• Bulletin of the Korean Mathematical Society
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• v.37 no.1
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• pp.153-169
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• 2000

Let E be a real q-uniformly smooth Banach space which admits a weakly sequentially continuous duality map, and K a nonempty closed convex subset of E. Let T : K -> K be a mapping such that $F(T)\;=\;{x\;{\in}\;K\;:\;Tx\;=\;x}\;{\neq}\;0$ and (I - T) satisfies the accretive-type condition: $\;{\geq}\;{\lambda}$\mid$$\mid$x-Tx$\mid$$\mid$^2$, for all $x\;{\in}\;K,\;x^*\;{\in}\;F(T)$ and for some ${\lambda}\;>\;0$. The weak and strong convergence of the Ishikawa iteration method to a fixed point of T are investigated. An application of our results to the approximation of a solution of a certain linear operator equation is also given. Our results extend several important known results from the Mann iteration method to the Ishikawa iteration method. In particular, our results resolve in the affirmative an open problem posed by Naimpally and Singh (J. Math. Anal. Appl. 96 (1983), 437-446).