Solutions and anti-periodic solutions for impulsive differential equations and inclusions containing Atangana-Baleanu fractional derivative of order $\zeta \in (1, 2)$ in infinite dimensional Banach spaces

1.   Introduction

It has been recognized that the dynamics of complex real-world problems are better described using fractional calculus. Fractional calculus has many applications in engineering ^{[2,3,4,5,6,7]}, in environmental, and biological studies ^{[8,9,10,11,12]}. As an extension to Newtonian derivatives, researchers have proposed different concepts of fractional derivatives and integrals, each of which generalizes the concept of differentiation and integration of integer order. The best known fractional operators are those of Riemann-Liouville and Caputo. These operators use a singular kernel. The problems arising from the presence of singular kernel were overcome by introducing fractional operators with non singular kernels. Caputo et al. ^[13] proposed a definition based on the exponential function. Atangana and Baleanu ^[14] generalized the Caputo fractional operators using kernels based on the Mittag-Leffler function.

Although Atangana and Baleanu's derivative is not the left inverse of the corresponding Atangana and Baleanu's integral (Lemma 1 and Remark 2 below), there are many applications of Atangana and Baleanu's fractional derivative to differential equations ^{[15,16,17,18,19]}. Many researchers obtained results regarding the existence of solutions for fractional differential equations and inclusions involving Atangana and Baleanu derivative in finite dimensional spaces ^{[20,21,22,23,24]}. Recently, Al Nuwairan et al. ^[25] investigated the existence of solutions for non-local impulsive differential equations and inclusions with Atangana and Baleanu derivative of order $\zeta \in (0, 1)$ in infinite dimensional spaces.

Impulsive differential equations and impulsive differential inclusions have been an object of interest with wide applications to physics, biology, engineering, medicine, industry, and technology. The impulsive differential equations provide appropriate models for processes that change their state rapidly and cannot be modeled using the ordinary differential equations. An example of such a process is the motion of an elastic ball bouncing vertically on a surface. The moments of the impulses are the times when the ball touches the surface and rapidly changes its velocity. For some applications of impulsive differential equations, we see ^[27]. Xu et al. ^[28] studied the exponential stability of stochastic nonlinear delay systems subject to multiple periodic impulses. For further results on the existence of solutions or mild solutions for impulsive differential equations and inclusions, we refer to ^{[29,30,31,32,33]}.

Kaslik et al. ^[34] showed that unlike the integer order derivative, the fractional-order derivative of a periodic function cannot be a function with the same period. This implies the non-existence of periodic solutions for a wide class of fractional-order differential systems on bounded intervals. Thus, much attention has been devoted to the study of anti-periodic solutions or $S$-asymptotically $w$-periodic solution. Fractional differential equations with anti-periodic conditions have been applied to the study of blood flow, chemical engineering, underground water flow, and population dynamics. The anti-periodic solutions to various fractional differential equations and inclusions are investigated by several authors ^{[36,37,38,39,40]} and papers cited therein. Very recently, Abdeljawad et al. ^[41] proposed a higher-order extension of Atangana–Baleanu fractional operators. For more recent results on fractional differential equations, we refer the reader to ^[42,43,44].

Notation 1.1. Throughout this paper, we use the following notation:

● For $b > 0$, let $J = [0, b] \subset \mathbb{R}$. Let $m$ be a natural number, $0\leq k\leq m, N_{k} = \{k, k+1, \ldots, m\}, 0 = \iota_{0} < \iota_{1}\leq < \iota_{2}\leq \iota_{3}\cdots < \iota_{m+1} = b$ be a partition of $J$, $J_{0} = [0, \iota_1],$ and $J_{k} = (\iota_{k}, \iota_{k+1}], k\in N_1.$

● $E$ is a reflexive real Banach space, $z_{0}, z_{1}$ are elements of $E$.

● $AC(J, E)$ is the Banach space of absolutely continuous functions from $J$ to $E$.

● $H^{1}((a, b), E)$ is the Sobolev space $\{z\in L^{2}((a, b), E):z^{ \prime}\in L^{2}((a, b), E)\}.$

● $PC(J, E)$ is the Banach space defined as

$PC(J, E) = \{z:J\rightarrow E, z\in H^{1}(J_{k}, E): z(\iota_{k}^{+}) \mathit{\text{and}}z(\iota_{k}^{-})\mathit{\text{exist with}} z(\iota_{k}) = z(\iota_{k}^{-})\}, \forall k\in N_{1}\}.$

The norm on $PC(J, E)$ is given by $||z||_{PC(J, E)} = \sup \{||z(\iota)||: \iota \in J\}.$

● $PCH^{1}(J, E) = \{z\in PC(J, E):z_{|J_{k}}\in H^{1}((\iota_{k}, \iota_{k+1}), E), \forall k\in N_{1}\}.$

● $PCH^{2}(J, E) = \{z\in PC(J, E):z_{|J_{k}}^{\prime}\in H^{1}((\iota_{k}, \iota_{k+1}), E), \forall k\in N_{1}\}.$ The spaces $PCH^{1}(J, E)$ and $PCH^{2}(J, E)$ are Banach spaces endowed with the norms

Recently, it was shown in ^[20,21,23] that the following fractional differential equation:

is equivalent to the fractional integral equation:

where $\zeta \in (1, 2)$ and $^{ABC}D_{0, \iota}^{\zeta }$ is the Atangana-Baleanu fractional derivative in the Caputo sense of order $\zeta$ with lower limit at $0, w:J\rightarrow \mathbb{R}$ is continuous function satisfying $w(0) = 0$ and $z_0, z_1$ are fixed points. We claim that the assumption of continuity of $w$ is not enough as it does not assure that the function $z$ in Eq (1.2) satisfies $z^{\prime }\in H^{1}((0, b))$. Thus, it does not guarantee that $z$ has Atangana-Baleanu fractional derivative of order $\zeta$. Without differentiability, $z$ would not be a solution for Eq (1.1).

