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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 $T_0\left(x\right)=x$, $T_{s+t}=T_s\left(T_t\left(x\right)\right)$, 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.

In this paper we give a characterization of $\sigma$-order continuity of modular function spaces $L_{\varrho }$ in terms of the existence of best approximants by elements of order closed sublattices of $L_{\varrho }$. We consider separately the case of Musielak–Orlicz spaces generated by non-$\sigma$-finite measures. Such spaces are not modular function spaces and the proofs require somewhat different methods.