Measure pseudo almost automorphic solution to second order fractional impulsive neutral differential equation

1.   Introduction

The idea about almost periodic functions prompted to varied fundamental generalization within the year 1924-25 by H.Bohr ^[7]. The notion of almost automorphic (in short $\mathcal{AA}$) function is one of its crucial generalization by S. Bochner ^[6]. The concept of weighted pseudo almost automorphic (in short $\mathcal{WPAA}$) functions is one of the further important generalization of $\mathcal{AA}$ introduced by Blot et.al. ^[4]. These functions are a lot of typical and complex than weighted pseudo almost periodic functions. In 2012, Blot, Cieutat and Ezzinbi ^[5] applied the abstract measure theory to define an ergodic function and established fundamental properties of measure pseudo almost automorphic functions (in short $\mu_1-\mathcal{PAA}$), and thus the classical theories of pseudo almost automorphic functions and weighted pseudo almost automorphic functions become particular cases of this approach. After that, the $\mu_1-\mathcal{PAA}$ function has been developed in different ways, see for instance ^[10,21,26] and references therein.

The Dirac delta functions and "leaps" are two main directions in mathematical theory of impulsive differential equations. For describing the impulsive effects, the Dirac delta functions are a fundamental mathematical tool. In 1960, second direction of research "leaps" processes with some results for the solutions of stability was given by V. D. Milman and A. D. Myshkis ^[24]. In reality, many processes and phenomena are affected by short-term external factors. While comparing to total duration of phenomena and processes, this duration is negligible and therefore they form the impulses. Ecology, population dynamics, epidemiology, pharmacokinetics, economics, mechanics, control theory and other fields of science are all concerned in the dynamical states developed by such "leaps and bounds", see the monographs ^[3,18] and the articles ^[12,13,17].

Fractional calculus deals with integro-differential equations can be considered as a branch of mathematical physics which has been effectively developed and plays a very important role in distinct fields such as biophysics, mechanics, electro chemistry, notable control theory and visco elasticity and so on. The fractional calculus is a generalization of the traditional calculus, but with a much wider applicability. The fractional methodology is suitable for a lot of applications in image processing, complex system dynamics and nonlinear dynamics. Thus it leads to the sustained interest in studying the theory of fractional differential equations ^{[2,8,14,15,16,20]}.

The authors Wang and Agarwal ^[25] investigated the existence of piecewise weighted pseudo almost automorphic mild solutions to impulsive $\nabla$-dynamic equations. Chang and Feng ^[11] study the existence and uniqueness of measure pseudo almost automorphic solutions of the fractional differential equations. Our main results can be described as generalization of work in ^[11,25]. Motivated by the works ^{[5,11,21,25,26]} the main purpose of this article is to establish the piecewise $\mu_1 - \mathcal{ PAA}$ properties for the following impulsive fractional neutral differential equation

where $1 < \alpha < 2$ and $A: D(A) \subset Y \rightarrow Y$ is a linear densely defined operator of sectorial type on a complex Banach Space $(Y, \|\cdot\|)$. The functions $I_{j}: Y \rightarrow Y, G_{1}: \mathcal{R} \times Y \rightarrow Y, g_{1}: \mathcal{R} \times Y \rightarrow Y$ is a $\mu_1-\mathcal{PAA}$ function in $t$ for each $z \in Y$ satisfying suitable conditions. $\Delta z(t)|_{t = t_{j}} = z(t_{j}^{+}) - z(t_{j}^{-}), (j = 1, 2, ...), 0 = t_{0} < t_{1} < ... < t_{n} < ...$. Here $z(t_{j}^{+})$ and $z(t_{j}^{-})$ represent right and left limits of $z(t)$ at $t = t_{j}$ respectively. The fractional derivative $D^{\alpha}_t$ is considered as Caputo's sense.

