Given an injective closed linear operator A defined in a Banach space X, and writing CFDtα the Caputo–Fabrizio fractional derivative of order α∈(0,1), we show that unique solution abstract Cauchy problem (∗)CFDtαu(t)=Au(t)+f(t),t≥0, where f is continuously differentiable, given by first u′(t)=Bαu(t)+Fα(t),t≥0;u(0)=−A−1f(0), family bounded operators Bα constitutes Yosida approximation Fα(t)→f(t)...

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a m...

In this paper, we define an operator function as a series of operators corresponding to the Taylor representing complex variable. previous papers, considered case when has decomposition in Laurent with infinite principal part and finite regular part. Our central challenge is improve result having entire satisfying special condition growth regularity. As application, consider opportunity broaden...