On approximation of fixed points of multivalued pseudocontractive mappings in Hilbert spaces

Résumé

Abstract It is proved that if a multivalued k-strictly pseudocontractive mapping S of Chidume et al. (Abstr. Appl. Anal. 2013:629468, 2013) is of type one, then $I-S$ I − S is demiclosed at zero. Also, under this condition, the Mann (respectively, Ishikawa) sequence weakly (respectively, strongly) converges to a fixed point of a multivalued k-strictly pseudocontractive (respectively, pseudocontractive) mapping S without the condition that the fixed point set of S is strict, where S is of type one if for any pair $r,g \in D(S)$ r , g ∈ D ( S ) , $$\|u-v\|\leq\Phi(Sr,Sg) \quad\mbox{for all } u\in P_{S}r, v\in P_{S}g, $$ ∥ u − v ∥ ≤ Φ ( S r , S g ) for all u ∈ P S r , v ∈ P S g , and Φ denotes the Hausdorff metric. The results obtained give a partial answer to the problem of the removal of the strict fixed point set condition, which is usually imposed on multivalued mappings. Thus, the results extend, complement, and improve the results on multivalued and single-valued mappings in the contemporary literature.

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