Caputo-Fabrizio fractional differential equations with instantaneous impulses

1.   Introduction

Fractional differential equations have recently been applied in various areas. For some fundamental results in the theory of fractional calculus and fractional differential equations we refer the reader to ^{[1,3,4,19,25,27,29]}, and the references therein. Some new aspects of the Caputo-Fabrizio derivative can be seen in ^[11,21] and the references therein. For some applications of a such derivative, we refer to ^[5,15].

Differential equations involving impulses effects; appear as a natural description of observed evolution phenomena of several real world problems ^{[12,16,18,26]}. Many physical situations are modeled by impulsive differential equations, for example problems in optimal control theory and problems in threshold theory in Biology. Major developments in the theory of impulsive fractional differential equations have been developed in the last years; see the books ^[1,26], the papers ^{[1,2,7,6,17,20,24,28]}, and the references therein.

Recently, in ^[4,8,13], the measure of noncompactness was applieded to some classes of functional Riemann-Liouville or Caputo fractional differential equations in Banach spaces. See also the classical monographs ^[9,10].

In this paper first we investigate the existence of solutions for the following Cauchy problem of Caputo-Fabrizio impulsive fractional differential equations

where $I_0 = [0, t_1], \ I_k = (t_k, t_{k+1}]; \ k = 1, \cdots, m; \ 0 = t_0 < t_1 < \dots < t_{m} < t_{m+1} = T, \ u_0\in{\Bbb R}, \ f:I_k\times {\Bbb R}\to {\Bbb R}; \ k = 0, \ldots, m, \ L_{k}: {\Bbb R}\to {\Bbb R}; \ k = 1, \ldots, m$ are given continuous functions, $^{CF}D_{t_k}^{r}$ is the Caputo-Fabrizio fractional derivative of order $r\in(0, 1).$

Next, by using the measure of noncompactness, we discuss the existence of solutions for problem (1.1), when $u_0\in E, \ f:I_k\times E\to E; \; \ k = 0, \ldots, m, \ L_{k}:E\to E; \ k = 1, \ldots, m$ are given functions, and $(E\|\cdot\|, )$ is a real or complex Banach space.

2.   Preliminaries

Let $I: = [0, T]; \ T > 0,$ and $C(I): = C(I, {\Bbb R})$ be the Banach space of all continuous functions from $I$ into ${\Bbb R}$ with the norm

By $L^{1}(I)$ we denote the Banach space of measurable function $u:I\to {\Bbb R}$ with are Lebesgue integrable, equipped with the norm

As usual, $AC(I)$ denotes the space of all absolutely continuous functions from $I$ into ${\Bbb R}.$

Let ${\cal M}_X$ denote the class of all bounded subsets of a metric space $X.$

Definition 2.1. ^[10] Let $X$ be a complete metric space. A map $\mu:{\cal M}_X\to[0, \infty)$ is called a measure of noncompactness on $X$ if it satisfies the following properties for all $B, B_1, B_2 \in{\cal M}_X.$

(a) $\mu(B) = 0$ if and only if $B$ is precompact (Regularity),

(b) $\mu(B) = \mu(\overline{B})$ (Invariance under closure),

(c) $\mu(B_1\cup B_2) = { \max}\{\mu(B_1), \mu(B_2)\}$ (Semi-additivity).

Definition 2.2. (^[10]). Let $X$ be a Banach space and let $\Omega_{X}$ be the family of bounded subsets of $E.$ The Kuratowski measure of noncompactness is the map $\mu: \Omega_{X} \to [0, \infty)$ defined by

where $M \in \Omega_{E}.$

The measure $\mu$ satisfies the following properties

$(1)$ $\mu(M) = 0 \Leftrightarrow \overline{M}$ is compact ($M$ is relatively compact).

