Iterative Algorithms with Variable Coefficients for Asymptotically Strict Pseudocontractions

Abstract

We introduce and study some new CQ-type iterative algorithms with variable coefficients for asymptotically strict pseudocontractions in real Hilbert spaces. General results for asymptotically strict pseudocontractions are established. The main result extends the previous results.

1. Introduction

Let be a real Hilbert space, a nonempty closed convex subset of , a self-mapping of and

Recall that a mapping is called to be nonexpansive if

(1.1)

is called to be asymptotically nonexpansive [1] if there exists a sequence with and such that

(1.2)

is called to be an asymptotically -strict pseudocontraction, if there exist such that

(1.3)

for all and all integers

As , asymptotically -strict pseudocontraction is asymptotically nonexpansive.

In [2], Nakajo and Takahashi studied the iterative approximation of fixed points of nonexpansive mappings and proved the following strong convergence theorem.

Theorem 1 A.

Let be a nonempty closed convex subset of a Hilbert space and let be a nonexpansive mapping of into itself such that . Suppose is given by

(1.4)

where is the metric projection from C onto and is chosen so that Then, converges strongly to , where is the metric projection from C onto

Such algorithm in (1.4) is referred to be the (CQ) algorithm in [3], due to the fact that each iterate is obtained by projecting onto the intersection of the suitably constructed closed convex sets and It is known that the (CQ) algorithm in (1.4) is of independent interest, and the (CQ) algorithm has been extended to various mappings by many authors (cf., e.g., [3–11]).

Very recently, by extending the (CQ) algorithm, Takahashi et al. [9] studied a family of nonexpansive mappings and gave some good strong convergence theorems. Kim and Xu [5] extended the (CQ) algorithm to study asymptotically -strict pseudocontractions and established the following interesting result with the help of some boundedness conditions.

Theorem 1 B.

Let be a closed convex subset of a Hilbert space and let be an asymptotically -strict pseudocontractions for some Assume that the fixed point set of is nonempty and bounded. Let be the sequence generated by the following (CQ) algorithm:

(1.5)

where

(1.6)

Assume that control sequence is chosen so that Then converges strongly to .

It is our purpose in this paper to try to obtain some new fixed point theorems for asymptotically strict pseudocontractions without the boundedness conditions as in Theorem B. Motivated by Nakajo and Takahashi [2], Takahashi et al. [9], and Kim and Xu [5], we introduce and study certain new CQ-type iterative algorithms with variable coefficients for asymptotically strict pseudocontractions in real Hilbert spaces. Our results improve essentially the corresponding results of [5].

2. Results and Proofs

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

(2.1)

where is the unique point in with the property

(2.2)

Lemma 2.2.

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

(2.3)

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:

(2.4)

where

(2.5)

if is a sequence in such that and

(2.6)

then

(2.7)

In particular,

(2.8)

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:

(2.9)

where

(2.10)

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),

(2.11)

for all , , and all integers

By (2.9) and (2.11), we deduce that for each ,

(2.12)

Therefore,

(2.13)

Next, we prove by induction that

(2.14)

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,

(2.15)

Denoting , we have for all and

(2.16)

where The definition of shows that , that is, This implies that

(2.17)

Thus is increasing. Since is bounded, exists and

(2.18)

Step 3.

We prove that

The definition of shows that , that is,

(2.19)

By (2.19) and the definition of in (2.9), we deduce that

(2.20)

Further, we have

(2.21)

Thus, (2.19) and (2.21) imply that

(2.22)

Noticing , we get

(2.23)

From and (2.22), it follows that

(2.24)

Step 4.

We prove that

(2.25)

By Lemma 2.3 and the definition of , we obtain

(2.26)

where

(2.27)

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:

(2.28)

where

(2.29)

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.