Impulsive differential equations involving general conformable fractional derivative in Banach spaces

This paper deals with two classes of impulsive equations involving the general conformable fractional derivative in Banach spaces: (1) impulsive Sobolev-type integro-differential equations with the general conformable fractional derivative, (2) impulsive delay evolution equations with the general conformable fractional derivative. By combining the generalized Laplace transform and the properties of the general conformable fractional derivative, we present a proper definition of mild solutions for the impulsive integro-differential equations with the general conformable fractional derivative. In view of this definition, we obtain a new existence theorem of (ω,c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\omega ,c)$$\end{document}-periodic solutions for a normal fractional inhomogeneous evolution equation with the general conformable fractional derivative (Theorem 2.3) which will be used to study the (ω,c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\omega ,c)$$\end{document}-periodic solutions for the impulsive delay evolution equations with the general conformable fractional derivative. Then we establish existence and uniqueness theorems for the impulsive integro-differential equations with the general conformable fractional derivative. Next, we derive existence theorems of (ω,c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\omega ,c)$$\end{document}-periodic solutions for the impulsive delay evolution equations involving the general conformable fractional derivative. Finally, applications are also given to illustrate our abstract results.