Convergence theorems for uniformly quasi-Lipschitzian mappings (Q1774697)

In the present paper the authors prove several results on convergence of the modified Ishikawa iterative sequence with errors (denoted by \(\{x_n\}\)) for asymptotically (and respectively, uniformly) quasi-Lipschitzian mappings (denoted by \(T\)) in metric spaces (especially, in Banach spaces). The authors prove that sequence \(\{x_n\}\) converges to a fixed point \(p\in \text{Fix}(T)\neq\emptyset\) if and only if \(\liminf_{n\to\infty}\,d(x_n, \text{Fix}(T))= 0\). The present results generalize and improve the corresponding results of \textit{W. V. Petryshyn} and \textit{T. E. Williamson} [J. Math. Anal. Appl. 43, 459--497 (1973; Zbl 0262.47037)], \textit{M. K. Ghosh} and \textit{L. Debnath} [J. Math. Anal. Appl. 207, 96--103 (1997; Zbl 0881.47036)], \textit{Q. Liu} [J. Math. Anal. Appl. 259, 1--7 (2001; Zbl 1033.47047)], and many others.