# Abstract Impulsive Volterra Integro-Differential Inclusions

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- (P1):
- $k\left(t\right)$ is Laplace transformable, i.e., it is locally integrable on $[0,\infty )$ and there exists $\beta \in \mathbb{R}$ such that $\tilde{k}\left(\lambda \right):=\left(\mathcal{L}k\right)\left(\lambda \right):={lim}_{b\to \infty}{\int}_{0}^{b}{e}^{-\lambda t}k\left(t\right)dt:={\int}_{0}^{\infty}{e}^{-\lambda t}k\left(t\right)dt$ exists for all $\lambda \in \mathbb{C}$ with $\Re \lambda >\beta $. Put $abs\left(k\right):=$ inf $\{\Re \lambda :\tilde{k}\left(\lambda \right)\phantom{\rule{4.pt}{0ex}}\mathrm{exists}\}$.
- (P2):
- $k\left(t\right)$ satisfies (P1) and $\tilde{k}\left(\lambda \right)\ne 0$, $\Re \lambda >\beta $ for some $\beta \ge abs\left(k\right)$.

#### 2.1. Solution Operator Families Subgenerated by MLOs

- (1)
- $D\left(\mathcal{A}\right):=\{u\in X:\mathcal{A}u\ne \varnothing \}$ is a linear submanifold of X;
- (2)
- $\mathcal{A}u+\mathcal{A}v\subseteq \mathcal{A}(u+v)$ for $u,\phantom{\rule{4pt}{0ex}}v\in D\left(\mathcal{A}\right)$ and $\lambda \mathcal{A}u\subseteq \mathcal{A}\left(\lambda u\right)$ for $\lambda \in \mathbb{C}$ and $u\in D\left(\mathcal{A}\right).$

**Lemma**

**1.**

- (i)
- $R\left(C\right)\subseteq R(\gamma -\mathcal{A})$;
- (ii)
- ${(\gamma -\mathcal{A})}^{-1}C$ is a single-valued linear continuous operator on $X.$

**Definition**

**1.**

- (i)
- (ii)
- A solution of (1) is any pre-solution $u(\xb7)$ of (1) satisfying additionally that there exist functions ${u}_{\mathcal{B}}\in C([0,T]:E)$ and ${u}_{a,\mathcal{A}}\in C([0,T]:E)$ such that ${u}_{\mathcal{B}}\left(t\right)\in \mathcal{B}u\left(t\right)$ and ${u}_{a,\mathcal{A}}\left(t\right)\in \mathcal{A}{\int}_{0}^{t}a(t-s)u\left(s\right)ds$ for $t\in [0,T]$, as well as$${u}_{\mathcal{B}}\left(t\right)\in {u}_{a,\mathcal{A}}\left(t\right)+\mathcal{F}\left(t\right),\phantom{\rule{1.em}{0ex}}t\in [0,T].$$
- (iii)
- A strong solution of (1) is any function $u\in C\left(\right[0,T]:X)$ satisfying that there exist two continuous functions ${u}_{\mathcal{B}}\in C([0,T]:E)$ and ${u}_{\mathcal{A}}\in C([0,T]:E)$ such that ${u}_{\mathcal{B}}\left(t\right)\in \mathcal{B}u\left(t\right)$, ${u}_{\mathcal{A}}\left(t\right)\in \mathcal{A}u\left(t\right)$ for all $t\in [0,T]$, and$${u}_{\mathcal{B}}\left(t\right)\in (a\ast {u}_{\mathcal{A}})\left(t\right)+\mathcal{F}\left(t\right),\phantom{\rule{1.em}{0ex}}t\in [0,T].$$

**Definition**

**2.**

- (i)
- $\mathcal{A}$ is said to be a subgenerator of a (local, if $\tau <\infty $) mild $(a,k)$-regularized $({C}_{1},{C}_{2})$-existence and uniqueness family ${({R}_{1}\left(t\right),{R}_{2}\left(t\right))}_{t\in [0,\tau )}\subseteq L(X,E)\times L\left(E\right)$ if the mappings $t\mapsto {R}_{1}\left(t\right)y,$ $t\ge 0$ and $t\mapsto {R}_{2}\left(t\right)x,$ $t\in [0,\tau )$ are continuous for every fixed $x\in E$ and $y\in X,$ as well as the following conditions hold:$$\left(\underset{0}{\overset{t}{\int}}a(t-s){R}_{1}\left(s\right)y\phantom{\rule{0.166667em}{0ex}}ds,{R}_{1}\left(t\right)y-k\left(t\right){C}_{1}y\right)\in \mathcal{A},\phantom{\rule{4pt}{0ex}}t\in [0,\tau ),\phantom{\rule{4pt}{0ex}}y\in Xand$$$$\underset{0}{\overset{t}{\int}}a(t-s){R}_{2}\left(s\right)y\phantom{\rule{0.166667em}{0ex}}ds={R}_{2}\left(t\right)x-k\left(t\right){C}_{2}x,whenevert\in [0,\tau )and(x,y)\in \mathcal{A}.$$
- (ii)
- Let ${\left({R}_{1}\left(t\right)\right)}_{t\in [0,\tau )}\subseteq L(X,E)$ be strongly continuous. We say that $\mathcal{A}$ is a subgenerator of a (local, if $\tau <\infty $) mild $(a,k)$-regularized ${C}_{1}$-existence family ${\left({R}_{1}\left(t\right)\right)}_{t\in [0,\tau )}$ if and only if (2) holds.
- (iii)
- Let ${\left({R}_{2}\left(t\right)\right)}_{t\in [0,\tau )}\subseteq L\left(E\right)$ be strongly continuous. $\mathcal{A}$ is said a subgenerator of a (local, if $\tau <\infty $) mild $(a,k)$-regularized ${C}_{2}$-uniqueness family ${\left({R}_{2}\left(t\right)\right)}_{t\in [0,\tau )}$ if and only if (3) holds.

