On the generalized successive approximations method
The authors discuss the question of global convergence of the iterative process \(x_{n+1}=f(x_ n,x_{n-1}),\) \(n\in {\mathbb{N}}\), \(x\in [0,1]\) where f is a continuous function from \([0,1]^ 2\) into [0,1]. Any point \((x,y)\in [0,1]^ 2\) is (i) a fixed point of f if \(x=y=f(x,y)\) and (ii) a periodic point of f of period two if \(f(y,x)=x\neq y=f(x,y).\) The authors show that if f is decreasing with respect to the first variable, and there does not exist a point (x,y) of \([0,1]^ 2\) such that there are no periodic points, then one cannot find a point \((x,y)\in [0,1]^ 2\) such that \(f(y,x)\leq x<y\leq f(x,y).\) Use of this result shows that f has a unique fixed point. The authors then present necessary conditions for global convergence when \(f\in A_ 1\cap A_ 2\) and necessary and sufficient conditions for global convergence when \(f\in A_ 1\cap S\). \(A_ 1(A_ 2)\) is the set of all continuous functions from \([0,1]^ 2\) into [0,1] which are decreasing with respect to the first (second) variable and S is the set of all continuous functions from \([0,1]^ 2\) into [0,1] which have no periodic point of order two.
scientific article; zbMATH DE number 4109959