Existence andUniqueness ofMild Solutions to ImpulsiveNonlocal Cauchy Problems

In this paper, a class of nonlocal impulsive differential equation with conformable fractional derivative is studied. By utilizing the theory of operators semigroup and fractional derivative, a new concept on a solution for our problem is introduced.We used some fixed point theorems such as Banach contraction mapping principle, Schauder’s fixed point theorem, Schaefer’s fixed point theorem, and Krasnoselskii’s fixed point theorem, and we derive many existence and uniqueness results concerning the solution for impulsive nonlocal Cauchy problems. Some concrete applications to partial differential equations are considered. Some concrete applications to partial differential equations are considered.


Introduction
Fractional differential equations have gained popularity due to their applications in many domains of science and engineering [1][2][3]. In consequence, many researchers pay attention to form a simple and best definition of fractional derivative. Recently, a new definition of fractional derivative named conformable fractional derivative has been introduced in [4]. is novel fractional derivative is very easy and satisfies all the properties of the standard one. In short time, many studies and discussion related to conformable fractional derivative have appeared in several areas of applications [1][2][3][4][5][6][7][8][9][10].
One of the main novelties of this paper is the concept on a mild solution for system (1). en, using some fixed point theorems such as Banach contraction mapping principle and Schauder's fixed point theorem, we derive many existence and uniqueness results concerning the mild solution for system (1) under the different assumptions on the nonlinear terms.
As a second problem, we discuss in Section 4, a nonlocal impulsive differential equation with conformable fractional derivative where A, f, y k are defined as above, g is a given function and constitutes a Cauchy problem. e condition x(0) � x 0 + g(x) represents the nonlocal condition [11]. For good effect of this condition, we refer to [12,13]. We adopt the ideas given in [14][15][16] and obtained some new existence and uniqueness results for system (2) under the different assumptions on the nonlocal terms. e content of this paper is organized as follows. In Section 2, we recall some preliminary facts on the conformable fractional calculus. Sections 3 and 4 are devoted to prove the main result. At last, some interesting examples are presented to illustrate the theory.

Preliminary
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper, and we recall some concepts on conformable fractional calculus.
Let L b (X) be the Banach space of all linear and bounded operators on X.
into X, endowed with the norm ‖x‖ C � sup t∈J ‖x(t)‖. We also introduce the set of functions x is continuous from left and has right hand limits at t ∈ t 1 , t 2 , . . . , t n , endowed with the norm where it is easy to see that (PC(J, X), ‖.‖ PC ) is a Banach space.
Theorem 1 (Krasnoselskii's fixed point theorem). Assume that K is a closed bounded convex subset of a Banach space X. Furthermore, assume that Γ 1 and Γ 2 are mappings from K into X such that , for all u, v ∈ K (2) Γ 1 is a contraction (3) Γ 2 is continuous and compact en, Γ 1 + Γ 2 has a fixed point in K.
Definition 1 (see [4]). Let α ∈ ]0, 1]. e conformable fractional derivative of order α of a function x(.) for t > 0 is defined as For t � 0, we adapt the following definition: e fractional integral I α (.) associated with the conformable fractional derivative is defined by Theorem 2 (see [4]). If x(.) is a continuous function in the domain of I α (.), then we have Definition 2 (see [2]). 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 [5]). 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 fraction Laplace transform on the conformable fractional derivative, respectively. Proposition 1 (see [5]). If x(.) is a differentiable function, then we have the following results: According to [6], we have the following remark.

Main Results
Now, we give the main contribution results.
To obtain the uniqueness of mild solution, we will need the following assumption.

Case 1.
We suppose that f is Lipschitz.
Let us list the following hypotheses: For brevity, let us take By using the following Duhamel formula (see [7]), we can introduce the following definition of the mild solution for system (1).
Theorem 3. If (T(t)) t>0 is compact and (HA) − (HF1) are satisfied, then Cauchy problem (1) has a unique mild solution on J, provided that Proof. Let x 0 ∈ X be fixed. Define an operator Journal of Mathematics By our assumptions and Lemma 1, Γ is well defined on PC(J, X).
Step 1. We prove that Γx ∈ PC(J, X) for x ∈ PC(J, X). Claim 1. For 0 ≤ τ < t ≤ t 1 , taking into account the imposed assumptions and applying Lemma 2, we obtain where we use the inequality t α − τ α ≤ (t − τ) α . e first and second terms tend to zero as t ⟶ τ. Moreover, it is obvious that the last terms tend to zero too as t ⟶ τ. us, we can deduce that Γx ∈ PC([0, t 1 ], X).

Claim 2.
For t 1 ≤ τ < t < t 2 , keeping in mind our assumptions and applying Lemma 2 again, we have As t ⟶ τ, the right-hand side of the above inequality tends to zero. us, we can deduce that Γx ∈ PC([t 1 , t 2 ], X).
Step 2. We show that Γ is the contraction on PC(J, X).