In this paper, we provide

$(1)$ A more precise result regarding the relation between the fractional differential equation (1.1) and the fractional integral equation (1.2) in a real Banach space $E$ (Lemma 3.1).

$(2)$ Two results (Theorems 3.1 and 3.2) concerning the existence and uniqueness of solutions for the following boundary value problem containing Atangana-Baleanu fractional derivative

where $f(0, z(0)) = 0$.

$(3)$ A formula (given in Lemma 4.1) for the relation between the boundary value problem

and the integral equation

$(4)$ A formula for the solutions to the following impulsive boundary value problem involving Atangana-Baleanu fractional derivative of order $\zeta \in (1, 2)$:

where $f(0, z(0)) = 0,$ and $I_{i}, \overline{I_{i}}:E\rightarrow E$ are continuous functions (Lemma 4.2). We also establish two results concerning the existence and uniqueness of the solution of (1.4) (Theorems 4.1 and 4.2).

$(5)$ The sufficient conditions for the existence of anti-periodic solution to the following impulsive differential equation involving Atangana-Baleanu fractional derivative of order $\zeta \in (1, 2)$

where $f(0, z(0)) = 0,$ (Theorem 5.1).

$(6)$ The sufficient conditions for the existence of solutions to the impulsive differential inclusion

where $F$ is a multi-valued function satisfying $F(0, z_{0}) = \{0\}$ (Theorem 6.1).

Remark 1.1. Previously, the authors in ^[29] had investigated problems (1.3) and (1.4) with the Atangana-Belearn derivative replaced with Caputo's. Also in ^[29,40] problems (1.5) and (1.6) were studied using Caputo derivative without impulses. Saha et al. ^[42] established the existence of solutions for problem (1.1) in finite dimensional spaces with the boundary conditions $z(0) = z_0, ^{ABC}D_{0, \iota} z(b) = z_1.$ Indeed, the vast majority of published research on the existence of solutions to differential equations involving Atangana-Baleanu fractional derivative are restricted to finite-dimensional spaces ^{[20,21,22,23,24,45]}. Up to the authors knowledge, there has been no published research on anti-periodic solutions.

The contribution of this paper can be summarized as follows:

$(1)$ In Lemma 3, we obtained a precise relationship between the fractional differential Equation (1.1) and the corresponding integral Equation (1.2). We showed in detail that the continuity assumption on the nonlinear term, used earlier, e.g., Theorem 3.6 in ^[21] and Lemma 2 in ^[23], is insufficient and should be replaced with the requirement that $w$ lies in the space $H^1((a, b), E)$.

$(2)$ As to our knowledge, Theorem 5.1 showing the existence of an anti-periodic solution for the impulsive fractional differential equation (1.5), with Attange-Baleanu fractional derivative of order $\zeta \in (1, 2)$, has not previously appeared in literature.

$(3)$ To our knowledge, there has been no published results on the existence of solutions for impulsive differential equations containing Atangana-Baleanu fractional derivative of order $\zeta \in (1, 2)$, or on the existence of anti-periodic solutions for differential equations containing Atangana-Baleanu fractional derivative.

The paper is organized as follows. In the second section, we recall the basic facts and concepts needed for the following sections. In Section 3, we present two existence and uniqueness results for the solution to problem (1.3). Section 4 studies the existence and uniqueness of solutions to problem (1.4), and Section 5 is devoted to showing the existence of solutions to problem (1.5). In Section 6, we prove the existence of solutions for problem (1.6). Three examples are given in the last section to illustrate the obtained results.

2.   Preliminaries and notations

Definition 2.1. ^[14,19] Let $a < b$ be two real numbers, and $\zeta \in (0, 1)$. The Atangana-Baleanu fractional derivative for a function $z\in H^{1}((a, b), E)$ in the Caputo sense and in the Riemann-Liouville sense of order $\zeta$ with lower limit at $a$ are defined by

and

where $M(\zeta) > 0$ is a normalized function satisfying $M(0) = M(1) = 1,$ and $E_{\zeta } = E_{\zeta, 1}$ is the Mittag-Leffler function given by:

Definition 2.2. ^[14,19] Let $a < b$ be two real numbers, and $\zeta \in (0, 1)$. The Atangana-Baleanu fractional integral for a function $z\in H^{1}((a, b), E)$ of order $\zeta$ with lower limit at $a$ is given by

The following lemma was proved in ^[14,19] for $E = \mathbb{R}.$ It can be generalized to a Banach space $E$ with little changes in the proof.

