The shooting method applied to a cyclic inequality

Abstract:

It is known that the cyclic sum \[ {S_n}({\mathbf {x}}) = \sum \limits _{i = 1}^{i = n} {{x_i}/({x_{i + 1}} + {x_{i + 2}})} \] where ${x_{n + 1}} = {x_1}$, ${x_{n + 2}} = {x_2}$, ${x_i} \geqslant 0$, $({x_{i + 1}} + {x_{i + 2}}) > 0$, can be made smaller than $n/2$ for $n \geqslant 24$. The value of $\lambda (n) = \lim \inf \;(n \to \infty )\;{S_n}/n$ is investigated by the shooting method for two-point boundary value problems, and the analytical result $\lim (n \to \infty )\;\lambda (n) \leqslant 0.49457$ is proved. The inherent difficulty in a straight-forward minimization of ${S_n}({\mathbf {x}})$ is mentioned.