Korovkin type theorems for weakly nonlinear and monotone operators

TL;DR

This paper extends Korovkin's approximation theorems to weakly nonlinear and monotone operators on Banach lattices, demonstrating convergence in various modes including almost everywhere, measure, and $L^{p}$-norm.

Contribution

It introduces new Korovkin-type theorems for weakly nonlinear and monotone operators on Banach lattices, broadening classical approximation theory.

Findings

Proves convergence almost everywhere for the operators.

Establishes convergence in measure and $L^{p}$-norm.

Includes several illustrative examples.

Abstract

In this paper we prove analogues of Korovkin's theorem in the context of weakly nonlinear and monotone operators acting on Banach lattices of functions of several variables. Our results concern the convergence almost everywhere, the convergence in measure and the convergence in $L^{p}$-norm. Several results illustrating the theory are also included.