Fixed Point Iterations Using Infinite Matrices
نویسنده
چکیده
Let £ be a closed, bounded, convex subset of a Banach space X, /: E —»E. Consider the iteration scheme defined by x"« = xQ e E, x , = ñx ), x = 2" na ,x., nal, where A is a regular weighted mean n + l ' n n * = 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, E a nonempty closed bounded, convex subset of X, /: E —» £ possessing at least one fixed point in E, and A an infinite matrix. Given the iteration scheme (1) *o = *oeE' (2) *n + l=^*J» « = 0,1,2,..., fl (3) x = F a ,x., «=1,2, 3, •••» n *-*Ä nK « fe=0 it is reasonable to ask what restrictions on the matrix A are necessary and/or sufficient to guarantee that the above iteration scheme converges to a fixed point of/. During the past few years several mathematicians have obtained results using iteration schemes of the form (1 )—(3) for certain classes of infinite matrices. In this paper we establish generalizations of several of these results as well as point out some of the duplication and overlap of the work of these authors. An infinite matrix A is called regular if it is limit preserving over c, the space of convergent sequences; i.e., if x e c, xn —» /, then Aß(x)= 2fcan^xfc—» /. A matrix is called triangular if it has only zeros above the main diagonal, and a triangle if it is triangular and all of its main diagonal entries are nonzero. We shall confine our attention to regular triangular matrices A satisfying (4) 0<a . <1, », *-0, 1, 2, •••, Received by the editors November 21, 1972 and, in revised form, October 1, 1973. AMS (MOS) subject classifications (1970). Primary 26A18, 40G05, 40G99, 47H10.
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