Approximating the zeros of accretive operators by the Ishikawa iteration process
Given a Banach space \(X\) and an operator \(A:X\to X\), the Ishikawa iteration \(\{x_n\}\) is defined by \(x_{n+1}=x_n-\alpha_n Ay_n-\alpha_n\beta_n Ax_n\), \(y_n=x_n-\beta_n Ax_n\). A main result states, if \(X\) is uniformly smooth, \(A\) is bounded and accretive, and the equation \(Ax=0\) has at least a solution, then the iteration process converges to a solution to the equation. Given a convex, closed subset \(C\) of \(X\) and an operator \(T:C\to C\), the Ishikawa iteration is rephrased to \(x_{n+1}=(1-\alpha_n)x_n+\alpha_n Ty_n\), \(y_n=(1-\beta_n)x_n+\beta_n Tx_n\). Another main result states, if \(X\) is uniformly smooth, \(C\) is bounded, \(T\) is nonexpansive, and the equation \(x=Tx\) has at least a solution, then the iteration process converges to a solution to the equation.
scientific article; zbMATH DE number 1370618