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Vietnam Journal of Mathematics 40:4(2012)
515-525
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Viscosity Approximation Method
for Lipschitzian Pseudocontraction Semigroups in
Banach Spaces
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Duong Viet Thong
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Faculty
of Economics Mathematics, National Economics
University, 207 Giai Phong,
Hai Ba
Trung, Hanoi, Vietnam
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October 27, 2010
April 22, 2012
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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 {αn}, {tn}
satisfy some appropriate conditions, the iterative process given by
xn=αnf(xn)+(1-αn)T(tn)xn for n \in N,
converges strongly to p\in F, which is
the unique solution in F to the
following variational inequality:
<(f-I)p,j(x-p)> ≤ 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].
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2000 Mathematics
Subject Classification. 47H09, 47H10, 47H20.
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Keywords.
Lipschitzian pseudocontraction semigroup, demiclosed principle, common
fixed point, Opial's condition, implicit iteration process.
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Established
by Vietnam Academy of Science and Technology &
Vietnam Mathematical Society
Published by
Springer since January 2013
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