The rest of this work is organized as follows. In Section 2, we define some definitions, terminologies, previous results, basic properties of $\mu_1 - \mathcal{ PAA}$ functions and assumptions. In Section 3, we investigate the important results which are needed to prove the main results. In Section 4, we establish the existence of $\mu_1 - \mathcal{ PAA}$ mild solutions to the model (1.1). In Section 5, we provide an example to illustrate our results.

2.   Preliminaries

In this section, we review few notations, definitions and Lemmas which will be utilized throughout this paper.

2.1.   Terminology and definitions

In this segment, we define basic definitions.

Let $(Y, \left \Vert \cdot \right\Vert_{Y})$ be a Banach space, $T$ be a subset of $Y$. The symbol $C(\mathcal{R}, Y)$(resp $C(\mathcal{R} \times T, Y)$) stands for the set of all continuous function from $\mathcal{R}$ to $Y$(resp from $\mathcal{R} \times T$ to $Y$) and $PC(\mathcal{R}, Y)$(resp $PC(\mathcal{R} \times T, Y)$) stands for set of all piecewise continuous function from $\mathcal{R}$ to $Y$(resp from $\mathcal{R} \times T$ to $\mathcal{R}$).

The fractional integral of order $q > 0$ in the Riemann–Liouville sense is defined as

also, the fractional derivative of function $f$ of order $q > 0$ in the Caputo sense is defined as

where $\Gamma(q)$ is a gamma function.

Moreover, the Riemann-Liouville definition entails physically unacceptable initial conditions (fractional order initial conditions); conversely of the Liouville-Caputo representation where the initial conditions are expressed in terms of integer-order derivatives having direct physical significance. The Caputo definition of fractional derivatives not only provides initial conditions with clear physical interpretation but it is also bounded, meaning that the derivative of a constant is equal to 0. Further, Caputo fractional derivative has lots of applications in real world problems, such as Groundwater flowing within an unconfined aquifer, measles epidemiological autonomous dynamical system etc.,

Definition 2.1. ^[1] A sequence $t_{1} : \mathbb{Z^{+}} {\rightarrow} Y$ is said to be $\mathcal{AA}$ sequence, if $t_{1}$ is bounded and for every sequence of integer numbers $\{ j'_{n}\}$, there exist a sub-sequence $\{j_{n}\} \subseteq \{ j'_{n}\}$ such that

is well defined and

for each $j \in \mathbb{Z^{+}}$. Denote this collection of sequences by $AA^{o}_{S}(\mathbb{Z}, Y)$.

Definition 2.2. ^[1] A piecewise continuous bounded function $G_{1} \in PC(\mathcal{R}, Y)$ is said to be $\mathcal{AA}$ if

● sequence of impulsive moments $\{t_{j} \}$ is an $\mathcal{AA}$ sequence

● for each sequence of real numbers $\{ u_{n}\}$, there exist a sub-sequence $\{u_{n_{k}}\} \subseteq \{ u_{n}\}$ such that

is well defined and

Denote this collection of functions by $AA^{o}_{\Omega}(\mathcal{R}, Y).$

Definition 2.3. ^[1] A piecewise continuous bounded function $G_{1} \in PC(\mathcal{R} \times T, Y)$ is said to be $\mathcal{AA}$ in compact subsets of $Y$ in $t$ uniformly for $t_{1}$ if

● sequence of impulsive moments $\{t_{j} \}$ is an $\mathcal{AA}$ sequence

● for each compact set $Q \subseteq Y$ and every sequence of real numbers $\{ u_{n}\}$, there exist a sub-sequence $\{u_{n_{k}}\} \subseteq \{ u_{n}\}$ such that

is well defined and

Denote this collection of functions by $AA^{o}_{\Omega}(\mathcal{R} \times T, Y).$

We denote $\mathcal{M}_1$ by the set of all positive measures $\mu_1$ on $\mathcal{B}_1,$ where $\mathcal{B}_1$ is the Lebesgue $\sigma$-field of $\mathcal{R}$ satisfying, $\mu_1([a, b]) < \infty,$ and $\mu_1(\mathcal{R}) = +\infty$ for all $b, a \in \mathcal{R} \, \, (b\geq a)$.

