Iterative Algorithms with Variable Coefficients for Asymptotically Strict Pseudocontractions
© The Author(s). 2010
Received: 8 October 2009
Accepted: 22 January 2010
Published: 28 January 2010
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.
is called to be asymptotically nonexpansive  if there exists a sequence with and such that
In , Nakajo and Takahashi studied the iterative approximation of fixed points of nonexpansive mappings and proved the following strong convergence theorem.
Theorem 1 A.
Such algorithm in (1.4) is referred to be the (CQ) algorithm in , 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.  studied a family of nonexpansive mappings and gave some good strong convergence theorems. Kim and Xu  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.
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 , Takahashi et al. , and Kim and Xu , 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 .
2. Results and Proofs
Throughout this paper,
We divide the proof into five steps.
By (2.18), (2.24), and (2.26), we know that (2.25) holds.
Theorem 2.4 improves [5, Theorem ] since the condition that is satisfied and the boundedness of is dropped off.
The authors are very grateful to the referee for his/her valuable suggestions and comments. The work was supported partly by the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805). This work is dedicated to W. Takahashi.
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