We prove a result on the existence and uniqueness of the solution of a new feature-preserving nonlinear nonlocal diffusion equation for signal denoising for the one-dimensional case. The partial differential equation is based on a novel diffusivity coefficient that uses a nonlocal automatically detected parameter related to the local bounded variation and the local oscillating pattern of the noisy input signal.

We study the isomorphic structure of $(\sum {\ell }_{q})_{c_{0}}\ (1< q<\infty )$ and prove that these spaces are complementably homogeneous. We also show that for any operator T from $(\sum {\ell }_{q})_{c_{0}}$ into ${\ell }_{q}$ , there is a subspace X of $(\sum {\ell }_{q})_{c_{0}}$ that is isometric to $(\sum {\ell }_{q})_{c_{0}}$ and the restriction of T on X has small norm. If T is a bounded linear operator on $(\sum {\ell }_{q})_{c_{0}}$ which is $(\sum {\ell }_{q})_{c_{0}}$ -strictly singular, then for any $\epsilon>0$ , there is a subspace X of $(\sum {\ell }_{q})_{c_{0}}$ which is isometric to $(\sum {\ell }_{q})_{c_{0}}$ with $\|T|_{X}\|<\epsilon $ . As an application, we show that the set of all $(\sum {\ell }_{q})_{c_{0}}$ -strictly singular operators on $(\sum {\ell }_{q})_{c_{0}}$ forms the unique maximal ideal of $\mathcal {L}((\sum {\ell }_{q})_{c_{0}})$ .