Definition 2.4. $\phi:\mathcal{R}\rightarrow Y,$ a bounded continuous function, is said to be $\mu_1$-ergodic if

where $\mu_1\in \mathcal{M}_1.$ Denote this collection of functions by $\kappa(\mathcal{R}, Y, \mu_1)$.

Definition 2.5. Let $\mu_1\in \mathcal{M}_1$. A piecewise continuous bounded function $G_1:\mathcal{R}\rightarrow Y$ is said to be $\mu_1$-$\mathcal{PAA}$ if $G_1$ is written in the form, $G_1 = H_1 + H_2,$ where $H_1\in \mathcal{AA}_\Omega^o(\mathcal{R}, Y)$ and $H_2\in \kappa(\mathcal{R}, Y, \mu_1).$ Denote collection of such functions as $\mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$ .

Remark 2.1. Define the positive measure $\mu_1$ by

where $\rho$ is a nonnegative $\mathcal{B}_1$-measurable function and $dt$ denotes the Lebesgue measure on $\mathcal{R}$. With respect to the Lebesgue measure on $\mathcal{R}$, the function $\rho$ in (2.1) is called the Radon-Nikodym derivative of $\mu_1$. In this case, $\rho$ is locally Lebesgue-integrable on $\mathcal{R}$ and $\int_{-\infty}^{+\infty}\rho(t)dt = +\infty$ if and only if its positive measure $\mu_1\in \mathcal{M}_1$.

A bounded sequence $h_1:\mathbb{Z}\rightarrow Y$ is said to be in $\kappa_S(\mathbb{Z}, Y, \mu_1)$ for $\mu_1\in \mathcal{M}_1$ if $-l_1 = t_1 < t_2 < ... < t_n = l_1$ be a sequence of real numbers then

Definition 2.6. A bounded sequence $x:\mathbb{Z}\rightarrow Y$ is said to be $\mu_1$-$\mathcal{PAA}$ sequence if it can be decomposed as $x = x_1+x_2$ where $x_1 \in AA^{o}_{S}(\mathbb{Z}, Y)$ and $x_2 \in \kappa_S(\mathbb{Z}, Y, \mu_1).$ Denote collection of such functions as $\mathcal{PAA}_S^o(\mathbb{Z}, Y, \mu_1)$.

2.2.   Previous results

In this section, we present some preliminary results which are needed in the sequel.

Proposition 2.1. ^[5] Let $\mu_1\in \mathcal{M}_1.$ Then $(\kappa(\mathcal{R}, Y, \mu_1), \Vert.\Vert_\infty)$ is a Banach space.

Proposition 2.2. ^[5] For $\varsigma \in \mathcal{R}$ and $\mu_1\in \mathcal{M}_1,$ we denote $\mu_{1\varsigma}$ be the positive measure on $(\mathcal{R}, \mathcal{B}_1)$ defined by

We give the following assumption from $\mu_1 \in \mathcal{M}_1$:

$(H)$ For all $\varsigma \in \mathcal{R}$ and a bounded interval $I_1$, there exist $\beta > 0$ such that

Theorem 2.1. ^[5] Let $I_1$ be a bounded interval (eventually $I_1 = \emptyset$) and $\mu_1 \in \mathcal{M}_1$. Assume that $G_1\in PC(\mathcal{R}, Y).$ Then the following statements are equivalent.

● $G_1\in \kappa(\mathcal{R}, Y, \mu_1).$

● $\lim_{l_1\rightarrow +\infty} \dfrac{1}{\mu_1([-l_1, l_1]\backslash{I_1})} \int_{[-l_1, l_1]\backslash{I_1}}\Vert G_1(t)\Vert d\mu_1(t) = 0$

● For any $\varepsilon > 0, \lim_{l_1\rightarrow +\infty} \dfrac{\mu_1(\lbrace{t\in[-l_1, l_1]\backslash{I_1}:\Vert G_1(t)\Vert > \varepsilon}\rbrace)}{\mu_1([-l_1, l_1]\backslash{I_1})} = 0.$

