Ishikawa iteration procedures with errors for certain nonlinear operator without Lipschitz assumption (Q2746280)

Properties of the Ishikawa iteration with error NEWLINE\[NEWLINEy_n= (1-\beta_n)x_n + \beta_nTx_n+v_n,\quad x_0\in K,NEWLINE\]NEWLINE NEWLINE\[NEWLINEx_{n+1} (1-\alpha_n)x_n + \alpha_nTy_n+u_n,\quad n=0,1,2,\ldots, NEWLINE\]NEWLINE and its generalized form NEWLINE\[NEWLINEy_n= \widehat\alpha_nx_n + \widehat \beta_nTx_n+\widehat\gamma_nv_n, \quad x_0\in K,NEWLINE\]NEWLINE NEWLINE\[NEWLINEx_{n+1}= \alpha_nx_n + \beta_nTy_n+\gamma_nu_n, \quad n=0,1,2,\ldots.NEWLINE\]NEWLINE are discussed. Here \(K\) is a subset of a linear space \(X\), and \(T\) is a \(\phi\)-strong pseudo-contractive nonlinear operator from \(K\) into \(X\), \(\alpha_n, \beta_n, \widehat\alpha_a, \widehat\beta_n, \gamma, \widehat\gamma_n\) are proper constants and \(u_n, v_n\) represent errors in the computation. The author relaxed some conditions for convergence and stability of these iterations to a fixed point of \(T\).