The iterative solutions of split common fixed point problem for asymptotically nonexpansive mappings in Banach spaces

In this paper, we propose an iteration algorithm for finding a split common fixed point of an asymptotically nonexpansive mapping in the frameworks of two real Banach spaces. Under some suitable conditions imposed on the sequences of parameters, some strong convergence theorems are proved, which also solve some variational inequalities that are closely related to optimization problems. The results here generalize and improve the main results of other authors.

Common fixed points of monotone ρ-nonexpansive semigroup in modular spaces

In this paper, we consider the class of monotone ρ-nonexpansive semigroups and give existence and convergence results for common fixed points. First, we prove that the set of common fixed points is nonempty in uniformly convex modular spaces and modular spaces. Then we introduce an iteration algorithm to approximate a common fixed point for the same class of semigroups.

Fixed point results for Geraghty quasi-contraction type mappings in dislocated quasi-metric spaces

In this paper, fixed point results for a newly introduced Geraghty quasi-contraction type mappings are proved in more general metric spaces called T-orbitally complete dislocated quasi-metric spaces. Geraghty quasi-contraction type mappings generalize, among others, Ciric’s quasi-contraction mappings and other Geraghty quasi-contractive type mappings in the literature. Fixed point results are obtained without imposing a continuity condition on the mapping, thereby further generalizing some other related work in the literature. An example is given to show the validity of results obtained.

A short and sharpened way to approach fixed point results involving fuzzy $\mathcal{H}$-contractive mappings

In the present paper, we adopt a short and sharpened approach to prove fixed point results involving fuzzy $\mathcal{H}$ H -contractive mappings utilized in (Wardowski, Fuzzy Sets Syst. 125:245–252, 2013) and other related articles. In this process, we are able to relax some conditions utilized by earlier authors which in turn yields affirmative answers to some open questions raised by earlier authors.

K-Correspondences, USCOs, and fixed point problems arising in discounted stochastic games

We establish a fixed point theorem for the composition of nonconvex, measurable selection valued correspondences with Banach space valued selections. We show that if the underlying probability space of states is nonatomic and if the selection correspondences in the composition are K-correspondences (meaning correspondences having graphs that contain their Komlos limits), then the induced measurable selection valued composition correspondence takes contractible values and therefore has fixed points. As an application we use our fixed point result to show that all nonatomic uncountable-compact discounted stochastic games have stationary Markov perfect equilibria – thus resolving a long-standing open question in game theory.