Remark 2.2. The fact that $\mu_1([-l_1, l_1]) = \mu_1([-l_1, l_1]\backslash{I_1})+\mu_1(I_1)$ for $l_1$ sufficiently large and from $\mu_1 \in \mathcal{M}_1$, we deduce that $\lim_{l_1\rightarrow +\infty}\mu_1([-l_1, l_1]\backslash{I_1}) = +\infty.$

Definition 2.7. ^[5] Let $\mu_2$ and $\mu_3 \in \mathcal{M}_1$. $\mu_2$ is said to be equivalent to $\mu_3(\mu_2\sim \mu_3)$ if there exists constants $\beta, \alpha > 0$ and a bounded interval $I_1$ (eventually $I_1 = \emptyset)$ such that $\beta\mu_2(C)\geq\mu_{3}(C)\geq\alpha \mu_2(C),$ for $C\in \mathcal{B}_1$ satisfying $C\cap I_1 = \emptyset.$

Theorem 2.2. Let $\mu_2$, $\mu_3 \in \mathcal{M}_1$. If $\mu_2$ and $\mu_3$ are equivalent, then $\kappa(\mathcal{R}, Y, \mu_2) = \kappa(\mathcal{R}, Y, \mu_3)$ and $\mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_2) = \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_3).$

Lemma 2.1. ^[5] Let $\mu_1\in \mathcal{M}_1.$ The measures $\mu_2 \sim \mu_3$ are equivalent for all $\varsigma \in \mathcal{R}$ if and only if $\mu_1$ satisfies $(H)$.

Theorem 2.3. ^[5] Assume $\mu_1\in\mathcal{M}_1$ and $(H)$ holds. If $\kappa(\mathcal{R}, Y, \mu_1)$ is translation invariant, then $\mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$ is also translation invariant.

Theorem 2.4. ^[5] Let $\mu_1\in \mathcal{M}_1$ and $G_1 = H_1+H_2\in \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$, where $H_1 \in AA^{o}_{\Omega}(\mathcal{R}, Y)$ and $H_2\in \kappa(\mathcal{R}, Y, \mu_1)$. If $\mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$ is translation invariant, then

Theorem 2.5. ^[5] Assume that $\mu_1\in \mathcal{M}_1$ and $\mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$ is translation invariant. Then $(\mathcal{PAA}_\Omega^o (\mathcal{R}, Y, \mu_1), \Vert.\Vert_\infty)$ is a Banach space.

Theorem 2.6. ^[9] Let $G_{1} = H_1 + H_2 \in \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$, where $\mu_1\in \mathcal{M}_1$. Assume that $G_{1}(t, z)$ and $H_1(t, z)$ are uniformly continuous on any bounded subset $K_{\ast} \subset T$ uniformly in $t \in \mathcal{R}.$ If $\Phi_{\ast} \in \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$ then $G_{1}(\cdot, \Phi_{\ast}(\cdot)) \in \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1).$

Theorem 2.7. (Krasnoselskii, ^[22]) Let $\mathbb{X}$ be a convex closed nonempty subset of a Banach space $(Y, \|\cdot\|)$. Suppose that $A_2$ and $B_2$ map $\mathbb{X}$ into $Y$ such that

● $A_2x+B_2y\in \mathbb{X}\quad (\forall \, \, x, y\in \mathbb{X})$

● $A_2$ is continuous and $A_2\mathbb{X}$ is contained in a compact set

● $B_2$ is a contraction mapping.

Then there is a $y\in \mathbb{X}$ with $A_2y+B_2y = y$.

