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

Abstract

This paper deals with the existence of mild solutions for the following Cauchy problem: , where is the so-called conformable fractional derivative. The linear part A is the infinitesimal generator of a uniformly continuous semigroup on a Banach space X, f and are given functions. The main result is proved by using the Darbo–Sadovskii fixed point theorem without assuming the compactness of the family and the Lipshitz condition on the nonlocal part .

1. 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 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 [1–4] 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 [5–9]. 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]. This new fractional derivative quickly becomes the subject of many contributions in several areas of science [11–22]. 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 is the conformable fractional derivative of the order . The linear part A is the infinitesimal generator of a uniformly continuous semigroup on a Banach space . For more details about semigroup theory, we refer to [23]. The nonlinear part is a given function. The initial condition means the nonlocal condition [24]. For physical interpretations of this condition, we can see [25, 26]. The nonlocal condition attracts the attention of many authors in several works [27, 28]. The vector is an element of X and is a given function, with is the space of continuous functions defined from into X. Throughout this paper, we endow the space with the norm . It is well known that the space is a Banach space. We also denote by the norm in the space 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 and the Lipshitz condition on the nonlocal part .

The 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.

2. Preliminaries

Recalling some preliminary facts on the conformable fractional calculus.

Definition 1 (see [10]). Let . The conformable fractional derivative of order α of a function for is defined as follows:For , we adopt the following definition:The fractional integral associated with the conformable fractional derivative is defined by

Theorem 1 (see [10]). If is a continuous function in the domain of , then we have

Definition 2 (see [8]). 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 [11]). The fractional Laplace transform of order of a function is defined byThe 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 is a differentiable function, then we have the following results:According to [15], we have the following remark.

Remark 1. For two functions and , we haveNow, 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 asThe following lemma presents some basic properties of the Hausdorff measure of noncompactness.

Lemma 1 (see [29, 30]). Let X be a Banach space and be bounded. Then, the following properties hold.(1)B is precompact if and only if (2), where and mean the closure and convex hull of B, respectively;(3), where (4), where (5)(6) for any , when X be a real Banach space;(7)If the operator is Lipschitz continuous with constant k ≥ 0 then we have for any bounded subset , where Y is another Banach space and ρ represents the Hausdorff measure of noncompactness in Y.

Definition 5 (see [30]). The operator is said to be a σ-contraction if there exists a positive constant such that for any bounded closed subset .

Lemma 2 (see [29, 30] (Darbo–Sadovskii theorem)). Let be a bounded, closed, and convex set. If is a continuous and σ-contraction operator. Then, Q has at least one fixed point in B.

Lemma 3 (see [31, 32]). Let be a bounded set, then there exists a countable set such that .
We denote by the Hausdorff measure of noncompactness in the space of continuous functions defined from into X.

Lemma 4 (see [33]). Let be a countable set, then(1) is Lebesgue integral on (2), where

Lemma 5 (see [29]). Let be bounded and equicontinuous, then(1) is continuous on (2)

3. 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

Then, 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 is called a mild solution of the Cauchy problem (2) ifTo obtain the existence of mild solutions, we will need the following assumptions:  The function is continuous, and for all there exists a function such that , for all   The function is continuous, for all   The function is continuous and compact.  There exist positive constants a and b such that , for all   There exists a positive constant L such that , for any countable set and

Theorem 2. Assume that hold, then the Cauchy problem (2) has at least one mild solution provided that

Proof. In order to use the Darbo–Sadovskii fixed point theorem, we put for and define the operator byThe proof will be given in four steps.

Step 1. Prove that there exists a radius such that .
Let , we haveTaking the supremum, we obtainUsing assumption , we deduce thatHence, it suffices to consider δ as a solution of the following inequality:Precisely, we can choose δ such that

Step 2. Prove that is continuous.
Let such that in . We haveThen, by using a direct computation, we obtainUsing assumption , we get and as .
The Lebesgue dominated convergence theorem proves that as . According to continuity of the function , we deduce that . Hence, is continuous.

Step 3. Prove that is equicontinuous.
For and such that . We haveBy using assumptions and , we obtainThe above inequality combined with the uniform continuity of the family proves that is equicontinuous on .

Step 4. Prove that is a -contraction operator.
Let , then by Lemma 3 there exists a countable set such that . Hence, Γ(D₀) becomes a countable subset of . Thus, Lemma 3 proves that . Since is bounded and equicontinuous, then by using Lemma 5, we obtainThen, one hasBy using point of Lemma 1, we deduce thatSince is compact, then is relatively compact. Hence, using point of Lemma 1 in the above inequality, we obtainIn view of Lemma 4, we getNext, point of Lemma 1 shows thatBy using assumption , we obtainHence, by using a direct computation combined with point of Lemma 5, we obtainIn consequence, we haveSince , then is a -contraction operator.
In conclusion, Lemma 2 shows that has at least one fixed point, which is a mild solution of the Cauchy problem (2).

Remark 2. We note that Theorem 2 improves Theorem 3 in [18] because in Theorem 2 we have not imposed the compactness of the family and the Lipshitz condition on the nonlocal part .

4. Conclusion

Without imposing the compactness condition on the semigroup family and the Lipschitz condition on the nonlocal condition, we have proved the existence of mild solutions for a class of conformable-fractional differential equations with nonlocal conditions in a Banach space. The main result is obtained by means of semigroup theory combined with the Darbo–Sadovskii fixed point theorem.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.