A Counterexample to a Theorem of Xu

Chidume, 1988, Fixed point iterations for certain classes of nonlinear mappings, Appl. Anal., 27, 19, 10.1080/00036818808839722 Ćirić, 1974, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc., 45, 267, 10.2307/2040075 Ishikawa, 1974, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44, 147, 10.1090/S0002-9939-1974-0336469-5 Mann, 1953, Mean value methods in iteration, Proc. Amer. Math. Soc., 4, 506, 10.1090/S0002-9939-1953-0054846-3 Naimpally, 1983, Extensions of some fixed point theorems of Rhoades, J. Math. Anal. Appl., 96, 437, 10.1016/0022-247X(83)90052-5 Qihou, 1987, On Naimpally and Singh's open questions, J. Math. Anal. Appl., 124, 159, 10.1016/0022-247X(87)90031-X Qihou, 1990, A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings, J. Math. Anal. Appl., 146, 301, 10.1016/0022-247X(90)90303-W Rhoades, 1974, Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc., 196, 161, 10.1090/S0002-9947-1974-0348565-1 Rhoades, 1976, Comments on two fixed point iteration methods, J. Math. Anal. Appl., 56, 741, 10.1016/0022-247X(76)90038-X Rhoades, 1994, Convergence of an Ishikawa-type iteration scheme for a generalized contraction, J. Math. Anal. Appl., 185, 350, 10.1006/jmaa.1994.1253 Xu, 1992, A note on the Ishikawa iteration scheme, J. Math. Anal. Appl., 167, 582, 10.1016/0022-247X(92)90225-3