Buthinah A Bin Dehaish

Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces

Let Lρ be a uniformly convex modular function space with a strong Opial property. Let T:C→C be an asymptotic pointwise nonexpansive mapping, where C is a ρ-a.e. compact convex subset of Lρ. In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point of T. In addition, we prove that if C is compact in the strong sense, then both processes converge strongly to a fixed point.MSC:47H09, 47H10.

Hybrid viscosity approximation methods for general systems of variational inequalities in Banach spaces

Let X be a uniformly convex and 2-uniformly smooth Banach space. In this paper, we propose an implicit iterative method and an explicit iterative method for solving a general system of variational inequalities (in short, GSVI) in X based on Korpelevich’s extragradient method and viscosity approximation method. We show that the proposed algorithms converge strongly to some solutions of the GSVI under consideration. When X is a 2-uniformly smooth Banach space with weakly sequentially continuous duality mapping, we also propose two methods, which were inspired and motivated by Korpelevich’s extragradient method and Mann’s iterative method. Furthermore, it is also proven that the proposed algorithms converge strongly to some solutions of the considered GSVI.MSC:49J30, 47H09, 47J20.

Fixed point theorems for generalized contractive type multivalued maps

Without using the concept of Hausdorff metric, we prove some results on the existence of fixed points for generalized contractive multivalued maps with respect to u-distance. Consequently, several known fixed point results are either generalized or improved.MSC:47H10, 54H25.

Fibonacci–Mann Iteration for Monotone Asymptotically Nonexpansive Mappings in Modular Spaces

In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci–Mann iteration process, introduced recently by Alfuraidan and Khamsi, defined by xn+1 = tnT(xn) + (1− tn)xn, for n ∈ N, when T is a monotone asymptotically nonexpansive self-mapping.

Approximate Fixed Points

In this chapter, we present some of the known results about the concept of approximate fixed points of a mapping. In particular, we discuss some new results on approximating fixed points of monotone mappings. Then we conclude this chapter with an application of these results to the case of a nonlinear semigroup of mappings. It is worth mentioning that approximate fixed points are useful when a computational approach is involved. In particular, most of the approximate fixed points discussed in this chapter are generated by an algorithm that allows its computational implementation.

On the convergence of iteration processes for semigroups of nonlinear mappings in modular function spaces

Let C be a ρ-bounded, ρ-closed, convexsubset of a modular function space . We investigate the problem of constructingcommon fixed points for asymptotic pointwise nonexpansive semigroups of mappings, i.e. a family such that, , and , where , for every .MSC: 47H09, 46B20, 47H10, 47E10.

Common fixed points for pointwise Lipschitzian semigroups in modular function spaces

AbstractLet C be a ρ-bounded, ρ-closed, convex subset of a modular function space Lρ. We investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups of nonlinear mappings Tt:C→C, i.e. a family such that T0(f)=f, Ts+t(f)=Ts∘Tt(f) and ρ(T(f)−T(g))≤αt(f)ρ(f−g), where lim supt→∞αt(f)≤1 for every f∈C. In particular, we prove that if Lρ is uniformly convex, then the common fixed point is nonempty ρ-closed and convex.MSC:47H09, 46B20, 47H10, 47E10.