On a singularity concept for kernels of nonlinear integral operators (Q2730726)

Let a family of nonlinear integral operators \((T_w)\), where \(w>0\), be defined by NEWLINE\[NEWLINE(T_w f)= \int_\Omega K_w(t, f(t+ s)) d\mu(t),\quad s\in\Omega,NEWLINE\]NEWLINE for \(f\) belonging to a modular function space \(L^\rho(\Omega)\), where \(K_w:\Omega\times \mathbb{R}\to \mathbb{R}\), \(w> 0\), form a nonlinear kernel. Until now it was usually supposed that \((K_w)\) satisfy a generalized Lipschitz condition with a function \(\psi(t,|u-v|)\) of the increment \(|u-v|\) independent of the index \(w\). In the present paper, a modular approximation theorem is proved by means of \((T_w)\), where \(\psi_w\) depends on \(w\). This result is obtained by introducing the notion of a \(\rho\)-uniformly equi-continuous net of functions in \(L^\rho(\Omega)\) and modifying the definition of singularity of \((K_w)\). Also, the connection between both notions of singularity is investigated.