Contractivity in certain D‐R spaces

Abstract

Let X be an arbitrary vector space (without topology) together with a pair of endomorphisms D , RCLO(X) such that D R = I . If the kernel Z of D is nontrivial, the product space X,: = )(Z has a corresponding pair of endomorphisms Do, Ro E L,(X,,) with DoRo = lo and we can find a homomorphic representation @: X -X o such that @D=DO@ and @R= Ro@ [3]. This certain problems in S, for instance abstract differential equations [I], can be transformed into associated problems in the representation space X, and analysed, taking advantage of the simpler structure of X, as a linear sequence space. In particular, the space X o is endowed with a nontrivial structural compatible topology such that X, is metrizable and Ex,, converges iff (2,) -0.Using an appropriate notion of contractivity

Fibre Contraction Principle with respect to an Iterative Algorithm

“…Actually, an -space is any set endowed with a structure implying a notion of convergence for sequences. For example, Hausdorff topological spaces, metric spaces, generalized metric spaces in Perov's sense (i.e., ( , ) ∈ R + ), generalized metric spaces in Luxemburg's sense (i.e., ( , ) ∈ R + ∪ {+∞}), -metric spaces (i.e., ( , ) ∈ , where is a cone in an ordered Banach space), gauge spaces, 2-metric spaces, --spaces ( [2,3]), probabilistic metric spaces, syntopogenous spaces are such -spaces. For more details see Fréchet [4], Blumenthal [5], and Rus [1].…”