Fixed Point Theorems for Nonexpansive Type Mappings in Banach Spaces

In this paper, we present some fixed point results for a class of nonexpansive type and α-Krasnosel’skiĭ mappings. Moreover, we present some convergence results for one parameter nonexpansive type semigroups. Some non-trivial examples have been presented to illustrate facts.The authors thanks the Basque Government for its support through Grant IT1207-19

Stability and Convergence Results Based on Fixed Point Theory for a Generalized Viscosity Iterative Scheme

Es reproducción del documento publicado en http://dx.doi.org/10.1155/2009/314581A generalization of Halpern's iteration is investigated on a compact convex subset of a smooth Banach space. The modified iteration process consists of a combination of a viscosity term, an external sequence, and a continuous nondecreasing function of a distance of points of an external sequence, which is not necessarily related to the solution of Halpern's iteration, a contractive mapping, and a nonexpansive one. The sum of the real coefficient sequences of four of the above terms is not required to be unity at each sample but it is assumed to converge asymptotically to unity. Halpern's iteration solution is proven to converge strongly to a unique fixed point of the asymptotically nonexpansive mapping.Ministerio de Educación (DPI2006-00714; Gobierno Vasco (GIC07143-IT-269-07 y SAIOTEK S-PE08UN15

Strong convergence theorems for nonexpansive semigroup in Banach spaces

AbstractLet K be a nonempty closed convex subset of a reflexive and strictly convex Banach space E with a uniformly Gâteaux differentiable norm, and F={T(t):t>0} a nonexpansive self-mappings semigroup of K, and f:K→K a fixed contractive mapping. The strongly convergent theorems of the following implicit and explicit viscosity iterative schemes {xn} are proved.xn=αnf(xn)+(1−αn)T(tn)xn,xn+1=αnf(xn)+(1−αn)T(tn)xn. And the cluster point of {xn} is the unique solution to some co-variational inequality

On the asymptotic behavior of families of nonlinear mappings and some weak convergence theorems (Study on Nonlinear Analysis and Convex Analysis)

In this paper, we study the asymptotic behavior of orbits of nonexpansive semigroups in Banach spaces. We also establish a weak convergence theorem for two normally 2-generalized hybrid mappings and we give some convergence theorems

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