Weighted piecewise pseudo almost automorphic functions with applications to abstract impulsive ∇-dynamic equations on time scales

Almost automorphic functions, which are more general than almost periodic functions, were introduced by Bochner in relation to some aspects of differential geometry (see [1–3]). For more details as regards this topic we refer to the recent books [4–6], where the authors gave important overviews about the theory of almost automorphic functions and their applications to differential equations. Almost automorphic and pseudo almost automorphic solutions in the context of differential equations had been studied by several authors [7–21]. N’Guérékata [13] and Xiao [15, 21] with their collaborators established the existence and uniqueness theorems of pseudo almost automorphic solutions to some semilinear abstract differential equations. Recently, Blot et al. [22] introduced the concept of weighted pseudo almost automorphic functions, which generalizes the concept of weighted pseudo almost periodicity [23–26], and the author proved some interesting properties of the space of weighted pseudo almost automorphic functions like the completeness and the composition theorem, which have many applications in the context of differential equations. For other contributions to the study of weighted pseudo almost automorphy, we refer the reader to [27–30] and references therein.

On the other hand, the theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his PhD thesis in 1988 [31] in order to unify continuous and discrete analysis. This theory represents a powerful tool for applications to economics, population models, and quantum physics among others. In fact, the progressive field of dynamic equations on time scales contains links to and extends the classical theory of differential and difference equations. For instance, by choosing the time scale to be the set of real numbers, the general result yields a result for differential equations. In a similar way, by choosing the time scale to be the set of integers, the same general result yields a result for difference equations. However, since there are many other time scales than just the set of real numbers or the set of integers, one has a much more general result. For these reasons, based on the concept of almost periodic time scales proposed in [32, 33], the concept of weighted pseudo almost automorphic functions on almost periodic time scales was formally introduced by Wang and Li (2013) in [34]. Moreover, some first results were proven which concern the weighted pseudo almost automorphic mild solution to abstract Δ-dynamic equations on time scales. In addition, by using the results obtained in [32, 33], Lizama and Mesquita [35] presented some new results about basic properties of almost automorphic functions on time scales and proved the existence and uniqueness of an almost automorphic solution to a class of Δ-dynamic equations.

For another thing, many phenomena in nature are characterized by the fact that their states are subject to sudden changes at certain moments and therefore can be described by impulsive system (see [36, 37]). Many evolution processes, optimal control models in economics, stimulated neural networks, population models, artificial intelligence, and robotics are characterized by the fact that at certain moments of time they undergo abrupt changes of state. The existence of almost periodic solutions of abstract impulsive differential equations has been considered by many authors; see [38–41].

where $A\in {\mathit{PC}}_{\mathrm{ld}}(\mathbb{T},\mathbb{X})$ is a linear operator in the Banach space $\mathbb{X}$ and $f\in {\mathit{PC}}_{\mathrm{ld}}(\mathbb{T}\times \mathbb{X},\mathbb{X})$, $x^{\varrho }=x(\varrho (t))$. f, $I_i$, $t_i$ satisfy suitable conditions that will be established later and $\mathbb{T}$ is an almost periodic time scale. In addition, the notations $x(t_i^{+})$ and $x(t_i^{-})$ represent the right-hand and the left-hand side limits of $x(\cdot )$ at $t_i$, respectively. In addition, some useful lemmas are obtained and the exponential stability of weighted piecewise pseudo almost automorphic mild solutions is also considered. Finally, we apply these obtained results to study a class of ∇-partial differential equations on time scales.

In the following, we will introduce some basic knowledge of time scales which is very useful to the proof of our relative results.

where $a,b\in \mathbb{T}$ with $a<\varrho (b)$.

The graininess function is defined by $\nu :\mathbb{T}\to [0,\mathrm{\infty })$: $\nu (t):=t-\varrho (t)$, for all $t\in \mathbb{T}$.

