Iterative solution of nonlinear equations involving K-accretive operator equations (Q2707038)

This article deals with Mann and Ishikawa types iterations for the equations \(Ax= f\) and \(Ax+ Kx+f\) where \(A\) is a Lipschitz strongly \(K\)-accretive operator in a real Banach space \(E\) (this means NEWLINE\[NEWLINE\langle Ax- Ay, j_p(Kx- Ky)\rangle\geq \mu\|Kx- Ky\|^pNEWLINE\]NEWLINE with \(j_px\in E^*: \langle x,j_px\rangle=\|x\|^p\), \(\|j_px\|=\|x\|^{p-1}\)), \(D(A)\subseteq D(K)\), \(f\in E\).NEWLINENEWLINENEWLINEThe main results are theorems about the convergence of iterations NEWLINE\[NEWLINEKx_{n+1}= Kx_n+ \alpha_n(f- GKy_n+ v_n),\;Ky_n= Kx_n+ \beta_n(f- GKx_n+ u_n)\;(n= 0,1,\dots)NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\begin{multlined} Kx_{n+ 1}= Kx_n+ \alpha_n(f- Kx_n- GKy_n+ v_n),\\ y_n= Kx_n+ \beta_n(f- Kx_n GKx_n+ u_n)\quad (n= 0,1,\dots)\end{multlined}NEWLINE\]NEWLINE (\(G= AK^{-1}\), \(u_n\) and \(v_n\) are sequences of errors, \(\alpha_n\), \(\beta_n\) are fixed number sequences) to the unique solution of the corresponding equation.