 
Summary: A family of third order Steffensen's
methods on Banach spaces
Determining the zeros of a function has attracted the attention of pure
and applied mathematicians for centuries. Solving equations numerically is
a classical problem. General problems may be formulated in terms of Ţnd
ing zeros. The roots of a nonlinear equation cannot in general be expressed
in closed form. Thus, in order to solve nonlinear equations, we have to use
approximate methods. One of the most important techniques to study these
equations is the use of iterative processes, starting from an initial approx
imation x0, called pivot, successive approaches (until some predetermined
convergence criterion is satisŢed) xi are computed , i = 1, 2, . . . , with the
help of certain iteration function : E E:
xi+1 := (xi), i = 0, 1, 2 . . . (1)
The advance of computational techniques has allowed the development
of some more complicated iterative methods in order to obtain greater order
of convergence.
For locating the root of a operator Steffensen's method is an iterative
process achieving quadratic convergence without employing derivatives.
In this lecture, we present a family of third order generalized Steffensen's
type methods. The main adventage of these methods is they do not need
