17-03-2014, 08:55 PM
Abstract
The aim of this paper is to study the semilocal convergence
of a third order iterative method used for finding fixed
points of nonlinear operator equations in Banach spaces.
This convergence is established under the assumption that
the first Fr´echet derivative of the involved operator satisfies
the Lipschitz continuity condition. The existence and
uniqueness regions along with a priori error bounds for a
fixed point are derived. The R−order of the method is also
shown to be equal to three. Finally, an integral equation is
worked out with our method and with a Newton-like method
and results are compared. It is observed that our method
gives superior existence and uniqueness regions.
Keywords: Stirling-like method, Lipschitz continuity condition,
Fr´echet derivative.
1. Introduction
The computation of fixed points of nonlinear operator
equations in Banach spaces is one of the most important
and challenging problems in numerical analysis and applied
mathematics. A large number of researchers [1–3, 7, 10, 11]
have studied this problem extensively and developed several
methods along with their convergence analysis. This has
motivated us to consider the problem of approximating a
fixed point x? of a nonlinear operator equation
x = F(x) (1)
where, F :
X ! X be a nonlinear operator on an
open convex subset
of a Banach space X with values in
itself. This problem is equivalent to finding solutions of the
nonlinear operator equations given by G(x) = x−F(x) = 0.
Recently, development of higher order multi points iterative
methods using information at a number of points
and without involving higher order derivatives have gained
importance to solve nonlinear equations in Banach spaces. It
is clear that higher the order of the method, higher will be the
rate of convergence. The convergence analysis of the multi
point methods [6, 9, 12] is established based on the Lipschitz
continuity condition on the second Fr´echet derivative of the
involved operator. It is not difficult to show that there are
certain problems in which the second Fr´echet derivative may
not exit or becomes unbounded. Ezquerro et al. [4, 5] used
a third order Newton-like method for solving (1) under the
assumption that the first Fr´echet derivative satisfies Lipschitz
continuity condition and established its convergence analysis
with R-order equal to three.
In this paper, the convergence analysis of a third order
Stirling-like method is established for finding a fixed point
in Banach spaces under the assumption that the first Fr´echet
derivative of the involved operator satisfies the Lipschitz
continuity condition. The existence and uniqueness regions
along with a priori error bounds for a fixed point are derived.
The R−order of the method is also shown to be equal to
three. An integral equation is worked out with our method
and the results are compared with those obtained in Ezquerro
et al [4] to show the efficacy of our work.
The aim of this paper is to study the semilocal convergence
of a third order iterative method used for finding fixed
points of nonlinear operator equations in Banach spaces.
This convergence is established under the assumption that
the first Fr´echet derivative of the involved operator satisfies
the Lipschitz continuity condition. The existence and
uniqueness regions along with a priori error bounds for a
fixed point are derived. The R−order of the method is also
shown to be equal to three. Finally, an integral equation is
worked out with our method and with a Newton-like method
and results are compared. It is observed that our method
gives superior existence and uniqueness regions.
Keywords: Stirling-like method, Lipschitz continuity condition,
Fr´echet derivative.
1. Introduction
The computation of fixed points of nonlinear operator
equations in Banach spaces is one of the most important
and challenging problems in numerical analysis and applied
mathematics. A large number of researchers [1–3, 7, 10, 11]
have studied this problem extensively and developed several
methods along with their convergence analysis. This has
motivated us to consider the problem of approximating a
fixed point x? of a nonlinear operator equation
x = F(x) (1)
where, F :
X ! X be a nonlinear operator on an
open convex subset
of a Banach space X with values in
itself. This problem is equivalent to finding solutions of the
nonlinear operator equations given by G(x) = x−F(x) = 0.
Recently, development of higher order multi points iterative
methods using information at a number of points
and without involving higher order derivatives have gained
importance to solve nonlinear equations in Banach spaces. It
is clear that higher the order of the method, higher will be the
rate of convergence. The convergence analysis of the multi
point methods [6, 9, 12] is established based on the Lipschitz
continuity condition on the second Fr´echet derivative of the
involved operator. It is not difficult to show that there are
certain problems in which the second Fr´echet derivative may
not exit or becomes unbounded. Ezquerro et al. [4, 5] used
a third order Newton-like method for solving (1) under the
assumption that the first Fr´echet derivative satisfies Lipschitz
continuity condition and established its convergence analysis
with R-order equal to three.
In this paper, the convergence analysis of a third order
Stirling-like method is established for finding a fixed point
in Banach spaces under the assumption that the first Fr´echet
derivative of the involved operator satisfies the Lipschitz
continuity condition. The existence and uniqueness regions
along with a priori error bounds for a fixed point are derived.
The R−order of the method is also shown to be equal to
three. An integral equation is worked out with our method
and the results are compared with those obtained in Ezquerro
et al [4] to show the efficacy of our work.