Error bounds for approximation solutions to nonlinear equations of strongly accretive operators in uniformly smooth Banach spaces
Let \(X\) be a real Banach space with norm \(|\cdot |\) and dual \(X^*\). The normalized duality mapping from \(X\) to \(X^*\) is defined by \(J(x)=\{x^*\in X^*: \langle x,x^*\rangle=|x|^2=|x^*|^2\}\), \(x\in X\). A mapping \(T\) with domain \(D(T)\) and range \(R(T)\) in \(X\) is said to be strongly accretive, if for each \(x,y\) in \(D(T)\), there exists a positive number \(k\) such that \(\langle Tx-Ty,j\rangle\geq k|x-y|^2\) holds for some \(j\in J(x-y)\). In this paper \(x\) is a uniformly smooth Banach space and \(T: X\to X\) is a strongly accretive operator. The author gives the error bounds for approximation solutions of the nonlinear equation \(Tx=f\) generated by both Mann and Ishikawa iteration processes. The related results deal with the error bounds for the iterative approximation of the fixed points of \(T\), where \(T: K\to K\), is a strictly pseudocontractive mapping and \(K\) is a nonempty convex subset of \(X\).
scientific article; zbMATH DE number 1031553