Itai Shafrir

Nonexpansive iterations in hyperbolic spaces

ONE OF THE most active research areas in nonlinear functional analysis is the asymptotics of nonexpansive mappings. Most of the results, however, have been obtained in normed linear spaces. It is natural, therefore, to try to develop a theory of nonexpansive iterations in more general infinite-dimensional manifolds. This is the purpose of the present paper. More specifically, we propose the class of hyperbolic spaces as an appropriate background for the study of operator theory in general, and of iterative processes for nonexpansive mappings in particular. This class of metric spaces, which is defined in Section 2, includes all normed linear spaces and Hadamard manifolds, as well as the Hilbert ball and the Cartesian product of Hilbert balls. In Section 3 we introduce co-accretive operators and their resolvents, and present some of their properties. In the fourth section we discuss the concept of uniform convexity for hyperbolic spaces. Section 5 is devoted to two new geometric properties of (infinite-dimensional) Banach spaces. Theorem 5.6 provides a characterization of Banach spaces having these properties in terms of nonlinear accretive operators. In Sections 6, 7 and 8 we study explicit, implict and continuous iterations, repectively, using the same approach in all three sections. We illustrate this common approach with the following special case. Let C be a closed convex subset of a hyperbolic space (X, p), let T: C --f C be a nonexpansive mapping, and let x be a point in C. In order to study the iteration (T”x: n = 0, 1,2, . . .), we set z,, = (1 (l/n))x 0 (l/n)T”x, K = clco(zj;j I l), and d = inf(p(y, Ty): y E C). The first step is to show that p(x, K) = lim p(x, T”x)/n = d. This leads to the convergence “+m of lz,) when X is uniformly convex and to the weak convergence of (z,,] when X is a Banach space which is reflexive and strictly convex. When T is an averaged mapping we are also able to establish the following triple equality. For all k 2 1,

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The approximate fixed point property in Banach and hyperbolic spaces

S2

Elliptic and parabolic problems

, on a bounded domain

On a discrete variational problem involving interacting particles

\Omega\subset\R2

Coaccretive operators and firmly nonexpansive mappings in the Hilbert ball

and on all of

Uniqueness of positive solutions for singular problems involving the p-laplacian

\R2

ASYMPTOTICS OF EIGENSTATES OF ELLIPTIC PROBLEMS WITH MIXED BOUNDARY DATA ON DOMAINS TENDING TO INFINITY

. In some cases we also address the question of existence of minimizers.

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