On the existence of mild solutions for nonlocal differential equations of the second order with conformable fractional derivative

Abstract In the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019), authors have used Krasnoselskii fixed point theorem for showing existence of mild solutions an abstract class conformable fractional differential equations form: $\frac{d^{\alpha }}{dt^{\alpha }}[\frac{d^{\alpha }x(t)}{dt^{\alpha }}]=Ax(t)+f(t,x(t))$ d ? t [ x ( ) ] = A + f , , $t\in [0,\tau ]$ ? 0 ? subject to nonlocal conditions $x(0)=x_{0}+g(x)$ g and }x(0)}{dt^{\alpha }}=x_{1}+h(x)$ 1 h where }(\cdot)}{dt^{\alpha }}$ ? is derivative order $\alpha \in\, ]0,1]$ A infinitesimal generator a cosine family $(\{C(t),S(t)\})_{t\in \mathbb{R}}$ { C S } R on Banach space X . The elements $x_{0}$ $x_{1}$ are two vectors f g h given functions. present paper continuation 2019) use Darbo–Sadovskii proving same result [Theorem 3.1] without assuming compactness $(S(t))_{t>0}$ > any Lipschitz functions