {Proof} For $f, f_0 \in L^2(I)$ we have that $$\| Gf-Gf_0\|_{1,2} \le c_0\| SGf-SGf_0 \|_2= c_0\| f-f_0 \|_2$$ and the continuity of $G$ follows. Moreover, setting $u=G(f)$ and $v = G(0)$ it follows that $$\| (u-v)''\|_{2} \le \frac 1{inf p(t)} ( \| f - p'(u'-v') - (g(t,u,u')-g(t,v,v'))\|_2 $$ and the result follows from the compactness of the imbedding $H^2(I)\hookrightarrow H^1(I)$. \hfill\rect \rm \medskip We'll apply theorem 2 in order to find solutions for the general Dirichlet problem $$\text {(2)} \cases Su=f(t,u,u') \qquad \qquad\text { in } (0,\a) &\\ u(0)=u_0, \quad u(\a)=u_\a & \endcases $$ \lema {Theorem 4} Let $f$ be continuous and $g$ satisfy (G1)-(G2). We'll assume that the growing condition $$|f(t,u,x)| \le c|(u,x)| + d \tag{F}$$ holds for some constant $c < {1\over c_0}$. Then (2) is solvable in $H^2(I)$. \demost {Proof} Using (F), we may prove that the operator $T:H^1(I)\to H^1(I)$ given by $ T\overline u = G(f(\cdot, \overline u,\overline u'))$ is well defined and compact. >From lemma 1, $$\| T\overline u - T0\|_{1,2} \le c_0 \left( \| f(t,0,0)\|_2 + c \| \overline u\|_{1,2} + \a^{1/2}d \right) $$ Letting $R\to \infty$ it follows that $T(B_R(0))\subset B_R(0)$, which completes the proof. \hfill\rect \rm \medskip In the next theorem we'll state some properties of the set of solutions of the semilinear equation $Su=f$. For $f \in L^2(I)$ and $g$ continuous let us consider the set $$\Cal S_{f,g}=\{ u\in H^2(I): (pu')'+g(t,u,u')=f\}$$ Then we have: \lema {Theorem 5} Let $g,\overline g$ satisfy (G1) and (G2) and $f, \overline f \in L^2(I)$. Then $\Cal S_{f,g}$ is homeomorphic to $\Cal S_{\overline f,\overline g}$ for the $H^2$-norm. \lema {Remark: } In particular, $\Cal S_{f,g}\simeq \R^2$. For example, $\Cal S_{0,0} = \{ u = x + y\int_0^t \frac 1p : x,y\in \R\}$. \demost {Proof of theorem 5} For any $u\in \Cal S_{f,g}$ we consider $Tu = \varphi$, where $$\varphi (t) = \frac {u(\alpha)-u(0)}\alpha t + u(0).$$ By theorem 2, $T: \Cal S_{f,g}\to span \{ 1, t\} $ is bijective. The continuity of $T$ is clear since $H^2(I) \hookrightarrow C(\overline I)$; on the other hand, if $\varphi_n\to \varphi$, for $u_n=T^{-1}(\varphi_n)$, $u=T^{-1}(\varphi)$ and $w_n= u_n-u$ we have: $$0 = \int (Su_n-S u)(u_n-u) \le \left[ pw_nw_n'\right]_0^\a + \int (c_2-p(1-{c_1\over\lambda_1}))(w_n')^2 +\left[\widetilde{\psi}(w_n) \right]_0^\a $$ $$+ \int rw_nw_n' \le -\delta \int p(w_n')^2 +\left[ pw_nw_n'\right]_0^\a + \left[\widetilde{\psi}(w_n) \right]_0^\a + \left[{w_n^2\over 2}\right]_0^\a$$ But $w_n(0) = \varphi_n(0)-\varphi(0)\to 0$, $w_n(\alpha) = \varphi_n(\alpha)-\varphi(\alpha)\to 0$. On the other hand, by lemma 1 $\| w_n\|_{1,2}$ is bounded and from the inequalities $$\|w_n''\|_2 \le \frac 1{\inf p(t)} \| g(t,u_n, u_n') - g(t,u, u') - p'w_n'\|_2 \tag3$$ $$ \|g(\cdot ,u_n,u_n')-g(\cdot ,u,u')\|_2 \le \| g_0 (u_n) \|_2 + k\| u_n'\|_2 + \|g(\cdot ,u,u')\|_2 \tag4$$ we conclude that $w_n$ is bounded in $H^2(I)$. Hence, $\| w_n'\|_\infty$ is bounded, and $$\delta \int p(w_n')^2\le \left[ pw_nw_n'\right]_0^\a + \left[\widetilde{\psi}(w_n) \right]_0^\a + \left[{w_n^2\over 2}\right]_0^\a \to 0.