2.3.   Assumptions

To prove the main results, we consider the following assumptions:

(H1) $G_1:\mathcal{R}\times Y \rightarrow Y$. Let $L_{G_1} > 0$ be such that $\Vert G_1(t, z_{1})-G_1(t, z_{2})\Vert \leq L_{G_1} \Vert z_{1}-z_{2}\Vert$, $t\in \mathcal{R}$, $z_1, z_2\in Y.$

(H2) The sequence $I_{j}$ is $\mu_1-\mathcal{PAA}$ and there exists $L_{1} > 0$ such that $\Vert I_{j}(z)-I_{j}(t_{1})\Vert\leq L_{1}\Vert z-t_{1} \Vert, j\in \mathbb{Z}$ and $z, t_{1}\in Y.$

(H3) $g_1:\mathcal{R}\times Y\rightarrow Y$. Let $L_g > 0$ be such that $\Vert g_1(t, z)-g_1(t, t_{1})\Vert \leq L_g \Vert z-t_{1} \Vert,$ $t\in \mathcal{R}$ and $z, t_{1}\in Y$.

3.   Useful results

In this section, we present the important results which are needed to prove the main results.

Lemma 3.2. If a bounded sequence $\lbrace \varphi(n)\rbrace_{n\in \mathbb{Z}} \in \kappa(\mathcal{R}, Y, \mu_1)$, then there exists a uniformly continuous function $g\in \kappa(\mathcal{R}, Y, \mu_1)$ such that $g(t_n) = \lbrace\varphi(n)\rbrace_{n\in \mathbb{Z}}, \quad t_n\in \mathcal{R}.$

Proof. We define a function $g(t) = \varphi(n)+ \int_{n}^{t}H_2(t, \varphi(t))dy$ for $t\in [n, n+1).$ If $\lbrace\varphi(n)\rbrace_{n\in \mathbb{Z}}\in \kappa(\mathcal{R}, Y, \mu_1)$, then it follows from the boundedness of $H_2$ that $g(t)$ is bounded on $\mathcal{R}.$ From Theorem 2.6, we have $H_{2}(., \varphi(.))\in \kappa(\mathcal{R}, Y, \mu_1)$. Let the set $B_j = \lbrace t\in [-l_1, l_1]:u_1(t)\in O_j\rbrace$ is open in $[-l_1, l_1]$ and $[-l_1, l_1] = U_{j = 1}^m B_j.$ Let $E_1 = B_1, E_j = B_j\setminus \cup_{j = 1}^{k-1}B_j\quad(2\leq k\leq m).$ Then $E_{i}\cap E_{j} = \phi$ when $i\neq j, 1\leq i, j\leq m.$ Now we have,

as $l_1\rightarrow \infty,$ we deduce that $\lim_{l_1\rightarrow \infty} \frac{1}{\mu_1[-l_1, l_1]}\int_{[-l_1, l_1]}\Vert g(t)\Vert d\mu_1 = 0.$ That is $\quad g \in \kappa(\mathcal{R}, Y, \mu_1)$.

Theorem 3.1. Let $I_{j} : Y {\rightarrow} Y$ be a $\mu_1-\mathcal{PAA}$ sequence and satisfying (H2). If $\phi \in$$\mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$ then $I_{j}(\phi(t_{j}))$ is a $\mu_1$-$\mathcal{PAA}$ sequence.

Proof. Since $\phi = p_{1}+p_{2}$, where $p_{1} \in \mathcal{AA}_\Omega^o(\mathcal{R}, Y),$ $p_{2} \in \kappa(\mathcal{R}, Y, \mu_1)$, it follows that $I_{j}(\phi(t_{j})) = I_{j}(p_1(t_{j}))+I_{j}(p_2(t_{j}))$. By [23, Lemma 3.2], $I_{j}(p_1(t_{j}))$ is an $\mathcal{AA}$ sequence. Now it remains to show that $I_{j}(p_2(t_{j}))\in \kappa(\mathcal{R}, Y, \mu_1)$. Since $p_2 \in \kappa(\mathcal{R}, Y, \mu_1)$, we have $\lim_{l_1\rightarrow \infty} \dfrac{1}{\mu_1([-l_1, l_1])} \int_{[-l_1, l_1]}\Vert p_2(t)\Vert d\mu_1(t) = 0.$

Let $-l_1 = t_1 < t_2 < .... < t_n = l_1$ be a sequence of real numbers, we have

Thus we obtain

Taking the limit when $l_1\rightarrow \infty,$ from Eq 3.1, we obtain $p_2(t_j)\in \kappa(\mathbb{Z}, Y, \mu_1)$.

