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

Abstract

In this work, we prove the existence and uniqueness of mild solution of the fractional conformable Cauchy problem with nonlocal condition. We obtained these results by applying the fixed point theorems precisely to the fixed point theorem of Krasnoselskii and Banach’s fixed point theorem. At the end, we provide application.

1. 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 [1–9]. 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]. This 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 is the conformable fractional derivative of . is a sectorial operator which generates a strongly analytic semigroup on a Banach space . For more details about semigroup theory, we refer to [7]. We denote by the banach space of continuous function from into with the norm . For our study, the functions and and and also .

is nonlocal condition; this notion has been a hot topic in recent years. Their association to classical problems has brought a lot of improvement at the level of modeling, thus making it more realistic. The 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. The 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. The main results are based on semigroup theory combined with the Krasnoselskii fixed point theorem.

The 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.

2. Preliminaries

Definition 1. The conformable fractional derivative of x of order at is defined asWhen the limit exists, we say that is -differentiable at .
If is -differentiable and exists, then defineThe -fractional integral of a function is defined by

Theorem 1. If is a continuous function in the domain of , then

Defintion 2. The fractional Laplace transform of order of is defined byThe fractional Laplace transform of conformable fractional derivative is given by the following proposition.

Proposition 1. If is differentiable, thenFractional powers of an operator.

Definition 3. Let be a sectorial operator defined on a Banach space , such that ; for , we note by the operator defined by

Definition 4. Let be a sectorial operator defined on a Banach space , such that . We define the family of operators as follows: , and for ,

Theorem 2. If is the infinitesimal generator of an analytic semigroup and if , then(1) is a Banach space with the norm for every .(2) for all and .(3)For every , we have .

We assume that M is a closed bounded convex subset of a Banach space E:(i) for each .(ii) is continuous and compact.(iii) is contraction.

Then, there exists such that .

We end these preliminaries with the notion of sectorial operator.

Definition 3. is said to be sectorial operator of type if there exists , , and as follows:(1) is closed and linear operator.(2), the resolvent of exists.(3), ,where .

Theorem 2. densely sectorial operator generates a strongly analytic semigroup . Moreover,with being a suitable path .
Now, we give the main contribution results.

3. Main Results

Before presenting our main results, we introduce the following assumptions:(H1) is continuous and there exists a constant such that for all and for .(H2) is continuous and for all , there exists a function such that .(H3); there exists a constant such that , for all with for .(H4).H(5)There exists a constant such thatfor all and for .

Existence of mild solution:

Applying the Laplace transform to equation (2), we obtain

Then,

Hence,

Then,

Hence,

Therefore,

According to inverse fractional Laplace transform, we find the formula

Then, we obtain

Defintion 1. We say that is a mild solution of equation (2) if the following assertion is true:

Theorem 3. If is compact and are satisfied, then problem (2) has at least one mild solution, provided that

Proof. Choosinglet , for , and define the operators and by

Claim 1. We prove that is contraction on . We haveThen,Since , so is a contraction on .

Claim 2. We prove that for every .

Claim 3. We will prove that is continuous on
Let such that .
We will prove that .
We haveBy , we have and . According to Lebesgue dominated convergence theorem, we obtain

Claim 4. We prove that is compact.

Step 1. We prove that is relativement compact in .
For some fixed , let and define the operator byAccording to compactness of , the set is relatively compact in .
We haveTherefore, is relatively compact in . It is clear that is compact.
Finally, is relativement compact in for all .

Step 2. We prove that is equicontinuous.
Let such that . We haveThen,Then,We conclude that , are equicontinuous at . By using Arzela Ascoli theorem, we obtain that is compact. Finally, the Krasnoselskii theorem helps us to complete the proof.
Uniqueness of mild solution.

Theorem 4. Assume that hold, then the Cauchy problem (2) has a unique solution, provided that

Proof. Define operator Next, let , we haveThen,Hence,Therefore,Since , as a consequence, has a unique fixed point in .

4. Application

We consider the conformable fractional Cauchy problem of the following form:

With , we define operator by and with the norm .(i) is class and .(ii) is continuous and there exists a constant such that(iii) is class and .

For with and , we define , , , and .

The problem takes the following form:

and , on the other hand,

From the equality of Poincare, we havewhich implies that hypothesis is verified

Let and . We know thatand hypothesis is verified.

Let ; we have ,

From the equality of Poincare, we have

Then, is verified.

Moreover, . As a consequence, (41) has a unique fixed point in .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.