Convergence theorems for finite family of a general class of multi-valued strictly pseudo-contractive mappings
C.E. Chidume, Mmaduabuchi Okpala, Abdulmalik Usman Bello, Patrice Ndambomve · Fixed Point Theory and Applications · 2015
Abstract Let H be a real Hilbert space, K a nonempty subset of H, and $T:K\rightarrow \mathit{CB}(K)$ T : K → CB ( K ) a multi-valued mapping. Then T is called a generalizedk-strictly pseudo-contractive multi-valued mapping if there exists $k\in[0,1)$ k ∈ [ 0 , 1 ) such that, for all $x,y\in D(T)$ x , y ∈ D ( T ) , we have $D^{2}(Tx,Ty)\leq\|x-y\|^{2}+kD^{2}(Ax,Ay)$ D 2 ( T x , T y ) ≤ ∥ x − y ∥ 2 + k D 2 ( A x , A y ) , where $A:=I-T$ A : = I − T , and I is the identity operator on K. A Krasnoselskii-type algorithm is constructed and proved to be an approximate fixed point sequence for a common fixed point of a finite family of this class of maps. Furthermore, assuming existence, strong convergence to a common fixed point of the family is proved under appropriate additional assumptions.