**Definition**

**3.**

- (i)
- Assume that $\tau \in (0,\infty ],$ $a\in {L}_{loc}^{1}\left([0,\tau )\right),$ $a\ne 0,$ $k\in C\left(\right[0,\tau \left)\right),$ $k\ne 0,$ $\mathcal{A}:E\to P\left(E\right)$ is an MLO, $C\in L\left(E\right)$ is injective and $C\mathcal{A}\subseteq \mathcal{A}C.$ We say that a strongly continuous operator family ${\left(R\left(t\right)\right)}_{t\in [0,\tau )}\subseteq L\left(E\right)$ is an $(a,k)$-regularized C-resolvent family with a subgenerator $\mathcal{A}$ if and only if ${\left(R\left(t\right)\right)}_{t\in [0,\tau )}$ is a mild $(a,k)$-regularized C-uniqueness family having $\mathcal{A}$ as subgenerator, $R\left(t\right)\mathcal{A}\subseteq \mathcal{A}R\left(t\right)$ and $R\left(t\right)C=CR\left(t\right)$ for all $t\in [0,\tau )$.
- (ii)
- If $\tau =\infty ,$ then we say that ${\left(R\left(t\right)\right)}_{t\ge 0}$ is exponentially bounded (bounded) if and only if there exists $\omega \in \mathbb{R}$ ($\omega =0$) such that the family $\{{e}^{-\omega t}R\left(t\right):t\ge 0\}$ is bounded.

## 3. Abstract Impulsive Differential Inclusions of Integer Order

**Definition**

**4.**

- (i)
- By a pre-solution of ${\left(ACP\right)}_{n;1}$ on $[0,T]$ we mean any function $u(\xb7)$ which is n-times continuously differentiable on the intervals $[0,{t}_{1}),$ $({t}_{1},{t}_{2}),$ $({t}_{2},{t}_{3}),...,$ $({t}_{l},T],$ the right derivatives ${lim}_{t\to {t}_{i}+}{u}^{\left(j\right)}\left(t\right)$ exist for $0\le j\le n$ and $1\le i\le l,$ the left derivatives ${lim}_{t\to {t}_{i}-}{u}^{\left(j\right)}\left(t\right)$ exist for $0\le j\le n$ and $1\le i\le l+1,$ and the requirements of ${\left(ACP\right)}_{n;1}$ hold.A solution of ${\left(ACP\right)}_{n;1}$ on $[0,T]$ is any pre-solution $u\left(t\right)$ of ${\left(ACP\right)}_{n;1}$ on $[0,T]$ which additionally satisfies that there exists a function ${u}_{\mathcal{A}}:[0,T]\to X$ such that ${u}_{\mathcal{A}}\left(t\right)\in \mathcal{A}u\left(t\right)$ for $t\in [0,T]\backslash \{{t}_{1},{t}_{2},...,{t}_{l}\}$, ${u}^{\left(n\right)}\left(t\right)={u}_{\mathcal{A}}\left(t\right)+f\left(t\right)$ for $t\in [0,T]\backslash \{{t}_{1},{t}_{2},...,{t}_{l}\}$, the right limits ${lim}_{t\to {t}_{i}+}{u}_{\mathcal{A}}\left(t\right)$ exist for $1\le i\le l$ and the left limits ${lim}_{t\to {t}_{i}-}{u}_{\mathcal{A}}\left(t\right)$ exist for $1\le i\le l+1.$
- (ii)
- Suppose that $0\equiv {t}_{0}<{t}_{1}<...<{t}_{l}<{t}_{l+1}<...<+\infty $ and the sequence ${\left({t}_{l}\right)}_{l}$ has no accumulation point. By a (pre-)solution of ${\left(ACP\right)}_{n;1}$ on $[0,\infty )$ we mean any function $u(\xb7)$ which satisfies that, for every $l\in \mathbb{N}$ and $T\in ({t}_{l},{t}_{l+1}),$ the function ${u}_{\left|\right[0,T]}(\xb7)$ is a (pre-)solution of ${\left(ACP\right)}_{n;1}$ on $[0,T].$

