Fixed-Point Theorems in Generalized Modular Metric Spaces

Modular function spaces are one of the unique conditions of modular vector spaces which were defined by Nakano [37] in 1950. Later on, Khamsi, Kozlowski, and Reich [28] introduced the fixed-point principle in modular function spaces in 1990. Chistyakov introduced concept of a modular metric space in 2011 [14]. Abdou and Khamsi introduced fixed-point theory into the modular metric spaces using different techniques from the viewpoint of Chistyakov [14, 15], the similar approach continues in this part as they used in [1]. In this chapter, the Banach Contraction Principle and Ćirić Quasi-contraction are proven in Generalized Modular Metric Spaces (briefly GMMS). The usual topology is defined on these spaces, and then, using Nadler [36] and Edelstein’s results in [1], two fixed-point theorems are given for a multivalued contractive-type map in the construction of modular metric spaces. They are Caristi and Feng-Liu types in GMMS with their applications as in [42] and [43]. © 2021, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.