Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph

Abstract

Abstract This article presents the notions of extended b-gauge space ( U , Q φ ; Ω ) \left(U,{Q}_{\varphi ;\Omega }) and extended J φ ; Ω {{\mathcal{J}}}_{\varphi ;\Omega } -families of generalized extended pseudo-b-distances on U U . Furthermore, we look at these extended J φ ; Ω {{\mathcal{J}}}_{\varphi ;\Omega } -families on U U and define the extended J φ ; Ω {{\mathcal{J}}}_{\varphi ;\Omega } -sequential completeness. We also look into some fixed and periodic point theorems for set-valued mappings in the new space with a graph that does not meet the completeness condition of the space. Additionally, this article includes some examples to explain the corresponding results and highlights some important consequences of our obtained results.