Fixed point theorems in locally convex spaces

Let C be a convex subset of a nuclear locally convex space that is also an -F-space. Suppose T:C → C is nonexpansive and {υ_n} is given by the Mann iteration process. It is shown that if {υ_n} is bounded, T has a fixed point. Also, a sequence {y_n} can be constructed such that y_n→y weakly where Ty = y. If C is a linear subspace and T is linear, then lim y_n = y.