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

Abstract

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