- Open Access
Erratum to "Iterative Methods for Variational Inequalities over the Intersection of the Fixed Points Set of a Nonexpansive Semigroup in Banach Spaces"
© Issa Mohamadi. 2011
- Received: 22 February 2011
- Accepted: 24 February 2011
- Published: 14 March 2011
The original article was published in Fixed Point Theory and Applications 2010 2011:620284
In my recent published paper  to prove Lemmas 3.1 and 5.1, an inequality involving the single-valued normalized duality mapping from into has been used that generally turns out there is no certainty about its accuracy. In this erratum we fix this problem by imposing additional assumptions in a way that the proofs of the main theorems do not change.
It is well known that Hilbert spaces, and for , are 2-uniformly smooth.
Throughout the paper we suggest to impose one of the following conditions:
(a) the Banach space is 2-uniformly smooth;
(b) there exists a constant for which satisfies the following inequality:
for all .
If is -Lipschitzian, then satisfies (3) and is norm-to-norm uniformly continues that suffices to guarantee that is 2-uniformly smooth. For more results concerning -Lipschitzian normalized duality mapping see .
Note that since every uniformly smooth Banach space has a Gateaux differentiable norm and each nonempty, bounded, closed, and convex subset of has common fixed point property for nonexpansive mappings, we have in . So, when is 2-uniformly smooth, we can remove these two conditions from Theorems 3.2, 4.2, and 5.2 in .
Considering the above discussion to complete our paper, we reprove Lemmas 3.1 and 5.1 of  here with some little changes.
Lemma 3.1 (see ).
is a contraction on for every .
With no loss of generality we can take ; therefore, if , then we have ; that is, is a contraction, and the proof is complete.
Also Lemma 5.1, which is easily proved in the same way as Lemma 3.1, will be as follows.
Lemma 5.1 (see ).
is a contraction on .
With the new imposed conditions and considering the above lemmas, the following corrections should be done in :
(1)in Theorem 3.2 and Theorem 4.2, ;
(2)in Theorem 5.2, , where ;
(3)in Remark 5.3, , where .
Also in [1, Corollary 4.3] the real Banach space does not necessarily need to have a uniformly Gateaux differentiable norm.
To avoid any ambiguity in terminology note also that -strongly monotone mappings in Banach spaces are usually called -strongly accretive.
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