Definition 2.1 ([42])

Definition 2.2 ([42])

A function $p:\mathbb{T}\to \mathbb{R}$ is called ν-regressive provided $1-\nu (t)p(t)\ne 0$ for all $t\in {\mathbb{T}}_k$. The set of all regressive and ld-continuous functions $p:\mathbb{T}\to \mathbb{R}$ will be denoted by ${\mathcal{R}}_{\nu }={\mathcal{R}}_{\nu }(\mathbb{T})={\mathcal{R}}_{\nu }(\mathbb{T},\mathbb{R})$. We define the set ${\mathcal{R}}_{\nu }^{+}={\mathcal{R}}_{\nu }^{+}(\mathbb{T},\mathbb{R})=\{p\in {\mathcal{R}}_{\nu }:1-\nu (t)p(t)>0,\mathrm{\forall }t\in \mathbb{T}\}$.

Definition 2.3 ([42])

Lemma 2.1 ([42])

Lemma 2.2 ([43])

For each $t_0\in \mathbb{T}$ in $\mathbb{T}\mathrm{\setminus }{\mathbb{T}}_k$ the single-point set $\{t_0\}$ is ∇-measurable and its ∇-measure is given by ${\mu }_{\mathrm{\nabla }}(\{t_0\})=t_0-\varrho (t_0)$.

Lemma 2.3 ([43])

For more details of time scales and ∇-measurability, one is referred to [42, 43]. For more on time scales, see [44–49].

Definition 2.4 ([32–34])

Remark 2.4 Definition 3.1 introduced in [35] is the same as the concept of almost periodic time scales proposed in [32, 34], and $\mathbb{T}$ is also called an invariant time scale under translations in [35].

After these preparations, in the next section, we will introduce the concept of weighted piecewise pseudo almost automorphic functions on time scales in a Banach space and some of their basic properties are investigated.

In the following, we will give the definition of ld-piecewise continuous functions on time scales.

Definition 3.1 We say $\phi :\mathbb{T}\to \mathbb{X}$ is ld-piecewise continuous with respect to a sequence $\{{\tau }_i\}\subset \mathbb{T}$ which satisfy ${\tau }_i<{\tau }_{i+1}$, $i\in \mathbb{Z}$, if $\phi (t)$ is continuous on ${({\tau }_i,{\tau }_{i+1}]}_{\mathbb{T}}$ and ld-continuous on $\mathbb{T}\mathrm{\setminus }\{{\tau }_i\}$. Furthermore, ${({\tau }_i,{\tau }_{i+1}]}_{\mathbb{T}}$ are called intervals of continuity of the function $\phi (t)$.

For convenience, ${\mathit{PC}}_{\mathrm{ld}}(\mathbb{T},\mathbb{X})$ denotes the set of all ld-piecewise continuous functions with respect to a sequence $\{{\tau }_i\}$, $i\in \mathbb{Z}$. Similar to Definition 3.1, we can also introduce the concept of functions which belong to ${\mathit{PC}}_{\mathrm{rd}}(\mathbb{T},\mathbb{X})$.

Throughout the paper, we denote by $\mathbb{X}$ a Banach space; let $\mathfrak{B}$ be the set consisting of all sequences ${\{t_i\}}_{i\in \mathbb{Z}}$ such that $\theta ={inf}_{i\in \mathbb{Z}}(t_{i+1}-t_i)>0$. For ${\{t_i\}}_{i\in \mathbb{Z}}\in \mathfrak{B}$, let ${\mathit{BPC}}_{\mathrm{ld}}(\mathbb{T},\mathbb{X})$ be the space formed by all bounded ld-piecewise continuous functions $\varphi :\mathbb{T}\to \mathbb{X}$ such that $\varphi (\cdot )$ is continuous at t for any $t\notin {\{t_i\}}_{i\in \mathbb{Z}}$ and $\varphi (t_i)=\varphi (t_i^{-})$ for all $i\in \mathbb{Z}$; let Ω be a subset of $\mathbb{X}$ and let ${\mathit{BPC}}_{\mathrm{ld}}(\mathbb{T}\times \mathrm{\Omega },\mathbb{X})$ be the space formed by all bounded piecewise continuous functions $\varphi :\mathbb{T}\times \mathrm{\Omega }\to \mathbb{X}$ such that, for any $x\in \mathrm{\Omega }$, $\varphi (\cdot ,x)\in {\mathit{BPC}}_{\mathrm{ld}}(\mathbb{T}\times \mathbb{X},\mathbb{X})$. For any $t\in \mathbb{T}$, $\varphi (t,\cdot )$ is continuous at $x\in \mathrm{\Omega }$.

