We study an implicit predictor-corrector iteration process for finitely many asymptotically quasi-nonexpansive self-mappings on a nonempty closed convex subset of a Banach space. We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. In the case is a uniformly convex Banach space and the mappings are asymptotically nonexpansive, we verify the weak (resp., strong) convergence of this iteration process to a common fixed point of these mappings if Opial's condition is satisfied (resp., one of these mappings is semicompact). Our results improve and extend earlier and recent ones in the literature.

Necessary conditions of optimality are derived for multiobjective optimal control problems with free end-time, in which the dynamics constraint is modeled as a nonconvex differential inclusion. The obtained results cover some previous results on necessary conditions for multiojective and single objective optimal control problems.

Let <i>T</i> be a compact disjointness preserving linear operator from <i>C</i>₀(<i>X</i>) into <i>C</i>₀(<i>Y</i>), where <i>X</i> and <i>Y</i> are locally compact Hausdorff spaces. We show that <i>T</i> can be represented as a norm convergent countable sum of disjoint rank one operators. More precisely, <i>T</i> = _<i>n</i> <i></i> <i>h_n</i> for a (possibly finite) sequence {<i>x_n</i> }_<i>n</i> of distinct points in <i>X</i> and a norm null sequence {<i>h_n</i> }_<i>n</i> of mutually disjoint functions in <i>C</i>₀(<i>Y</i>). Moreover, we develop a graph theoretic method to describe the spectrum of such an operator ( 2009 WILEYVCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Let be a locally compact Hausdorff space. We show that any local -linear map (where" local" is a weaker notion than -linearity) between Banach -modules are" nearly -linear" and" nearly bounded". As an application, a local -linear map between Hilbert -modules is automatically -linear. If, in addition, contains no isolated point, then any -linear map between Hilbert -modules is automatically bounded. Another application is that if a sequence of maps between two Banach spaces" preserve -sequences"(or" preserve ultra- -sequences"), then is bounded for large enough and they have a common bound. Moreover, we will show that if is a bijective" biseparating" linear map from a" full" essential Banach -module into a" full" Hilbert -module (where is another locally compact Hausdorff space), then is" nearly bounded"(in fact, it is automatically bounded if or contains no isolated point) and there exists a homeomorphism such that ( ).

We study disjointness preserving (quasi-) n-shift operators on C 0 (X), where X is locally compact and Hausdorff. When C 0 (X) admits a quasi-n-shift T, there is a countable subset of X= X{} equipped with a tree-like structure, called -tree, with exactly n joints such that the action of T on C 0 (X) can be implemented as a shift on the -tree. If T is an n-shift, then the -tree is dense in X and thus X is separable. By analyzing the structure of the -tree, we show that every (quasi-) n-shift on c 0 can always be written as a product of n (quasi-) 1-shifts. Although it is not the case for general C 0 (X) as shown by our counter examples, we can do so after dilation.