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Approximation of fixed points of nonexpansive mappings and solutions of variational inequalities. (English) Zbl 1149.47055

The main result of the paper is a convergence theorem for the iterative scheme
\[ x_{n+1}=(1-\sigma) x_n + \sigma [T x_n-\delta G(T x_n)],\quad n\geq 0, \]
used to approximate the unique solution of the variational inequality
\[ VI(G,K): \left\langle G x^*,j_q(y-x^*)\right\rangle \geq 0\quad \forall y \in K. \] Here \(E\) is real \(q\)-uniformly smooth Banach space, \(T:E\rightarrow E\), \(G:E\rightarrow E\) are self-mappings, and \(j_q(.)\) denotes the generalized duality mapping.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J40 Variational inequalities
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References:

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