Comment on "A Strong Convergence of a Generalized Iterative Method for Semigroups of Nonexpansive Mappings in Hilbert Spaces"
© Farman Golkarmanesh and Saber Naseri. 2011
Received: 23 January 2011
Accepted: 3 March 2011
Published: 15 March 2011
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 .
In , Saeidi and Naseri established a strong convergence theorem for a semigroup of nonexpansive mappings, as follows.
Theorem 1 (Saeidi and Naseri ).
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 .
Theorem 2 (Piri and Vaezi ).
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.
- Saeidi S, Naseri S: Iterative methods for semigroups of nonexpansive mappings and variational inequalities. Mathematical Reports 2010,12(62)(1):59–70.MathSciNetMATHGoogle Scholar
- Piri H, Vaezi H: A strong convergence of a generalized iterative method for semigroups of nonexpansive mappings in Hilbert spaces. Fixed Point Theory and Applications 2010, 2010:-16.Google Scholar
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