Erratum to "Iterative Methods for Variational Inequalities over the Intersection of the Fixed Points Set of a Nonexpansive Semigroup in Banach Spaces"

In my recent published paper [1] 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.

We recall that a uniformly smooth Banach space is -uniformly smooth for if and only if there exists a constant such that, for all ,

(1)

for more details see [2]. Therefore, if , then there exists a constant such that

(2)

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:

(3)

for all .

Remark 1.1.

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 [3].

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 [1]. So, when is 2-uniformly smooth, we can remove these two conditions from Theorems 3.2, 4.2, and 5.2 in [1].

Considering the above discussion to complete our paper, we reprove Lemmas 3.1 and 5.1 of [1] here with some little changes.

Lemma 3.1 (see [1]).

Either let be a real Banach space, and let be the single-valued normalized duality mapping from into satisfing (3) or let be a 2-uniformly smooth real Banach space. Assume that is -strongly monotone and -Lipschitzian on . Then

(4)

is a contraction on for every .

Proof.

If satisfies (3), considering the inequality

(5)

for all , we have

(6)

Clearly, the same inequality holds if is a 2-uniformly smooth real Banach space. Thus, we obtain

(7)

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 [1]).

Either let be a real Banach space, and let be the single-valued normalized duality mapping from into satisfing (3), or let be a 2-uniformly smooth real Banach space. Assume that is -strongly monotone and -Lipschitzian on . If , where , then

(8)

is a contraction on .

With the new imposed conditions and considering the above lemmas, the following corrections should be done in [1]:

(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.