Existence of Solution for a Conformable Fractional Cauchy Problem with Nonlocal Condition

We replace the partial derivative by a fractional derivative [1, 2] because fractional derivatives have been proven to be a very good way to model many phenomena with memory in various fields of science and engineering [1–9]. Consequently, several researchers are working to form the best definition of the fractional derivative. A new definition called conformal fractional derivative is introduced in [7]. ,is new fractional derivative becomes the subject of many contributions in several areas of science [9]. Motivated by the better effect of the fractional derivative and simple properties of the conformable fractional derivative, we consider model (1) in this work; we are going to study Cauchy problem with fractional derivative. Precisely, we consider fractional Cauchy problem of the following form:


Introduction
Many dynamic processes in physics, biology, economics, and other fields of application can be governed by differential evolution equations of neutral type of the following form: We replace the partial derivative by a fractional derivative [1,2] because fractional derivatives have been proven to be a very good way to model many phenomena with memory in various fields of science and engineering [1][2][3][4][5][6][7][8][9]. Consequently, several researchers are working to form the best definition of the fractional derivative. A new definition called conformal fractional derivative is introduced in [7].
is new fractional derivative becomes the subject of many contributions in several areas of science [9]. Motivated by the better effect of the fractional derivative and simple properties of the conformable fractional derivative, we consider model (1) in this work; we are going to study Cauchy problem with fractional derivative. Precisely, we consider fractional Cauchy problem of the following form: where d α /dt α is the conformable fractional derivative of α ∈ (0, 1). A is a sectorial operator which generates a strongly analytic semigroup (T(t)) on a Banach space X. For more details about semigroup theory, we refer to [7]. ).
x(0) + g(x) � x 0 is nonlocal condition; this notion has been a hot topic in recent years. eir association to classical problems has brought a lot of improvement at the level of modeling, thus making it more realistic. e nonlocal condition joined the main equation instead of the classical initial condition which is necessary to model well and write mathematically, physical phenomena like in electronics, in mechanics of materials, or in biomathematics in the way closest to the reality of many phenomena in multiple disciplines.
e nonlocal condition means that the initial condition depends on some future times.
In this paper, we prove the existence of mild solution of conformable fractional differential equations with nonlocal condition. e main results are based on semigroup theory combined with the Krasnoselskii fixed point theorem. e content of this paper is organized as follows. In Section 2, we recall some preliminary facts on the conformable fractional calculus and Section 3 is devoted to prove the main result.
In this section, we recall some concepts on conformable fractional calculus. Definition 1. e conformable fractional derivative of x of order α at t > 0 is defined as

Preliminaries
When the limit exists, we say that e (α)-fractional integral of a function x is defined by

Theorem 1. If x is a continuous function in the domain of I α , then
Defintion 2. e fractional Laplace transform of order α of x is defined by e fractional Laplace transform of conformable fractional derivative is given by the following proposition.
Fractional powers of an operator.
Definition 3. Let A be a sectorial operator defined on a Banach space X, such that Reσ(A) > 0; for α > 0, we note by A − α the operator defined by Definition 4. Let A be a sectorial operator defined on a Banach space X, such that Reσ(A) > 0. We define the family of operators (A α ) α ≥ 0 as follows: A 0 � I X , and for α > 0, is the infinitesimal generator of an analytic semigroup (T(t)) t≥0 and if 0 ∈ ρ(A), then We assume that M is a closed bounded convex subset of a Banach space E: en, there exists y ∈ M such that y � Ay + By. We end these preliminaries with the notion of sectorial operator.

Theorem 2.
A densely sectorial operator generates a strongly analytic semigroup (T(t)) t≥0 . Moreover, 2 International Journal of Differential Equations with gamma being a suitable path λ ∉ w + S θ . Now, we give the main contribution results.

Main Results
Before presenting our main results, we introduce the following assumptions: for all x 1 , x 2 ∈ D(A α ) and for 0 ≤ t ≤ a.

Existence of mild solution:
Applying the Laplace transform to equation (2), we obtain en, Hence, en, Hence, International Journal of Differential Equations erefore, According to inverse fractional Laplace transform, we find the formula en, we obtain (20) Defintion 1. We say that x ∈ C([0, a], D(A α )) is a mild solution of equation (2) if the following assertion is true: Theorem 3. If (T(t)) t>0 is compact and (H1) − (H4) are satisfied, then problem (2) has at least one mild solution, provided that Proof. Choosing let B r � x ∈ X‖x‖ α ≤ r , for x ∈ B r , and t ∈ [0, a] define the operators Γ 1 and Γ 2 by International Journal of Differential Equations Claim 1. We prove that Γ 1 is contraction on B r . We have en, Since Claim 3. We will prove that Γ 2 is continuous on B r Let x n ∈ B r such that lim n⟶∞ x n � x.
Step 1. We prove that Γ 2 (x)(t)|x ∈ B r is relativement compact in X. For some fixed t ∈]0, a[, let ε ∈]0, t[ and define the operator Γ ε 2 by According to compactness of (T(t)) t>0 , the set Γ ε 2 (x)(t)|x ∈ B r is relatively compact in X.
We have erefore, Γ 2 (x)(t)|x ∈ B r is relatively compact in X. It is clear that Γ 2 (x)(0)|x ∈ B r is compact.