Semigroups of Nonlinear Mappings in Modular Function Spaces

Let us recall that a family $\{T_t\}_{t \geq 0}$ of mappings forms a semigroup if $T_0(x)=x$ , and $T_{s+t}=T_s(T_t(x))$ . Such a situation is quite typical in mathematics and applications. For instance, in the theory of dynamical systems, the modular function space $L_{\rho}$ would define the state space and the mapping $(t,x)\rightarrow T_t(x)$ would represent the evolution function of a dynamical system. The question about the existence of common fixed points, and about the structure of the set of common fixed points, can be interpreted as a question whether there exist points that are fixed during the state space transformation T t at any given point of time t, and if yes - what the structure of a set of such points may look like. In the setting of this chapter, the state space may be an infinite dimensional. Therefore, it is natural to apply these result to not only to deterministic dynamical systems but also to stochastic dynamical systems. Because of the wide body of potential applications, the theory of semigroups of nonlinear mappings in modular function spaces, initiated in 1992 paper by Khamsi [106], has become recently a subject of an intensive development, see [9, 55, 56, 135, 139, 140, 141]. There is, however, a lot of space for future research in this area.