Existence and Uniqueness of Mild Solutions to Impulsive Nonlocal Cauchy Problems

Abstract

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.

1. Introduction

Fractional differential equations have gained popularity due to their applications in many domains of science and engineering[1–3]. 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]. This 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 [1–10].

Motivated by the abovementioned works, we consider the following impulsive differential equation with conformable fractional derivative:where is the so-called fractional conformable derivative [4]. This novel fractional derivative attracts the attention of many authors in various domains of science [1–10]. is the generator of a -semigroup on a Banach space , is continuous, , are the element of , and , and represent respectively the right and left limits of at .

One of the main novelties of this paper is the concept on a mild solution for system (1). Then, 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 derivativewhere , , are defined as above, is a given function and constitutes a Cauchy problem. The condition represents the nonlocal condition [11]. For good effect of this condition, we refer to [12, 13]. We adopt the ideas given in [14–16] and obtained some new existence and uniqueness results for system (2) under the different assumptions on the nonlocal terms.

The 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.

2. 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 be the Banach space of all linear and bounded operators on . For a -semigroup on , we set . Let be the Banach space of all -valued continuous functions from into , endowed with the norm . We also introduce the set of functionsendowed with the normwhere it is easy to see that is a Banach space.

Theorem 1. (Krasnoselskii’s fixed point theorem). Assume that is a closed bounded convex subset of a Banach space . Furthermore, assume that and are mappings from into such that(1), for all (2) is a contraction(3) is continuous and compactThen, has a fixed point in .

Definition 1. (see [4]). Let . The conformable fractional derivative of order of a function for is defined asFor , we adapt the following definition:The fractional integral associated with the conformable fractional derivative is defined by

Theorem 2. (see [4]). If is a continuous function in the domain of , then we have

Definition 2. (see [2]). The Laplace transform of a function is defined byIt 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]). The fractional Laplace transform of order of a function is defined byThe 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 is a differentiable function, then we have the following results:According to [6], we have the following remark.

Remark 1. For two functions and , we have

Lemma 1. A measurable function is Bochner integrable, if is Lebesuge integrable.

Lemma 2. For and , we have .

3. 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 is Lipschitz.
Let us list the following hypotheses:
: is the infinitesimal generator of a compact semigroup in .
: is continuous and there exists a constant and a real-valued function such thatFor brevity, let us takeBy using the following Duhamel formula (see [7]), we can introduce the following definition of the mild solution for system (1).

Definition 4. We say that a function is called a mild solution of Cauchy problem,if satisfies

Theorem 3. If is compact and are satisfied, then Cauchy problem (1) has a unique mild solution on , provided that

Proof. Let be fixed. Define an operator : byBy our assumptions and Lemma 1, is well defined on .

Step 1. We prove that for .

Claim 1. For , taking into account the imposed assumptions and applying Lemma 2, we obtainwhere we use the inequality . The first and second terms tend to zero as . Moreover, it is obvious that the last terms tend to zero too as . Thus, we can deduce that .

Claim 2. For , keeping in mind our assumptions and applying Lemma 2 again, we haveAs , the right-hand side of the above inequality tends to zero. Thus, we can deduce that .
Similarly, we can also obtain that . That is, .

Step 2. We show that is the contraction on .

Claim 1. For each , it comes from our assumptions thatIn general, for each , using our assumptions again,thusHence, condition (17) allows us to conclude in view of the Banach contraction mapping principle that has a unique fixed point which is just the unique mild solution of system (1).

Case 2. is not Lipschitz.
We make the following assumptions:
: is continuous and maps a bounded set into a bounded set.
: the function is continuous, and for all , there exists a function such that , for all .
: for each , there exists a constant such thatwhere

Theorem 4. Suppose that , , and are satisfied. Then, for every , system (1) has at least a mild solution on .

Proof. Let be fixed. We introduce that map bywhereFor each , ,For each , ,Noting the condition , we see that .

Step 1. We prove that is a continuous mapping from to .
In order to derive the continuity of , we only check that and are all continuous.
For this purpose, we assume that in ; it comes from the continuity of that , as .
Noting thatby means of Lebesgue dominated convergence theorem, we obtain that as . It is easy to see that, for each ,Thus, is continuous. On the contrary, it is obvious that is continuous. Since and are continuous, is also continuous.

Step 2. We show that is a compact operator, or and are compact operators.
The compactness of is clear since it is a constant map.
Compactness of :

Claim 1. We prove that is relatively compact in .
For some fixed , let , , and define the operator byWe can write as follows:According to the compactness of , the set is relatively compact in . Using , we haveTherefore, is relatively compact in . It is clear that is compact. Finally, is relatively compact in , for all .

Claim 2. We prove that is equicontinuous.
Let such that . We haveHence,We conclude that , are equicontinuous at . By using the Arzela–Ascoli theorem, we obtain that is compact. Now, Schauder’s fixed point theorem implies that has a fixed point, which gives rise to a mild solution.

4. 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 , which satisfies the following integral equation:

Case 1. is Lipschitz.
To end this section, we make the following assumption:
: and there exists a constant such that
.

Theorem 5. Let , , and be satisfied. Then, for every , system (2) has a unique mild solution on , provided that

Proof. Define the operator byIt is obvious that is well defined on .

Step 1. We prove that , for .
For , by our assumptions,As , the right-hand side of the above inequality tends to zero. Recalling Step 1 in Theorem 3, we know that .

Step 2. is the contraction.
We only take , then we haveso we getwhereHence, condition (38) allows us to conclude, in view of the Banach contraction mapping principle again, that has a unique fixed point which is the mild solution of system (2).

Theorem 6. Suppose that , , and are satisfied. If , then system (2) has at least a mild solution on .

Proof. ChooseConsider . Define the operators on bywhereand is the same as the operator , defined in Theorem 4.
It suffices to proceed exactly the steps of the proof in Theorem 4, while replacing by to obtain that are continuous and compact. We want to use Krasnoselkii’s fixed point theorem. Thus, to complete the rest proof of this theorem, it suffices to show that is a contraction mapping and that if , then . Indeed, for any , we haveSince , we can deduce thatNext, for any ,Therefore, we can deduce that is the contraction from . Moreover, 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 .
Case 2: is not Lipschitz.
: and maps bounded sets into bounded sets.
: for each , there exists a constant such thatwhere

Theorem 7. Suppose that , , , and are satisfied. Then, for every , system (2) has at least a -mild solution on .

Proof. Define an operator on bywhereand is the same as defined in Theorem 4. Thus, we need to check that is compact. Observing the expression of the , we only check that, for each , the set is precompact in since is compact and . On the other hand, the equicontinuity of can be shown using the same idea.
Therefore, is also a compact operator. By Schauder’s fixed point theorem again, has a fixed point, which gives rise to a mild solution.

4.1. Example

In this section, some interesting examples are presented to illustrate the theory. Consider the following impulsive fractional differential equations with nonlocal conditions:

Let . Define , for , where . Then, is the infinitesimal generator of a strongly continuous semigroup in . Moreover, is also compact and , .

Case 1. Define
.
Clearly, f: [0, 1]  × X ⟶ X are continuous functions,
, with .
It is obvious that satisfies with .
makes the assumptions in Theorem 5 satisfied. Therefore, equation (54) has a unique mild solution on , provided thatCase 2. Define .
Clearly, , with .
makes the assumptions in Theorem 6 satisfied. Therefore, equations (54) has at least one mild solution on , provided that .
Case 3. Define .
.
Clearly, and are continuous and map a bounded set into a bounded set.
makes the assumptions in Theorem 7 satisfied, for large . Therefore, equation (54) has at least one mild solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest.