© Yuan Qing et al. 2011
Received: 21 November 2010
Accepted: 8 February 2011
Published: 1 March 2011
We use to denote the unique fixed point of , which yields that . In 1967, Browder  proved the following theorem.
In , Moudafi proposed a viscosity approximation method which was considered by many authors [2–8]. If is a Hilbert space, is a nonexpansive mapping and is a contraction, he proved the following theorems.
For each , we denote by the resolvent of , that is, . Note that if is -accretive, then is nonexpansive and , for all . We also denote by the Yosida approximation of , that is, . It is known that is a nonexpansive mapping from to .
where is a real sequence , is a contractive mapping, and is a nonexpansive mapping with a fixed point. Strong convergence theorems of fixed points are obtained in a uniformly smooth Banach space; see  for more details.
Very recently, Zegeye and Shahzad  studied the common zero problem of a family of -accretive mappings. To be more precise, they proved the following result.
where is a real sequence which satisfies the following conditions: ; ; or and with for for and . If every nonempty, closed, bounded convex subset of has the fixed point property for a nonexpansive mapping, then converges strongly to a common solution of the equations for .
where with for , and is a real sequence in . It is proved that the sequence generated in the iterative algorithms (1.17) and (1.18) converges strongly to a common zero point of a finite family of -accretive mappings in reflexive Banach spaces, respectively.
for , , , where , then (see ).
In order to prove our main results, we also need the following lemmas.
The first part of the next lemma is an immediate consequence of the subdifferential inequality, and the proof of the second part can be found in .
Lemma 2.2 (see ).
Let be a Banach space satisfying a weakly continuous duality map, let be a nonempty, closed, convex subset of , and let be a nonexpansive mapping with a fixed point. Then, is demiclosed at zero, that is, if is a sequence in which converges weakly to and if the sequence converges strongly to zero, then .
Lemma 2.3 (see ).
Let be a nonempty, closed, convex subset of a strictly convex Banach space . Let , , be a family of -accretive mappings such that . Let be real numbers in such that and , where . Then, is nonexpansive and .
Lemma 2.4 (see ).
3. Main Results
We need the strong convergence of the implicit algorithm (1.17) to prove the strong convergence of the explicit algorithm (3.14).
Let be a strictly convex and reflexive Banach space which has a weakly continuous duality map with the gauge . Lek be a nonempty, closed, convex subset of and a contractive mapping. Let be a family of -accretive mappings with . Let for each . For any , let be generated by the algorithm (1.18), where with for , , and is a sequence in which satisfies the following conditions: and . Assume also that
As an application of Theorems 3.1 and 3.2, we have the following results for a single mapping.
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