Iterative process to \(\varphi\)-hemicontractive operator and \(\varphi\)-strongly accretive operator equations (Q5932108)

This article deals with the following Ishikawa type iterations \[ x_{n+1}= a_nx_n+ b_n Ty_n+ c_nu_n,\quad y_n= a_n' x_n+ b_n' Tx_n+ c_n v_n,\quad n= 1,2,\dots \] for a uniformly continuous \(\phi\)-hemicontractive operator \(T: K\to K\) with bounded range on a nonempty closed convex subset of a real Banach space \(E\); the sequences \(\{a_n\}\), \(\{b_n\}\), \(\{c_n\}\), \(\{a_n'\}\), \(\{b_n'\}\), \(\{c_n'\}\) of numbers from \([0,1]\) satisfy the conditions (1) \(a_n+ b_n+ c_n= a_n'+ b_n'+ c_n'= 1\), (2) \(b_n\to 0\), \(b_n'\to 0\), \(c_n'\to 0\), (3) \(\sum^\infty_{n=1} b_n= \infty\), (4) \(c_n= o(b_n)\), the sequences \(\{u_n\}\), \(\{v_n\}\) from \(K\) are bounded. The main result is the strong convergence of these iterations to the unique fixed point of \(T\).