Lemma 2.1. Let$\ z\in H^{1}((a, b), E),$ $\zeta \in (0, 1)$ and $\iota\in J.$

i. $^{ABR}D_{a, \iota}^{\zeta }(^{AB}I_{a, \iota}^{\zeta }z(\iota)) = z(\iota)$ and $^{AB}I_{a, \iota}^{\zeta }(^{ABR}I_{a, \iota}^{\zeta }z(\iota)) = z(\iota).$

ii. $^{ABC}D_{a, \iota}^{\zeta }(^{AB}I_{a, \iota}^{\zeta }z(\iota)) = z(\iota)-z(a)E_{\zeta }\left(\frac{-\zeta (\iota-a)^{\zeta }}{1-\zeta }\right).$

iii. $^{ABC}I_{a, \iota}^{\zeta }(^{ABC}I_{a, \iota}^{\zeta }z(\iota)) = z(\iota)-z(a).$

iv. $^{ABR}D_{a, \iota}^{\zeta }z(\iota) = ^{ABC}D_{0, \iota}^{\zeta }z(\iota)+\frac{M(\zeta) }{1-\zeta }z(a)E_{\zeta }\left(\frac{-\zeta (\iota-a)^{\zeta }}{1-\zeta }\right).$

v. $^{ABR}D_{a, \iota}^{\zeta }c = cE_{\zeta }(\frac{-\zeta }{1-\zeta } (\iota-a)^{\zeta }), ^{ABC}D_{a, \iota}^{\zeta }c = 0$, for a constant $c$.

Remark 2.1. Note that the second assertion of Lemma 2.1 implies that $^{ABC}D_{a, \iota}^{\zeta }(^{AB}I_{a, \iota}^{\zeta }z(\iota))\neq z(\iota)$, unless $z(a) = 0.$ Thus, we can not drop the assumption that $f(0, z(0)) = 0$ in problems (1.3)–(1.5) and that $F(0, z_{0}) = {0}$ in problem (1.6).

Definition 2.3. ^[14,19] Let $\zeta \in (n, n+1), n\in\mathbb{N}$ and $z:[a, b]\rightarrow E\ $ with $z^{(n)}\in H^{1}((a, b), E)$. The left Atangana-Baleanu fractional derivative of $z,$ in the Caputo sense and in the Riemann-Liouville sense of order $\zeta$ with lower limit at $a$ are defined by

and

Definition 2.4. ^[14,19] Let $\zeta \in (n, n+1), n\in\mathbb{N}$ and $z:[a, b]\rightarrow E$ with $z^{(n)}\in H^{1}((a, b), E)$. The left Atangana-Baleanu fractional integral for $z,$ of order $\zeta$ with lower limit at $a$, is defined by

As in ^[20,22], one can prove the following lemma.

Lemma 2.2. Let $\zeta \in (1, 2)$ and $z:J\rightarrow E$ with $z^{\prime}\in H^{1}((a, b), E)$. For any $\iota\in [a, b]$,

$(1)$ $^{ABR}D_{a, \iota}^{\zeta }\left(^{AB}I_{a, \iota}^{\zeta }z(\iota)\right) = z(\iota).$

$(2)$ $^{ABC}D_{a, \iota}^{\zeta }\left(^{AB}I_{a, \iota}^{\zeta}z(\iota)\right) = ^{ABC}D_{a, \iota}^{\zeta-1}\left(\frac{d}{d\iota}\left(I \left(^{AB}I_{0, \iota}^{\zeta -1}z(\iota)\right)\right)\right) = ^{ABC}D_{a, \iota}^{\zeta-1}\left(^{AB}I_{0, \iota}^{\zeta -1}z(\iota)\right)\\ = z(\iota)-z(a)E_{\zeta -1}\left(\frac{-(\zeta-1)(\iota-a)^{\zeta-1 }}{2-\zeta }\right).$

$(3)$ $^{AB}I_{a, \iota}^{\zeta}\left(^{ABC}D_{a, \iota}^{\zeta}z(\iota)\right) = z(\iota)-c_{0}-c_{1}(\iota-a).$

We end this section by listing some assumptions that are used later.

Assumptions 2.1. Let $f:J\times E\rightarrow E$ be a function, we assume the following:

- $(A_{1}):$ For any $\delta > 0$ there is $L_{\delta }\ > 0$ such that for any $x, y\in E$ with $||x||\leq \delta, \ ||y||\leq \delta$ and any $s, \iota\in J$, we have

- $(A_{2}):$ There is $\sigma > 0$ such that for any $x, y\in E$, we have

- $(A_{3}):$ For every $i\in N_{1},$ the functions $I_{i}, \overline{I_{i}}:E\rightarrow E$ are continuous, compact and there exist positive constants $h_{i}\, \overline{h_{i}}\ (i = 1, 2, .., m)$ such that

- $(A_{4}):$ For every $i\in N_{1}$, there exists positive constants $\delta _{i}\, \eta _{i},$ such that

and

3.   Existence solutions of problem (1.3)

In this section, we stat and prove the relationship between the fractional differential Equation (1.1) and the fractional integral Equation (1.2) in a reflexive Banach space $E$.

Lemma 3.1. Let $\zeta \in (1, 2).$

$(1)$ If $w:J\rightarrow E$ is continuous and $z:J\rightarrow E$ is a solution to Eq (1.1), then $z$ satisfies the integral equation (1.2).

$(2)$ If $w \in H^{1}((0, b), E)$ with $w(0) = 0$ and $z$ satisfies Eq (1.2), then $z^{\prime}\in H^{1}((0, b), E)$ and $z$ is a solution to Eq (1.1).

Proof.

