In this paper, we prove Browder's convergence theorem for G-nonexpansive mappings in a Hilbert space with a directed graph. Moreover, we also prove strong convergence of the Halpern iteration process to a fixed point of G-nonexpansive mappings in a Hilbert space endowed with a directed graph. The main results obtained in this paper extend and generalize many well-known results in the literature. MSC: 47H04; 47H10

In this paper, we introduce the concepts of graph-preserving multi-valued mapping and a new type of multi-valued weak G-contraction on a metric space endowed with a directed graph G. We prove some coincidence point theorems for this type of multi-valued mapping and a surjective mapping g : X → X under some conditions. Several examples for these new concepts and some examples satisfying all conditions of our main results are also given. Our main results extend and generalize many coincidence point and fixed point theorems in partially ordered metric spaces. MSC: 47H04; 47H10

In this paper, we introduce a split best proximity point and equilibrium problem, and find a solution of the best proximity point problem such that its image under a given bounded linear operator is a solution of the equilibrium problem. We construct an iterative algorithm to solve such problem in real Hilbert spaces and obtain a weak convergence theorem. Finally, we also give an example to illustrate our result.

A new concept of set-valued Mizoguchi-Takahashi G-contractions is introduced in this paper and some fixed point theorems for such mappings in b-metric spaces endowed with directed graphs are established under some sufficient conditions. Our results improve and extend those of [20] and [24]. We also give some examples supporting our main results. As an applications, we prove the existence of fixed points for multivalued mappings satisfying generalized MT-contractive condition in-chainable b-metric spaces and the existence of a solution for some integral equations.

We give a category-free order theoretic variant of a key result in Auinger and Szendrei (J Pure Appl Algebra 204(3):493–506, 2006) and illustrate how it might be used to compute whether a finite X-generated group H admits a canonical dual prehomomorphism into the Margolis–Meakin expansion M(G) of a finite X-generated group G. We show that for G the Klein four-group a suitable H must be of exponent 6 at least and recapture a result of Szakács.