Ordered Convex Metric Spaces

The aim of this article is to introduce a new notion of ordered convex metric spaces and study some basic properties of these spaces. Several characterizations of these spaces are proven that allow making geometric interpretations of the new concepts.

On convergence theorems for single-valued and multi-valued mappings in p-uniformly convex metric spaces

We prove ∆-convergence and strong convergence theorems of an iterative sequence generated by the Ishikawa’s method to a fixed point of a single-valued quasi-nonexpansive mappings in p-uniformly convex metric spaces without assuming the metric convexity assumption. As a consequence of our single-valued version, we obtain a result for multi-valued mappings by showing that every multi-valued quasi-nonexpansive mapping taking compact values admits a quasi-nonexpansive selection whose fixed-point set of the selection is equal to the strict fixed-point set of the multi-valued mapping. In particular, we immediately obtain all of the convergence theorems of Laokul and Panyanak [Laokul, T.; Panyanak, B. A generalization of the (CN) inequality and its applications. Carpathian J. Math. 36 (2020), no. 1, 81–90] and we show that some of their assumptions are superfluous.

Fixed point theorems for enriched Ćirić-Reich-Rus contractions in Banach spaces and convex metric spaces

The main aim of this paper is to establish some fixed point theorems for enriched Ćirić-Reich-Rus contractions in Banach spaces and convex metric spaces, respectively.

Note on Common Fixed Point Theorems in Convex Metric Spaces

In the present paper, we pointed out that there is a gap in the proof of the main result of Rouzkard et al. (The Bulletin of the Belgian Mathematical Society 2012). Then after, utilizing the concept of (E.A.) property in convex metric space, we obtained an alternative and correct version of this result. Finally, it is clarified that in the theory of common fixed point, the notion of (E.A.) property in the set up of convex metric space develops some new dimensions in comparison to the hypothesis that a range set of one map is contained in the range set of another map.

Convergence theorems for generalized hemicontractive mapping in p-uniformly convex metric space

In this paper, we introduce and study an Ishikawa-type iteration process for the class of generalized hemicontractive mappings in 𝑝-uniformly convex metric spaces, and prove both Δ-convergence and strong convergence theorems for approximating a fixed point of generalized hemicontractive mapping in complete 𝑝-uniformly convex metric spaces. We give a surprising example of this class of mapping that is not a hemicontractive mapping. Our results complement, extend and generalize numerous other recent results in CAT(0) spaces.