Strong Convergence Theorems for Lipschitzian Demicontraction Semigroups in Banach Spaces (original) (raw)
Related papers
Convergence theorem for finite family of lipschitzian demi-contractive semigroups
Fixed Point Theory and Applications, 2011
Let E be a real Banach space and K be a nonempty, closed, and convex subset of E. Let {J i } N i=1 be a finite family of Lipschitzian demi-contractive semigroups of K, with sequences of bounded measurable functions L i : [0, ∞) (0, ∞) and bounded functions l i : [0, ∞) (0, ∞), respectively, where J i := {T i (t) : t ≥ 0}, i = 1,2, ..., N. Strong convergence theorem for common fixed point for finite family {J i } N i=1 is proved in a real Banch space. As an application, a new convergence theorem for finite family of Lipschitzian demi-contractive maps is also proved. Mathematics subject classification (2000) 47H09, 47J25
Israel Journal of Mathematics, 1979
We study a general condition on A that guarantees the strong convergence of the semigroup generated by-A and of related implicit and explicit iterative schemes to a zero of A. Rates of convergence are also obtained. In Hilbert space this condition has been recently introduced by A. Pazy. We also establish strong convergence under the assumption that the interior of A-~0 is not empty. In Hilbert space this result is due to H. Brezis.
On the Convergence of Iteration Processes for Semigroups of Nonlinear Mappings in Banach Spaces
Springer proceedings in mathematics & statistics, 2013
Let C be a bounded, closed, convex subset of a uniformly convex Banach space X. We investigate the convergence of the generalized Krasnosel'skii-Mann and Ishikawa iteration processes to common fixed points of pointwise Lipschitzian semigroups of nonlinear mappings T t : C → C. Each of T t is assumed to be pointwise Lipschitzian, that is, there exists a family of functions α t : C → [0, ∞) such that T t (x) − T t (y) ≤ α t (x) x − y for x, y ∈ C.
Journal of Linear and Topological Algebra, 2021
It is our purpose in this paper to introduce the concept of α-demicontractive semigroup. Also, we construct a new implicit iterative scheme for approximating the common fixed points of α-demicontractive semigroup. We prove strong convergence of our new iterative scheme to the common fixed points of α-demicontractive semigroup in Banach spaces. Our result is an improvement and generalization of several well known results in the existing literature.
Implicit Iteration Process for Lipschitzian α – Hemicontraction Semigroups
2020
In this paper, the concepts of α–demicontractive semigroup and α–hemicontractive semigroup are considered. We study the strong and weak convergence of an implicit iterative scheme to the common fixed points of Lipschitzian α–hemicontractive semigroup. The result presented in this paper extend, generalize, improve and unify the results of several well known authors in the literature. keyword: Banach space, Fixed point, Normalized duality mapping, Implicit iterative scheme, Strong convergence, weak convergence, α-hemicontractive semigroup.
International Journal of Mathematics and Mathematical Sciences, 2012
Let E be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let J {T t : t ≥ 0} be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of E, with functions u, v : 0, ∞ → 0, ∞. Let F : F J ∩ t≥0 F T t / ∅ and f : K → K be a weakly contractive map. For some positive real numbers λ and δ satisfying δ λ > 1, let G : E → E be a δ-strongly accretive and λ-strictly pseudocontractive map. Let {t n } be an increasing sequence in 0, ∞ with lim n → ∞ t n ∞, and let {α n } and {β n } be sequences in 0, 1 satisfying some conditions. Strong convergence of a viscosity iterative sequence to common fixed points of the family J of uniformly asymptotically regular asymptotically nonexpansive semigroup, which also solves the variational inequality G − γf p, j p − x ≤ 0, for all x ∈ F, is proved in a framework of a real Banach space.