Application of iterative processes of \(R\)-order at least three to operators with unbounded second derivative (Q870229)

The paper investigates the iterative solution of a nonlinear operator equation in a Banach space. The authors try to present in a unified form a lot of well-known iterative methods (Chebyshev method, Halley method, super-Halley method, Euler method, Ostrowski method) by introducing a new linear operator. Thus, a family of Newton-like methods that includes the methods mentioned above is obtained. A semilocal convergence result for this family is stated and proved. The assumptions required for the second and for the third Fréchet-derivative of the operator are weaker. The convergence result is applied for solving a boundary value problem for a second order nonlinear ordinary differential equation.