Convergence Theorems for Fixed Points of Multivalued Mappings in Hilbert Spaces

Let be a real Hilbert space and a nonempty closed convex subset of . Suppose is a multivalued Lipschitz pseudocontractive mapping such that . An Ishikawa-type iterative algorithm is constructed and it is shown that, for the corresponding sequence , under appropriate conditions on the iteration parameters, holds. Finally, convergence theorems are proved under approximate additional conditions. Our theorems are significant improvement on important recent results of Panyanak (2007) and Sastry and Babu (2005). 1. Introduction Let be a nonempty subset of a normed space . The set is called proximinal (see, e.g., [1–4]) if for each there exists such that where for all . It is known that every nonempty closed convex subset of a real Hilbert space is proximinal. Let and denote the families of nonempty closed bounded subsets and nonempty proximinal bounded subsets of , respectively. The Hausdorff metric on is defined by for all . Let be a multivalued mapping on . A point is called a fixed point of if and only if . The set is called the fixed point set of . A multivalued mapping is called Lipschitzian if there exists such that In (3), if , is said to be a contraction, and is called nonexpansive if . Existence theorem for fixed point of multivalued contractions and nonexpansive mappings using the Hausdorff metric have been proved by several authors (see, e.g., Nadler Jr. [5], Markin [6], and Lim [7]). Later, an interesting and rich fixed point theory for such maps and more general maps was developed which has applications in control theory, convex optimization, differential inclusion, and economics (see Gorniewicz [8] and references cited therein). Several theorems have been proved on the approximation of fixed points of multivalued nonexpansive mappings (see, e.g., [1–4, 9, 10] and the references therein) and their generalizations (see, e.g., [11, 12]). Sastry and Babu [2] introduced the following iterative scheme. Let be a multivalued mapping and let be a fixed point of . The sequence of iterates is given for by where is a real sequence in (0,1) satisfying the following conditions:(i);(ii). They also introduced the following sequence: where , are real sequences satisfying the following conditions:(i);(ii);(iii). Sastry and Babu called the process defined by (4) a Mann iteration process and the process defined by (5) where the iteration parameters satisfy conditions (i), (ii), and (iii) an Ishikawa iteration process. They proved in [2] that the Mann and Ishikawa iteration schemes for a multivalued map with fixed point converge to a fixed point of under certain