Comment on "A Strong Convergence of a Generalized Iterative Method for Semigroups of Nonexpansive Mappings in Hilbert Spaces"

Abstract

Piri and Vaezi (2010) introduced an iterative scheme for finding a common fixed point of a semigroup of nonexpansive mappings in a Hilbert space. Here, we present that their conclusions are not original and most parts of their paper are picked up from Saeidi and Naseri (2010), though it has not been cited.

Let be a semigroup and the Banach space of all bounded real-valued functions on with supremum norm. For each , the left translation operator on is defined by for each and . Let be a subspace of containing 1 and let be its topological dual. An element of is said to be a mean on if . Let be -invariant, that is, for each . A mean on is said to be left invariant if for each and . A net of means on is said to be asymptotically left invariant if for each and , and it is said to be strongly left regular if for each , where is the adjoint operator of . Let be a nonempty closed and convex subset of . A mapping is said to be nonexpansive if , for all . Then is called a representation of as nonexpansive mappings on if is nonexpansive for each and for each . The set of common fixed points of is denoted by .

If, for each , the function is contained in and is weakly compact, then, there exists a unique point of such that for each . Such a point is denoted by . Note that is a nonexpansive mapping of into itself and , for each .

Recall that a mapping with domain and range in a normed space is called -strongly accretive if for each , there exists such that

((1))

is called -strictly pseudocontractive if for each , there exists such that

((2))

for some .

In [1], Saeidi and Naseri established a strong convergence theorem for a semigroup of nonexpansive mappings, as follows.

Theorem 1 (Saeidi and Naseri [1]).

Let be a nonexpansive semigroup on such that . Let be a left invariant subspace of such that , and the function is an element of for each . Let be a left regular sequence of means on and let be a sequence in such that and . Let , and let be generated by the iterative algorithm

((3))

where:  is a contraction with constant and is strongly positive with constant (i.e., , for all ). Then, converges in norm to which is a unique solution of the variational inequality , . Equivalently, one has .

Afterward, Piri and Vaezi [2] gave the following theorem, which is a minor variation of that given originally in [1], though they are not cited [1] in their paper.

Theorem 2 (Piri and Vaezi [2]).

Let be a nonexpansive semigroup on such that . Let be a left invariant subspace of such that , and the function is an element of for each . Let be a left regular sequence of means on and let be a sequence in such that and . Let and be generated by the iteration algorithm

((4))

where:  is a contraction with constant and is -strongly accretive and -strictly pseudocontractive with , , and . Then, converges in norm to which is a unique solution of the variational inequality , . Equivalently, one has .

The following are some comments on Piri and Vaezi's paper.

(i)It is well known that for small enough 's, both of the mappings and in Theorems 1 and 2 are contractive with constants and , respectively. In fact what differentiates the proofs of these theorems is their use of different constants, and the whole proof of Theorem 1 has been repeated for Theorem 2.

(ii)In Hilbert spaces, accretive operators are called monotone, though, it has not been considered, in Piri and Vaezi's paper.

(iii)Repeating the proof of Theorem 1, one may see that the same result holds for a strongly monotone and Lipschitzian mapping. A -strict pseudocontractive mapping is Lipschitzian with constant .

(iv)The proof of Corollary 3.2 of Piri and Vaezi's paper is false. To correct, one may impose the condition .

(v)The constant , used in Theorem 2, should be chosen in .