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On a singularity concept for kernels of nonlinear integral operators. (English) Zbl 0982.47033
Let a family of nonlinear integral operators \((T_w)\), where \(w>0\), be defined by \[ (T_w f)= \int_\Omega K_w(t, f(t+ s)) d\mu(t),\quad s\in\Omega, \] 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.

47G10 Integral operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
41A25 Rate of convergence, degree of approximation
47J20 Variational and other types of inequalities involving nonlinear operators (general)
41A35 Approximation by operators (in particular, by integral operators)
47A58 Linear operator approximation theory
46A80 Modular spaces
47A35 Ergodic theory of linear operators
47B34 Kernel operators