Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces

A new algorithm for two finite families of demicontractive mappings and equilibrium problems

“…We also apply our result to obtain a strong convergence theorem for a finite family of k-strictly pseudocontractive mappings and a finite family of β-strictly pseudononspreading mappings. Our results improve and extend the corresponding results of Osilike and Isiogugu [10], Plibtieng and Punpaeng [17], Takahashi and Takahashi [16], Dhompongsa [22], and many others.…”

“…The following lemma was proved in [10] for k-strictly pseudononspreading mappings. The statement is true for demicontractive mappings as well.…”

“…Recently, Osilike and Isiogugu [10] introduced a new class of mappings in a Hilbert space, which is called the class of k-strictly pseudononspreading mappings. A mapping T: C → C is said to be k-strictly pseudononspreading if there exists k ∈ [0, 1) such that T x − Ty 2 ≤ x − y 2 + k x − T x − (y − Ty) 2 …”

“…We mention in passing that the class of k-strictly pseudononspreading mappings is more general than the class of nonspreading mappings (see [10]). In [10], Osilike and Isiogugu proved a strong convergence theorem somewhat related to a Halpern-type iteration algorithm for a k-strictly pseudononspreading mapping in Hilbert spaces. Theorem 1.1 ([10]).…”

Simultaneous iteration for variational inequalities over common solutions for finite families of nonlinear problems

“…for all x ∈ C, y ∈ Fix(T ); (4) monotone if x − y, T x − T y 0 for all x, y ∈ C; (5) γ-strongly monotone if there exists γ > 0 such that x − y, T x − T y γ x − y 2 for all x, y ∈ C; (6) pseudocontractive if T x − T y 2 x − y 2 + x − T x − (y − T y) 2 for all x, y ∈ C; (7) k-demicontractive if Fix(T ) = ∅ and there exists −∞ < k < 1 such that T x − y 2 x − y 2 + k T x − x 2 for all x ∈ C and for all y ∈ Fix(T ); (8) k-strictly pseudononspreading [20] if there exists k ∈ (0, 1) such that T x − T y 2 x − y 2 + k x − T x − (y − T y) 2 + x − T x, y − T y for all x, y ∈ C; (9) firmly nonexpansive if T x − T y 2 + (I 1 − T )x − (I 1 − T )y 2 x − y 2 for all x, y ∈ C; (10) directed if Fix(T ) = ∅, and T x − y, T x − x 0 for all x ∈ C and for all y ∈ Fix(T ); (11) demiclosed if for each sequence {x n } and x in C with x n x and (I − T )x n → 0 implies that (I − T )x = 0; (12) α-averaged if there exist α ∈ (0, 1) and a nonexpansive mapping S : C → H 1 such that T = (1 − α)I + αS; (13) hemicontinuous if, for all x, y ∈ C, the mapping g : [0, 1] → H 1 , defined by g(t) = T (tx + (1 − t)y) is continuous with respect to weak topology on H 1 ; (14) quasi-pseudocontractive if Fix(T ) = ∅ and T x − y 2…”