On Convergence of Random Products of Mapping in Metric Spaces

Start: 10/31/2012 - 4:15pm
End  : 10/31/2012 - 5:15pm


Amine Khamsi, University of Texas, El Paso


 Many problems in mathematics and physical sciences uses a technique known as search for a common fixed point. Indeed let X be a real Hilbert space and suppose T1,...,TN are pairwise distinct self-mappings of some closed nonempty subset D of X. Suppose further that the set of fixed points, Fix(Ti) = {x ∈ D : Ti(x) = x}, of each mapping Ti is nonempty and that C = Fix(T1) ∩ ··· ∩ Fix(TN) ̸= ∅. The aim is to find a common fixed point of these mappings. One frequently employed approach is the following: Let r be a random mapping for {1,...,N} , i.e., a surjective mapping from N onto {1,...,N} that takes each value in {1,...,N} infinitely often. Then generate a random sequence (xn)n by taking x0 ∈ D arbitrary, and
xn+1 = Tr(n)(xn),
for all n ≥ 0. And hope that this sequence converges to some point in C. We also speak of a random or unrestricted product (resp. iteration). In general, this random product fails to have any good convergence behavior. The first positive results were done in the case when D = X and each mapping Ti, is the projection onto some closed convex nonempty subset Ci of X ; hence Fix(Ti) = Ci, i = 1,···,N. The problem of finding a common fixed point is then the famous Convex Feasibility Problem. Combettes (1997) gave some interesting applications of this problem. In this talk we will discuss the convergence of this random iterative sequence in metric spaces.

Millikan 134, Pomona College

Khamsi.pdf144.78 KB