Theorem 23.2.41C4C3411B46C3

Proof. Fix $k<0$, put $c=-k$, and let $f_0=\min F>0$ and $f_1=\max F$. The constant functions $f_0/c$ and $f_1/c$ are a subsolution and a supersolution, respectively, of $C'=C^{-m}(F-cC)$. By comparison, the forward flow of this equation preserves the compact interval $I_ k=[f_0/c,f_1/c]$ of positive numbers, and existence and uniqueness for smooth ordinary differential equations give a continuous time-two map $I_ k\to I_ k$. Its difference from the identity is nonnegative at the left endpoint and nonpositive at the right one, so the intermediate value theorem gives a fixed point; if $f_0=f_1$, the constant $C=f_0/c$ is the solution. Since the equation is periodic, the solution through a fixed point extends to a smooth positive periodic solution. It is unique by lemma 23.2.40 and satisfies

By lemma 23.2.40, the period $L(k)=-\int _0^2F/C_ k^{m+1}\, \mathrm dr$ is strictly increasing in $k$. It is also continuous. Indeed, for $k$ in a compact neighborhood of any $k_0<0$, (23.2.42) gives uniform positive lower and finite upper bounds for $C_ k$, and the equation then bounds all derivatives. Any sequence $k\to k_0$ therefore has smoothly convergent subsequences, and the equation and uniqueness identify every limit with $C_{k_0}$. Thus $C_ k\to C_{k_0}$ uniformly, which gives continuity of the integral.

The same bounds imply $L(k)\to 0$ as $k\uparrow 0$ and $L(k)\to -\infty $ as $k\to -\infty $. Hence exactly one $k$ has period $L_0$. For this $k$, set $D=F/C_ k^ m$. The identities of lemma 23.2.37 show that $C_ k\alpha +D\beta $ has the required properties. Conversely, every metric of the form (23.2.36) with these properties gives a positive periodic solution of the same equation, so the ordering of the periods also proves uniqueness. $\square$