Projections of the sphere graph to the arc graph of a surface

Let [Formula: see text] be a compact surface, and [Formula: see text] be the double of a handlebody. Given a homotopy class of maps from [Formula: see text] to [Formula: see text] inducing an isomorphism of fundamental groups, we describe a canonical uniformly Lipschitz retraction of the sphere graph of [Formula: see text] to the arc graph of [Formula: see text]. We also show that this retraction is a uniformly bounded distance from the nearest point projection map.

Implicit iteration process of nonexpansive non-self-mappings

SupposeCis a nonempty closed convex subset of real Hilbert spaceH. LetT:C→Hbe a nonexpansive non-self-mapping andPis the nearest point projection ofHontoC. In this paper, we study the convergence of the sequences{xn},{yn},{zn}satisfyingxn=(1−αn)u+αnT[(1−βn)xn+βnTxn],yn=(1−αn)u+αnPT[(1−βn)yn+βnPTyn], andzn=P[(1−αn)u+αnTP[(1−βn)zn+βnTzn]], where{αn}⊆(0,1),0≤βn≤β<1andαn→1asn→∞. Our results extend and improve the recent ones announced by Xu and Yin, and many others.