Now from (H2),

We see that the sequence $I_{j}(\phi(t_{j}))$ is a $\mu_1$-$\mathcal{PAA}$ sequence.

Remark 3.3. In order to prove the main results, we need to present an important estimate from ^[19] as follows:

Lemma 3.3. Let $E_{\alpha}(t)$ be strongly continuous family of bounded linear operators satisfying (3.2). If $z \in AA^{0}_{\Omega}(\mathcal{R}, Y)$ and $u_{0} : \mathcal{R} {\rightarrow} Y$ is defined by

then $u_{0}(\cdot) \in AA^{0}_{\Omega}(\mathcal{R}, Y)$.

Proof. Since $z$ is $\mathcal{AA}$, there exists a subsequence $\{ u_{n_k}\}$ of $\{ u_{n}\}$ such that $h_{1}(t) = \lim_{n {\rightarrow} \infty} z(t + u_{n_{k}})$ is well defined for every $t \in \mathcal{R}$. We consider

And

Since $z \in AA^{0}_{\Omega}(\mathcal{R}, Y),$ $\lim_{n {\rightarrow} \infty} z(t + u_{n_{k}}) = h_{1}(t_{j})$ for every $j \in \mathbb{Z}.$ Therefore, for any $t > t_{j},$ $j \in \mathbb{Z}$, by Lebesgue's dominated convergence theorem, we obtain

Therefore, $u_{0} (\cdot) \in AA^{0}_{\Omega}(\mathcal{R}, Y)$.

Lemma 3.4. Let $f = g_2+h\in \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$ with $g_2\in\mathcal{AA}_\Omega^o(\mathcal{R}, Y)$ and $h\in\kappa(\mathcal{R}, Y, \mu_1)$. Then

Proof. Now, let $u_0(t) = \int_{-\infty}^{t}E_{\alpha}(t-s)\gamma_1(s)ds + \int_{-\infty}^{t}E_{\alpha}(t-s)\gamma_2(s)ds < /italic > < italic > = u_1(t)+v_1(t),$

where $u_1(t) = \int_{-\infty}^{t}E_{\alpha}(t-s)\gamma_1(s)ds$ and $v_1(t) = \int_{-\infty}^{t}E_{\alpha}(t-s)\gamma_2(s)ds$

Let $(\tau_{n}^{'})$ be an arbitrary sequence on $\mathcal R$. Since $\gamma_{1} \in AA^{o}_{\Omega}(\mathcal{R}, Y)$, there exists a subsequence $(\tau_{n})$ of $(\tau_{n}^{'})$ such that

and

Define $\bar{u_1}(t) = \int_{-\infty}^{t} E_{\alpha}(t - s)\bar{\gamma}_{1}(s)ds$.

Consider,

where $\gamma_{1_{n}}(u) = \gamma_{1}(u+ \tau_{n}), n = 1, 2, ...$

Now, we have

and since $E_{\alpha}(\cdot)z$ is continuous, we get $E_{\alpha}(t-u)\gamma_{{1}_{n}}(u)\rightarrow E_{\alpha}(t-u)\overline{\gamma_{1}}(u)$ as $n \rightarrow \infty$ for any $t\geq u$ and for all fixed $u\in \mathcal{R}$. Then by using the Lebesgue's dominated convergence theorem,

By this way we can show that,

Therefore $u_1(t)\in AA_{\Omega}^{0}(\mathcal R, Y)$.

Next we show $v_{1}\in \kappa(\mathcal{R}, {Y}, \mu_1)$.