$(1)$ By applying $^{AB}I_{0, \iota}^{\zeta }$ to both sides of Eq (1.1) and using the definition of $^{AB}I_{0, \iota}^{\zeta },$ the third assertion of Lemma 2.2, and Definition (2.4), we obtain that for any $\iota\in [0, b]$

From the boundary conditions $z(0) = z_{0}$ and $z(b) = z_{1},$ it follows that $c_{0} = z_{0}$ and

i.e.,

Substituting the values of $c_{0}$ and $c_{1}$ into (3.1), we obtain

$(2)$ Assume that $w\in H^{1}((0, b), E)$ with $w(0) = 0$, and that Eq (1.2) holds. Clearly $z(0) = z_{0}$ and $z(b) = z_{1}.$ Moreover,

where $c_{0} = z_{0}\ $and $c_{1}\ $is given by (3.2). Since $\zeta > 1,$ Eq (3.3) gives us that

where $I_{0, \iota}^{\ \zeta -1}$ is the Riemann-Liouville fractional integral of order $\zeta -1$. Since $w\in H^{1}(J, E)$, $\zeta -1\in (0, 1)$, and $E$ is reflexive, $w$ has a Bochner integrable derivative $w ^{\prime}$ almost everywhere, and

This implies that

i.e., $I_{0, \iota}^{\ \zeta -1}w(\iota)$ is the primitive of a Bochner integrable function, hence is absolutely continuous. Thus Eq (3.4) is valid for every $\iota\in J.$ Moreover,

giving us that $z^{\prime}\in H^{1}((0, b), E).$ Equation (3.2) implies

Finally, by the second assertion of Lemma 2.2,

$^{ABC}D_{0, \iota}^{\zeta}z(\iota) = ^{ABC}D_{0, \iota}^{\zeta }\ ^{AB}I_{0, \iota}^{\zeta }w(\iota) = w(\iota)-w(0)E_{\zeta-1}(\frac{-(\zeta-1) \iota^{\zeta-1 }}{2-\zeta }) = w(\iota), \ \iota\in J$.

□

Remark 3.1. Note that

$(1)$ The first assertion of Lemma 3.1 has been proved in Lemma 2 in ^[23] for the case where $E = \mathbb{R}$.

$(2)$ The solution formula of problem (1.1) does not follow from the first assertion of Lemma 3.1, nor from Lemma 2 in ^[23].

$(3)$ The assumption $w(0) = 0$ cannot be omitted in the second assertion of Lemma 3.1 since

$(4)$ If $w$ is continuous and not in $H^{1}((0, b), E)$, then Eq (3.4) does not imply the existence of $z^{(2)}$. Therefore, without the assumption $w\in H^{1}((0, b), E)$, there is no guarantee that $^{ABC}D_{0, \iota}^{\zeta }z(\iota)$ exists.

$(5)$ Lemma 3.1 gives a more accurate statement of Lemma 2 in ^[23] and generalizes it to the infinite dimensional case.

The results in Lemma 3.1 can be summarized as follows.

Lemma 3.2. Let $w\in H^{1}((0, b), E)$ with $w(0) = 0$. A function $z:J\rightarrow E$ is a solution of problem (1.1) if and only if

Theorem 3.1. Let $f:J\times E\rightarrow E$ be a function. If $(A_{1})$ holds, then problem $ (1.3) has a unique solution provided that $f(0, z_{0}) = 0$ and there is $r > 0$ such that

Proof. Define $T:C(J, E)\rightarrow C(J, E)$ by

Using the Schauder fixed point theorem, we will show that $T$ has a unique fixed point. Set $B_{0} = \{z\in C(J, E):\Vert z\Vert _{C(J, E)}\leq r\}.$

● Step $1$: $T(B_{0})\subseteq B_{0}.$ Let $z\in B_{0}$. It follows that from $(A_{1})\ $

From (3.6)–(3.8), one has

showing that $T(B_{0})\subseteq B_{0}$.

● Step $2$: $T(B_{0})$ is equicontinuous. Let $z\in B_{0}\ $and $\iota, \iota+\lambda \in J.$ Using (3.7), we obtain

Since $\zeta -1\ > 0, \Vert T(z)(\iota+\lambda)-T(z)(\iota)\Vert \rightarrow 0$ when $\lambda \rightarrow 0,$ independently of $z,$ proving the assertion.

● Step $3$: For $n\geq1$, let $B_{n} = \overline{conv}T(B_{n-1}),$ and $B = \cap_{n\geq 0}B_{n}$.

Let $B, B_{n}$ be as defined above, then the set $B$ is a non empty compact subset of $C(J, E)$. It follows from Step 1, that $B_{n}\subseteq B_{n-1},$ $n\geq1.$ By Cantor intersection property ^[47], it is enough to show that

where $\chi _{C(J, E)}$ is the Hausdorff measure of noncompactness on $C(J, E)$ ^[48].

Let $n\geq 1$ be a fixed natural number and $\varepsilon > 0$. By Lemma 3 in ^[49], there exists a sequence $(z_{k}),$ $k\geq 1$ in $B_{n-1}$ such that

Since $B_{n}$ is equicontinuous, inequality (3.10) becomes

Let $\iota\in J\ $be fixed. In view of (3.8)

It follows that

We also have that

Since $\zeta > 1,$ Eqs (3.12) and (3.13) give

Using (3.11) and (3.14), we obtain that

This inequality yields that

The inequality in (3.6) implies that $4b L_{r} \left[\frac{(2-\zeta)}{M(\zeta -1)}+\frac{b^{\zeta }(\zeta -1)}{M(\zeta -1)\Gamma (\zeta)}\right] < 1,$ and thus, (3.15) implies (3.9).

