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Multivalued pseudo-contractive mappings defined on unbounded sets in Banach spaces

Claudio H. Morales — 1992

Commentationes Mathematicae Universitatis Carolinae

Let X be a real Banach space. A multivalued operator T from K into 2 X is said to be pseudo-contractive if for every x , y in K , u ∈ T ( x ) , v ∈ T ( y ) and all r > 0 , ∥ x - y ∥ ≤ ∥ ( 1 + r ) ( x - y ) - r ( u - v ) ∥ . Denote by G ( z , w ) the set { u ∈ K : ∥ u - w ∥ ≤ ∥ u - z ∥ } . Suppose every bounded closed and convex subset of X has the fixed point property with respect to nonexpansive selfmappings. Now if T is a Lipschitzian and pseudo-contractive mapping from K into the family of closed and bounded subsets of K so that the set G ( z , w ) is bounded for some z ∈ K and some w ∈ T ( z ) , then T has a fixed point in K .

Fixed point approximation under Mann iteration beyond Ishikawa

Anthony Hester;  Claudio H. Morales — 2020

Commentationes Mathematicae Universitatis Carolinae

Consider the Mann iteration x n + 1 = ( 1 - α n ) x n + α n T x n for a nonexpansive mapping T : K → K defined on some subset K of the normed space X . We present an innovative proof of the Ishikawa almost fixed point principle for nonexpansive mapping that reveals deeper aspects of the behavior of the process. This fact allows us, among other results, to derive convergence of the process under the assumption of existence of an accumulation point of { x n } .

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