$(2)$ $\mu(M) = \mu(\overline{M}).$

$(3)$ $M_1\subset M_2 \Rightarrow \mu(M_1)\leq \mu(M_2).$

$(4)$ $\mu(M_1+M_2)\leq \mu(M_1)+\mu(M_2).$

$(5)$ $\mu(cM) = |c|\mu(M), \ c\in{\Bbb R}.$

$(6)$ $\mu(\operatorname{conv} M) = \mu(M).$

Definition 2.3. ^[14] The Caputo-Fabrizio fractional integral of order $0 < r < 1$ for a function $h\in L^{1}(I)$ is defined by

where $M(r)$ is normalization constant depending on $r.$ For example, taking $M(r) = \frac{2}{2-r},$ we have

Definition 2.4. ^[14] The Caputo-Fabrizio fractional derivative of order $0 < r < 1$ for a function $h\in AC(I)$ is defined by

Note that $^{CF}D_0^{r}h = 0$ if and only if $h$ is a constant function.

For $M(r) = \frac{2}{2-r}$, one has

Lemma 2.5. Let $h\in L^{1}(I).$ Then the linear Cauchy problem

has a unique solution given by

where

Proof. Suppose that $u$ satisfies (2.1). From Proposition 1 in ^[22]; the equation

implies that

Thus from the initial condition $u(0) = u_0,$ we obtain

Hence we get (2.2).

For our purpose we will need the following fixed point theorems:

Theorem 2.6. (Schauder's fixed point theorem ^[9]). Let $X$ be a Banach space, $D$ be a bounded closed convex subset of $X$ and $T:D\to D$ be a compact and continuous map. Then $T$ has at least one fixed point in $D.$

Theorem 2.7. (Monch's fixed point theorem ^[23]). Let $D$ be a bounded, closed and convex subset of a Banach space such that $0\in D,$ and let $N$ be a continuous mapping of $D$ into itself. If the implication

holds for every subset $V$ of $D,$ then $N$ has a fixed point.

3.   Main results

In this section, we present some results concerning the existence of solutions for the problem (1.1). Consider the Banach space

with the norm

In the case when $E = {\Bbb R},$ we get

Definition 3.1. By a solution of the problem (1.1) we mean a function $u\in PC$ that satisfies $u(0) = u_0,$ $(^{CF}D_{t_k}^{r}u)(t) = f(t, u(t));$ for $t\in I_k, \ k = 0, \cdots, m,$ and

Lemma 3.2. Let $h:I\to E$ be a continuous function. A function $u\in PC$ is a solution of the fractional integral equation

if and only if u is a solution of the following problem

Proof. Assume $u$ satisfies (3.2). If $t\in I_0,$ then

Lemma 2.5 implies that

If $t\in I_1,$ then

Lemma 2.5 implies that

Thus

If $t\in I_2,$ then

Then, we obtain

If $t\in I_k,$ then again from Lemma 3.3 we get (3.1).

Conversely, suppose that $u$ satisfies (3.1). If $t\in I_0,$ then

Thus, $u(0) = u_0$ and using the fact that $^{CF}D^{r}_{t_k}$ is the left inverse of $(^{CF}I^{r}_{0}$ we get $(^{CF}D^{r}_{0}u)(t) = h(t).$

Now, if $t\in I_k; \ k = 1, \ldots, m,$ we get $(^{CF}D^{r}_{t_k}u)(t) = h(t).$ Also, we can easily show that

Hence, if $u$ satisfies (3.1) then we get (3.2).

As in the prove of the above Lemma, we can show the following one:

Lemma 3.3. A function $u\in PC$ is a solution of problem (1.1), if and only if u satisfies the following integral equation

where $c = u_0-a_r f(0, u_0).$

3.1.   Existence results in the scalar case

The following hypotheses will be used in the sequel.

$(H_{1})$ There exists a positive continuous function $p\in C(I_k); \ k = 0, \dots, m,$ such that

$(H_{2})$ There exists $q^\ast\geq 0$ such that

$(H_{3})$ For each bounded set $B\subset PC,$ the set $\{t\mapsto f(t, u(t)): u\in B, \ t\in I_k; \ k = 0, \dots, m \}$ is equicontinuous.