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Example**

**1.**

- (i)
- Let $\mathcal{A}=A$ be a closed single-valued linear operator and $\lambda \in \rho \left(A\right).$ Then it is well known that A is the integral generator of a distribution semigroup (distribution cosine function) if and only if for each $\tau >0$ there exists $n\in \mathbb{N}$ such that A is the integral generator of a local ${(\lambda -A)}^{-n}$-regularized semigroup (${(\lambda -A)}^{-n}$-regularized cosine function) on $[0,\tau )$; furthermore, there exists an injective operator $C\in L\left(X\right)$ such that A is the integral generator of a global C-regularized semigroup (C-regularized cosine function); cf. [29] for the notion and more details. Therefore, Theorem 1 and Corollary 1 can be successfully applied in the case that $n=1$ ($n=2$).
- (ii)
- Suppose that the sequence $\left({M}_{p}\right)$ of positive real numbers satisfies ${M}_{0}=1,$ (M.1), (M.2) and (M.3) as well as that a closed linear operator A generates a regular ultradistribution semigroup (regular ultradistribution cosine function) of $\left({M}_{p}\right)$-class; then there exists an injective operator $C\in L\left(X\right)$ such that A is the integral generator of a global C-regularized semigroup (C-regularized cosine function); cf. ([29], Section 3.5 and 3.6) for the notion and more details. Consequently, Theorem 1 and Corollary 1 can be successfully applied in the case that $n=1$ ($n=2$). For some important examples of (differential) operators generating ultradistribution semigroups (ultradistribution cosine functions), we refer the reader to ([29], Example 3.5.18, Example 3.5.23, Example 3.5.30(ii), Example 3.5.39).
- (iii)
- Suppose that $k\in \mathbb{N}$, ${a}_{\alpha}\in \mathbb{C}$, $0\phantom{\rule{-0.166667em}{0ex}}\le \phantom{\rule{-0.166667em}{0ex}}\left|\alpha \right|\phantom{\rule{-0.166667em}{0ex}}\le \phantom{\rule{-0.166667em}{0ex}}k$, ${a}_{\alpha}\ne 0$ for some α with $\left|\alpha \right|=k$, $P\left(x\right)={\sum}_{\left|\alpha \right|\le k}{a}_{\alpha}{i}^{\left|\alpha \right|}{x}^{\alpha}$, $x\in {\mathbb{R}}^{n}$, $\omega :={sup}_{x\in {\mathbb{R}}^{n}}\Re \left(P\left(x\right)\right)<+\infty $ (condition ([4], (W), p. 68) holds), and X is one of the spaces ${L}^{p}\left({\mathbb{R}}^{n}\right)$ ($1\le p\le \infty $), ${C}_{0}\left({\mathbb{R}}^{n}\right)$, ${C}_{b}\left({\mathbb{R}}^{n}\right)$, $BUC\left({\mathbb{R}}^{n}\right)$. Define$$P\left(D\right):=\sum _{\left|\alpha \right|\le k}{a}_{\alpha}{f}^{\left(\alpha \right)}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}D\left(P\left(D\right)\right):=\left\{f\in E:P\left(D\right)f\in E\phantom{\rule{4.pt}{0ex}}\mathrm{distributionally}\right\}.$$Then it is well known that the operator $P\left(D\right)$ generates an exponentially bounded C-regularized semigroup (C-regularized cosine function) with an appropriately chosen regularizing operator $C\in L\left(X\right),$ so that Corollary 1 can be successfully applied in the case that $n=1$ ($n=2$).

**Example**

**2.**

- (i)
- We can analyze the well-posedness of the abstract impulsive inclusion ${\left(ACP\right)}_{1;1}$ for the multivalued linear operators $\mathcal{A}$ satisfying the following condition:
- (P)
- There exist finite constants $c,\phantom{\rule{4pt}{0ex}}M>0$ and $\beta \in (0,1]$ such that$$\Psi :={\Psi}_{c}:=\left\{\lambda \in \mathbb{C}:\Re \lambda \ge -c\left(|\Im \lambda |+1\right)\right\}\subseteq \rho \left(\mathcal{A}\right)$$$$\parallel R(\lambda :\mathcal{A})\parallel \le M{\left(1+\left|\lambda \right|\right)}^{-\beta},\phantom{\rule{1.em}{0ex}}\lambda \in \Psi .$$

