Semilocal convergence on a family of root-finding multi-point methods in Banach spaces under relaxed continuity condition (Q513655)

The article deals with the following iterative scheme \[ \begin{aligned} x_{n+1} & = z_n - \Gamma_n[2F'(x_n) - F'(v_n)]\Gamma_nF(z_n), \\ z_n & = y_n - F'(v_n)^{-1}F(y_n), \\ y_n & = v_n - \bigg[\frac12 \, P(x_n) + \delta P(x_n)^2\bigg]\Gamma_nF(x_n)\end{aligned} \] \[ (\Gamma_n = F'(x_n)^{-1}, \quad P(x_n) = \Gamma_n[F'(x_n) - F'(v_n)], \quad v_n = x_n - \Gamma_nF(x_n), \quad \delta \in [-1,1]) \] of approximate solving the nonlinear operator equation \(F(x) = 0\) with the operator \(F\) between Banach spaces \(X\) and \(Y\). Under some cumbersome (however natural) conditions it is proved that the sequence \(x_n\) converges to a solution \(x_*\) of \(F(x) = 0\) with the order \(6 + q\) (\(q\) is a given number); the authors state that the method is more efficient than classical Chebyshev and Halley ones. In the end of the article there are two numerical examples that illustrate the method under consideration. The reviewer could not understand what the square in the term \(\bigg[\frac12 \, P(x_n) + \delta P(x_n)^2\bigg]\) means.