Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps

Abstract

Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, L p or p spaces, 1 < p < ∞), and K a nonempty closed convex (not necessarily bounded) subset of E. Let T : K → K be a k-strictly asymptotically pseudocontractive map with a nonempty fixed-point set. It is proved that (I − T ) is demiclosed at 0. Furthermore, weak and strong convergence of an averaging iteration method to a fixed point of T are proved.

Some Convergence Theorems for Contractive Type Mappings inCAT(0) Spaces

“…The existence of (common) fixed points of one mapping (or two mappings or family of mappings) is not known in many situations. So the approximation of fixed points of one or more nonexpansive, asymptotically nonexpansive, or asymptotically quasinonexpansive mappings by various iterations have been extensively studied in Banach spaces, convex metric spaces, (0) spaces, and so on (see, [2,6,8,9,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]). In this paper, we establish theorems of strong convergence for the Ishikawa-type (or two step, cf.…”

Strong Convergence Theorems for a Common Fixed Point of a Family of Asymptoticallyk-Strict Pseudocontractive Mappings

“…The iteration scheme (5) is called modified Mann's iterative processes which was introduced by Schu [7,8] and has been used by several authors (see, e.g., [3][4][5][9][10][11][12][13][14][15][16][17]). We observe that Liu [5] proved strong convergence of scheme (5) to a fixed point of asymptotically -strict pseudocontractive mapping with additional assumption that is completely continuous, where : → is said to be completely continuous if for every bounded sequence { }, there exists a subsequence, say { } of { } such that the sequence { } converges strongly to some element of the range of .…”

Convergence of the modified Mann's iterative method for asymptotically κ-strictly pseudocontractive mappings

“…It is shown in [5] that the class of asymptotically -strictly pseudocontractive mappings and the class of -strictly pseudocontractive mappings are independent.…”

“…Osilike et al [5] proved the convergence theorems of modified Mann iteration method in the framework of q-uniformly smooth Banach spaces which are also uniformly convex. They also obtained that a modified Mann iterative process {x n } converges weakly to a fixed point of T under suitable control conditions.…”