Let $\mathit{UPC}(\mathbb{T},\mathbb{X})$ be the space of all functions $\phi \in {\mathit{PC}}_{\mathrm{ld}}(\mathbb{T},\mathbb{X})$ such that ϕ satisfies the condition: for any $\epsilon >0$, there exists a positive number $\delta =\delta (\epsilon )$ such that if the points $t^{\prime }$, $t^{″}$ belong to the same interval of continuity of φ and $|t^{\prime }-t^{″}|<\delta$ implies $\parallel \phi (t^{\prime })-\phi (t^{″})\parallel <\epsilon$.

for each $k\in \mathbb{Z}$.

Similarly, we can also introduce the concept of piecewise almost automorphic functions which belong to ${\mathit{PC}}_{\mathrm{rd}}(\mathbb{T},\mathbb{X})$.

for each $\rho \in U$.

where $r\in \mathrm{\Pi }$ and $M_{r,\epsilon ,t_0}(\varphi ):=\{t\in {[t_0-r,t_0+r]}_{\mathbb{T}}:\parallel \varphi (t)\parallel \ge \epsilon \}$.

where $M:={sup}_{t\in \mathbb{T}}\parallel \varphi (t)\parallel <\mathrm{\infty }$ and $K:={sup}_{t\in \mathbb{T}}\rho (t)<\mathrm{\infty }$.

Therefore, ${lim}_{r\to \mathrm{\infty }}\frac{1}{m(r,\rho ,t_0)}{\int }_{t_0-r}^{t_0+r}\parallel \varphi (s)\parallel \rho (s)\mathrm{\nabla }s=0$, that is, $\varphi \in {\mathit{PAA}}_0(\mathbb{T},\rho )$. This completes the proof. □

Lemma 3.3 ${\mathit{PAA}}_0(\mathbb{T},\rho )$ is a translation invariant set of ${\mathit{BPC}}_{\mathrm{ld}}(\mathbb{T},\mathbb{X})$ with respect to Π if $\rho \in U_B$, i.e., for any $s\in \mathrm{\Pi }$, one has $\varphi (t+s):={\theta }_s\varphi \in {\mathit{PAA}}_0(\mathbb{T},\rho )$ if $\rho \in U_B$.

Again, using Lemma 3.2, one can get ${\theta }_s\varphi \in {\mathit{PAA}}_0(\mathbb{T},\rho )$ for any $s\in \mathrm{\Pi }$. This completes the proof. □

By Definition 3.4, one can easily get the following lemma.

Lemma 3.4 Let $\varphi \in \mathit{AA}(\mathbb{T},\mathbb{X})$, then the range of ϕ, $\varphi (\mathbb{T})$, is a relatively compact subset of $\mathbb{X}$.

Lemma 3.5 If $f=g+\varphi$ with $g\in \mathit{AA}(\mathbb{T},\mathbb{X})$, and $\varphi \in {\mathit{PAA}}_0(\mathbb{T},\rho )$, where $\rho \in U_B$, then $g(\mathbb{T})\subset \overline{f(\mathbb{T})}$.

