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On the iterative process \(x_{n+1}=f(x_ n,x_{n-1})\). (English) Zbl 0707.26002

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.

MSC:

26A18 Iteration of real functions in one variable
54H25 Fixed-point and coincidence theorems (topological aspects)
65H05 Numerical computation of solutions to single equations
90C30 Nonlinear programming
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References:

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