Approximate fixed points of nonexpansive mappings in unbounded sets

It follows from Banach’s fixed point theorem that every nonexpansive self-mapping of a bounded, closed and convex set in a Banach space has approximate fixed points. This is no longer true, in general, if the set is unbounded. Nevertheless, as we show in the present paper, there exists an open and everywhere dense set in the space of all nonexpansive self-mappings of any closed and convex (not necessarily bounded) set in a Banach space (endowed with the natural metric of uniform convergence on bounded subsets) such that all its elements have approximate fixed points.     

Generic power convergence of operators in banach spaces

Let K be a closed convex subset of a Banach space X. We consider complete metric spaces of self-mappings of K which are nonexpansive with respect to a convex function on X. We prove that the iterates of a generic operator in these spaces converge strongly. In some cases the limits do not depend on the initial points and are the unique fixed point of the operator.     

Generic Properties of Continuous Differential Inclusions and the Tonelli Method of Approximate Solutions

We study Cauchy problems for differential inclusions in Banach spaces and show that most such problems (in the sense of Baire’s categories) have solutions. We consider separately the cases where the point images of the right-hand side are compact and convex, and where they are merely bounded, closed and convex.     

Xanthones and Oxepino[2, 3‐b]chromones from Three Endophytic Fungi

Three new metabolites, microsphaeropsones A–C ( 1 – 3 ) with a unique oxepino[2,3‐b]chromen‐6‐one (ring‐enlarged xanthone) skeleton, were isolated from the endophytic fungus Microsphaeropsis species, co‐occurring with their putative biogenetic anthraquinoide precursors citreorosein ( 4 ) and emodin ( 5 ). From another Microsphaeropsis species, large amounts of fusidienol A ( 8 a ), smaller amounts of emodin ( 5 ), the known aromatic xanthones 9 a and 9 b , the new 3,4‐dihydrofusidienol A ( 8 b ), and the new aromatic xanthone 9 c were isolated. The endophyte Seimatosporium species produced a new aromatic xanthone, seimatoxanthone A ( 10 ), and 3,4‐dihydroglobosuxanthone A ( 12 ), closely related to α‐diversolonic ester ( 13 ) from Microdiplodia sp.. The structures were determined mainly by extensive 1D and 2D NMR experiments and supported by X‐ray single‐crystal analysis of 1 and the oxidation product 7 . The absolute configurations of the microsphaeropsones A–C ( 1 – 3 ) were established by comparison of the electronic and vibrational circular dichroism (ECD and VCD) spectra of 1 with time‐dependent DFT (TDDFT) and DFT calculations by using either the solid‐state structures or DFT‐optimized geometries as inputs. Preliminary studies indicated that 1 , 2 , and enone 7 showed antibacterial, fungicidal, and algicidal properties.     

Asymptotic behavior of resolvents of coaccretive operators in the Hilbert ball

We study the resolvents of coaccretive operators in the Hilbert ball, with special emphasis on the asymptotic behavior of their compositions and metric convex combinations. We consider the case where the given coaccretive operators share a common fixed point inside the ball, as well as the case where they share a common sink point on its boundary. We establish weak convergence in the former case and strong convergence in the latter. We also present two related convergence results for a continuous implicit scheme.