Convergence theorem for a countable family of multi-valued strictly pseudo-contractive mappings in Hilbert spaces

Abstract

A Krasnoselskii-type algorithm is constructed and proved to be an approximate fixed point sequence for a countable family of multi-valued strictly pseudo-contractive mappings in a real Hilbert space. Under some additional mild conditions, the sequence is proved to converge strongly to a common fixed point of the family. Our theorems complement and improve the results of Chidume and Ezeora [6], Abbas et al. [1], Chidume et al. [5] and a host of other important results.

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

“…Then, they proved strong convergence theorems for this class of mappings. The recursion formula used in [26] is of Krasnoselskii-type [36] which is known to be superior (see, e.g., Remark 4 in [26]) to the recursion formula of Mann [24] or Ishikawa [18]. In fact, they proved the following theorem.…”

“…Theorem CA1 ( Chidume et al [26]). Let K be a nonempty, closed and convex subset of a real Hilbert space H. Suppose that T : K → CB(K) is a multi-valued k-strictly pseudo-contractive mapping such that F (T ) = ∅.…”

“…They also proved in [26] that under some conditions or on the map T or on the set K, the sequence {x n } converges strongly to a fixed point of T .…”

“…Lemma 2.2 (chidume et. al., [26]) Let E be a reflexive real Banach space and let A, B ∈ CB(X). Assume that B is weakly closed.…”

“…Motivated by approxiting fixed of multivalued mappings, Chidume et al [26] introduced the following important class of multi-valued strictly pseudo-contractive mappings in real Hilbert spaces which is more general than the class of multivalued nonexpansive mappings.…”