Proof (1) For any $t\in \mathbb{T}\mathrm{\setminus }\{t_i\}$, $g(t)\in g(\mathbb{T})$, one has $g(t)=f(t)-\varphi (t)$. Since $g\in \mathit{AA}(\mathbb{T},\mathbb{X})$, there exists a sequence $\{{\alpha }_n\}\subset \mathrm{\Pi }$ such that $g(t+{\alpha }_n)\to g(t)$, $n\to \mathrm{\infty }$.

Lemma 3.6 The decomposition of a weighted piecewise pseudo almost automorphic function according to $\mathit{AA}\oplus {\mathit{PAA}}_0$ is unique for any $\rho \in U_B$.

Proof Assume that $f=g_1+{\varphi }_1$ and $f=g_2+{\varphi }_2$. Then $(g_1-g_2)+({\varphi }_1-{\varphi }_2)=0$. Since $g_1-g_2\in \mathit{AA}(\mathbb{T},\mathbb{X})$, and ${\varphi }_1-{\varphi }_2\in {\mathit{PAA}}_0(\mathbb{T},\rho )$, in view of Lemma 3.5, we deduce that $g_1-g_2=0$. Consequently, ${\varphi }_1-{\varphi }_2=0$, i.e. ${\varphi }_1={\varphi }_2$. This completes the proof. □

Theorem 3.7 For $\rho \in U_B$, $(\mathit{WPAA}(\mathbb{T},\rho ),{\parallel \cdot \parallel }_{\mathrm{\infty }})$ is a Banach space.

Proof Assume that ${\{f_n\}}_{n\in \mathbb{N}}$ is a Cauchy sequence in $\mathit{WPAA}(\mathbb{T},\rho )$. We can write uniquely $f_n=g_n+{\varphi }_n$. Using Lemma 3.5, we see that ${\parallel g_p-g_q\parallel }_{\mathrm{\infty }}\le {\parallel f_p-f_q\parallel }_{\mathrm{\infty }}$, from which we deduce that ${\{g_n\}}_{n\in \mathbb{N}}$ is a Cauchy sequence in $\mathit{AA}(\mathbb{T},\mathbb{X})$. Hence, ${\varphi }_n=f_n-g_n$ is a Cauchy sequence in ${\mathit{PAA}}_0(\mathbb{T},\rho )$. We deduce that $g_n\to g\in \mathit{AA}(\mathbb{T},\mathbb{X})$, ${\varphi }_n\to \varphi \in {\mathit{PAA}}_0(\mathbb{T},\rho )$, and finally $f_n\to g+\varphi \in \mathit{WPAA}(\mathbb{T},\rho )$. This completes the proof. □

Definition 3.5 Let ${\rho }_1,{\rho }_2\in U_{\mathrm{\infty }}$. One says that ${\rho }_1$ is equivalent to ${\rho }_2$, written ${\rho }_1\sim {\rho }_2$ if ${\rho }_1/{\rho }_2\in U_B$.

Theorem 3.8 Let ${\rho }_1,{\rho }_2\in U_{\mathrm{\infty }}$. If ${\rho }_1\sim {\rho }_2$, then $\mathit{WPAA}(\mathbb{T},{\rho }_1)=\mathit{WPAA}(\mathbb{T},{\rho }_2)$.

This completes the proof. □

Lemma 3.9 If $g\in \mathit{AA}(\mathbb{T}\times \mathbb{X},\mathbb{X})$ and $\alpha \in \mathit{AA}(\mathbb{T},\mathbb{X})$, then $G(t):=g(\cdot ,\alpha (\cdot ))\in \mathit{AA}(\mathbb{T},\mathbb{X})$.

Hence, $G\in \mathit{AA}(\mathbb{T},\mathbb{X})$. This completes the proof. □

Then $f(\cdot ,h(\cdot ))\in \mathit{WPAA}(\mathbb{T},\rho )$ if $h\in \mathit{WPAA}(\mathbb{T},\rho )$ and $h(\mathbb{T})\subset \mathrm{\Omega }$.

and this implies that $\mathrm{\Psi }\in {\mathit{PAA}}_0(\mathbb{T},\rho )$.