Strong Convergence Theorem of a Viscosity Process for a Finite Family of Total Asymptotically Nonexpansive Mappings

Strong Convergence Theorem for Finite Family of Asymptotically Nonexpansive in the Intermediate Sense Nonself Maps

“…Proof: The proof of the Lemma 2.3 follows from Lemmas 2.6, 2.7,2.8 and Remark 2.1 of Nnubia and Bishop(2020). Let K be a nonexpansive retract of a uniformly convex Banach space X with nonexpansive retraction P. Let T_i: K → E be a finite family of uniformly continuous asymptotically nonexpansive in the intermediate sense maps with sequences {u _{n,i}}_{n≥ 1}, {e _{n,i}}_{n ≥ 1} subset [0,+ ∞)$ such that ∑ ∞ {n = 0} {u_{n,j} ) < ∞, ∑ ∞ {n = 0} {e_{n,j} ) < ∞.…”

“…These classes of mappings have been studied by several authors (see e.g. Goebel and Kirk (1972), Lim and Xu (1994), Alber et al (2008) , Zegeye and Shahzad (2013), Nnubia and Bishop (2020)).…”

“…By defining the sequence µ_{n} := 1 + u_{n}, the family of generalized asymptotically nonexpansive mapping coincides with the family of asymptotically nonexpansive in the intermediate sense Mapping and hence from Theorems ofNnubia and Bishop (2020) we have the following corresponding theorems and corollaries for families of asymptotically nonexpansive in the intermediate sense Mapping. a nonexpansive retract of a Banach space X with nonexpansive retraction P. Let T_i: K → Xi iin I = {1,...,m} be a finite family of continuous asymptotically nonexpansive in the intermediate sense maps with sequences {u_ {n,j}}_{n ≥ 1}, {e _{n,j}}_{n ≥ 1} subset [0,+∞) such that ∑ ∞ {n = 0} {u_{n,j} < ∞, ∑ ∞ {n = 0} {e_{n,j} ) < ∞.…”