Then the degenerate semigroup ${\left(T\left(t\right)\right)}_{t>0}$ generated by $\mathcal{A}$ has an integrable singularity at zero but we can still apply the method obeyed in the proof of Theorem 1 if the function $f\left(t\right)$ satisfies the requirements of ([23], Theorem 3.7) and there exist vectors ${z}_{1},...,{z}_{k},...$ from the continuity set of the semigroup ${\left(T\left(t\right)\right)}_{t>0}$ such that ${z}_{k}\in \mathcal{A}{y}_{k},$ $k=1,...,l,....$ The established conclusion can be simply applied in the analysis of the following abstract impulsive Poisson heat equation in the space $X={L}^{p}(\Omega ):$$$\left\{\begin{array}{c}\frac{d}{dt}\left[m\left(x\right)v(t,x)\right]=(\Delta -b)v(t,x)+f(t,x),\phantom{\rule{1.em}{0ex}}t\ge 0,\phantom{\rule{4pt}{0ex}}x\in \Omega ;\hfill \\ v(t,x)=0,\phantom{\rule{1.em}{0ex}}(t,x)\in [0,\infty )\times \partial \Omega ,\hfill \\ m\left(x\right)v({t}_{k}+,x)-m\left(x\right)v({t}_{k}-,x)={f}_{k}\left(x\right),\phantom{\rule{4pt}{0ex}}k\in \mathbb{N},\hfill \\ m\left(x\right)v(0,x)={u}_{0}\left(x\right),\phantom{\rule{1.em}{0ex}}x\in \Omega ,\hfill \end{array}\right.$$ - (ii)
- Suppose that $A,\phantom{\rule{4pt}{0ex}}B$ and C are closed linear operators in $X,$ $D\left(B\right)\subseteq D\left(A\right)\cap D\left(C\right),$ ${B}^{-1}\in L\left(X\right)$ and the conditions ([23], (6.4)-(6.5)) are satisfied with some numbers $c>0$ and $0<\beta \le \alpha =1;$ cf. also ([30], Example 3.10.10). In ([23], Chapter VI), the following second order differential equation without impulsive conditions$$\frac{d}{dt}\left(C{u}^{\prime}\left(t\right)\right)+B{u}^{\prime}\left(t\right)+Au\left(t\right)=f\left(t\right),\phantom{\rule{4pt}{0ex}}t>0;\phantom{\rule{1.em}{0ex}}u\left(0\right)={u}_{0},\phantom{\rule{4pt}{0ex}}C{u}^{\prime}\left(0\right)=C{u}_{1}$$$$\frac{d}{dt}Mz\left(t\right)=Lz\left(t\right)+F\left(t\right),\phantom{\rule{4pt}{0ex}}t>0;\phantom{\rule{1.em}{0ex}}Mz\left(0\right)=M{z}_{0},$$$$M=\left[\begin{array}{cc}\phantom{\rule{4pt}{0ex}}I\phantom{\rule{4pt}{0ex}}& \phantom{\rule{4pt}{0ex}}O\\ O& \phantom{\rule{4pt}{0ex}}C\end{array}\right],\phantom{\rule{4pt}{0ex}}L=\left[\begin{array}{cc}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}O\phantom{\rule{4pt}{0ex}}& \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}I\\ -A& -B\end{array}\right],\phantom{\rule{4pt}{0ex}}{z}_{0}=\left[\begin{array}{c}{u}_{0}\\ {u}_{1}\end{array}\right]andF\left(t\right)=\left[\begin{array}{c}0\\ f\left(t\right)\end{array}\right]\phantom{\rule{4pt}{0ex}}(t0).$$The argumentation contained in the proof of ([23], Theorem 6.1) shows that the multivalued linear operator $({L}_{\left[D\right(B\left)\right]\times X}-\omega {M}_{\left[D\right(B\left)\right]\times X}){\left({M}_{\left[D\right(B\left)\right]\times X}\right)}^{-1}$ satisfies the condition (P) for a sufficiently large number $\omega >0,$ in the pivot space $\left[D\right(B\left)\right]\times X.$ Hence, this MLO generates a degenerate semigroup ${\left(T\left(t\right)\right)}_{t>0}$ in $\left[D\right(B\left)\right]\times X,$ having an integrable singularity at zero and exponentially decaying growth rate at infinity. Then we can apply ([23], Theorem 3.8, Theorem 3.9) in the analysis of existence and uniqueness of solutions of the abstract degenerate Cauchy problem without impulsive conditions:$$\begin{array}{c}\hfill \frac{d}{dt}Mz\left(t\right)=(L-\omega M)z\left(t\right)+F\left(t\right),\phantom{\rule{4pt}{0ex}}t>0;\phantom{\rule{1.em}{0ex}}Mz\left(0\right)=M{z}_{0},\end{array}$$Furthermore, we can apply Corollary 1 with $n=1,$ $u\left(t\right)=Mz\left(t\right),$ $t\ge 0$ and $\mathcal{A}=(L-\omega M){M}^{-1}$ in the analysis of the existence and uniqueness of the piecewise continuously differentiable solutions of the following second-order impulsive differential equation:$$\left\{\begin{array}{c}\frac{d}{dt}\left(C{u}^{\prime}\left(t\right)\right)+(2\omega C+B){u}^{\prime}\left(t\right)+\left(A+\omega B+{\omega}^{2}C\right)u\left(t\right)=f\left(t\right),\phantom{\rule{4pt}{0ex}}t>0;\hfill \\ u\left({t}_{k}+\right)-u\left({t}_{k}-\right)={y}_{k},\phantom{\rule{4pt}{0ex}}C\left[{u}^{\prime}\left({t}_{k}+\right)+\omega u\left({t}_{k}+\right)\right]-C\left[{u}^{\prime}\left({t}_{k}-\right)+\omega u\left({t}_{k}-\right)\right]={z}_{k},\phantom{\rule{4pt}{0ex}}k\in \mathbb{N},\hfill \\ u\left(0\right)={u}_{0},\phantom{\rule{4pt}{0ex}}C\left[{u}^{\prime}\left(0\right)+\omega {u}_{0}\right]=C{u}_{1}.\hfill \end{array}\right.$$As is well known, we can simply incorporate this result in the analysis of existence and uniqueness of piecewise continuously differentiable solutions of the following damped Poisson-wave type equation in the spaces $X:={H}^{-1}(\Omega )$ or $X:={L}^{p}(\Omega ):$$$\left\{\begin{array}{c}\frac{\partial}{\partial t}\left(m\left(x\right)\frac{\partial u}{\partial t}\right)+\left(2\omega m\left(x\right)-\Delta \right)\frac{\partial u}{\partial t}+\left(A(x;D)-\omega \Delta +{\omega}^{2}m\left(x\right)\right)u(x,t)=f(x,t),\hfill \\ t\ge 0,\phantom{\rule{4pt}{0ex}}x\in \Omega \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}};\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}u=\partial u/\partial t=0,\phantom{\rule{1.em}{0ex}}(x,t)\in \partial \Omega \times [0,\infty ),\hfill \\ u\left(x,{t}_{k}+\right)-u\left(x,{t}_{k}-\right)={y}_{k}\left(x\right),\phantom{\rule{4pt}{0ex}}k\in \mathbb{N},\hfill \\ C\left[(\partial u/\partial t)\left(x,{t}_{k}+\right)+\omega u\left(x,{t}_{k}+\right)\right]-C\left[(\partial u/\partial t)\left(x,{t}_{k}-\right)+\omega u\left(x,{t}_{k}-\right)\right]={z}_{k}\left(x\right),\phantom{\rule{4pt}{0ex}}k\in \mathbb{N},\hfill \\ u(0,x)={u}_{0}\left(x\right),\phantom{\rule{4pt}{0ex}}m\left(x\right)\left[(\partial u/\partial t)(x,0)+\omega {u}_{0}\right]=m\left(x\right){u}_{1}\left(x\right),\phantom{\rule{1.em}{0ex}}x\in \Omega ,\hfill \end{array}\right.$$

