Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

Abstract

Abstract. In this paper, we study the convergence of the sequence defined bywhere 0 ≤ αn ≤ 1 and T is a nonexpansive mapping from a closed convex subset of a Banach space into itself.

Variational Inequalities Approaches to Minimization Problems with Constraints of Generalized Mixed Equilibria and Variational Inclusions

“…In the present paper, we introduce two composite schemes for finding a solution of the CMP (8) with the constraints of finitely many GMEPs and finitely many variational inclusions for maximal monotone and inverse strongly monotone mappings in a real Hilbert space H. Strong convergence of the suggested algorithms are given. Our theorems complement, develop and extend the results obtained in [6,15,38], having as background [39][40][41][42][43][44][45][46].…”

A generalization and improvement of Chidume theorems for total asymptotically nonexpansive mappings in Banach spaces

“…As well as a sufficient condition for convergence of the iteration process to a common fixed point of mappings under the setting of uniformly convex Banach space is also established. The results presented in the article not only generalize and improve the corresponding results of Chidume et al [13][14][15] but also unify, extend and generalize the corresponding result of [3][4][5][6][7][9][10][11][12]19]. …”

“…Especially, recently Chidume and Ofoedu [13,14] introduced the following iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings in Banach spaces which extend and generalize the corresponding results of Kirk [3], Alber et al [4], Quan et al [5], Shahzad et al [6], Chang et al [9], Jung [10], Shioji et al [11], Suzuki [12], and Schu [19]. Theorem 1.3 [ [13,14]] Let E be a real Banach space, C be a nonempty closed convex subset of E and T i : C C, i = 1, 2, ..., m be m total asymptotically nonexpansive mappings with sequences {ν in }, {μ in }, i = 1, 2, ..., m, such that…”

“…Within the past 30 years, research on iterative approximation of common fixed points of nonexpansive mappings, asymptotically nonexpansive mappings and asymptotically quasi-nonexpansive mappings have been considered by many authors (see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the references therein).…”

On the computational content of convergence proofs via Banach limits

“…Under this assumption (in fact already under the assumption of the plain Cauchy property of (z n )), the proof of the strong convergence of the Halpern iteration (x n ) that results by our elimination of the use of Banach limits from the proof of Shioji & Takahashi [22] is basically constructive. Hence metatheorems for the constructive case [31] guarantee (and our proof displays this; see theorem 3.4) a uniform effective procedure to transform a rate of convergence for (z n ) into one for (x n ).…”