● Step $4$: The function $T_{|B}: B\rightarrow B$ is continuous. Assume that $z_{n}\rightarrow z$ in $B$. Note that for $n\geq 1$ and $\iota\in J,$ we have

Using $(A_{1}), \zeta > 1$, the inequality (3.8), and the Lebesgue dominated convergence theorem, we obtain that $T(z_{n})\rightarrow T(z)$.

It follows from Steps $(1)$ to $(4)$ and Schauder's fixed point theorem that there is $z\in B$ such that $z = T(z)$. That is,

where $w(\iota) = f(\iota, z(\iota)), \ \iota\in J.$

Next, we show that this function $z$ is a solution for problem (1.3). By Lemma 3.1, it is sufficient to show that $w\in H^{1}(J, E).$ Since $\zeta > 1,$ then

From $(A_{1}), w$ is absolutely continuous, and since $E$ is reflexive, the function $t\rightarrow I_{0, t}^{\zeta -1}w(t)$ is absolutely continuous. Hence $w\in H^{1}(J, E).$

To show the uniqueness of the solution, let $z, v\in C(J, E)$ be two solutions for problem (1.3) and $\iota\in J.\ $ Since $z, v$ are solutions, it follows from $(A_{1})$ that

Hence,

Since $\iota$ is arbitrary, it follows that

Inequality (3.6) implies $\frac{2L_{r}(2-\zeta)b}{M(\zeta -1)}+ \frac{2L_{r}(\zeta -1)b^{\zeta }}{M(\zeta -1)\Gamma (\zeta)} < 1$, consequently, $||z-v||_{C(J, E)} = 0$, and $z = v$. □

In the following, another existence and uniqueness result for solutions of problem (1.3) is obtained. Replacing the assumption $(A_{1})$ by $(A_{2})$ simplifies the inequality (3.6) enabling us to use the Banach fixed point theorem for contraction mappings instead of the Schauder fixed point.

Theorem 3.2. Let $f:J\times E\rightarrow E$. If $(A_{2})$ is satisfied, then problem (1.3) has a unique solution provided that $f(0, z_{0}) = 0$ and

Proof. Consider the function $\ T:C(J, E)\rightarrow C(J, E)$ defined by $(3.7).$ Let $z, v\in C(J, E).$ For any $\iota\in J$,

Since $\zeta > 1,$ this inequality together with $(A2)$ imply that

Thus,

Using (3.16), we obtain that $T$ is contraction, and hence has a unique fixed point. □

4.   Existence of solutions for problem (1.4)

The following lemmas will be used for deriving an existence result for solutions of problem $(1.4).$

Lemma 4.1.

$(1)$ If $w: J\rightarrow E$ is continuous $, a\in [0, b), z:J\rightarrow E$ be such that $z^{\prime}\in H^{1}((0, b), E)$ and

then for any $\iota\in J$,

$(2)$ If $a\in [0, b), w: J\rightarrow E$ be continuous with $w(0) = 0$ and $z:J\rightarrow E$ are such that (4.2) holds, then $z^{\prime}\in H^{1}((0, b), E)$ and $z$ is a solution for (4.1).

Proof.

$(1)$ Apply $^{AB}I_{0, \iota}^{\zeta }$ on both side of the equation $^{ABC}D_{0, \iota}^{\ \zeta }z(\iota) = w(\iota); \iota\in [0, b].$ As in the proof of first assertion of Lemma 3.1, we obtain for any $\iota\in [0, b]$

Using the boundary conditions $z(a) = z_{0}, z^{\prime}(a) = z_{1},$ we obtain

and

This gives that

From (4.3) and (4.4), we obtain

Substituting the values of $c_{0}\ $and $c_{1}$into $z(\iota),$ we obtain

Hence, Eq (4.2) is verified.

$(2)$ Suppose that $w: J\rightarrow E$ be continuous function with $w(a) = 0$ and Eq (4.2) holds. Clearly $z(a) = z_{0}$ and $z^{\prime}(a) = z_{1}.$ As in the proof of second assertion of Lemma 3.1, we can show that $z^{\prime}\in H^{1}((0, b), E).$ For any $\iota\in [a, b]$, we have

□

Remark 4.1. Following the same method, used in the above proof, a generalization of Theorem 4 in ^[21] can be derived for any Banach space.

Lemma 4.2. If $w\in PCH^{1}(J, E)$ with $w(0) = 0$ and $z:J\rightarrow E$ be a function satisfying

then $z\ \in PCH^{2}(J, \ E)$ and satisfies the impulsive fractional differential equation:

Note that for $k = 0$, in Eq (4.6), the sum $\sum_{i = 1}^{k}$ is an empty sum and conventionally, equals zero.

Proof. For any $\iota\in J_{0}$,

Clearly, $z(0) = z_{0},$ $z^{\prime}(0) = z_{1}$. Since $w\in PCH^{1}(J, E)$ and $w(0) = 0$, it follows by the second statement of Lemma 4.1, that $z$ is a solution for the fractional differential equation:

Let us define a function $v$ on $J_{1} = (\iota_{1}, \iota_{2}]$ by:

From the second assertion of Lemma 4.1, $v$ is a solution for the fractional differential equation:

Let $\iota\in J_{1}.$ We show that, $v(\iota) = z(\iota).$ From Eq (4.8), it follows that

and

By substituting the values of $z(\iota_{1}^{-})$ and $\ z^{\prime}(\iota_{1}^{-})$ into Eq (4.9), we obtain

Therefore, $v(\iota) = z(\iota), \forall \iota\in J_{1}$. Since

then $z$ is a solution for the fractional differential equation (4.10). By repeating the above steps for $J_{k}; k\in N_{2}$, the proof follows. □

Definition 4.1. A function $z\in PCH^{2}(J, E)$ is said to be a solution for problem $(1.4)$ if it has left Atangana-Baleanu fractional derivative of order $\zeta$ on each $J_{k}, \ k\in N_{1}$ and satisfies Eq (4.6).

