Mann-Type Viscosity Approximation Methods for Multivalued Variational Inclusions with Finitely Many Variational Inequality Constraints in Banach Spaces

We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature

On solving variational inequalities defined on fixed point sets of multivalued mappings in Banach spaces

AbstractWe are concerned with the problem of solving variational inequalities which are defined on the set of fixed points of a multivalued nonexpansive mapping in a reflexive Banach space. Both implicit and explicit approaches are studied. Strong convergence of the implicit method is proved if the space satisfies Opial's condition and has a duality map weakly continuous at zero, and the strong convergence of the explicit method is proved if the space has a weakly continuous duality map. An essential assumption on the multivalued nonexpansive mapping is that the mapping be single valued on its nonempty set of fixed points

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

Construction of minimum-norm fixed points of pseudocontractions in Hilbert spaces

Abstract An iterative algorithm is introduced for the construction of the minimum-norm fixed point of a pseudocontraction on a Hilbert space. The algorithm is proved to be strongly convergent. MSC:47H05, 47H10, 47H17

Approximation of common fixed points of a countable family of continuous pseudocontractions in a uniformly smooth Banach space

AbstractIn this paper, we introduce a new implicit iterative algorithm for finding a common element of a countable family of continuous pseudocontractions in a uniformly smooth Banach space. We obtain some strong convergence theorems under suitable conditions. Our results extend the recent results announced by many others

Modified Mann-Halpern Algorithms for Pseudocontractive Mappings

Modified Mann-Halpern algorithms for finding the fixed points of pseudocontractive mappings are presented. Strong convergence theorems are obtained