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A general viscosity approximation method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces. (English) Zbl 1169.47055

Summary: Let \(H\) be a real Hilbert space and \({\mathcal S}=\{T(s): 0\leq s<\infty\}\) be a nonexpansive semigroup on \(H\) such that \(F({\mathcal S})\neq\emptyset\). We consider a contraction \(f\) with coefficient \(0<\alpha<1\) and a strongly positive bounded linear operator \(A\) with coefficient \(\overline{\gamma}>0\). Let \(0<\gamma< \frac{\overline{\gamma}}{\alpha}\). It is proved that the sequences \(\{x_t\}\) and \(\{x_n\}\) generated by the iterative method
\[ x_t= t\gamma f(x_t)+(I-tA) \frac{1}{\lambda_t} \int_0^{\lambda_t} T(s)x_t\,ds, \]
and
\[ x_{n+1}= \alpha_n\gamma f(x_n)+(I-\alpha_nA) \frac{1}{t_n} \int_0^{t_n} T(s)x_n\,ds, \]
where \(\{t\},\{\alpha_n\}\subset (0,1)\) and \(\{\lambda_t\}\), \(\{t-n\}\) are positive real divergent sequences, converges strongly to a common fixed point \(\overline{x}\in F({\mathcal S})\) which solves the variational inequality \(\langle(\gamma f-A)\overline{x}, x-\overline{x}\rangle\leq 0\) for \(x\in F({\mathcal S})\).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H20 Semigroups of nonlinear operators
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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