Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications

Abstract

a b s t r a c tIn this paper, some properties of the generalized f -projection operator are proved in Banach spaces. Using these results, the strong convergence theorems for relatively nonexpansive mappings are studied in Banach spaces. As applications, the strong convergence of general H-monotone mappings in Banach spaces is also given. The results presented in this paper generalize and improve the main results of Matsushita and Takahashi (2005) [9].

A modified Mann iterative scheme by generalized f-projection for a countable family of relatively quasi-nonexpansive mappings and a system of generalized mixed equilibrium problems

“…Furthermore, we show that our new iterative scheme converges strongly to a common element of the aforementioned sets. Our results extend and improve the recent result of Li et al [26], Matsushita and Takahashi [35], Takahashi et al [43], Nakajo and Takahashi [41] and Shehu [45] and others.…”

“…They obtained a strong convergence theorem for finding an element in the fixed point set of T. The results of Li et al [26] extended and improved on the results of Matsushita and Takahashi [35].…”

“…Motivated by Li et al [26] we obtain the following result: Theorem 4.5. Let C be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space E with δ E (ε) ≥ kε 2 and r E (t) ≤ ct 2 for some c, k >0, and E* be the dual space of E. Let B : E → E * be a strictly monotone and b-Lipschitz x 0 and C 1 = C, we define the sequence {x n } as follows:…”

“…Recently, Li et al [26] introduced the following hybrid iterative scheme for approximation of fixed point of relatively nonexpansive mapping using the properties of generalized f-projection operator in a uniformly smooth real Banach space which is also uniformly convex:…”

A New Hybrid Iterative Scheme for Countable Families of Relatively Quasi‐Nonexpansive Mappings and System of Equilibrium Problems

“…Lemma 2.5 see Li et al 29 . Let E be a Banach space, and let f : E → Ê ∪ { ∞} be a lower semicontinuous convex functional.…”