Convergence of the variants of the Chebyshev-Halley iteration family under the Hölder condition of the first derivative
Let \(X,Y\) be (real or complex) Banach spaces, \(\Omega\subseteq X\) be an open subset and let \(F:\Omega\to Y\) be a nonlinear operator. Assume that the first derivative \(F'\) satisfies a Hölder condition like \(\|F'(x)-F'(y)\|\leq K\|x-y\|^p\), for all \(x,y\in B(x_0,R)\) (= a suitable closed ball in \(\Omega\)) and some \(p\in (0,1]\). For all such pairs \((x,y)\) define the linear operators \(H(x,y)=(1/\lambda)(F'(x))^{-1}[F'(x+\lambda(y-x))-F'(x)]\), \(Q(x,y)=-(1/2)H(x,y)[I+\alpha H(x,y)]^{-1}\); here, \(\lambda\in (0,1]\) and \(\alpha\in [0,1]\) are parameters. The variant of the Chebyshev-Halley iteration family with initial point \(x_0\) is introduced as (\(y_n=x_n-F'(x_n)^{-1}F(x_n)\), \(n\geq 0\); \(x_{n+1}=y_n+Q(x_n,y_n)(y_n-x_n)\), \(n\geq 0\)). Under the precise conditions, it is established that the sequence of iterates \((x_n)\) converges at a rate of order \(1+p\) to the unique solution \(x^*\) of \(F(x)=0\) in \(B(x_0,R)\). An application of these results to Hammerstein equations is also provided to illustrate the method.
scientific article; zbMATH DE number 5141783