Two initial intervals converge to different minima of a periodic objective

By [F3, F4, F5], $F^{\mathrm{\prime }}(x)=\sin x$. Then [F6] makes the gradient continuous and $1$-Lipschitz; [F9] gives $F(x)\ge 0$, and [F3, F8] give $F(0)=F(2\pi )=0$. Thus the hypotheses of [F2] hold with $L=1$ and $\alpha =1$, so every trajectory has gradient norm tending to zero.

Suppose $x_0\in (0,\pi )$. If $0<x_t\le x_0$, [F7] gives $\sin x_t>0$, and [F6] with $v=0$ gives $\sin x_t\le x_t$; hence $0\le x_{t+1}=x_t-\sin x_t<x_t$. If $x_t=0$, it stays at $0$. Induction shows $0\le x_{t+1}\le x_t\le x_0$, so [F10] gives a limit $\mathrm{\ell }\in [0,x_0]$. Given $\epsilon >0$, convergence gives an index after which both $\mathrm{\mid }{x}_t-\mathrm{\ell }\mathrm{\mid }<\epsilon \mathrm{/}2$ and $\mathrm{\mid }{x}_{t+1}-\mathrm{\ell }\mathrm{\mid }<\epsilon \mathrm{/}2$; hence $\mathrm{\mid }{x}_t-x_{t+1}\mathrm{\mid }\to 0$ and, by the update, $\sin x_t\to 0$. Also [F6] gives $\mathrm{\mid }\sin x_t-\sin \mathrm{\ell }\mathrm{\mid }\le \mathrm{\mid }{x}_t-\mathrm{\ell }\mathrm{\mid }\to 0$, so $\sin \mathrm{\ell }=0$. By [F8], the only such zero in $[0,x_0]$ is $0$, so $\mathrm{\ell }=0$.

Suppose $x_0\in (\pi ,2\pi )$. If $x_0\le x_t<2\pi$, put $u_t=2\pi -x_t\in (0,\pi )$. By [F7, F8, F9], $\sin x_t=-\sin u_t<0$, and [F6] gives $-\sin x_t=\sin u_t\le u_t=2\pi -x_t$. Therefore $x_t<x_{t+1}=x_t-\sin x_t\le 2\pi$; if $x_t=2\pi$, it remains there. Induction gives a nondecreasing sequence bounded above by $2\pi$, so [F10] gives a limit $\mathrm{\ell }\in [x_0,2\pi ]$. Convergence implies $\mathrm{\mid }{x}_{t+1}-x_t\mathrm{\mid }\to 0$ by the same two-term estimate used in step 1.2, so $\sin u_t\to 0$. Since $u_t=2\pi -x_t\to 2\pi -\mathrm{\ell }$, [F6] gives $\mathrm{\mid }\sin u_t-\sin (2\pi -\mathrm{\ell })\mathrm{\mid }\le \mathrm{\mid }{u}_t-(2\pi -\mathrm{\ell })\mathrm{\mid }\to 0$, hence $\sin (2\pi -\mathrm{\ell })=0$. By [F8], the only such zero with $0\le 2\pi -\mathrm{\ell }<2\pi -x_0<\pi$ is $0$. Thus $\mathrm{\ell }=2\pi$.