Perturbation for fractional-order evolution equation

Fractional calculus has gained a lot of importance during the last decades, mainly because it has become a powerful tool in modeling several complex phenomena from various areas of science and engineering. This paper gives a new kind of perturbation of the order of the fractional derivative with a study of the existence and uniqueness of the perturbed fractional-order evolution equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$^{C}D^{\alpha-\epsilon}_{0+}u(t)=A~^{C}D^{\delta}_{0+}u(t)+f(t),$\end{document}u(0)=uo, α∈(0,1), and 0≤ε, δ<α under the assumption that A is the generator of a bounded Co-semigroup. The continuation of our solution in some different cases for α, ε and δ is discussed, as well as the importance of the obtained results is specified.