- Research Article
- Open Access
Convergence of Iterative Sequences for Common Zero Points of a Family of -Accretive Mappings in Banach Spaces
© Yuan Qing et al. 2011
- Received: 21 November 2010
- Accepted: 8 February 2011
- Published: 1 March 2011
We introduce implicit and explicit viscosity iterative algorithms for a finite family of -accretive operators. Strong convergence theorems of the iterative algorithms are established in a reflexive Banach space which has a weakly continuous duality map.
- Banach Space
- Variational Inequality
- Convex Subset
- Iterative Algorithm
- Nonexpansive Mapping
where denotes the dual space of and denotes the generalized duality pairing. In the sequel, we denote a single-valued normalized duality mapping by .
We use to denote the unique fixed point of , which yields that . In 1967, Browder  proved the following theorem.
In a Hilbert space, as , converges strongly to a fixed point of , that is, closet to , that is, the nearest point projection of onto .
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.
where is a sequence of positive numbers tending to zero.
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.
exists for each in its unit sphere . It is said to be uniformly Fréchet differentiable (and is said to be uniformly smooth) if the limit in (2.1) is attained uniformly for .
for , , , where , then (see ).
where denotes the subdifferential in the sense of convex analysis. It also follows from (2.5) that is convex and .
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 .
Assume that has a weakly continuous duality map with the gauge .
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 ).
where and such that
(i) and ,
(ii)either or .
Then converges to zero.
Hence, we have .
It follows from Lemma 2.1 that is a positive-scalar multiple of . We, therefore, obtain that is a solution to (3.1).
which guarantees . So, (3.1) can have at most one solution. This completes the proof.
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
(ii) converges strongly to , where is the sequence generated by the implicity algorithm (1.17).
Then, converges strongly to , which solves the variational inequality (3.1).
where is a appropriate constant such that . In view of Lemma 2.4, we can obtain the desired conclusion easily. This completes the proof.
As an application of Theorems 3.1 and 3.2, we have the following results for a single mapping.
Then, converges strongly to a solution of the equations .
where is a sequence in which satisfies the following conditions: and . Also assume that
(ii) converges strongly to , where is the sequence generated by the implicity scheme (3.27) and .
converges strongly to a solution of the equation .
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