Then, the mapping is demiclosed at zero.
Then the mapping is demiclosed at zero, that is,, implies.
Then the mapping is demiclosed at zero, that is, imply that.
(ii If is a -strict pseudocontraction, then the mapping is demiclosed (at 0).
Since is a strict pseudocontraction mapping, by Lemma 2.1 we know that the mapping is demiclosed at zero.
Then the mapping is demiclosed on, where is the identity mapping; that is, in and imply that and.
(3.16) S is an asymptotical nonexpansive mapping, it is demiclosed at zero.
(Vert Wp-q Vert leq Vert p-q Vert ) for any (pin H), and (q in F(W)); this means that W is a quasi-nonexpansive mapping; W is demiclosed on H; that is, if ({zeta_{k}}subset H), (zeta_{k}rightharpoonupxi), and ((I-W zeta_{k} rI-W zeta_{k, then (xiin F(W)).
Since the k-demicontractive mapping S is demiclosed at θ 2, taking into account x n → q, A x n → A q, ∥ z n − x n ∥ → 0 and (3.13), we have A z n → A q. and A q ∈ F ( S ).
One of the fundamental and celebrated results in the theory of nonexpansive mappingsis Browder's demiclosed principle[1] which states that if X is a uniformly convex Banach space,C is a nonempty closed convex subset of X, and if is a nonexpansive nonself mapping, then is demiclosed at 0, that is, for any sequence in C if weakly and, then (where I is the identity mapping inX).
If (T : C rightarrow C) is a nonexpansive mapping with (F(T neq emptyset), then the mapping (I -T) is demiclosed at 0, i.e., if ({x_{n}}) is a sequence in C weakly converging to x, and if ({(I -T )x_{n}}) converges strongly to 0, then ((I -T )x = 0).
