Let and be a real -uniformly smooth Banach space, be a nonempty closed convex subset of and be a Lipschitz continuous mapping. Let and be bounded sequences in and and be real sequences in satisfying some restrictions. Let be the sequence generated from an arbitrary by the Ishikawa iteration process with errors: , , . Sufficient and necessary conditions for the strong convergence to a fixed point of is established.

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,

We show that a compact operator A is a multiple of a positive semi-definite operator if and only if (A B) W (A) W (B), for all (rank one) operators B. An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators A so that equivalence conditions hold.

The purpose of this paper is to construct two superimposed optimization methods for solving the mixed equilibrium problem and variational inclusion. We show that the proposed superimposed methods converge strongly to a solution of some optimization problem. Note that our methods do not involve any projection.

We describe the potential functions for N= 4B supersymmetric quantum mechanics with D (2, 1; ) symmetry.