Since $\gamma_2\in \kappa(\mathcal{R}, {Y}, \mu_1)$, we find $\frac{1}{\mu_1([-l_1, l_1])} \int_{[-l_1, l_1]} \left\|\gamma_2{(t-s)}\right\|d\mu_1(t) = 0$ for all $t \in \mathcal{R}$. Therefore $v_1(t) \in \kappa(\mathcal{R}, {Y}, \mu_1)$, by Lebesgue's dominated convergence theorem.

Theorem 3.2. If $G_{1} = H_{1} + H_{2}\in \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$ with $H_1 \in \mathcal{AA}_\Omega^o(\mathcal{R}, Y), H_2 \in \kappa(\mathcal{R}, {Y}, \mu_1)$.

Then $Q_{1}(t) = \int_{-\infty}^{t} E_{\alpha}(t-s)G_1(t) ds + \sum_{ t > t_{j}} E_{\alpha}(t- t_{j}) I_{j}(z(t_{j}))$ is a $\mu_1$-$\mathcal{PAA}$ function.

Proof. From Lemma 3.4, it follows that $\int_{-\infty}^{t} E_{\alpha}(t-s)G_{1}(t) ds \in \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$.

Next we show that $\sum_{t > t_{j}} E_{\alpha}(t-t_{j}) I_{j}(z(t_{j})) \in \mathcal{PAA}_\Omega^o(\mathcal{R}, {Y}, \mu_1).$ By Theorem 3.1, $I_{j}(z(t_{j})) \in \mathcal{PAA}_\Omega^o(\mathcal{R}, {Y}, \mu_1)$. Let $I_{j}(z(t_{j})) = \beta_{j} + \gamma_{j}$, where $\beta_{j} \in \mathcal{AA}_S^o(\mathbb{Z}, {Y})$ and $\gamma_{j} \in \kappa(\mathbb{Z}, {Y}, \mu_1)$, then

By Lemma 3.3, $R_{2}(t)\in \mathcal{AA}_\Omega^o(\mathcal{R}, {Y}).$ Next to show that $V_{2}(t)\in \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1).$

Since $\gamma_j\in \kappa(\mathbb{Z}, {Y}, \mu_1)$, by Lemma 3.2, there exists $g(t) = \gamma_j, \quad t\in[j, j+1)$ such that $g\in \kappa(\mathcal{R}, {Y}, \mu_1)$ and $g(j) = \gamma_j, j\in \mathbb{Z}.$ Then,

where $m_{1} = \{\min(t-t_{j}):0 < t-t_{j}\leq 1\}$. Since $g(t)\in \kappa(\mathcal{R}, Y, \mu_1)$, we have $V_2{(t)}\in \kappa(\mathcal{R}, Y, \mu_1)$. Thus $\sum_{t > t_{j}} E_{\alpha}(t-t_{j})I_{j}(z(t_{j})) \in \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$.

4.   Main results

Here we give the mild solution of our model (1.1).

Definition 4.8. A function $z:\mathcal{R} {\rightarrow} Y$ is said to be a mild solution of (1.1) if

Theorem 4.1. Suppose $(H1)-(H3)$ are satisfied then the model $(1.1)$ has a $\mu_1-\mathcal{PAA}$ solution $z$ on $\mathcal{R}$ provided $\left(L_{g}+\dfrac{CM \pi L_{G_{1}}|\omega|^{1 / \alpha}}{\alpha sin(\pi / \alpha)}\right) < 1.$

Proof. Let $B_{q_0}$ be the closed convex and bounded subset of $\mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$, where $B_{q_0} = \{z \in \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1) : \Vert z \Vert \leq q_0 \}$.

Now introduce the operator $\Gamma_{1} : B_{q_0} \rightarrow \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$ as follows:

We decompose $\Gamma_{1} = \Gamma^{*}_{1}+ \Gamma^{*}_{2}$ as

Step 1: For $z \in B_{q_0}$ implies $\Gamma^{*}_{1}z, \Gamma^{*}_{2}z \in \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$.

By Lemma 3.4 and Theorem 3.2 we have $\Gamma^{*}_{1}z, \Gamma^{*}_{2}z \in \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1)$.

