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Modified Mann iterations for nonexpansive semigroups in Banach space. (English) Zbl 1181.47067

Summary: Let \(E\) be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from \(E\) to \(E^*\), and \(C\) be a nonempty closed convex subset of \(E\). Let \(\{T(t): t \geq 0\}\) be a nonexpansive semigroup on \(C\) such that \(F:= \cap_{t\geq 0} \text{Fix}(T(t)) \neq \emptyset\), and \(f: C \rightarrow C\) be a fixed contractive mapping. If \(\{\alpha_n\}, \{\beta_n\}, \{a_n\}, \{b_n\}, \{t_n\}\) satisfy certain appropriate conditions, then we suggest and analyze the two modified iterative processes as:
\[ \begin{cases} y_n = \alpha _n x_n + (1 - \alpha _n )T(t_n )x_n, \\ x_n = \beta _n f(x_n ) + (1 - \beta _n )y_n;\end{cases} \]
\[ \begin{cases} u_0 \in C, \\ v_n = a_n u_n + (1-a_n) T (t_n) u_n, \\ u_{n+1}=b_n f(u_n) + (1-b_n)v_n. \end{cases} \]
We prove that the approximate solutions obtained from these methods converge strongly to \(q\in \bigcap_{t\geq 0}\text{Fix}(T(t))\), which is a unique solution in \(F\) to the following variational inequality:
\[ \langle(I-f)q,j(q-u)\rangle\leq 0\quad \forall u\in F. \]
Our results extend and improve the corresponding ones of T.Suzuki [Proc.Am.Math.Soc.131, No.7, 2133–2136 (2003; Zbl 1031.47038)] and T.H.Kim and H.-K.Xu [Nonlinear Anal., Theory Methods Appl.61, No.1–2 (A), 51–60 (2005; Zbl 1091.47055)] and R.D.Chen and R.M.He [Appl.Math.Lett.20, No.7, 751–757 (2007; Zbl 1161.47049)].

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H20 Semigroups of nonlinear operators
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
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References:

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