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Vietnam Journal of Mathematics 40:4(2012) 515-525

 

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 {αn}, {tn} satisfy some appropriate conditions, the iterative process given by

xnnf(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].

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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