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In this paper, we first prove a general fixed point theorem for nonlinear mappings in a Banach space. Then we prove a nonlinear mean convergence theorem of Baillons type and a weak convergence theorem of Manns type for 2-generalized nonspreading mappings in a Banach space.

Let T be a locally compact Hausdorff space, called base space. Suppose for each t in T there is a (real or complex) Banach space Et. A vector field x is an element in the product space t T Et, that is, x (t) Et, for all t T.

Let A , A be two regular commutative unital Banach algebras such that A is integral over A . In 2003, Dawson and Feinstein showed that the topological stable rank A whenever A . In this note, we investigate whether we will have A in general. For instance, when A is a commutative unital A -algebra, we show that A , and the equality holds at least when the integral extension is separable. In general, A and A have the same Bass stable ranks A .

Let A be a A -algebra with real rank zero. Let A and A be Hilbert A -modules with A being full. Suppose that A is a linear map preserving orthogonality, ie,