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Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point. (English) Zbl 1081.47069
Summary: The existence of fixed points of multivalued mappings that satisfy a certain contractive condition was proved by N. Mizoguchi and W. Takahashi [J. Math. Anal. Appl. 141, No. 1, 177–188 (1989; Zbl 0688.54028)]. An alternative proof of this theorem was given by P. Z. Daffer and H. Kaneko [J. Math. Anal. Appl. 192, No. 2, 655–666 (1995; Zbl 0835.54028)]. In the present paper, we give a simple proof of that theorem. Also, we define Mann and Ishikawa iterates for a multivalued map T with a fixed point p and prove that these iterates converge to a fixed point q of T under certain conditions. This fixed point q may be different from p. To illustrate this phenomenon, an example is given.

47J25Iterative procedures (nonlinear operator equations)
47H04Set-valued operators
47H10Fixed point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces
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