Weak Convergence Theorems for a Countable Family of Strict Pseudocontractions in Banach Spaces
© Prasit Cholamjiak and Suthep Suantai. 2010
Received: 2 June 2010
Accepted: 16 September 2010
Published: 20 September 2010
We investigate the convergence of Mann-type iterative scheme for a countable family of strict pseudocontractions in a uniformly convex Banach space with the Fréchet differentiable norm. Our results improve and extend the results obtained by Marino-Xu, Zhou, Osilike-Udomene, Zhang-Guo and the corresponding results. We also point out that the condition given by Chidume-Shahzad (2010) is not satisfied in a real Hilbert space. We show that their results still are true under a new condition.
Let and be a real Banach space and the dual space of , respectively. Let be a nonempty subset of . Let denote the normalized duality mapping from into given by , for all , where denotes the duality pairing between and . If is smooth or is strictly convex, then is single-valued.
It is obvious by the definition that
(1)every strictly pseudocontractive mapping is pseudocontractive,
(2)every -strictly pseudocontractive mapping is -Lipschitzian; see .
It is worth mentioning that the class of strict pseudocontractions includes properly the class of nonexpansive mappings. Moreover, we know from  that the class of pseudocontractions also includes properly the class of strict pseudocontractions. A mapping is called accretive if, for all , there exists such that . It is also known that is accretive if and only if is pseudocontractive. Hence, a solution of the equation is a solution of the fixed point of . Note that if , then is -strictly accretive if and only if is -strictly pseudocontractive.
where . If is a nonexpansive mapping with a fixed point and the control sequence is chosen so that , then the sequence defined by (1.8) converges weakly to a fixed point of ( this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm  ). However, if is a Lipschitzian pseudocontractive mapping, then Mann iteration defined by (1.8) may fail to converge in a Hilbert space; see .
In 1967, Browder-Petryshyn  introduced the class of strict pseudocontractions and proved existence and weak convergence theorems in a real Hilbert setting by using Mann's iteration (1.8) with a constant sequence for all . Respectively, Marino-Xu  and Zhou  extended the results of Browder-Petryshyn  to Mann's iteration process (1.8). To be more precise, they proved the following theorem.
Theorem 1.4 (see ).
Let be a closed convex subset of a real Hilbert space . Let be a -strict pseudocontraction for some , and assume that admits a fixed point in . Let a sequence be the sequence generated by Mann's algorithm (1.8). Assume that the control sequence is chosen so that for all and . Then converges weakly to a fixed point of .
Meanwhile, Marino, and Xu raised the open question: whether Theorem 1.4 can be extended to Banach spaces which are uniformly convex and have a Fréchet differentiable norm. Later, Zhou  and Zhang-Su , respectively, extended the result above to -uniformly smooth and -uniformly smooth Banach spaces which are uniformly convex or satisfy Opial's condition.
Theorem 1.6 (see ).
which is a contradiction.
It is known that one can extend his result from a single strict pseudocontraction to a finite family of strict pseudocontractions by replacing the convex combination of these mappings in the iteration under suitable conditions. The construction of fixed points for pseudocontractions via the iterative process has been extensively investigated by many authors; see also [18–22] and the references therein.
Our motivation in this paper is the following:
(1)to modify the normal Mann iteration process for finding common fixed points of an infinitely countable family of strict pseudocontractions,
(2)to improve and extend the results of Chidume-Shahzad  from a real uniformly smooth and uniformly convex Banach space to a real uniformly convex Banach space which has the Fréchet differentiable norm.
In this paper, we prove the weak convergence of a Mann-type iteration process (1.16) in a uniformly convex Banach space which has the Fréchet differentiable norm for a countable family of strict pseudocontractions under some appropriate conditions. The results obtained in this paper improve and extend the results of Chidume-Shahzad , Marino-Xu , Osilike-Udomene , Zhou , and Zhang-Guo  in some aspects.
We will use the following notation:
In the sequel, we will need the following lemmas.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Let be a real uniformly convex Banach space, a nonempty, closed, and convex subset of , and a continuous pseudocontractive mapping. Then, is demiclosed at zero, that is, for all sequence with and it follows that .
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Let be a real uniformly convex Banach space with a Fréchet differentiable norm. Let be a closed and convex subset of and a family of -Lipschitzian self-mappings on such that and . For arbitrary , define for all . Then for every , exists, in particular, for all and , .
Lemma 2.6 (see ).
Let be a closed and convex subset of a smooth Banach space . Suppose that is a family of -strictly pseudocontractive mappings from into with and is a real sequence in such that . Then the following conclusions hold:
Lemma 2.8 (see ).
3. Main Results
This completes the proof.
So we obtain the following results.
Now, we prove our main result.
As a direct consequence of Theorem 3.5, Lemmas 2.7 and 2.8 we also obtain the following results.
Theorems 3.5 and 3.6 extend and improve Theorems and of Chidume-Shahzad  in the following senses:
(i)from real uniformly smooth and uniformly convex Banach spaces to real uniformly convex Banach spaces with Fréchet differentiable norms;
(ii)from finite strict pseudocontractions to infinite strict pseudocontractions.
Using Opial's condition, we also obtain the following results in a real reflexive Banach space.
This is a contradiction, and thus the proof is complete.
The authors would like to thank the referees for valuable suggestions. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, and the Thailand Research Fund, Thailand. The first author is supported by the Royal Golden Jubilee Grant PHD/0261/2551 and by the Graduate School, Chiang Mai University, Thailand.
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