Abstract
Implicit Mann process and Halpern-type iteration have been extensively studied by many others. In this paper, in order to find a common fixed point of a countable family of nonexpansive mappings in the framework of Banach spaces, we propose a new implicit iterative algorithm related to a strongly accretive and Lipschitzian continuous operatorF:xn=αnγV(xn)+βnxn-1+((1-βn)I-αnμF)Tnxnand get strong convergence under some mild assumptions. Our results improve and extend the corresponding conclusions announced by many others.
Highlights
Let X be a real q-uniformly smooth Banach space with induced norm ‖ ⋅ ‖, q > 1
We prove that the implicit iterative process (15) has strong convergence and find the unique solution xof variational inequality:
In Proposition 10, under the demiclosed assumption and combined with Proposition 9, we find the unique solution of a variational inequality
Summary
Let X be a real q-uniformly smooth Banach space with induced norm ‖ ⋅ ‖, q > 1. Xu [6] proved that under certain appropriate conditions on {αn}, the sequence {xn} generated by (5) converges strongly to the unique solution x∗ ∈ C of the variational inequality: ⟨(I − f)x∗, x − x∗⟩ ≥ 0, for all x ∈ Fix(T), (where C = Fix(T)) in Hilbert spaces as well as in some Banach spaces.
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