Global iteration schemes for strongly pseudo-contractive maps. (English) Zbl 0901.47046

Summary: Suppose \(E\) is a real uniformly smooth Banach space, \(K\) is a nonempty closed convex and bounded subset of \(E\), and \(T:K\to K\) is a strong pseudo-contraction. It is proved that if \(T\) has a fixed point in \(K\) then both the Mann and the Ishikawa iteration process, for an arbitrary initial vector in \(K\), converge strongly to the unique fixed \(T\). No continuity assumption is necessary for this convergence. Moreover, our iteration parameters are independent of the geometry of the underlying Banach space and of any property of the operator.


47J25 Iterative procedures involving nonlinear operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
47J05 Equations involving nonlinear operators (general)
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