TRANSACTIONSOF THE AMERICANMATHEMATICAL SOCIETY Volume 196, 1974
FIXED POINTITERATIONSUSINGINFINITEMATRICES BY
B. E. RHOADES ABSTRACT. Let £ be a closed,
space
X, /: E —»E. Consider
bounded,
the iteration
convex
subset
scheme defined
of a Banach
by x"« = xQ e E,
x n + ,l = 'ñx n ), x n = 2"* = na ,x., nal, where A is a regular weighted mean 0 nk k o er matrix. For particular spaces X and functions /we show that this iterative scheme converges
to a fixed point of /.
Let X be a normed linear space,
set of X, /: E —»£ possessing matrix.
Given the iteration
E a nonempty closed
bounded, convex sub-
at least one fixed point in E, and A an infinite
scheme
(1)
*o = *oeE'
(2)
*n+l=^*J»
« = 0,1,2,...,
fl
(3)
xn = F a nK,x.,« *-*Ä
«=1,2,
3, •••»
fe=0
it is reasonable
sufficient
to ask what restrictions
to guarantee
on the matrix A are necessary
that the above iteration
scheme converges
and/or
to a fixed point
of/. During the past few years several
iteration
schemes
this paper we establish
generalizations
point out some of the duplication An infinite
/. A matrix is called
sequences; triangular
if it is triangular
shall confine our attention
(4)
by the editors
of several
of infinite
matrices.
of these results
In
as well as
regular if it is limit preserving
over c, the
i.e., if x e c, xn —» /, then Aß(x)=
2fcan^xfc—»
if it has only zeros above the main diagonal,
and all of its main diagonal
to regular triangular
0
-n*n?W 