Step 2: For $z_1, z_2 \in B_{q_0}$ implies $\Gamma^{*}_{1}z_1+ \Gamma^{*}_{2}z_2 \in B_{q_0}$.

Step 3: $\Gamma^{*}_{1}$ is contraction on $B_{q_0}.$

For each $t \in \mathcal{R}$, let $z_{1}, z_{2} \in B_{q_0}$ then by (H1), (H3) and (3.2), we have

Step 4: $\Gamma^{*}_{2}$ is continuous on $B_{q_0}$.

Let $\{z^n(t)\}_0^{\infty}\subseteq B_{q_0}$ with $z^n\rightarrow z$ in $B_{q_0}$ then by (H2) and (3.2), we have

As $n \rightarrow \infty$, $\Gamma^{*}_{2} z^{n} \rightarrow \Gamma^{*}_{2} z$.

Step 5: $\Gamma^{*}_{2}$ maps bounded sets into bounded sets.

It is enough to prove that for $t > 0$, there exist positive constant $\gamma$ such that, for each $z \in B_{q_0} = \{ z \in \mathcal{PAA}_\Omega^o(\mathcal{R}, Y, \mu_1) : \Vert z \Vert \leq q_{0}\}$ and we have $\Vert \Gamma^{*}_{2} z \Vert \leq \gamma$. Now for $t \in \mathcal{R}$,

Step 6: $\Gamma^{*}_{2} z$ maps bounded sets into equi-continuous sets.

Let $z \in B_{q_0}$ and for $t_{j} < \tau_{1} < \tau_{2} \leq t_{j+1}$, we receive

The right hand side does not depend on z and ${\rightarrow} 0$ as $\tau_{2} {\rightarrow} \tau_{1}$. Hence by utilizing the general form of Arzela-Ascoli theorem for equi-continuous function (Diethelm, [20, Theorem D.10]), we find that $\Gamma_{1}$ is relatively compact. Therefore the operator $\Gamma_{1}$ is compact. Now, by Theorem 2.7, the model $(1.1)$ admits at least one mild solution.

5.   Example

Consider the following model:

where $\beta$, $\varphi$ and $\varrho$ are positive constant. Let the Radon-Nikodym derivative $\rho$ of the measure $\mu_1$ be defined by $\rho(t) = e^{\sin t}$. Clearly $\mu_1$ satisfy (H). Take $Y = L^{2}([0, \pi])$ and define the operators A by $A \psi = \frac{\partial^2\psi}{\partial t^2}- \omega\psi, \psi \in D(A),$ where $D(A) = \{ \psi \in Y: \psi'' \in Y, \psi(0) = \psi(\pi)\} \subset Y$.

Let

It is not difficult to see that the function $G_{1}, g_1$ and $I_{j}$ are continuous function and $\mu_1-\mathcal{PAA}$ in $t$.

Hence

Also

Furthermore

Hence

Thus $G_{1}, g_1, I_{j}$ satisfies Lipschitz conditions with $L_{G_{1}} = 2 |\upsilon| = 1/10, L_{g} = 2|\varphi| = 1/10, L_{l} = 2|\varrho| = 1/10.$ Let $\omega = -1$ and $\alpha = 5/4$, then $\left(L_{g}+\dfrac{CM \pi L_{G_{1}}|\omega|^{1 / \alpha}}{\alpha \sin(\pi / \alpha)}\right) = 0.5276 < 1.$ Therefore by Theorem 4.1, the model (5.1) has $\mu_1-\mathcal{PAA}$ mild solution.

6.   Conclusions

In this paper, we investigate many important results on the new theory of measure pseudo almost automorphic functions with impulses. Those results have an important impact on the theory of systems. Numerous researchers are particularly involved in discussing the existence, stability and controllability results for various systems under different hypotheses. We assure that, these existence results can be further extended to stepanov type measure pseudo almost periodic and automorphic functions for integer and non-integer systems.

Conflict of interest

The authors declare no conflict of interest.