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Weak convergence of an iterative sequence for accretive operators in Banach spaces. (English) Zbl 1128.47056
The authors study the generalized variational inequality problem of finding \(u \in C\) such that \[ \langle Au, J(v-u) \rangle \geq 0\;\forall v \in C, \] where \(C\) is a nonempty closed convex subset of a smooth Banach space \(E\), \(A\) is an accretive operator of \(C\) into \(E\), \(J\) is the duality mapping of \(E\) into \(E^*\), and \(\langle \cdot, \cdot \rangle\) is the duality paring between \(E\) and \(E^*\). To solve this problem, the authors propose the following iterative scheme: \(x_1 =x \in C\) and \[ x_{n+1}=\alpha_{n}x_n +(1-\alpha_{n})Q_{C}(x_n -\lambda_{n}Ax_n) \] for \(n=1, 2, 3,\dots\), where \(Q_C\) is a sunny nonexpansive retraction from \(E\) onto \(C\), \(\{\alpha_n\}\) is a sequence in \([0,1]\), and \(\{\lambda_n\}\) is a sequence of real numbers.
For this iterative scheme, the authors establish a weak convergence result (Theorem 3.1) in a uniformly convex and \(2\)-uniformly smooth Banach space for an \(\alpha\)-inverse strongly accretive operator. Applications to finding a zero point of an inverse strongly accretive operator and to finding a fixed point of a strictly pseudocontractive mapping are given.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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