Iterative approximations of \(\phi\)-hemicontractive mappings without Lipschitz assumption (Q2721775)

The paper concerns solution approximation for the equation \(Tx=f\). Here \(T:E\to E\) is a continuous operator on the Banach space \(E\) and \(f\in E\). In addition, \(T\) is \(\varphi\)-strongly accretive for some \(\varphi:[0,+\infty)\to [0,+\infty)\), so the equation has a unique solution \(x\), which is also the unique fixed point of the \(\varphi\)-strongly pseudocontractive, continuous operator \(H=f+I-T\). The main result states an \(H\)-Ishikawa iterative sequence strongly converges to \(x\) provided that \(I-T\) has a bounded range and \(E\) is \(q\)-uniformly smooth for some \(q>1\).