$$ By Poincar\'e's inequality, $u_n\to u$ in $H^{1}(I) \hookrightarrow C(\overline I)$. By dominated convergence we obtain from (3) that $\|w_n''\|_2 \to 0$ and the result follows. \hfill\rect \tit {An application to a periodic problem} In this section we'll apply the previous results to the periodic problem $$\text {(Per)} \cases Su=f(t) \qquad \qquad\text { in } (0,\a) &\\ u(0)= u(\a), \quad u'(0)= u'(\a)& \endcases $$ Let $g$ satisfy (G1)-(G2) and assume that $p(0)=p(\a)$. From theorem 2, for any $s\in \R$ we may consider $u_s$ as the unique solution of the problem $$\cases Su=f(t) \qquad \qquad\text { in } (0,\a) &\\ u(0)= u(\a) = s & \endcases $$ Then $\Cal C = \{ u_s: s\in \R\}\subset \Cal S_{f,g}$ is an imbedded curve for the $H^2$-norm. Clearly $\Cal C$ is unbounded for $\| \cdot \|_\infty$. Let $I:H^2(I)\to \R$ given by $I(u)=\int_0^\alpha g(\cdot ,u,u')$. Then we have: \lema {Theorem 6} Let $g$ satisfy (G1)-(G2) and $f\in L^2$. Then the following statements are equivalent: i) (Per) admits (at least) a solution ii) There exist $u^+,u^- \in \Cal C$ such that $$ I(u^+) \ge \int_0^\alpha f \ge I(u^-)$$ \demost {Proof } Let $\tau:H^2(I)\to \R$ given by $\tau (u) = \int_0^\alpha f - I(u)$. Then $\tau$ is continuous, and as $\left[ pu'\right]_0^\a = \int_0^\a (pu')'$, it's clear that (Per) is solvable if and only if $\tau (u) = 0$ for some $u\in \Cal C$. As $\Cal C$ is connected, the result follows. \hfill\rect \lema {Remark:} >From lemma 1 we have the following apriori bound: $$\left( \int_I p(u_s')^2\right)^{1/2}\le \frac {\sqrt{\lambda_1}}{\delta} \| Su_s-Ss\|_2 = \frac {\sqrt{\lambda_1}}{\delta} \| f - g(\cdot, s, 0)\|_2 $$ Furthermore, as $u_s(t)= s + \int_0^t u_s'$, we obtain: $$\|u_s - s\|_\infty \le \alpha^{1/2} \| u_s'\|_2 \le \frac {(\lambda_1\alpha)^{1/2}}{\delta\inf p^{1/2}(t)} \| f - g(\cdot, s, 0)\|_2$$ In particular, if $g(\cdot ,s,0)$ satisfies the growth condition $$\|g(\cdot, s, 0)\|_2 \le c|s| + d \quad\text { for some }\quad c < \frac {\delta\inf p^{1/2}(t)}{(\lambda_1\alpha)^{1/2}}\tag{G3}$$ we obtain that $u_s(t) \in J_s$ for any $t$, where the interval $J_s$ is defined by $$J_s = \cases [(1- \widetilde c)s - \widetilde d, (1 + \widetilde c)s + \widetilde d] \qquad \text { if } s \ge 0 &\\ [(1 + \widetilde c)s - \widetilde d, (1 - \widetilde c)s + \widetilde d] \qquad \text { if } s < 0 & \endcases $$ with $$\widetilde c = \frac {c(\lambda_1\alpha)^{1/2}}{\delta\inf p^{1/2}(t)} < 1, \qquad \qquad \widetilde d = \frac {(\lambda_1\alpha)^{1/2}}{\delta\inf p^{1/2}(t)} (d+\| f\|_2).