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We study mapsϕof positive operators of the Schattenp-classes (1

The Opial property of Hilbert spaces and some other special Banach spaces is a powerful tool in establishing fixed point theorems for nonexpansive and, more generally, nonspreading mappings. Unfortunately, not every Banach space shares the Opial property. However, every Banach space has a similar Bregman-Opial property for Bregman distances. In this paper, using Bregman distances, we introduce the classes of Bregman nonspreading mappings and investigate the Mann and Ishikawa iterations for these mappings. We establish weak and strong convergence theorems for Bregman nonspreading mappings.

This special volume was originally conceived to provide authors and readers a publication to present the most recent advances in the study of applications of various fixed point theorems. Since 1909, when Luitzen Brouwer proved the first fixed point theorem named after him, fixed point theory has played very important roles in many different fields. We can find a lot of demonstrations in optimization theory, approximation theory, differential equations, variational inequalities, complementary problems, equilibrium theory, game theory, economics theory, and so forth. Fixed point theorems are developed for single-valued or set-valued mappings of metric spaces, topological vector spaces, posets and lattices, Banach lattices, . . .. Among the themes of fixed point theory, the topic of approximation of fixed points of mappings is particularly important because it is useful for proving the existence of fixed points ofmappings. It can be applied to prove the solvability of optimization problems, differential equations, variational inequalities, and equilibrium problems. Due to the importance of and the high volume of active research in the nonlinear analysis and optimization and, in particular, many new tools in studying them involving fixed point approximations, it is worthwhile to publish a special issue on this topic to highlight the recent advances in this field. The selection of the papers included in this volume, which has been based on a strict international peer review procedure, contains a representative list of papers with newfangled results which covers the different topics considered in our original proposal. In the papers in this special issue, the following topics are discussed.

Let $X$ be a real reflexive Banach space with a weakly continuous duality mapping ${J}_{\phi }$ . Let $C$ be a nonempty weakly closed star-shaped (with respect to $u$ ) subset of $X$ . Let $\scr F$ = $\{T(t):t\in [0,+\infty )\}$ be a uniformly continuous semigroup of asymptotically nonexpansive self-mappings of $C$ , which is uniformly continuous at zero. We will show that the implicit iteration scheme: ${y}_{n}={\alpha }_{n}u+(1-{\alpha }_{n})T({t}_{n}){y}_{n}$ , for all $n\in \Bbb N$ , converges strongly to a common fixed point of the semigroup $\scr F$ for some suitably chosen parameters $\{{\alpha }_{n}\}$ and $\{{t}_{n}\}$ . Our results extend and improve corresponding ones of Suzuki (2002), Xu (2005), and Zegeye and Shahzad (2009).

We study the abstract Cauchy problem for a class of integrodifferential equations in a Banach space with nonlinear perturbations and nonlocal conditions. By using MNC estimates, the existence and continuous dependence results are proved. Under some additional assumptions, we study the topological structure of the solution set.