**Example**

**3.**

- (i)
- $R\left({W}_{n}\left(z\right)\right)\subseteq {D}_{\infty}\left(P\left(A\right)\right),$$z\in \mathbb{C}$ and$$\overline{P\left(A\right)}\underset{0}{\overset{z}{\int}}{g}_{n}(z-s){W}_{n}\left(s\right)\overrightarrow{x}\phantom{\rule{0.166667em}{0ex}}ds={W}_{n}\left(z\right)\overrightarrow{x}-{C}_{m}\overrightarrow{x},\phantom{\rule{4pt}{0ex}}z\in \mathbb{C},\phantom{\rule{4pt}{0ex}}\overrightarrow{x}\in {E}^{m}.$$
- (ii)
- The mapping $z\mapsto {W}_{n}\left(z\right),$ $z\in \mathbb{C}$ is entire.

#### 3.1. The Abstract Impulsive Higher-Order Cauchy Problems

## 4. On Abstract Impulsive Fractional Differential Inclusions

**Definition**

**5.**

- (i)
- The function $u(\xb7)$ is $(m-1)$-times continuously differentiable on the intervals $[0,{t}_{1}),$ $({t}_{1},{t}_{2}),$ $({t}_{2},{t}_{3}),...,$ $({t}_{l},T];$
- (ii)
- The right derivatives ${lim}_{t\to {t}_{i}+}{u}^{\left(j\right)}\left(t\right)$ exist for $0\le j\le m-1$ and $1\le i\le l;$ the left derivatives ${lim}_{t\to {t}_{i}-}{u}^{\left(j\right)}\left(t\right)$ exist for $0\le j\le m-1$ and $1\le i\le l+1.$

**Definition**

**6.**

- (i)
- Suppose that $u\in {\mathrm{A}}_{\alpha}([0,T]:X).$ Then the Caputo fractional derivative ${\mathbf{D}}_{t}^{\alpha}u\left(t\right)$ is defined if and only if ${g}_{m-\alpha}\ast (u-{\sum}_{k=0}^{m-1}{u}_{k}{g}_{k+1})\in {W}^{m,1}((0,T):X)$, by (8).
- (ii)
- Suppose that $u\in {L}_{loc}^{1}([0,\infty ):X).$ Then the Caputo fractional derivative ${\mathbf{D}}_{t}^{\alpha}u\left(t\right)$ is defined for $t\ge 0$ if and only if for each $T>0$ we have ${u}_{\left|\right[0,T]}\in {\mathrm{A}}_{\alpha}([0,T]:X)$ and the Caputo fractional derivative ${\mathbf{D}}_{t}^{\alpha}u\left(t\right)$ is defined on the segment $[0,T].$