In the following theorem, we provide an existence result for problem (1.4).

Theorem 4.1. Let $f:J\times E\rightarrow E$ with $f(0, z_{0}) = 0$ and $I_{i}, \overline{I_{i}}:E\rightarrow E \ (i\in N_{1})$ be functions. If both Assumptions $(A_{1})$ and $(A_{3})$ are satisfied, then problem (1.4) has a unique solution provided that $f(0, z_{0}) = 0$, and there is $r > 0$ such that

where $h = \max \{\sum_{i = 1}^{m}h_{i}\, \sum_{i = 1}^{m}\overline{h_{i}}\}.$

Proof. Using Schauder's fixed point theorem, we show that the function $R:PC(J, E)\rightarrow PC(J, E)$ given by

has a fixed point. Set $B_{0} = \{z\in PC(J, E):\Vert z\Vert_{PC(J, E)}\leq r\}$. The remainder of the proof is similar to the steps used in proving Theorem (3.1), so we give it in outline.

● Step $1$: Let $z\in B_{0}$ and $\iota\in J_{k}, \ k = 0, 1, 2, .., m$. Using (2.1), (3.8), (4.12), and (4.13), we obtain that for $\iota\in J_{k}, k = 1, 2, .., m,$

from which we deduce that $R(B_{0})\subseteq B_{0}$.

● Step $2$: Let $Z = R(B_{0}).$ We claim that $Z$ is equicontinuous on every $J_{k}, k\in N_{0} = \{0, 1, 2, .., m\}$. Let $k \in N_{0}$ be fixed, $z\in B_{0}$ and $\iota, \iota+\lambda \in J_{k}$. Using (4.13), we get

Since $\zeta -1\ > 0,$ we have $\Vert R(z)(\iota+\lambda)-R(z)(\iota)\Vert\rightarrow 0$ \ as $\lambda \rightarrow 0,$\ independently of $z.$

● Step $3$: We show that $B = \cap _{n\geq 1}B_{n}$ is non-empty and compact in $PC(J, E),$ where $B_{n} = \overline{conv}\left(R(B_{n-1})\right), \forall n\geq 1$. By Step 1, it follows that $B_{n}, n\geq 1$ is a decreasing sequence of non-empty, closed convex and bounded subsets of $PC(J, E)$, and hence it is sufficient to show that

where $\chi _{PC(J, E)}$ is the Hausdorff measure of noncompactness on $PC(J, E)$.

Let $n\geq 1$ be a fixed natural number and $\varepsilon > 0$. In view of Lemma 3 in ^[49], there exists a sequence $(z_{k}), k\geq 1$ in $B_{n-1}$ such that

where $\chi_{i}$ is the Hausdorff measure of noncompactness on $C(\overline{J_{i}}, E).$ Since $R(B_{n-1})$ is equicontinuous, the above inequality becomes

where $\chi$ is the the Hausdorff measure of noncompactness on $E.$ Since $I_{i}, \overline{I_{i}}, i = 0, .., m,$ are compact, we have

Thus, as in (3.12)

from which,

Inequality (4.12) insures that $L_{r}\left(\frac{2-\zeta }{M(\zeta -1)}+ \frac{(\zeta -1)b^{\zeta }}{M(\zeta -1)\Gamma (\zeta)}\right) < 1,$ and Eq (4.14) follows.

● Step $4$: The function $R_{|B} : B\rightarrow B$ is continuous. Let $z_{n}\rightarrow z$ in $B$ and $y_{n} = R(z_{n}).$ The proof follows from the continuity of both $I_{i}, \overline{I_{i}}\; i = 0, 1, 2, .., m$, by following the same arguments in Step 4 of the proof of Theorem (3.1).

As a result of Steps $(1)$ to $(4)$ and Schauder's fixed point theorem, there is $z\in B\subseteq PC(J, E)$ such that $z = R(z).$

To show the uniqueness of the solution, let $z$ and $v$ be two solutions for problem (1.4). For $\iota\in J_{0}$, we have

Thus,

The inequality (4.12) gives

Thus, $\sup_{s\in J_{0}}||z(s)-v(s)|| = 0$, and hence $z(s) = v(s), \forall s\in J_{0}.$

Assume that $\iota\in J_{1}$. Because $z(\iota_{1}^{-}) = v(\iota_{1}^{-}),$ it yields that,

As above, we obtain that $z(\iota) = v(\iota), \forall \ \iota\in J_{1}$. By continuing in the same manner, we show that $z = v$. □

Next, we show that replacing the Assumptions $(A_{1}),$ and $(A_{3})$ in Theorem 4.1 by $(A_{2})$, and $(A_{4})$ simplifies (4.12). In fact this enable us to apply Banach fixed point theorem for contraction mappings instead of Schauder fixed point.

