‎$g$-continuous mapping‎

In this paper, we introduce a new type of graph contraction using a special class of functions and give a best proximity point theorem for this contraction in complete metric spaces endowed with a graph. Then we support our main theorem by a non-trivial example and give some consequences of it for usual graphs.

In this work, we define the notion of an algebraic distance in algebraic cone metric spaces defined by Niknam et al. [A. Niknam, S. Shamsi Gamchi and M. Janfada, Some results on TVS-cone normed spaces and algebraic cone metric spaces, Iranian J. Math. Sci. Infor. 9 (1) (2014), 71--80] and introduce some its elementary properties. Then we prove the existence and uniqueness of fixed point for a Banach contractive type mapping in algebraic cone metric spaces associated with an algebraic distance and endowed with a graph.

In this paper, we study the existence and uniqueness of fixed points for mappings with respect to a wt -distance in b-metric spaces endowed with a graph. Our results are significant, since we replace the condition of continuity of mapping with the condition of orbitally G-continuity of mapping and we consider b-metric spaces with graph instead of b -metric spaces, under which can be generalized, improved, enriched and unified a number of recently announced results in the existing literature. Additionally, we elicit all of our main results by a non-trivial example and pose an interesting two open problems for the enthusiastic readers.

In this paper‎, ‎we introduce a new type of graph contraction using a special class of functions and give a best proximity point theorem for this contraction in complete metric spaces endowed with a graph under two different conditions‎. ‎We then support our main theorem by a non-trivial example and give some consequences of best proximity point of it for usual graphs.