一个差分方程解的存在性问题研究

A Study on Existence of Solution of Difference Equation

考虑差分方程x_(1)+a/x_(2)=x_(2)+a/x_(3)=…=x_(n-1)+a/x_(n)=x_(n)++a/x_(1),其中a∈R^(+),n>2,x_(1),x_(2),…,x_(n)(n>2)互不相等。借助一个递推多项式列,研究上述差分方程的解的性质,进而得出了其有解的存在性、通解的个数与通解的显式表达式。

Considering the difference equation x_(1)+a/x_(2)=x_(2)+a/x_(3)=…=x_(n-1)+a/x_(n)=x_(n)++a/x_(1),where,a∈R^(+),n>2,x_(1),x_(2),…,x_(n)(n>2)are not equal to each other.With a recursive polynomial column,the properties of the solutions of the above difference equation are studied,and then the existence of the solutions,the number of general solutions and the explicit expressions of the general solutions are obtained.