Theorem 4.2. Let $f:J\times E\rightarrow E$ such that $f(0, z_{0}) = 0$ and $I_{i},$ $\overline{ I_{i}}:E\rightarrow E$\ $(i\in N_{1})$ be functions. If Assumptions $(A_{2})$ and $(A_{4})$ are satisfied, then problem (1.4) has a unique solution provided that

Proof. Let $R:PC(J, E)\rightarrow PC(J, E),$ be function given by Eq (4.13), and $z, v\in PC(J, E)$. For each $\iota\in J_{k}; k\in N_{1}$, we have

Thus, $R$ is contraction. By applying Banach fixed point theorem, we obtain that $R$ has a unique fixed point, and such point is a solution for problem (1.4). □

5.   Existence of solutions for problem (1.5)

To obtain the sufficient conditions for the existence of anti-periodic solution for problem (1.5), we consider the following problem:

Note that problem (5.1) can be obtained from (4.7) by setting $z_{0} = -z(b)$ and $z_{1} = -z^{\prime}(b)\ $. Therefore, the solution of (5.1) is given by Eq (4.6) after substituting the values of $z_{0}$ and $z_{1}$.

Lemma 5.1. Let $w\in PCH^{1}(J, E)$ with $w(0) = 0.$ The solution function of problem (5.1) is given by Eq (4.6), where $z_{0}, z_{1}$ are given as follows:

and

Proof. Using Eq (4.6) and the boundary conditions $z(0) = -z(b)$, $z^{\prime}(0) = -z^{\prime}(b)$, we obtain

So, Eq (5.3) is verified. Moreover,

i.e.,

This equation along with Eq (5.3) lead to

□

By substituting the values of $z_{0}$ and $z_{1}$ into Eq (4.6), we obtain the following

Corollary 5.1. Let $w\in PCH^{1}(J, E)$ with $w(0) = 0.$ The solution of system in (5.1) is given by:

As a result of Corollary 5.1, we state the following definition.

Definition 5.1. A function $z\in PCH^{2}(J, E)$ is said to be a solution for problem (1.5) if it has left Atangana-Baleanu fractional derivative of order $\zeta$ on each $J_{k},$ where $k\in N_{0},$ and satisfies the integral equation:

Theorem 5.1. Under the assumptions of Theorem 4.2, problem (1.5) has a unique solution provided that

Proof. Consider the function $\mathcal{R}: PC(J, E)\rightarrow PC(J, E)$ defined as:

Let $z, v\in PC(J, E)$, $\iota\in J_{k}, k\in N_{1}$. By the assumptions ($A_{2}$), ($A_{4}$), and above equality, we have

This equation along with (5.6) shows that $\mathcal{R}$ is a contraction. Therefore, problem (1.5) has a unique solution. □

6.   Existence of solutions for problem(1.6)

Definition 6.1. Let $\Lambda$ and $\Delta$ be two normed spaces. A multi-valued function $G:\Lambda \rightarrow 2^{E}$ with non-empty closed, bounded and convex values is called $\rho$-Lipschitz if

where $h$ is the Hausdorff distance.

For information about the multi-valued functions, we refer the reader to ^[50].

Lemma 6.1. (^[51], Theorem 7) Let $(\Omega, \sum, \mu)$ be a $\sigma-$ finite measure space and $(T, d)$ be a metric space. If $G:T\rightarrow 2^{L^{p}(\Omega, \mathbb{R}^{n})}, p\in \lbrack 1, \infty)$ is a $\rho$-Lipschitz multi-valued function with non-empty, closed, convex, bounded and decomposable values, then there is single-valued function $f:T\rightarrow 2^{L^{p}(\Omega, \mathbb{R}^{n})}$ such that $f(t)\in G(t), a.e,$ and $||f(z)-f(v)||\leq \xi _{G} \ \rho ||z-v||_{L^{p}(\Omega, \mathbb{R}^{n})}, \ \forall z, v\in T$, where $\xi _{G}\ $ is a positive real number.

In the following theorem, we provide sufficient conditions for the existence of solutions of problem (1.6).

Theorem 6.1. Let $F:J\times L^{2}(J, \mathbb{R})\rightarrow 2^{L^{2}(J, \mathbb{R})}$ be an $\rho$-Lipschitz multi-valued function with non-empty, closed, convex, bounded and decomposable values, and $I_{i}$, $\overline{I_{i}}:$ $L^{2}(J, \mathbb{R})\rightarrow L^{2}(J, \mathbb{R}), i\in N_{1}$ be functions such that

and

where $\delta _{i}\, \eta _{i},$ are positive real numbers, then problem $(1.6)$ has a solution provided that $F(0, z_{0}) = \{0\}$ and

Proof. Let $E = L^{2}(J, \mathbb{R})$. The set $T = J\times E$ is a complete metric space, where $d((\iota_1, x_1), (\iota_2, x_2)) = |\iota_1-\iota_2|+||x_1-x_2||_{E}$. By Lemma 6.1, there exists $f:$ $J\times E\rightarrow E$ satisfying $f(\iota, x)\in F(\iota, x), a.e.,$ and

By applying Theorem (4.2), the following fractional boundary problem

has a solution. Since $f(\iota, x)\in F(\iota, x) \ a.e$, we have $^{ABC}D_{0, \iota}^{\zeta}z(\iota)\in F(\iota, z(\iota))\ a.e.$ for $\ \iota\in J-\{\iota_{1}, \iota_{2}, \ldots, \iota_{m}\}$ which completes the proof. □

Example 6.1. Let $E$ $= L^{2}[0, \pi]$, $\zeta \in (1, 2)$, $\ J = [0, 1]$ and $z_{0}:[0, \pi]\rightarrow \mathbb{R}$ be the zero function. Define $f:J\times E \rightarrow E$ by

where $\lambda > 0.$ Thus, $f(0, z_{0})(\eta) = 0;\forall \eta \in \lbrack 0, \pi].$ For any $\varsigma _{1}, \varsigma _{2}\in E = L^{2}[0, \pi]$ and any $w_{1}, w_{2}\in J$, we have

Thus, $(A_{1})$ is satisfied with $L_{\delta } = 2\lambda \delta.$ By applying Theorem 3.1, we have that if there is $r > 0$ such that

then there is a unique function $z:[0, 1]\rightarrow L^{2}[0, \pi]$ satisfying the boundary value problem

(see relation (3.6)).