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Theorem**

**3.**

**Proof.**

## 5. Abstract Volterra Integro-Differential Inclusions with Impulsive Effects

**Definition**

**7.**

- (i)
- A function $u\in \mathcal{P}C\left(\right[0,T]:X)$ is said to be a pre-solution of (12) on $[0,T]$ if and only if $u(\xb7)$ is continuous on the set $[0,T]\backslash \{{t}_{1},...,{t}_{l}\},$ $(a\ast u)\left(t\right)\in D\left(\mathcal{A}\right)$ and $u\left(t\right)\in D\left(\mathcal{B}\right)$ for $t\in [0,T]\backslash \{{t}_{1},...,{t}_{l}\}$, as well as (12) holds.
- (ii)
- A solution of (12) on $[0,T]$ is any pre-solution $u(\xb7)$ of (12) on $[0,T]$ satisfying additionally that there exist functions ${u}_{\mathcal{B}}\in \mathcal{P}C([0,T]:E)$ and ${u}_{a,\mathcal{A}}\in \mathcal{P}C([0,T]:E)$, continuous on the set $[0,T]\backslash \{{t}_{1},...,{t}_{l}\},$ such that ${u}_{\mathcal{B}}\left(t\right)\in \mathcal{B}u\left(t\right)$ and ${u}_{a,\mathcal{A}}\left(t\right)\in \mathcal{A}{\int}_{0}^{t}a(t-s)u\left(s\right)ds$ for $t\in [0,T]\backslash \{{t}_{1},...,{t}_{l}\}$, as well as$${u}_{\mathcal{B}}\left(t\right)\in {u}_{a,\mathcal{A}}\left(t\right)+\mathcal{F}\left(t\right),\phantom{\rule{1.em}{0ex}}t\in [0,T]\backslash \{{t}_{1},...,{t}_{l}\}.$$
- (iii)
- A strong solution of (12) on $[0,T]$ is any function $u\in \mathcal{P}C\left(\right[0,T]:X),$ continuous on the set $[0,T]\backslash \{{t}_{1},...,{t}_{l}\},$ satisfying that there exist two functions ${u}_{\mathcal{B}}\in \mathcal{P}C([0,T]:E)$ and ${u}_{\mathcal{A}}\in \mathcal{P}C([0,T]:E),$ continuous on the set $[0,T]\backslash \{{t}_{1},...,{t}_{l}\},$ such that ${u}_{\mathcal{B}}\left(t\right)\in \mathcal{B}u\left(t\right)$, ${u}_{\mathcal{A}}\left(t\right)\in \mathcal{A}u\left(t\right)$ for all $t\in [0,T]\backslash \{{t}_{1},...,{t}_{l}\}$, and$${u}_{\mathcal{B}}\left(t\right)\in (a\ast {u}_{\mathcal{A}})\left(t\right)+\mathcal{F}\left(t\right),\phantom{\rule{1.em}{0ex}}t\in [0,T]\backslash \{{t}_{1},...,{t}_{l}\}.$$
- (iv)
- Suppose that $0\equiv {t}_{0}<{t}_{1}<...<{t}_{l}<{t}_{l+1}<...<+\infty $ and the sequence ${\left({t}_{l}\right)}_{l}$ has no accumulation point. By a (pre-)solution [solution, strong solution] of (13) we mean any function $u(\xb7)$ which satisfies that, for every $l\in \mathbb{N}$ and $T\in ({t}_{l},{t}_{l+1}),$ the function ${u}_{\left|\right[0,T]}(\xb7)$ is a (pre-)solution [solution, strong solution] of (12) on $[0,T].$

- (i)
- (ii)

**Theorem**

**4.**

- (i)
- Suppose $a\left(t\right)$ and $k\left(t\right)$ are kernels, $k\left(0\right)=1$, ${C}_{2}\in L\left(X\right)$ and $\mathcal{A}$ is a closed subgenerator of a mild $(a,k)$-regularized ${C}_{2}$-uniqueness family ${\left({R}_{2}\left(t\right)\right)}_{t\in [0,\tau )}\subseteq L\left(X\right)$, where $\tau >T.$ Define $\mathcal{F}\left(t\right):=0$ for $t\in [0,{t}_{1}]$ and $\mathcal{F}\left(t\right):={\sum}_{s=1}^{m}k(t-{t}_{s}){C}_{2}{y}_{s}$ if $t\in ({t}_{m},{t}_{m+1}]$ for some integer $m\in {\mathbb{N}}_{l}.$ Define also $u\left(t\right):=0$ for $t\in [0,{t}_{1}]$ and $u\left(t\right):={\sum}_{s=1}^{l}{R}_{2}(t-{t}_{s}){y}_{s}$ if $t\in ({t}_{m},{t}_{m+1}]$ for some integer $m\in {\mathbb{N}}_{l}.$ If ${y}_{1},...,{y}_{l}\in D\left(\mathcal{A}\right),$ then $u\left(t\right)$ is a unique strong solution of problem (12) on $[0,T],$ with the operator C replaced therein with the operator ${C}_{2}$.
- (ii)
- Suppose that $a\left(t\right)$ and $k\left(t\right)$ are kernels, $k\left(0\right)=1$, ${C}_{1}\in L(X,E)$ and $\mathcal{A}$ is a closed subgenerator of a mild $(a,k)$-regularized ${C}_{1}$-existence family ${\left({R}_{1}\left(t\right)\right)}_{t\in [0,\tau )}\subseteq L(X,E)$ such that ${R}_{1}\left(0\right)={C}_{1}$, where $\tau >T.$ Define $\mathcal{F}\left(t\right)$ and $u\left(t\right)$ in the same way as above, with the operator ${C}_{2}$ replaced therein with the operator ${C}_{1}$ and the elements ${y}_{1},...,{y}_{l}\in X.$ Then $u\left(t\right)$ is a solution of problem (12) on $[0,T]$, with the operator C replaced therein with the operator ${C}_{1}$.

