
Vietnam Journal of Mathematics 40:4(2012)
515525

Viscosity Approximation Method
for Lipschitzian Pseudocontraction Semigroups in
Banach Spaces

Duong Viet Thong^{}

Faculty
of Economics Mathematics, National
Economics University, 207 Giai Phong,
Hai Ba Trung,
Hanoi, Vietnam

October 27, 2010
April 22, 2012^{}

Abstract. Let E
be a real Banach space which admits a weakly sequentially continuous
duality mapping from E to E*, and K be a nonempty closed convex subset of E. Let {T(t):t\geq 0} be a Lipschitzian pseudocontractive
semigroup on K such that
$F:=\underset{t\geq 0}{\cap}\Fix(T(t))\ne \emptyset,$ and $f:K\to K$ be a
fixed contractive mapping. When {α_{n}}, {t_{n}}
satisfy some appropriate conditions, the iterative process given by
x_{n}=α_{n}f(x_{n})+(1α_{n})T(t_{n})x_{n } for n \in N,
converges strongly to p\in F, which is
the unique solution in F to the
following variational inequality:
<(fI)p,j(xp)> ≤ 0 \forall x\in F.
Our results presented in this paper
extend and improve recent results of R. Chen and H. He [1], Y. Song and R.
Chen [8], Xu [13].

2000 Mathematics
Subject Classification. 47H09, 47H10, 47H20.

Keywords.
Lipschitzian pseudocontraction semigroup, demiclosed principle, common
fixed point, Opial's condition, implicit iteration process.


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by Vietnam Academy of Science and Technology &
Vietnam Mathematical Society
Published by
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