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

In 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: dα dtα [ dαx(t) dtα ] = Ax(t) + f (t, x(t)), t ∈ [0,τ ] subject to the nonlocal conditions x(0) = x0 + g(x) and d αx(0) dtα = x1 + h(x), where dα (·) dtα is the conformable fractional derivative of order α ∈ ]0, 1] and A is the infinitesimal generator of a cosine family ({C(t), S(t)})t∈R on a Banach space X . The elements x0 and x1 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 and any Lipschitz conditions on the functions g and h. MSC: 34A08; 47D09


Introduction
Classical derivatives appear in several mathematical models in various areas of science such as physics, engineering, biology, finance, and so on. However, there are many phenomena that may not depend only on the time moment but also on the former time history, which cannot be modeled utilizing the classical derivatives. For this reason, many authors try to replace the classical derivatives with the so-called fractional derivatives in numerous contributions [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], because it has been proven that this last kind of derivatives is a very good way to describe processes with memory. According to the literature of fractional calculus, it is remarkable that there are many approaches to defining fractional derivatives, and each definition has advantages compared to others [18][19][20][21][22]. In consequence, many researchers have paid attention to propose new fractional derivatives in order to deal better with modeling of evolutionary phenomena [23,24]. In the work [23], the authors proposed the so-called fractional conformable derivative, which quickly became the subject of many research papers [1,. For example, in [1] the authors proved the existence of mild solutions for the following nonlocal conformable fractional Cauchy problem: which is a natural extension of the works [55,56] in the frame of the conformable fractional derivative, where A is the infinitesimal generator of a cosine family {C(t), S(t)} t∈R on a Banach space (X, · ) and d α (·) dt α presents the conformable fractional derivative of order α ∈ ]0, 1]. The elements x 0 and x 1 are two fixed vectors in X, and f : mean the nonlocal conditions, which can be applied in physics with better effects than the classical initial conditions [57][58][59]. We note that the existence result given in [1, Theorem 3.1] for Cauchy problem (1.1) has been proved by using the Krasnoselskii fixed point theorem, under the following assumptions: The family (S(t)) t∈R is compact for all t > 0. However, there are many concrete applications in which the above assumptions are difficult to realize. Indeed, for X = L 2 ( , R) with is a bounded domain in R n (n ≥ 1), many authors [60][61][62] have considered the following nonlocal condition: which does not satisfy Lipschitz condition (A 1 ), where K is a kernel with the following conditions: (C 1 ) K(t, ξ , σ , ·) is a continuous function for almost every (t, ξ , σ ) ∈ [0, T] × × . (C 2 ) K(·, ·, ·, r) is a measurable function for each fixed r ∈ R.
Motivated by this discussion, in the present work we use the Darbo-Sadovskii fixed point theorem in order to prove the existence of mild solutions for Cauchy problem (1.1) without assuming the Lipschitz conditions imposed in (A 1 ), (A 2 ) and the compactness of the family (S(t)) t>0 .
The rest of this paper is organized as follows. In Sect. 2, we briefly recall some tools related to the conformable fractional calculus, the cosine family of linear operators, and the Hausdorff measure of noncompactness. Section 3 is devoted to proving the main result.

Preliminaries
We recall some preliminary facts on the conformable fractional calculus.
provided that the limits exist.
The conformable fractional integral I α (·) of a function x(·) is defined by

Theorem 2.2 ([25]) If x(·) is a differentiable function, then we have
Now, we present some definitions concerning the cosine family of linear operators.

Definition 2.2 ([55]) A one parameter family (C(t)) t∈R of bounded linear operators on a
Banach space X is called a strongly continuous cosine family if and only if: We also define the sine family (S(t)) t∈R associated with the cosine family (C(t)) t∈R as follows: The infinitesimal generator A of a strongly continuous cosine family ((C(t)), (S(t))) t∈R on X is defined by We end these preliminaries by some concepts on the Hausdorff measure of noncompactness.

Definition 2.3 ([63, 64])
The Hausdorff measure of noncompactness σ of a bounded set B in a Banach space X is defined as follows: The following lemma presents some basic properties of the Hausdorff measure of noncompactness.  x(t) . It is well known that the space (C, | · | c ) is a Banach space. (σ (D(t))).

Main result
According to [1], we have the following definition.

Definition 3.1 A function x ∈ C is called a mild solution of Cauchy problem (1.1) if
To obtain the existence of mild solutions, we will need the following assumptions: (H 1 ) The function f (t, ·) : X − → X is continuous, and for all r > 0, there exists a function Proof In order to use the Darbo-Sadovskii fixed point theorem, we define the operator We also consider the ball B r := {x ∈ C, |x| c ≤ r}, where The proof will be given in four steps.
Step 1: Prove that (B r ) ⊂ B r .
For x ∈ C, we have Using assumptions (H 1 ), (H 5 ), and (H 6 ), we get Taking the supremum, we obtain | (x)| c ≤ r, and this shows that (B r ) ⊂ B r .
The above inequality combined with assumption (H 8 ) shows that (B r ) is equicontinuous on [0, τ ].
Step 4: Prove that : B r − → B r is a σ c -contraction operator. Let D ⊂ B r , then by Lemma 2.3, there exists a countable set D 0 such that D 0 = {x n } ⊂ D. Hence, (D 0 ) becomes a countable subset of (D). Thus, Lemma 2.3 proves that σ c ( (D)) ≤ 2σ c ( (D 0 )). Since (D 0 ) is bounded and equicontinuous, then by using the second point of Lemma 2.5, we obtain