Convergence theorems for generalized projections and maximal monotone operators in Banach spaces

Abstract

We study a sequence of generalized projections in a reflexive, smooth, and strictly convex Banach space. Our result shows that Mosco convergence of their ranges implies their pointwise convergence to the generalized projection onto the limit set. Moreover, using this result, we obtain strong and weak convergence of resolvents for a sequence of maximal monotone operators.

A new projection and convergence theorems for the projections in Banach spaces

“…Theorem 3.4 (Ibaraki et al [11]). Let E be a smooth, reflexive, and strictly convex Banach space and let C 1 , C 2 , C 3 , .…”

“…In 2003, Ibaraki et al [11] proved the following two theorems for the generalized projections. Theorem 3.4 (Ibaraki et al [11]).…”

“…. be nonempty closed convex subsets of E. If C 0 = M-lim n C n exists and nonempty, then C 0 is a closed convex subset of E and, for each x ∈ E, C n x converges weakly to C 0 x. Theorem 3.5 (Ibaraki et al [11]). Let E be a smooth Banach space and let E * have a Fréchet differential norm.…”

“…In addition, we know convergence theorems for sequences of sets concerning metric projections, generalized projections and sunny nonexpansive retractions (see [11,14,22]). …”

A hybrid iteration scheme for equilibrium problems and common fixed point problems of generalized quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces

“…If lim n ∞ j(x n ,y n ) = 0, then lim n ∞ ∥x n -y n ∥ = 0. Lemma 1.3 [21]Let E be a smooth, strictly convex and reflexive Banach space having the Kadec-Klee property. Let {K n } be a sequence of nonempty closed convex subsets of E.…”