Existence results of mild solutions for impulsive fractional differential equations with almost sectorial operators (Q2050839)

In this article, there is considered the following initial value problem for semilinear impulsive fractional evolution equations \[\begin{cases} & ^C D^\alpha x(t) + A x(t)= f(t,x(t)), t \in I=[0,T], t \neq t_k , \\ & \Delta x |_{t=t_k}= I_k(x(t_k^-)), t=t_k, k=1,2, \dots, m , \\ & x(0)=x_0, \end{cases} \tag{1}\] where \(^C D^\alpha \) is the standard Caputo's fractional derivative of order \(\alpha, 0 < \alpha < 1\) and \(A : D(A) \subset \mathbb{X} \to \mathbb{X}\) is an almost sectorial operator on a Banach space \(\mathbb{X}\), \(0 < t_1 < t_2 < \dots < t_m = T\), \(I_k \in C(\mathbb{X}, \mathbb{X}), k = 1, 2, \dots , m\) and \(\Delta x|_{t=t_k}=x(t_k^+)-x(t_k^-)\) with \(x(t_k^+)\) and \(x(t_k^-)\) representing the right and left limits of \(x(t)\) at \(t = t_k\) respectively. As first main result, using the Banach's contraction principle, the author obtains sufficient conditions for the existence and uniqueness of piecewise continuous mild solutions of (1). The second main result is the obtained via Schauder's fixed point theorem and presents sufficient conditions for the existence of at least one piecewise continuous mild solution of the same initial value problem. An example for the obtained first result is given too. For the entire collection see [Zbl 1468.65003].