**Proof.**

**Example**

**4.**

- (i)
- Suppose that Ω is a bounded domain in ${\mathbb{R}}^{n},$ $b>0,$ $m\left(x\right)\ge 0$ a.e. $x\in \Omega $, $m\in {L}^{\infty}(\Omega )$ and $1<p<\infty .$ Let B be the multiplication in ${L}^{p}(\Omega )$ with $m\left(x\right),$ and let $A=\Delta -b$ act with the Dirichlet boundary conditions. Then our analysis from ([28], Example 3.2.23) shows that that there exists an operator ${C}_{1}\in L\left({L}^{p}(\Omega )\right)$ such that the MLO $\mathcal{A}=-A{B}^{-1}$ is a subgenerator of an entire $({g}_{1},{g}_{1})$-regularized ${C}_{1}$-existence family. Consider now the following degenerate Volterra integral equation associated to the abstract backward Poisson heat equation in the space $X={L}^{p}(\Omega )$:$$\left(PR\right):\left\{\begin{array}{c}m\left(x\right)v(t,x)={u}_{0}\left(x\right)+{\int}_{0}^{t}(-\Delta +b)v(s,x)\phantom{\rule{0.166667em}{0ex}}ds,\hfill \\ t\in [0,\infty )\backslash \{{t}_{1},...,{t}_{l},...\},\phantom{\rule{4pt}{0ex}}x\in \Omega ;\hfill \\ v(t,x)=0,\phantom{\rule{1.em}{0ex}}(t,x)\in [0,\infty )\times \partial \Omega ,\hfill \\ m\left(x\right)v\left({t}_{l}+,x\right)-m\left(x\right)v\left({t}_{l}-,x\right)={C}_{1}\left[{f}_{1}\left(x\right)+....+{f}_{l}\left(x\right)\right],\hfill \\ x\in \Omega ,\phantom{\rule{4pt}{0ex}}t\in ({t}_{l},{t}_{l+1}]\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}(l\in \mathbb{N}),\hfill \end{array}\right.$$
- (ii)
- Our analysis from ([28], Example 3.10.7) and the second equality in ([1], (1.21)) shows that we can similarly analyze the following degenerate Volterra integral equation closely connected with the inverse generator problem and the abstract backward Poisson heat equation in the space $X={L}^{p}(\Omega )$:$$\left\{\begin{array}{c}(\Delta -b)v(t,x)=(\Delta -b)v(0,x)+t{\left(\frac{d}{ds}\left[(\Delta -b)v(s,x)\right]\right)}_{s=0}\hfill \\ +{\int}_{0}^{t}{g}_{\alpha}(t-s)m\left(s\right)v(s,x)\phantom{\rule{0.166667em}{0ex}}ds,\hfill \\ t\in [0,\infty )\backslash \{{t}_{1},...,{t}_{l},...\},\phantom{\rule{4pt}{0ex}}x\in \Omega ;\hfill \\ v(t,x)=0,\phantom{\rule{1.em}{0ex}}(t,x)\in [0,\infty )\times \partial \Omega ,\hfill \\ (\Delta -b)v\left({t}_{l}+,x\right)-(\Delta -b)v\left({t}_{l}-,x\right)={C}_{1}\left[{f}_{1}\left(x\right)+....+{f}_{l}\left(x\right)\right],\hfill \\ x\in \Omega ,\phantom{\rule{4pt}{0ex}}t\in ({t}_{l},{t}_{l+1}]\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}(l\in \mathbb{N}),\hfill \end{array}\right.$$

**Definition**

**8.**

- (i)
- The mapping $t\mapsto R\left(t\right)x$, $t\ge 0$ is continuous for every fixed element $x\in D\left(B\right)$.
- (ii)
- There exist $M\ge 1$ and $\omega \ge 0$ such that $\parallel R\left(t\right)\parallel \le M{e}^{\omega t},$ $t\ge 0.$
- (iii)
- For every $\lambda \in \mathbb{C}$ with $\Re \lambda >\omega $ and $\tilde{k}\left(\lambda \right)\ne 0$, the operator $B-\tilde{a}\left(\lambda \right)A$ is injective, $C\left(R\left(B\right)\right)\subseteq R(B-\tilde{a}\left(\lambda \right)A)$ and$$\tilde{k}\left(\lambda \right){(B-\tilde{a}\left(\lambda \right)A)}^{-1}CBx={\int}_{0}^{\infty}{e}^{-\lambda t}R\left(t\right)x\phantom{\rule{0.166667em}{0ex}}dt,\phantom{\rule{1.em}{0ex}}x\in D\left(B\right).$$

**Lemma**

**2.**

- (i)
- Suppose that ${v}_{0}\in D\left(B\right)$ and the following condition holds:
- (i.1)
- for every $x\in D\left(B\right)$, there exista function $h(\lambda ;x)\in LT-E$ and a number ${\omega}_{0}>\omega $ such that $h(\lambda ;x)=\tilde{k}\left(\lambda \right)B{(B-\tilde{a}\left(\lambda \right)A)}^{-1}CBx$ provided $\tilde{k}\left(\lambda \right)\ne 0$ and $\Re \lambda >{\omega}_{0}$.

Then the function $u\left(t\right)=R\left(t\right){v}_{0}$, $t\ge 0$ is a mild solution of (14) with $f\left(t\right)=k\left(t\right)CB{v}_{0}$, $t\ge 0$ and without impulsive effects. The uniqueness of mild solutions holds if we suppose additionally that $CB\subseteq BC$ and the function $k\left(t\right)$ satisfies (P2). - (ii)
- Suppose that $CB\subseteq BC$, ${v}_{0}\in D\left(A\right)\cap D\left(B\right)$ and the following condition holds:
- (ii.1)
- for every $x\in E$, there exist a function $h(\lambda ;x)\in LT-E$ and a number ${\omega}_{1}>\omega $ such that $h(\lambda ;x)=\tilde{k}\left(\lambda \right)B{(B-\tilde{a}\left(\lambda \right)A)}^{-1}Cx$ provided $\tilde{k}\left(\lambda \right)\ne 0$ and $\Re \lambda >{\omega}_{1}$.

