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On the united theory of the family of Euler-Halley type methods with cubical convergence in Banach spaces. (English) Zbl 1057.65033
Let $E$ and $F$ be real or complex Banach spaces and $f$ a nonlinear twice differentiable operator from the subset $D\subseteq E$ into $F$. The family of Euler-Halley iterations with the parameter $\lambda\in [0,2]$ for solving an operator equation $f(x)= 0$ is defined as follows: $$x_{n+1}= T_{f,\lambda}(x_n)= x_n+ u_f(x_n)+ v_{f,\lambda}(x_n),\quad n= 0,1,\dots,$$ where $$\gather u_f(x)= - f'(x)\cdot f(x),\\ v_{f,\lambda}(x)= 1/2\cdot f'(x)^{-1}\cdot f''(x)\cdot u_f(x)\cdot Q_{f,\lambda}(x)\cdot u_f(x),\\ Q_{f,\lambda}(x)= (I+ \lambda/2\cdot f'(x)^{-1}\cdot f''(x)\cdot u_f(x))^{-1}.\endgather$$ This family includes as special cases the well-known Euler method $(\lambda= 0)$, the Halley method $(\lambda= 1)$, and the convex acceleration of Newton’s method or super-Halley method $(\lambda= 2)$. An operator $f$ is called to satisfy the center Lipschitz condition in the ball $B(x_0,r)$ with the $L$ average, where $L$ is a positive integrable function on the interval $[0,R]$ for some sufficient large number $R$, if $$\Vert f(x)- f(x_0)\Vert\le \int^{\rho(x)}_0 L(u)\,du\quad\text{for all }x\in B(x_0,r).$$ 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.

65J15Equations with nonlinear operators (numerical methods)
47J25Iterative procedures (nonlinear operator equations)