Abstract
In this paper, we approximate a fixed point of the semigroup of Lipschitzian mappings from a nonempty compact convex subset C of a smooth Banach space E into C with a uniform Lipschitzian condition and with respect to a finite family of sequences of left strong regular invariant means defined on an appropriate invariant subspace of . Our result extends the main results announced by several others. MSC:47H09, 47H10, 47J25.
Highlights
Let E be a real Banach space with the topological dual E∗, and let C be a nonempty closed and convex subset of E
If {xn}∞ n= is a sequence generated by x ∈ C and xn+ = αnf + βnxn + γnTμn xn, n ≥, the sequence {xn}∞ n= converges strongly to some z ∈ Fix(φ), the set of common fixed points of φ, which is the unique solution of the variational inequality (f – I)z, J(y – z) ≤, ∀y ∈ Fix(φ)
Theorem . [ ] Let S be a left reversible semigroup, and let φ = {Ts : s ∈ S} be a representation of S as a Lipschitzian mapping from a nonempty compact convex subset C of a smooth Banach space E into C, with uniform Lipschitzian constant lims K(s) ≤, and let f be an α-contraction on C for some < α
Summary
Let E be a real Banach space with the topological dual E∗, and let C be a nonempty closed and convex subset of E. [ ] Let S be a left reversible semigroup and φ = {Ts : s ∈ S} be a representation of S as a Lipschitzian mapping from a nonempty compact convex subset C of a smooth
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