Throughout this paper,
(i) means that converges weakly to
(ii) means that converges strongly to
(iii), that is, the weak -limit set of
(v) is the set of nonnegative integers.
The following lemmas are basic (cf., e.g.,  for Lemma 2.1, and  for Lemmas 2.2-2.3).
Let be a closed convex subset of a real Hilbert space . Given . Then if and only if
where is the unique point in with the property
Let be a closed convex subset of a real Hilbert space , , and . Suppose that satisfies
and . Then
Let be a closed convex subset of a Hilbert space and an asymptotically -strict pseudocontraction. Then
for each , satisfies the Lipschitz condition:
if is a sequence in such that and
is closed and convex so that the projection is well defined.
Let be a closed convex subset of a Hilbert space , an asymptotically -strict pseudocontraction for some , and Let be the sequence generated by the following CQ-type algorithm with variable coefficients:
the sequence is chosen so that , the positive real number is chosen so that , and is as in (1.3). Then converges strongly to .
We divide the proof into five steps.
We prove that is nonempty, convex and closed.
Clearly, both and are convex and closed, so is . Since is an asymptotically -strict pseudocontraction, we have by (1.3),
for all , , and all integers
By (2.9) and (2.11), we deduce that for each ,
Next, we prove by induction that
Obviously, , that is, (2.14) holds for . Assume that for some Then, (2.13) implies that and is well defined.
By Lemma 2.1, we get In particular, for each we have This together with the definition of , the inequality (2.14) holds for . So (2.14) is true.
We prove that
By the definition of and Lemma 2.1, we get Hence,
Denoting , we have for all and
where The definition of shows that , that is, This implies that
Thus is increasing. Since is bounded, exists and
We prove that
The definition of shows that , that is,
By (2.19) and the definition of in (2.9), we deduce that
Further, we have
Thus, (2.19) and (2.21) imply that
Noticing , we get
From and (2.22), it follows that
We prove that
By Lemma 2.3 and the definition of , we obtain
By (2.18), (2.24), and (2.26), we know that (2.25) holds.
Finally, by Lemma 2.3 and (2.25), we have . Furthermore, it follows from (2.16) and Lemma 2.2 that the sequence converges strongly to
Theorem 2.4 improves [5, Theorem ] since the condition that is satisfied and the boundedness of is dropped off.
Let be a closed convex subset of a Hilbert space , an asymptotically -strict pseudocontraction for some , and be nonempty and bounded. Let the sequence generated by the following CQ-type algorithm with variable coefficients:
the sequence is chosen so that and is as in (1.3). Then converges strongly to .
It is easy to see that in Theorem 2.6. Following the reasoning in the proof of Theorem 2.4 and using instead of , we deduce the conclusion of Theorem 2.6.