Fixed point theory for a class of generalized nonexpansive mappings

Abstract

In this paper we introduce two new classes of generalized nonexpansive mapping and we study both the existence of fixed points and their asymptotic behavior.

Fixed point theorems for (L)-type mappings in complete CAT(0) spaces

“…For instance, in 2008, Suzuki [28] defined a class of generalized nonexpansive mappings, which he called (C)-type mappings, whose set-valued version was defined and studied in [1,2,26,30]. In 2011, García-Falset et al [14] introduced two classes of single-valued generalized nonexpansive mappings called (C λ )-type mappings and (E µ )-type mappings, respectively, which both enlarged the family of (C)-type mappings. Again these new classes were generalized to the setvalued case in [3,9,12,17].…”

On the existence of ɛ-fixed points

“…The map I − F is said to be weakly demiclosed in ∈ WF(X ) if for every sequence ( ) in X which weakly converges to ( ) and such that lim →∞ ( F ( )) = 0, we have ∈ F ( ) (in [5] the strongly demiclosed property for multimaps is introduced).…”

“…As in the proof of Theorem 3.1, we consider the multimap G ε : H ε → P c (V ) defined in (5). The map G ε is partially closed.…”

A shrinking projection extragradient algorithm for equilibrium problem and fixed point problem

“…A multivalued mapping T : H → CB(H) is called (i) nonexpansive if h(T x, T y) ≤ x − y , x, y ∈ H,(ii) quasi-nonexpansive if F (T ) = ∅ and h(T x, T p) ≤ x − p for all x ∈ H and all p ∈ F (T ). Recently, J.Garcia-Falset, E. Llorens-Fuster and T. Suzuki [15], introduced a new generalization of the concept of a nonexpansive single valued mapping which called condition (E). Very recently, Abkar and Eslamian [1], modify the condition (E) for multivalued mappings as follows:…”