Convergence theorem for finite family of lipschitzian demi-contractive semigroups (original) (raw)
Related papers
Strong Convergence Theorems for Lipschitzian Demicontraction Semigroups in Banach Spaces
Fixed Point Theory and Applications, 2011
The purpose of this paper is to introduce and study the strong convergence problem of the explicit iteration process for a Lipschitzian and demicontraction semigroups in arbitrary Banach spaces. The main results presented in this paper not only extend and improve some recent results announced by many authors, but also give an affirmative answer for the open questions raised by Suzuki 2003 and Xu 2005 .
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.
Convergence theorems for semigroup of asymptotically nonexpansive mappings
Analele Universitatii Bucuresti Seria Matematica
In this paper, we prove the following results: Let K be a closed convex subset of a real Banach space E. Let T := {T (t) | t ∈ R+} be strongly continuous semigroup of asymptotically nonexpansive mappings from K into K such that F (T) := ∩ t∈R + F (T (t)) = ∅, where F (T (t)) = {x ∈ K | T (t)x = x} and R+ denotes the set of nonnegative real numbers. Then for arbitrary x0 ∈ K, the implicit iteration {xn} given by xn = αnxn−1 + (1 − αn) (T (tn)) n xn , n ≥ 0 converges weakly (strongly) to an element of F (T), where {αn}, {tn} are sequences of real numbers satisfying certain conditions.
Fixed Point Theory and Applications, 2012
The purpose of this article is to introduce two iterative algorithms for finding a common fixed point of a semigroup of asymptotically nonexpansive mappings which is a unique solution of some variational inequality. We provide two algorithms, one implicit and another explicit, from which strong convergence theorems are obtained in a uniformly convex Banach space, which admits a weakly continuous duality mapping. The results in this article improve and extend the recent ones announced by Li et al. (Nonlinear Anal. 70:3065-3071, 2009), Zegeye et al. (Math. Comput. Model. 54:2077-2086) and many others. MSC: 47H05; 47H09; 47H20; 47J25 Keywords: iterative approximation method; common fixed point; semigroup of asymptotically nonexpansive mapping; strong convergence theorem; uniformly convex Banach space
Common Fixed Points Approximation for Asymptotically Nonexpansive Semigroup in Banach Spaces
ISRN Mathematical Analysis, 2011
Let be a real Banach space satisfying local uniform Opial's condition, whose duality map is weakly sequentially continuous. Let be a uniformly asymptotically regular family of asymptotically nonexpansive semigroup of with function . Let and be weakly contractive map. Let be -strongly accretive and -strictly pseudocontractive map with . Let be an increasing sequence in and let and be sequences in satisfying some conditions. For some positive real number appropriately chosen, let be a sequence defined by , , , . It is proved that converges strongly to a common fixed point of the family which is also the unique solution of the variational inequality .
Strong Convergence Theorems for Semigroups of Asymptotically Nonexpansive Mappings in Banach Spaces
Abstract and Applied Analysis, 2013
Let be a real reflexive Banach space with a weakly continuous duality mapping. Let be a nonempty weakly closed star-shaped (with respect to) subset of. Let F = { () : ∈ [0, +∞)} be a uniformly continuous semigroup of asymptotically nonexpansive self-mappings of , which is uniformly continuous at zero. We will show that the implicit iteration scheme: = +(1−) () , for all ∈ N, converges strongly to a common fixed point of the semigroup F for some suitably chosen parameters { } and { }.
Fixed Point Theory and Applications, 2014
The purpose of this paper is to study the viscosity iterative schemes for approximating a fixed point of an asymptotically nonexpansive semigroup on a compact convex subset of a smooth Banach space with respect to a sequence {μ i,n } m,∞ i=1,n=1 of strongly asymptotic invariant means defined on an appropriate space of bounded real valued functions of the semigroup. Our results extend and improve the result announced by Lau et al. (Nonlinear Anal. 67(4):1211-1225) and many others.
Fixed point theorems for generalized Lipschitzian semigroups
International Journal of Mathematics and Mathematical Sciences, 2001
LetKbe a nonempty subset of ap-uniformly convex Banach spaceE,Ga left reversible semitopological semigroup, and𝒮={Tt:t∈G}a generalized Lipschitzian semigroup ofKinto itself, that is, fors∈G,‖Tsx−Tsy‖≤as‖x−y‖+bs(‖x−Tsx‖+‖y−Tsy‖)+cs(‖x−Tsy‖+‖y−Tsx‖), forx,y∈Kwhereas,bs,cs>0such that there exists at1∈Gsuch thatbs+cs<1for alls≽t1. It is proved that if there exists a closed subsetCofKsuch that⋂sco¯{Ttx:t≽s}⊂Cfor allx∈K, then𝒮with[(α+β)p(αp⋅2p−1−1)/(cp−2p−1βp)⋅Np]1/p<1has a common fixed point, whereα=lim sups(as+bs+cs)/(1-bs-cs)andβ=lim sups(2bs+2cs)/(1-bs-cs).
Structure of Common Fixed Point Set of Demicontinuous Asymptotically S-Nonexpansive Semigroups
2010
Every asymptotically nonexpansive mapping is uniformly continuous, but this fact is not true for asymptotically S-nonexpansive mappings in general. In a Banach space X, by constructing a sequence {xn} defined in Browder’s technique for a demicontinuous asymptotically S-nonexpansive semigroup T = {T (t) : t ≥ 0} of mappings from C ⊂ X into itself with function k(·), we prove that the common fixed point set F (S) ∩ ∩t>0F (T (t)) is the sunny nonexpansive retract of F (S), where F (S) is the fixed point set of a weakly continuous mapping S form C into itself. Under the assumption on uniform convexity of X, we prove that the common fixed point set F (S)∩∩t>0F (T (t)) is closed and convex.