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Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. (English) Zbl 0895.47048

It is the object of the present paper to show that if \(X\) is a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable, \(C\) is a bounded closed convex subset of \(X\), and \(T:C\to C\) is a nonexpansive mapping, then for any initial data \(x_0\) in \(C\) the Ishikawa iterates \(\{x_n\}\) defined by \[ x_{n+1}= t_nT (s_nTx_n+ (1-s_n)x_n)+ (1-t_n)x_n, \qquad n=0,1,2,\dots, \] where \(\{t_n\}\) and \(\{s_n\}\) are chosen so that \(\sum_n t_n(1-t_n)\) diverges, \(\sum_n s_n(1-t_n)\) converges, and \(\varlimsup_n s_n<1\), converge weakly to a fixed point of \(T\). This generalizes a theorem of Reich.

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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