Claim 1.
For each t ∈ [0, t 1 ], it comes from our assumptions that In general, for each t ∈ (t k , t k+1 ], using our assumptions again, Hence, condition (17) allows us to conclude in view of the Banach contraction mapping principle that Γ has a unique fixed point x ∈ PC(J, X) which is just the unique mild solution of system (1).

Case 2. f is not Lipschitz.
We make the following assumptions: (HF2): f: J × X ⟶ X is continuous and maps a bounded set into a bounded set.
(HF3): the function f(t, .): X ⟶ X is continuous, and for all r > 0, there exists a function (C1): for each x 0 ∈ X, there exists a constant r > 0 such that Proof. Let x 0 ∈ X be fixed. We introduce that map where Noting the condition (C1), we see that Γ: Υ Γ ⟶ Υ Γ .
Step 1. We prove that Γ is a continuous mapping from Υ Γ to Υ Γ .
In order to derive the continuity of Γ, we only check that Γ 1 and Γ 2 are all continuous.
For this purpose, we assume that x n ⟶ x in Υ Γ ; it comes from the continuity of f that Noting that by means of Lebesgue dominated convergence theorem, we obtain that t 0 s α− 1 ‖f(s, x n (s)) − f(s, x(s))‖ds ⟶ 0, as n ⟶ ∞. It is easy to see that, for each t ∈ J, us, Γ 1 is continuous. On the contrary, it is obvious that Γ 2 is continuous. Since Γ 1 and Γ 2 are continuous, Γ is also continuous.

Journal of Mathematics
Step 2. We show that Γ is a compact operator, or Γ 1 and Γ 2 are compact operators. e compactness of Γ 2 is clear since it is a constant map. Compactness of Γ 1 : For some fixed t ∈ [0, b[, let ε ∈ ]0, t[, x ∈ B r , and define the operator Γ ε 1 by We can write Γ ε 1 as follows: According to the compactness of (T(t)) t>0 , the set Γ ε 1 (x)(t)| x ∈ B r is relatively compact in X. Using (HF3), we have Claim 2. We prove that Γ 1 (B r ) is equicontinuous.
Let t 1 , t 2 ∈ ]0, b] such that t 1 < t 2 . We have Hence, We conclude that Γ 1 (x), x ∈ B r are equicontinuous at t ∈ [0, b]. By using the Arzela-Ascoli theorem, we obtain that Γ 2 is compact. Now, Schauder's fixed point theorem implies that Γ has a fixed point, which gives rise to a mild solution.

Existence Results for Impulsive Nonlocal Cauchy Problems
In this section, we extend the results obtained in Section 3 to nonlocal problems for impulsive conformable fractional evolution equations. More precisely, we will prove the existence and uniqueness of the mild solutions for system (2). As we all known, the nonlocal conditions have a better effect on the solution and are more precise for physical measurements than the classical initial condition alone.
Definition 5. By a mild solution of system (2), we mean that a function x ∈ PC(J, X), which satisfies the following integral equation: It is obvious that F is well defined on PC(J, X).
Step 1. We prove that Fx ∈ PC(J, X), for x ∈ PC(J, X). For 0 ≤ τ < t ≤ t 1 , by our assumptions, As t ⟶ τ, the right-hand side of the above inequality tends to zero. Recalling Step 1 in eorem 3, we know that Fx ∈ PC(J, X).
Step 2. F is the contraction.
We only take t ∈ (t k , t k+1 ], then we have so we get where μ ′ ≔ M L g + T * . Hence, condition (38) allows us to conclude, in view of the Banach contraction mapping principle again, that F has a unique fixed point x ∈ PC(J, X) which is the mild solution of system (2). Theorem 6. Suppose that (HA), (HF3), and (Hg1) are satisfied. If ML g < (1/2), then system (2) has at least a mild solution on J.
Proof. Choose Consider B σ � x ∈ PC(J, X)|‖x‖ PC ≤ σ . Define the operators N on B σ by Journal of Mathematics where and N 3 is the same as the operator Γ 2 , defined in eorem 4. It suffices to proceed exactly the steps of the proof in eorem 4, while replacing B r by B σ to obtain that N 2 + N 3 are continuous and compact. We want to use Krasnoselkii's fixed point theorem. us, to complete the rest proof of this theorem, it suffices to show that N 1 is a contraction mapping and that if x, y ∈ B σ , then N 1 x + (N 2 + N 3 )y ∈ B σ . Indeed, for any x ∈ B σ , we have Since ML g < (1/2), we can deduce that Next, for any t ∈ (t k , t k+1 ], x, y ∈ C((t k , t k+1 ], X), erefore, we can deduce that N 1 is the contraction from ML g < 1. Moreover, N 2 + N 3 is compact and continuous. Hence, by the well-known Krasnoselskii's fixed point theorem, we can conclude that system (2) has at least one mild solution on J.
Case 2: g is not Lipschitz.