Let and be real or complex Banach spaces and a nonlinear twice differentiable operator from the subset into . The family of Euler-Halley iterations with the parameter for solving an operator equation is defined as follows:
This family includes as special cases the well-known Euler method , the Halley method , and the convex acceleration of Newton’s method or super-Halley method .
An operator is called to satisfy the center Lipschitz condition in the ball with the average, where is a positive integrable function on the interval for some sufficient large number , if
For the operators satisfying this condition the united convergence theorem for the family of Euler-Halley iterations is guaranteed and the cubical speed of such iteration is proved.