Throughout this paper,

(i) means that converges weakly to

(ii) means that converges strongly to

(iii), that is, the weak -limit set of

(iv).

(v) is the set of nonnegative integers.

The following lemmas are basic (cf., e.g., [6] for Lemma 2.1, and [5] for Lemmas 2.2-2.3).

Lemma 2.1.

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

Lemma 2.2.

Let be a closed convex subset of a real Hilbert space , , and . Suppose that satisfies

and . Then

Lemma 2.3.

Let be a closed convex subset of a Hilbert space and an asymptotically -strict pseudocontraction. Then

for each , satisfies the Lipschitz condition:

where

if is a sequence in such that and

then

In particular,

is closed and convex so that the projection is well defined.

Theorem 2.4.

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:

where

the sequence is chosen so that , the positive real number is chosen so that , and is as in (1.3). Then converges strongly to .

Proof.

We divide the proof into five steps.

Step 1.

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 ,

Therefore,

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.

Step 2.

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

Step 3.

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

Step 4.

We prove that

By Lemma 2.3 and the definition of , we obtain

where

By (2.18), (2.24), and (2.26), we know that (2.25) holds.

Step 5.

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

Remark 2.5.

Theorem 2.4 improves [5, Theorem ] since the condition that is satisfied and the boundedness of is dropped off.

Theorem 2.6.

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:

where

the sequence is chosen so that and is as in (1.3). Then converges strongly to .

Proof.

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.