Nonlocal Conformable-Fractional Differential Equations with a Measure of Noncompactness in Banach Spaces

+is paper deals with the existence of mild solutions for the following Cauchy problem: dαx(t)/dtα � Ax(t) + f(t, x(t)), x(0) � x0 + g(x), t ∈ [0, τ], where d α(.)/dtα is the so-called conformable fractional derivative. +e linear part A is the infinitesimal generator of a uniformly continuous semigroup (T(t))t≥0 on a Banach spaceX, f and g are given functions.+emain result is proved by using the Darbo–Sadovskii fixed point theorem without assuming the compactness of the family (T(t))t>0 and the Lipshitz condition on the nonlocal part g.


Introduction
Many dynamical processes in physics, biology, economics, and other areas of applications can be governed by abstract ordinary differential evolution equations of the following form: Unfortunately, the classical derivative _ x(t) appearing in equation (1) is local and cannot model the dynamical processes with memory. Hence, in order to avoid this shortcoming of classical derivative, many authors try to replace the classical derivative by a fractional derivative [1][2][3][4] because fractional derivatives have been proved that they are a very good way to model many phenomena with memory in various fields of science and engineering [5][6][7][8][9]. In consequence, many researchers pay attention to form the best definition of fractional derivative. Recently, a novel definition named conformable fractional derivative is introduced in [10].
is new fractional derivative quickly becomes the subject of many contributions in several areas of science [11][12][13][14][15][16][17][18][19][20][21][22]. Motivated by the better effect of the fractional derivative and simple properties of the conformable fractional derivative, we consider model (1) in the framework of conformable fractional calculus. Precisely, we study the following Cauchy problem: where d α (.)/dt α is the conformable fractional derivative of the order α ∈ ]0, 1]. e linear part A is the infinitesimal generator of a uniformly continuous semigroup (T(t)) t≥0 on a Banach space (X, ‖.‖). For more details about semigroup theory, we refer to [23]. e nonlinear part f: [0, τ] × X ⟶ X is a given function. e initial condition x(0) � x 0 + g(x) means the nonlocal condition [24]. For physical interpretations of this condition, we can see [25,26]. e nonlocal condition attracts the attention of many authors in several works [27,28]. e vector x 0 is an element of X and g: C ⟶ X is a given function, with C is the space of continuous functions x(.) defined from [0, τ] into X. roughout this paper, we endow the space C with the norm |x| c � sup t∈[0,τ] ||x(t)||. It is well known that the space (C, |.| c ) is a Banach space. We also denote by |.| the norm in the space L(X) of bounded operators defined form X into itself.
Our goal in this paper is to prove the existence of mild solutions for the Cauchy problem (2) by means of the Darbo-Sadovskii fixed point theorem without assuming the compactness of the family (T(t)) t>0 and the Lipshitz condition on the nonlocal part g. e content of this paper is organized as follows. In section 2, we recall some preliminary facts on the conformable fractional calculus and measure of noncompactness. Section 3 is devoted to prove the main result.

Preliminaries
Recalling some preliminary facts on the conformable fractional calculus.
Definition 1 (see [10]). Let α ∈ ]0, 1]. e conformable fractional derivative of order α of a function x(.) for t > 0 is defined as follows: For t � 0, we adopt the following definition: e fractional integral I α (.) associated with the conformable fractional derivative is defined by Theorem 1 (see [10]). If x(.) is a continuous function in the domain of I α (.), then we have Definition 2 (see [8]). e Laplace transform of a function x(.) is defined by It is remarkable that the above transform is not compatible with the conformable fractional derivative. For this, the adapted transform is given by the following definition.
Definition 3 (see [11]). e fractional Laplace transform of order α ∈ ]0, 1] of a function x(.) is defined by e following proposition gives us the actions of the fractional integral and the fractional Laplace transform on the conformable fractional derivative, respectively. Proposition 1 (see [11]). If x(.) is a differentiable function, then we have the following results: According to [15], we have the following remark.
Remark 1. For two functions x(.) and y(.), we have Now, we recall some concepts on the Hausdorff measure of noncompactness.
Definition 4 (see [29,30]). For a bounded set B in a Banach space X, the Hausdorff measure of noncompactness σ is defined as σ(B) � inf ε > 0: B can be covered by a finite number of balls with radii ε . (11) e following lemma presents some basic properties of the Hausdorff measure of noncompactness.
Lemma 1 (see [29,30]). Let X be a Banach space and B, C ⊆ X be bounded. en, the following properties hold. Definition 5 (see [30]). e operator Q: D(Q) ⊆ X ⟶ X is said to be a σ-contraction if there exists a positive constant k < 1 such that σ(Q(B)) ≤ kσ(B) for any bounded closed subset B ⊆ D(Q).
Lemma 2 (see [29,30] (Darbo-Sadovskii theorem)). Let B ⊂ X be a bounded, closed, and convex set. If Q: B ⟶ B is a continuous and σ-contraction operator. en, Q has at least one fixed point in B.

We denote by σ c the Hausdorff measure of noncompactness in the space C of continuous functions x(.) defined from [0, τ] into X.
Lemma 4 (see [33]). Let D 0 ≔ x n ⊂ C be a countable set, then Lemma 5 (see [29]). Let D ⊂ C be bounded and equicontinuous, then

Main Result
We first give the definition of mild solutions for the Cauchy problem (2). To do so, applying the fractional Laplace transform in equation (2), we obtain en, one has Using the inverse fractional Laplace transform combined with Remark 1, we obtain Motivate by the above calculus, we can introduce the following definition.

Definition 6.
A function x ∈ C is called a mild solution of the Cauchy problem (2) if To obtain the existence of mild solutions, we will need the following assumptions: Proof. In order to use the Darbo-Sadovskii fixed point theorem, we put B r ≔ x ∈ C, |x| c ≤ r for r > 0 and define the operator Γ: C ⟶ C by e proof will be given in four steps.
Step 1. Prove that there exists a radius δ > 0 such that Taking the supremum, we obtain Using assumption (H 4 ), we deduce that Hence, it suffices to consider δ as a solution of the following inequality: Precisely, we can choose δ such that Journal of Mathematics Step 2. Prove that Γ: en, by using a direct computation, we obtain Using assumption (H 1 ), we get ‖s α− 1 [f(s, x n (s))− f(s, x(s))]‖ ≤ 2μ δ (s)s α− 1 and f(s, x n (s)) ⟶ f(s, x(s)) as n ⟶ +∞.