MathDB

For a given real number

a

and a positive integer

n

, prove that: i) there exists exactly one sequence of real numbers

x_0,x_1,\ldots,x_n,x_{n+1}

such that

\begin{cases} x_0=x_{n+1}=0,\\ \frac{1}{2}(x_i+x_{i+1})=x_i+x_i^3-a^3,\ i=1,2,\ldots,n.\end{cases}

ii) the sequence

x_0,x_1,\ldots,x_n,x_{n+1}

in i) satisfies

|x_i|\le |a|

where

i=0,1,\ldots,n+1

.
Liang Yengde

Problem Statement