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Iterative Algorithms with Variable Coefficients for Asymptotically Strict Pseudocontractions
Fixed Point Theory and Applications volume 2010, Article number: 948529 (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.
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
is called to be asymptotically nonexpansive  if there exists a sequence with and such that
is called to be an asymptotically -strict pseudocontraction, if there exist such that
for all and all integers
As , asymptotically -strict pseudocontraction is asymptotically nonexpansive.
In , 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
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 , 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.
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:
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 , 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,
(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.
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.
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1972,35(1):171–174. 10.1090/S0002-9939-1972-0298500-3
Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003,279(2):372–379. 10.1016/S0022-247X(02)00458-4
Martinez-Yanes C, Xu H-K: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis: Theory, Methods & Applications 2006,64(11):2400–2411. 10.1016/j.na.2005.08.018
Ge CS, Liang J: Convergence theorems of new Ishikawa iterative procedures with errors for multi-valued -hemicontractive mappings. Communications in Mathematical Analysis 2009,7(1):12–20.
Kim T-H, Xu H-K: Convergence of the modified Mann's iteration method for asymptotically strict pseudocontractions. Nonlinear Analysis: Theory, Methods & Applications 2008,68(9):2828–2836. 10.1016/j.na.2007.02.029
Marino G, Xu H-K: Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,329(1):336–346. 10.1016/j.jmaa.2006.06.055
Osilike MO, Udomene A, Igbokwe DI, Akuchu BG: Demiclosedness principle and convergence theorems for -strictly asymptotically pseudocontractive maps. Journal of Mathematical Analysis and Applications 2007,326(2):1334–1345. 10.1016/j.jmaa.2005.12.052
Qin XL, Cho YJ, Kang SM, Shang MJ: A hybrid iterative scheme for asymptotically -strict pseudocontractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):1902–1911. 10.1016/j.na.2008.02.090
Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2008,341(1):276–286. 10.1016/j.jmaa.2007.09.062
Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(1):45–57. 10.1016/j.na.2007.11.031
Zegeye H, Shahzad N: Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups. Nonlinear Analysis: Theory, Methods & Applications 2008,69(12):4496–4503. 10.1016/j.na.2007.11.005
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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Ge, C., Liang, J. & Xiao, T. Iterative Algorithms with Variable Coefficients for Asymptotically Strict Pseudocontractions. Fixed Point Theory Appl 2010, 948529 (2010). https://doi.org/10.1155/2010/948529
- Hilbert Space
- Point Theorem
- Iterative Algorithm
- Nonnegative Integer
- Variable Coefficient