Convergence in $BV_\varphi$ by nonlinear Mellin-type convolution operators

Abstract

In this paper we establish convergence results for a family $\mathbb{T}$ of nonlinear integral operators of the form: $$(T_{w}f)(s) = \int_0^{+\infty}K_{w}(t,f(st))dt = \int_0^{+\infty} L_{w}(t)H_{w}(f(st))dt, \ s \in \mathbb{R}_0^+,$$ where $f \in Dom \mathbb{T}$, $Dom \mathbb{T}$ being the class of all the measurable functions $f:\mathbb{R}_0^+ \rightarrow \mathbb{R}$ such that $T_{w}f$ is well defined as Lebesgue integral for every $s \in \mathbb{R}_0^+$. For the above family of nonlinear Mellin type operators, under suitable singularity assumptions on the kernels $\mathbb{K} = \{K_{w}\}$, we state a convergence result of type $\mathrm{lim}_{w \rightarrow + \infty}V_{\varphi}[\mu(T_{w}f - f)] = 0$, for some constant $\mu > 0$ and for every $f$ belonging to a suitable subspace of $BV_\varphi$-functions.