$$\mathcal {H}^+$$-multivalued contractions and their application to homotopy theory

Kikkawa and Suzuki (Nonlinear Anal 69:2942–2949, 2008) and Kikkawa and Suzuki (Fixed Point Theory Appl, 2008, Art. ID 649749) proved some fixed point results that are generalizations of Kannan’s, Nadler’s and Suzuki’s fixed point theorems. Here, we present fixed point results of this kind for multivalued mappings in the setting of \(\mathcal {H}^+\)-metric spaces. The theorems provided allow upgrading of some known results which is shown by examples. Moreover, we give a homotopy result as an application of our main theorem.     

Meir–Keeler-type conditions in abstract metric spaces

Various Meir–Keeler-type conditions for mappings acting in abstract metric spaces are presented and their connections are discussed. Results about associated symmetric spaces, obtained in [S. Radenovi?, Z. Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, Banach J. Math. Anal. 5 (2011), 38–50] are used to show that the regularity condition for the underlying cone can be dropped in some fixed point results that have appeared recently.     

Remarks on “Quasi-contraction on a cone metric space”

Recently, D. Ili? and V. Rako?evi? [D. Ili?, V. Rako?evi?, Quasi-contraction on a cone metric space, Appl. Math. Lett. (2008) doi:10.1016/j.aml.2008.08.011] proved a fixed point theorem for quasi-contractive mappings in cone metric spaces when the underlying cone is normal. The aim of this paper is to prove this and some related results without using the normality condition.     

Coupled common fixed point theorems for w-compatible mappings in ordered cone metric spaces

We establish coupled coincidence point results for mixed g-monotone mappings under general contractive conditions in partially ordered cone metric spaces over solid cones. We also present results on existence and uniqueness of coupled common fixed points. Our results generalize, extend and unify several well known comparable results in the literature. To illustrate our results and to distinguish them from the earlier ones, we equip the paper with examples.     

On Some Problems of Grothendieck Concerning (F) and (DF) Spaces

This is an expository article devoted to the answers to the 10 open problems of the fundamental paper of Grothendieck (Summa Res. Math. 1954; 3:57–123), some of which were given in the recent years.     

Coupled fixed point results for mappings without mixed monotone property

It is shown that a mixed monotone property in coupled fixed point results can be replaced by another property which is automatically satisfied in the case of a totally ordered space, the case which is the most important in applications. Hence, these results can be applied in a much wider class of problems.     

Common positive solutions for two non-linear matrix equations using fixed point results in b-metric-like spaces

Aequationes mathematicae - We consider the system of non-linear matrix equations (NMEs) of the form $$\begin{aligned} {\mathcal {X}}={\mathcal {B}}_{1} + \sum _{i=1}^{k}{\mathcal...