On the iterative process \(x_{n+1}=f(x_ n,x_{n-1})\) (Q919467)

Summary: In this paper we consider the iterative process \(x_{n+1}=f(x_ n,x_{n-1})\), \(n\in N\), where f is a continuous function from \([0,1]^ 2\) in \([0,1]\) and we prove that the condition of the non existence of a pair \((x,y)\) of distinct points of \([0,1]\) such that \(f(x,y)=y\) and \(f(y,x)=x,\) obviously necessary for the global convergence, is sufficient if f is decreasing with respect to both variables and whatever the point \((x_ 1,x_ 0)\) of \([0,1]^ 2\) be, the following implications: \[ \max \{x_ 0,x_ 1\}<x_ 2\Rightarrow \min \{x_ 0,x_ 1\}<\min \{x_ 3,x_ 4\}; \] \[ x_ 2<\min \{x_ 0,x_ 1\}\Rightarrow \max \{x_ 3,x_ 4\}<\max \{x_ 0,x_ 1\}, \] are true.