$$ \rm \lema {Corollary 7} Let $f\in L^2(I)$ and assume that $g$ satisfies (G1), (G2) and (G3). Furthermore, we'll assume that there exist $s^+, s^- \in \R$, $\varphi^+, \varphi^- \in L^1(I)$ and $k_1,k_2\in H^1(I)$ such that \qquad $i) g(t,u,x) \ge \varphi^+(t) + k_1'(t)u + k_1(t)x \quad\text { for any } u\in J_{s^+}, t\in I, x\in \R$ \qquad $ii) g(t,u,x) \le \varphi^-(t) + k_2'(t)u + k_2(t)x \quad\text { for any } u\in J_{s^-}, t\in I, x\in \R $ \qquad $iii) \int_0^\a \varphi^+ + (k_1(\a)-k_1(0))s^+ \ge \int_0^\a f \ge \int_0^\a \varphi^- + (k_2(\a)-k_2(0))s^- \ge (k_2(\a)-k_2(0))s^-$. Then (Per) admits at least a solution. \demost{Proof} >From the previous remark, $u_{s^+}(t)\in J_{s^+}$, and then $$\int_0^\a g(t, u_{s^+}, u_{s^+}') \ge \int_0^\a \varphi^+ + k_1'u_{s^+} + k_1 u_{s^+}' = \int_0^\a \varphi^+ + ( k_1(\a)-k_1(0))s^+ \ge \int_0^\a f$$ In the same way, $\int_0^\a g(t, u_{s^-}, u_{s^-}') \le \int_0^\a f$ and the result follows from theorem 6. \hfill\rect \medskip As a consequence of corollary 7 we obtain conditions on $g$ for nonresonance: \lema {Corollary 8} Let $g$ satisfy (G1), (G2) and (G3) and assume that there exist $k_1, k_2\in H^1(I)$ and $l_1,l_2: \R \to \R$ such that one of the following assumptions holds: \medskip Assumption A) \qquad\qquad $i) g(t,u,x) \ge l_1(u) + k_1'(t)u + k_1(t)x $ for $x\in \R, u>0$ \qquad\qquad $ii) g(t, u,x) \le l_2(u) + k_2'(t)u + k_2(t)x $ for $x\in \R, u<0$ \noi and \qquad\qquad $iii) l_1(u) \ge a_1u+b_1$ and $ l_2(u) \le a_2u+b_2$, with $$a_i(1-\widetilde c)\a + k_i(\a) > k_i(0)$$ \medskip Assumption B) \qquad\qquad $i) g(t,u,x) \le l_1(u) + k_1'(t)u + k_1(t)x$ for $x\in \R, u>0$ \qquad\qquad $ii) g(t,u,x) \ge l_2(u) +k_2'(t)u + k_2(t)x $ for $x\in \R, u<0$ \noi and \qquad\qquad $iii) l_1(u) \le a_1u+b_1$ and $ l_2(u) \ge a_2u+b_2$, with $$a_i(1+\widetilde c)\a + k_i(\a) < k_i(0)$$ \medskip Then (Per) admits at least one solution for any $f\in L^2$. Moreover, for every $f$ the set of solutions of (Per) is contained in a compact arc of $H^2(I)$. \demost{Proof} Under assumption A), we obtain for $s>0$ that $$l_1(u_s) \ge a_1u_s(t)+b_1 \ge a_1(1-\widetilde c) s - a_1\widetilde d +b_1$$ and $$l_2(u_{-s}) \le a_2u_{-s}(t)+b_2 \le -a_2(1-\widetilde c) s + a_2\widetilde d + b_2$$ Thus, it suffices to consider $$\varphi^+ \equiv a_1(1-\widetilde c) s - a_1\widetilde d +b_1, \qquad \varphi^- \equiv -a_2(1-\widetilde c) s + a_2\widetilde d +b_2$$ since in that case $\int_0^\a \varphi^+ + (k_1(\a)-k_1(0))s = \left( a_1(1-\widetilde c)\a +(k_1(\a)-k_1(0)\right) s +\a (- a_1\widetilde d +b_1)\to +\infty$ $\int_0^\a \varphi^- - (k_2(\a)-k_2(0))s = -\left( a_2(1-\widetilde c)\a +(k_2(\a)-k_2(0)\right) s +\a ( a_2\widetilde d +b_2)\to -\infty$ and conditions of corollary 7 hold taking $s$ large enough. Furthermore, the same computations show that every solution of (Per) belongs to the arc $\{ u_{\overline s}: -s\le \overline s\le s\}$. The proof is analogous under assumption B). \hfill\rect \lema {Remark:} The nonresonance (i.e. solvability of (Per) for any $f$) in Corollary 8 is due only to the fact that $I:\Cal C \to \R$ is onto. \rm \medskip In an similar way we can state the following existence result