309 research outputs found
Strong Convergence Theorems of the General Iterative Methods for Nonexpansive Semigroups in Banach Spaces
Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E*. Let S={T(s):0≤s1 and γ a positive real number such that γ<1/α(1-1-δ/λ). When the sequences of real numbers {αn} and {tn} satisfy some appropriate conditions, the three iterative processes given as follows: xn+1=αnγf(xn)+(I-αnF)T(tn)xn, n≥0, yn+1=αnγf(T(tn)yn)+(I-αnF)T(tn)yn, n≥0, and zn+1=T(tn)(αnγf(zn)+(I-αnF)zn), n≥0 converge strongly to x̃, where x̃ is the unique solution in Fix(S) of the variational inequality 〈(F-γf)x̃,j(x-x̃)〉≥0, x∈Fix(S). Our results extend and improve corresponding ones of Li et al. (2009) Chen and He (2007), and many others
On unification of the strong convergence theorems for a finite family of total asymptotically nonexpansive mappings in Banach spaces
In this paper, we unify all know iterative methods by introducing a new
explicit iterative scheme for approximation of common fixed points of finite
families of total asymptotically -nonexpansive mappings. Note that such a
scheme contains as a particular case of the method introduced in [C.E. Chidume,
E.U. Ofoedu, \textit{Inter. J. Math. & Math. Sci.} \textbf{2009}(2009) Article
ID 615107, 17p]. We construct examples of total asymptotically nonexpansive
mappings which are not asymptotically nonexpansive. Note that no such kind of
examples were known in the literature. We prove the strong convergence theorems
for such iterative process to a common fixed point of the finite family of
total asymptotically nonexpansive and total asymptotically nonexpansive
mappings, defined on a nonempty closed convex subset of uniformly convex Banach
spaces. Moreover, our results extend and unify all known results.Comment: 22 pages, Journal of Applied Mathematics (in press
Alternative iterative methods for nonexpansive mappings, rates of convergence and application
Alternative iterative methods for a nonexpansive mapping in a Banach space
are proposed and proved to be convergent to a common solution to a fixed point
problem and a variational inequality. We give rates of asymptotic regularity
for such iterations using proof-theoretic techniques. Some applications of the
convergence results are presented
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