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.
Langevin Equations with Generalized Proportional Hadamard–Caputo Fractional Derivative