In this book, Goro Shimura provides the most comprehensive generalizations of this type by stating several reciprocity laws in terms of abelian varieties, theta functions, and modular functions of several variables, including Siegel modular functions.
This example is a typical application of the methods of modular function spaces to the theory of nonlinear integral and differential equations.
The purpose of this article is to give an outline of fixed point theory for mappings defined on some subsets of modular function spaces which are natural generalization of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others.
The purpose of this paper is to give an outline of a fixed point theory for monotone-contraction mappings defined on some subsets of modular function spaces which are natural generalizations of both function and sequence variants of many important spaces like the Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces, and many others [1].
The purpose of this paper is to give outlines of the Knaster-Kuratowski-Mazurkiewicz theory for mappings defined on some subsets of modular function spaces which are natural generalization of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others.
In the context of modular function spaces, convergence to fixed points of some iterative algorithms, applied to asymptotic pointwise nonexpansive mappings, was proven by Bin Dehaish and Kozlowski in [22].
The importance of applications of nonexpansive mappings in modular function spaces lies in the richness of structure of modular function spaces, that is, besides being Banach spaces (or F-spaces in a more general setting --are equipped with modular equivalentsetting --aremetric notionsetting --arerequippeded with almodularequivalentsnvergence and cofvergenormin submeasure.
The importance of applications of nonexpansive mappings in modular function spaces lies in the richness of structure of modular function spaces that besides being Banach spaces (or F-spaces in a more general settings) are equipped with modular equivalents of norm or metric notions, and also are equipped with almost everywhere convergence and convergence in submeasure.
In the following theorem, we recall some of the properties of modular function spaces that will be used later on in this paper.
In the context of modular function spaces, Khamsi discussed in [12] the existence of nonlinear semigroups in Musielak-Orlicz spaces and considered some applications to differential equations.