Set

Theorem 3.4. Assume that the hypotheses $(H_{1})-(H_{3})$ hold. If

then the problem (1.1) has at least one solution defined on $I.$

Proof. Consider the operator $N:PC\rightarrow PC$ defined by:

Clearly, the fixed points of the operator $N$ are solutions of the problem (1.1).

Let $R > 0,$ such that

and consider the ball $B_R: = B(0, R) = \{w\in \|w\|_{PC}\leq R\}.$

For each $t\in I_0,$ and $u\in PC,$ we have

On the other hand, for each $t\in I_k:\ k = 1, \dots, m,$ and $u\in PC,$ we have

Hence, for $t\in I,$ and $u\in PC,$ we get

This proves that $N(B_R)\subset B_R.$ We shall show that the operator $N:B_R\to B_R$ satisfies all the assumptions of Theorem 2.6. The proof will be given in two steps.

Step 1. $N:B_R \to B_R$ is continuous.

Let $\{u_{n}\}_{n\in {{\mathbb N}}}$ be a sequence such that $u_n\rightarrow u$ in $B_R.$ Then, for each $t\in I_0,$ we have

Since $u_n\rightarrow u\text{ as } n\rightarrow \infty$ and $f$ is continuous, then by using the Lebesgue dominated convergence theorem, (3.6) implies

Also, for each $t\in I_k; \ k = 1, \dots, m,$ we have

Again, we get the continuity of our operator $N.$

Step 2. $N(B_R)$ is bounded and equicontinuous.

Since $N(B_R)\subset B_R$ and $B_R$ is bounded, then $N(B_R)$ is bounded.

Next, let $\tau_1, \tau_2\in I_k; \ k = 0, \dots, m;$ such that $t_k\leq\tau_1 < t\leq \tau_2\leq t_{k+1}$ and let $u\in B_R.$ Then, from the continuity of $f,$ and $(H_{3}),$ we get

Hence, $N(B_R)$ is bounded and equicontinuous.

As a consequence of the above two steps, together with the Ascoli-Arzelá theorem, we can conclude that $N:B_R\to B_R$ is continuous and compact. From an application of Theorem 2.6, we deduce that $N$ has a fixed point $u$ which is a solution of problem (1.1).

3.2.   Existence results in Banach spaces

The following hypotheses will be used in the sequel.

$(H_{4})$ The function $t\mapsto f(t, u)$ is measurable on $I_k; \ k = 0, \dots, m,$ for each $u\in E,$ and the function $u\mapsto f(t, u)$ is continuous on $E$ for each $t\in I_k; \ k = 0, \dots, m,$

$(H_{5})$ There exists a positive continuous function ${\overline p}\in C(I_k); \ k = 0, \dots, m,$ such that

$(H_{6})$ For each bounded set $B\subset E$ and for each $t\in I_k; \ k = 0, \dots, m,$ we have

$(H_{7})$ There exists ${\overline q}^\ast\geq 0,$ such that

and, for each bounded set $B\subset E; \ k = 0, \dots, m,$ we have

$(H_8)$ For each bounded set $B_1\subset PC,$ the set $\{t\mapsto f(t, u(t)): u\in B_1, \ t\in I_k; \ k = 0, \dots, m \}$ is equicontinuous.

Set

Theorem 3.5. Assume that the hypotheses $(H_{4})-(H_{8})$ hold. If

then the problem (1.1) has at least one solution defined on $I.$

Proof. Consider the operator $N:PC\rightarrow PC$ defined in (3.5), and let $B_R\subset PC$ be the ball centered at the origin with radius $R\geq\frac{\|c\|+\rho}{1-\rho}.$ For each $t\in I_0,$ and $u\in PC,$ we have

Next, for each $t\in I_k:\ k = 1, \dots, m,$ and $u\in PC,$ we get

Hence, for $t\in I,$ and $u\in PC,$ we get

This proves that $N$ transforms the ball $B_R$ into itself.

We prove in three steps that $N:B_R\to B_R$ satisfies all the assumptions of Theorem 2.7.

Step 1. $N:B_R \to B_R$ is continuous.

