Convexity and boundedness relaxation for fixed point theorems in modular spaces

[EN] Although fixed point theorems in modular spaces have remarkably applied to a wide variety of mathematical problems, these theorems strongly depend on some assumptions which often do not hold in practice or can lead to their reformulations as particular problems in normed vector spaces. A recent trend of research has been dedicated to studying the fundamentals of fixed point theorems and relaxing their assumptions with the ambition of pushing the boundaries of fixed point theory in modular spaces further. In this paper, we focus on convexity and boundedness of modulars in fixed point results taken from the literature for contractive correspondence and single-valued mappings. To relax these two assumptions, we seek to identify the ties between modular and b-metric spaces. Afterwards we present an application to a particular form of integral inclusions to support our generalized version of Nadler’s theorem in modular spaces. The authors gratefully acknowledge the reviewer and the editor for their useful observations and recommendations. Lael, F.; Shabanian, S. (2021). Convexity and boundedness relaxation for fixed point theorems in modular spaces. Applied General Topology. 22(1):91-108. https://doi.org/10.4995/agt.2021.13902 M. Abbas, F. Lael and N. Saleem, Fuzzy b-metric spaces: Fixed point results for ψ-contraction correspondences and their application, Axioms 9, no. 2 (2020), 1-12. https://doi.org/10.3390/axioms9020036 A. Ait Taleb and E. Hanebaly, A fixed point theorem and its application to integral equations in modular function spaces, Proceedings of the American Mathematical Society 128 (1999), 419-426. https://doi.org/10.1090/S0002-9939-99-05546-X M. R. Alfuraidan, Fixed points of multivalued mappings in modular function spaces with a graph, Fixed Point Theory and Applications 42 (2015), 1-14. https://doi.org/10.1186/s13663-015-0292-7 A. H. Ansari, T. Dosenovic, S. Radenovic, N. Saleem, V. Sesum-Cavic and J. Vujakovic, C-class functions on some fixed point results in ordered partial metric spaces via admissible mappings, Novi Sad Journal of Mathematics 49, no. 1 (2019), 101-116. https://doi.org/10.30755/NSJOM.07794 A. H. Ansari, J. M. Kumar and N. Saleem, Inverse-C-class function on weak semi compatibility and fixed point theorems for expansive mappings in G-metric spaces, Mathematica Moravica 24, no. 1 (2020), 93-108. https://doi.org/10.5937/MatMor2001093H A. Aghajani, M. Abbas and J. R. Roshan, Common fixed point of generalized weak contractive