Global convergence of the method of successive approximations on \(S^ 1\)
Let \(S^ 1\) denote the unit circumference and \(C(S^ 1,S^ 1)\) the class of continuous functions defined in \(S^ 1\) with range in \(S^ 1\). The author considers the recursion defined by \(z_{n+1}=f(z_ n),\) where \(z_ 0\in S^ 1\) and \(f\in C(S^ 1,S^ 1)\). He shows that the sequence \(z_ n\) is convergent for every choice of \(z_ 0\) if and only if (i) f has at least one fixed point, (ii) f has no periodic points of even period.
scientific article; zbMATH DE number 3903216
scientific article