The inequality (6.3) is equivalent to

where$\ \omega = \frac{(2-\zeta)}{M(\zeta -1)}+\frac{(\zeta -1)}{M(\zeta -1)\Gamma (\zeta)}.$

If $16 \ \lambda \ \omega (||z_{1}||+2\omega) < 1,$ then the equation $4\omega \lambda r^{2}-r+||z_{1}||+2\omega = 0$ has two positive solutions, namely

Therefore, (6.5) will be satisfied for $r \in (r_{1}, r_{2})$. Thus, inequality (6.3) has a solution provided the following inequality

holds. The last inequality will hold if we choose $\lambda$ sufficiently small.

Example 6.2. Let $E$ $= L^{2}[0, \pi]$, $\zeta \in (1, 2)$, $\ J = [0, 1]$. Define $f:J\times E\rightarrow E$ by:

where, $\sigma > 0$. Let $z_{0}:[0, \pi] \rightarrow \mathbb{R},$ be the zero function and $z_{1} \in L^{2}[0, \pi]$ be a fixed function. Note that $f(0, z_{0})(\eta) = 0, \ \forall \eta \in \lbrack 0, \pi]$. For every $\varsigma _{1}, \varsigma _{2}\in E = L^{2}[0, \pi]\ $ and every $w_{1}, w_{2}\in J$, we have

Thus, $(A_{2})$ is satisfied. By applying Theorem 3.2, there is a unique $z:[0, 1]\rightarrow L^{2}[0, \pi]$ such that $z^{\prime}\in H^{1}((0, 1), L^{2}[0, \pi])$ and $z$ satisfies the boundary value problem:

provided the following condition is satisfied

The last inequality holds by choosing $\sigma$ sufficiently small.

Example 6.3. Let $E, \zeta, J, f$ as be in Example 6.2, $\iota_{0} = 0, \iota_{1}$ $= \frac{1}{2}\ $ and $\iota_{2} = 1$. Define $I_{1}(x) = \lambda \ Proj_{K}(x),$ $\overline{I}_{1}(x) = \lambda \ Proj_{Z}(x), \forall x\in E,$ where $\lambda > 0,$ $K$ and $Z$ are convex and compact subset of $L^{2}[0, \pi]$ and $\lambda > 0$. Then, $(A_{2})$ is satisfied. Also, (2.2) and (2.3) are verified with $\delta _{1} = \eta _{1} = \lambda$. By applying Theorem 4.2, the problem

has a solution provided that

This inequality will hold by choosing $\sigma$ and $\lambda$ sufficiently small.

In a similar manner, many examples of the application of Theorems 5.1 and 6.1 can be provided.

7.   Discussion and conclusions

The relationships between some boundary value problems involving the Atangana-Baleanu fractional derivative ($AB$) of order $\zeta \in (1, 2)$ and the corresponding fractional integral equations were obtained in infinite dimensional Banach spaces with or without impulses. We showed that the continuity assumption on the nonlinear term is insufficient and must be replaced by membership in the space $H^{1}((a, b), E)$. The sufficient conditions for the existence of solutions for differential equations and inclusions involving $AB$ fractional derivative in the presences of instantaneous impulses in infinite dimensional Banach spaces were established. Additionally, the sufficient conditions for the existence and uniqueness of solutions and anti-periodic solutions for differential equations and inclusions containing $AB$ fractional derivative of order $\zeta \in (1, 2)$ in the presences of instantaneous impulses in infinite dimensional Banach spaces were obtained.

The major contributions of this work can be summarized as follows:

$(1)$ A modified formula for relationship between the solution of problem (1.1) and the corresponding integral equation (1.2) is derived.

$(2)$ A new class of boundary value for differential equations and inclusions containing AB derivative with instantaneous impulses in infinite dimensional Banach spaces were formulated with and without impulsive effects.

$(3)$ The existence/uniqueness of solutions and anti-periodic solutions for the considered problems and the corresponding inclusions were proved.

We provide methods to deal with differential equations and differential inclusions in infinite dimensional Banach spaces. i.e., to extend the results in ^{[19,20,21,22,23,24,45,46]} to infinite dimensional spaces. The methods used in this paper can help researchers to generalize many of the results cited above in the presence of impulsive effects, infinite dimensional Banach spaces, and when the right hand side of the equation is a multi-valued function. We suggest the following topics for future research

● Study the existence of $S$-asymptotically $w$-periodic solutions for problems (1.3) and (1.4).

● Extend the recent work by Saha et al.^[42] to infinite dimensional Branch spaces.

● Generalize the present work to the case where the Atangana and Baleanu's derivative is replaced by the Atangana and Baleanu's derivative with respect to another function.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors acknowledge the Deanship of Scientific Research at King Faisal University for the financial support.

Fund

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant No. GRANT3812].

Conflict of interest

The authors declare no conflicts of interest.