Abstract
Abstract In this paper, we propose two new algorithms for finding a common fixed point of a nonexpansive semigroup in Hilbert spaces and prove some strong convergence theorems for nonexpansive semigroups. Our results improve and generalize the corresponding results given by Shimizu and Takahashi (J. Math. Anal. Appl. 211:71-83, 1997), Shioji and Takahashi (Nonlinear Anal. TMA 34:87-99, 1998), Lau et al. (Nonlinear Anal. TMA 67:1211-1225, 2007) and many others. MSC:47H05, 47H10, 47H17.
Highlights
Let H be a real Hilbert space with the inner product ·, · and the norm ·
We denote by Fix(T(s)) the set of fixed points of T(s) and by Fix(S) the set of all common fixed points of S, i.e., Fix(S) = s≥ Fix(T(s))
Approximation of fixed points of nonexpansive mappings by a sequence of finite means has been considered by many authors; see, for instance, [ – ]
Summary
Let H be a real Hilbert space with the inner product ·, · and the norm ·. Recall that a family S := {T(s)}s≥ of mappings of C into itself is called a nonexpansive semigroup if it satisfies the following conditions: (S ) T( )x = x for all x ∈ C; (S ) T(s + t) = T(s)T(t) for all s, t ≥ ; (S ) T(s)x – T(s)y ≤ x – y for all x, y ∈ C and s ≥ ; (S ) for each x ∈ H, s → T(s)x is continuous. It is known that Fix(S) is closed and convex [ , Lemma ].
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
