Iterative solutions to nonlinear equations of the accretive type in Banach spaces (Q2785274)

This article deals with Ishikawa approximations NEWLINE\[NEWLINEx_{n+1}= (1-\alpha_n) x_n+\alpha_n Sy_n+u_n,\quad y_n=(1-\beta_n) x_n+\beta_n Sx_n+v_n\;(n=0,1,2, \dots)NEWLINE\]NEWLINE for a Lipschitz and strongly accretive operator \(T:X\to X(\|Tx-Ty \|\leq l\|x-y\|\) and \(\langle Tx-Ty,\;j(x-y) \rangle\geq k \|x-y \|^2)\) in an arbitrary real Banach space \(X\). Here, \(Sx=f+(I-T) x\), \(\alpha_n\), \(\beta_n\) are real sequences and \(u_n\), \(v_n\) are two sequences in \(X\). The authors present some theorems about the strong convergence of \(x_n\) to the solution of \(Tx=f\). For example, this convergence holds if the following conditions are true: NEWLINE\[NEWLINE\sum^\infty_{n=0} \alpha_n=\infty,\;0\leq \alpha_n, \beta_n\leq 1,{k-L(L+1) \beta_n-L(L+1) (1+\beta_nl) \alpha_n\over 1-(1-k)\alpha_n}\geq t\;(L=1+l),NEWLINE\]NEWLINE where \(t\in(0,1)\), \(\lim_{n\to\infty}\|u_n |=0\), \(\sum^\infty_{n=0}|v_n\|<\infty\). The case of the equation \(x+Tx=f\) is also considered.NEWLINENEWLINENEWLINEThe results of the article improve some theorems of L. S. Liu, C. E. Chidume, C. E. Chidume and M. O. Osilike, K. K. Tan and H. K. Xu, L. Deng, L. Deng and X. P. Ding, and others.