University of MaraghehSahand Communications in Mathematical Analysis2322-580713120190201The Existence Theorem for Contractive Mappings on $wt$-distance in $b$-metric Spaces Endowed with a Graph and its Application1153432210.22130/scma.2018.89571.471ENKamal FallahiDepartment of Mathematics, Payame Noor University, Tehran, Iran.0000-0003-3400-4424Dragan SavicPrimary School ''Kneginja Milica", Beograd, Serbia.Ghasem Soleimani RadDepartment of Mathematics, Payame Noor University, Tehran, Iran.0000-0002-0758-2758Journal Article20180717In 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.http://scma.maragheh.ac.ir/article_34322_c1a6f4a5bb424cbcce7290bf293886ca.pdf