Convergence Theorems for Fixed Points of Multivalued Mappings in Hilbert Spaces

Abstract

Let H be a real Hilbert space and K a nonempty closed convex subset of H. Suppose T:K→CB(K) is a multivalued Lipschitz pseudocontractive mapping such that F(T)≠∅. An Ishikawa-type iterative algorithm is constructed and it is shown that, for the corresponding sequence {xn}, under appropriate conditions on the iteration parameters, lim infn→∞⁡ d (xn,Txn)=0 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).

Mann and Ishikawa-Type Iterative Schemes for Approximating Fixed Points of Multi-valued Non-Self Mappings

“…[7] from singlevalued mapping to multi-valued mapping. 2) Theorem 3.11 extends Corollary 10 of Djitte and Sene [8] from self mappings to non-self mappings.…”

“…More recently, Djitte and Sene [8] introduced the following Ishikawatype iterative scheme for multi-valued Lipschitz pseudocontractive mapping T. Let {x n } be a sequence generated from arbitrary x 1 ∈ K by…”

“…Then, they proved the following theorem. Theorem DS [8] Let K be a nonempty, closed and convex subset of a real Hilbert space H and T : K → CB(K) be a multi-valued Lipschitz pseudocontractive mapping with F (T ) = ∅ and T p = {p}, ∀p ∈ F (T ). Let {x n } be a sequence defined by (1.4).…”

Strong convergence theorems for a common fixed point of a finite family of multivalued mappings in certain Banach spaces

“…It is easily seen that any multivalued nonexpansive mapping is k-strictly pseudocontractive for any k ∈ (0, 1). Moreover the inverse is not true (see,e.g., Djitte and Sene [17]).…”

Iterative Algorithms for a Finite Family of Multivalued Quasi-Nonexpansive Mappings

“…Several papers deal with the problem of approximating fixed points of multivalued nonexpansive mappings (see, e.g., [8][9][10][11][12] and the references therein) and their generalizations (see, e.g., [13][14][15]).…”