Krasnoselskii-type algorithm for fixed points of multi-valued strictly pseudo-contractive mappings

Abstract

Let q > 1 and let K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Let T : K → CB(K) be a multi-valued strictly pseudo-contractive map with a nonempty fixed point set. A Krasnoselskii-type iteration sequence {x n } is constructed and proved to be an approximate fixed point sequence of T, i.e., lim n→∞ d(x n , Tx n ) = 0. This result is then applied to prove strong convergence theorems for a fixed point of T under additional appropriate conditions. Our theorems improve several important well-known results. MSC: 47H04; 47H06; 47H15; 47H17; 47J25

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

“…With this definition at hand, Chidume et al [27] proved some strong convergence theorems for approximating fixed points multi-valued k-strictly pseudocontrcative mappings defined on q-uniformly smooth space.…”

“…Lemma 2.4 (chidume et. al., [27]) Let q > 1, E be a q-uniformly smooth real Banach space, k ∈ (0, 1). Suppose T : D(T ) ⊂ E → CB(E) is a multivalued map with F (T ) = ∅, and such that for all x ∈ D(T ),…”

“…Lemma 2.5 (chidume et. al., [27]) Let K be a nonempty closed subset of a real Banach space E and let T : K → P (K) be a k-strictly pseudocontractive mapping. Assume that for every x ∈ K, T x is weakly closed.…”

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]).…”

“…Suppose that ̸ = 0 and that ( ) = { } for all ∈ . Let { } be the sequence defined by (14). Then, for all = 1, .…”