On the Existence of Common Fixed Points for Semigroups of Nonlinear Mappings in Modular Function Spaces

Let $C$ be a $\rho$-bounded, $\rho$-closed, convex subset of a modular function space $L_{\rho }$. We investigate the existence of common fixed points for semigroups of nonlinear mappings $T_t:C\to C$, i.e. a family such that $T_0(x)=x$, $T_{s+t}=T_s(T_t(x))$, where each $T_t$ is either $\rho$-contraction or $\rho$-nonexpansive. We also briefly discuss existence of such semigroups and touch upon applications to differential equations.