Let $\{u_{n}\}_{n\in {{\mathbb N}}}$ be a sequence such that $u_n\rightarrow u$ in $B_R.$ Then, for each $t\in I_0,$ we have

Since $u_n\rightarrow u\text{ as } n\rightarrow \infty$ and $f$ is continuous, then (3.6) implies

by the Lebesgue dominated convergence theorem.

Also, for each $t\in I_k; \ k = 1, \dots, m,$ we get

Hence, we get the continuity of our operator $N.$

Step 2. $N(B_R)$ is bounded and equicontinuous.

Since $N(B_R)\subset B_R$ and $B_R$ is bounded, then $N(B_R)$ is bounded.

Next, let $\tau_1, \tau_2\in I_k; \ k = 0, \dots, m;$ such that $t_k\leq\tau_1 < t\leq \tau_2\leq t_{k+1}$ and let $u\in B_R.$ Then, we have

From the continuity of $f,$ and $(H_8)$ the right-hand side of the above inequality tends to zero as $\tau_{1}\longrightarrow \tau_{2},$ and such convergence is uniform in $u \in B_R.$ Hence, $N(B_R)$ is bounded and equicontinuous.

Step 3. The implication (2.3) holds.

Now let $V$ be a subset of $B_R$ such that $V\subset \overline{N(V)}\cup\{0\},$ $V$ is bounded and equicontinuous and therefore the function $t\to v(t) = \mu(V(t))$ is continuous on $I.$ By $(H_6)$ and the properties of the measure $\mu,$ for each $t\in I_0,$ we have

Thus

Also, for each $t\in I_k; \ k = 1, \dots, m,$ we get

Hence

From (3.7), we get $\|v\|_{PC} = 0,$ that is $v(t) = \beta(V(t)) = 0,$ for each $t\in I,$ and then $V(t)$ is relatively compact in $E.$ From the Ascoli-Arzelà theorem, $V$ is relatively compact in $B_R.$ We conclude by Theorem 2.7 that $N$ has a fixed point which is a solution of (1.1).

4.   Examples

Example 1. Consider the problem of impulsive Caputo-Fabrizio fractional differential equation

where

and

Clearly, the function $f$ is continuous.

For each $t\in[0, 1],$ we have

and

Hence, the hypothesis $(H_{1})$ is satisfied with

and $(H_{2})$ is satisfied with

We shall show that condition (3.4) holds with $T = 1.$ Indeed; if we assume, for instance, that the number of impulses $m = 3,$ then we have

Simple computations show that all conditions of Theorem 3.4 are satisfied. It follows that the problem (4.1) has at least one solution on $[0, 1].$

Example 2. Let

be the Banach space with the norm

Consider the problem of Caputo-Fabrizio fractional impulsive differential equation

where $u = (u_{1}, u_{2}, \ldots, u_{n}, \ldots), \ f = (f_{1}, f_{2}, \ldots, f_{n}, \ldots),$

and $c_r = \frac{1}{1+a_r +b_r}.$

For each $u\in E$ and $t\in[0, 1],$ we have

and

Hence, the hypothesis $(H_5)$ is satisfied with $p^\ast = 2c_re^{-5},$ and $(H_7)$ is satisfied with $q^\ast = \frac{c_r}{3}e^{-4}.$ We shall show that condition (3.7) holds with $T = 1.$ Indeed; if we assume, for instance, that the number of impulses $m = 3,$ then we have

Simple computations show that all conditions of Theorem 3.5 are satisfied. It follows that the problem (4.2) has at least one solution on $[0, 1].$

5.   Conclusions

In this paper, we provided some sufficient conditions ensuring the existence of solutions for functional fractional differential equations with instantaneous impulses; involving the Caputo-Fabrizio fractional derivative. The techniqued used are the fixed point theory and the measure of noncompactness.

Acknowledgments

The work of Juan J. Nieto has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain, co-financed by the European Fund for Regional Development (FEDER) corresponding to the 2014–2020 multiyear financial framework, project MTM2016-75140-P and by Xunta de Galicia under grant ED431C 2019/02.

Conflict of interest

The authors declare no conflict of interests.