of periodic solutions of a Duffing equation $$u'' + g(u) = f(t,u,u') \tag{D}$$ \lema {Theorem 9} Let $g:\R\to \R$ be continuous such that $\frac {g(u)-g(v)}{u-v} \le c_1$, $|g(s)| \le c_2|s| + d$ with $c_1 + c_2\pi <\left( \frac \pi\a \right)^2$ and $f: \overline I\times \R^2\to \R$ continuous, bounded and continuously differentiable with respect to $u$ and $u'$. We'll assume that $\frac {\partial f}{\partial u'}\le 0$, $c_1 - \frac {\partial f}{\partial u} \le c_3$ for some constant $c_3< \left( \frac {\pi}\a\right)^2$ and that $$\lim_{x\to \pm \infty} g(x)sg(x) = +\infty \quad\text{ or } \quad \lim_{x\to \pm \infty} g(x)sg(x) = -\infty $$ Then (D) admits at least one $\a$-periodic solution. Moreover, the set of $\a$-periodic solutions of (D) is contained in a compact arc of $H^2(I)$. \demost{Proof} >From the previous theorem it will suffice to see that a version of theorem 6 holds also for this case. Indeed, from theorem 4 we have that the problem $$(*_s)\cases Su=f(t,u,u') \qquad \qquad\text { in } (0,\a) &\\ u(0)= u(\a) = s & \endcases $$ is solvable, and if $u,v$ are solutions of ($*_s$) we have by mean value theorem that $$Su - Sv - \frac {\partial f}{\partial u'}(t,\xi_1,\xi_2)(u-v)' -\frac {\partial f}{\partial u}(t,\xi_1,\xi_2)(u-v) = 0$$ for some $\xi\in L^\infty(I)\times L^\infty(I)$. If $\overline S u := Su - \frac {\partial f}{\partial u'}(t,\xi_1,\xi_2)u' -\frac {\partial f}{\partial u}(t,\xi_1,\xi_2)u$, then $\overline S$ satisfies the conditions of lemma 1. Hence $\| u - v\|_2 \le c\|\overline Su - \overline Sv\|_2 = 0$, and uniqueness for ($*_s$) follows. Then the curve $\Cal C = \{ u_s: u_s \text{ solves ($*_s$)}\}$ is well defined, and its continuity can be proved in the same way of theorem 5. As in the previous corollary, $I:\Cal C \to \R$ is onto, and $I(u_s)sg(s)\to +\infty$ (resp. $I(u_s)sg(s)\to -\infty$). As $f$ bounded, the proof is complete. \hfill\rect \tit{References} [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. [D] Dolph, C.L.: Nonlinear integral equations of the Hammerstein type, Trans. Amer. Math. Soc. 66 (1949), 289-307. [Du] Duffing, G.: Erzwungene Schwingungen bei ver\"anderlicher Eigenfrequenz, Vieweg. Braunschweig, 1918. [H] Hamel, G.: \"Uber erzwungene Schwingungen bei endlichen Amplituden. Math. Ann., 86 (1922), 1-13. [L] Lichtenstein, L.: \"Uber einige Existenzprobleme der Variationsrechnung. Methode der unendlichvielen Variabeln, J.Reine Angew. Math. 145 (1915), 24-85. [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. [P] Picard, E.: Sur l'application des m\'ethodes d'approximations succesives \`a l'\'etude de certaines \'equations diff\'erentielles ordinaires, J.Math. Pures Appl. 9 (1893), 217-271. [S] Severini, C.: Sopra gli integrali delle equazione differenziali del secondo ordine con valori prestabiliti in due punti dati, Atti R. Acc. Torino 40 (1904-5), 1035-40. \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 ---------------0004041715246--