Then the function $u\left(t\right)=R\left(t\right){v}_{0}$, $t\ge 0$ is a strong solution of (14) with $f\left(t\right)=k\left(t\right)CB{v}_{0}$, $t\ge 0$ and without impulsive effects. The uniqueness of strong solutions holds if we suppose additionally that the function $k\left(t\right)$ satisfies (P2).

**Theorem**

**5.**

- (i)
- Suppose that the requirements of Lemma 2(i) hold and $k\left(0\right)=1.$ Define $\mathcal{F}\left(t\right):=0$ for $t\in [0,{t}_{1}]$ and $\mathcal{F}\left(t\right):={\sum}_{s=1}^{m}k(t-{t}_{s})CB{y}_{s}$ if $t\in ({t}_{m},{t}_{m+1}]$ for some integer $m\in {\mathbb{N}}_{l}.$ Define also $u\left(t\right):=0$ for $t\in [0,{t}_{1}]$ and $u\left(t\right):={\sum}_{s=1}^{l}R(t-{t}_{s}){y}_{s}$ if $t\in ({t}_{m},{t}_{m+1}]$ for some integer $m\in {\mathbb{N}}_{l}.$ If ${y}_{1},...,{y}_{l}\in D\left(B\right),$ then $u\left(t\right)$ is a mild solution of problem (14) on $[0,T].$ The uniqueness of mild solutions of problem (14) holds if we suppose additionally that $CB\subseteq BC$ and the function $k\left(t\right)$ satisfies (P2).
- (ii)
- Suppose that the requirements of Lemma 2(ii) hold and $k\left(0\right)=1.$ Define $\mathcal{F}\left(t\right)$ and $u\left(t\right)$ as above. If ${y}_{1},...,{y}_{l}\in D\left(A\right)\cap D\left(B\right),$ then $u\left(t\right)$ is a strong solution of problem (14) on $[0,T].$ The uniqueness of strong solutions of problem (14) holds if we suppose additionally that the function $k\left(t\right)$ satisfies (P2).

## 6. Conclusions and Final Remarks

- (1).
- It seems very plausible that we can similarly analyze the well-posedness of the abstract incomplete Cauchy inclusions with impulsive effects; for more details about the subject, we refer the reader to ([28], Section 2.7, Section 3.9).
- (2).
- (3).
- Let us finally mention that we have not considered here the abstract impulsive Volterra integro-differential inclusions on the line as well as the existence and uniqueness of discontinuous almost periodic (automorphic) type solutions for certain classes of the abstract impulsive Cauchy problems on the line; see e.g., ([29], pp. 51–53) for more details about this subject in non-degenerate case. Our work almost completely belongs to the realm of pure mathematics, and numerical simulations and illustrations will appear somewhere else.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bazhlekova, E. Fractional Evolution Equations in Banach Spaces. Ph.D. Thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 2001. [Google Scholar]
- Diethelm, K. The Analysis of Fractional Differential Equations; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; North-Holland Mathematics Studies; Elsevier Science B. V.: Amsterdam, The Netherlands, 2006; Volume 204. [Google Scholar]
- Kostić, M. Abstract Volterra Integro-Differential Equations; CRC Press: Boca Raton, FL, USA, 2015. [Google Scholar]
- Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Bainov, D.; Simeonov, P. Impulsive Differential Equations: Periodic Solutions and Applications; Wiley: New York, NY, USA, 1993. [Google Scholar]
- Bainov, D.; Simeonov, P. Oscillation Theory of Impulsive Differential Equations; International Publications: Geneva, Switzerland, 1998. [Google Scholar]
- Faree, T.A.; Panchal, S.K. Existence of solution for impulsive fractional differential equations via topological degree method. J. Korean Soc. Ind. Appl. Math.
**2021**, 25, 16–25. [Google Scholar] - Halanay, A.; Wexler, D. Qualitative Theory of Impulse Systems; Mir: Moscow, Russia, 1971. [Google Scholar]
- Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S. Theory of Impulsive Differential Equations; World Scientific Publishing Co. Pte. Ltd.: Singapore, 1989. [Google Scholar]
- Samoilenko, A.M.; Perestyuk, N.A. Impulsive Differential Equations; World Scientific: Singapore, 1995. [Google Scholar]
- Stamov, G.T. Almost Periodic Solutions of Impulsive Differential Equations; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Wang, J.R.; Xiang, X.; Wei, W.; Chen, Q. Bounded and periodic solutions of semilinear impulsive periodic system on Banach spaces. Fixed Point Theory Appl.
**2008**, 2008, 401947. [Google Scholar] [CrossRef] [Green Version] - Wang, J.; Fečkan, M. A general class of impulsive evolution equations. Topol. Meth. Nonlinear Anal.
**2015**, 46, 915–934. [Google Scholar] [CrossRef] - Zhang, X.; Liu, Z.; Yang, S.; Peng, Z.; He, Y.; Wei, L. The right equivalent integral equation of impulsive Caputo fractional-order system of order ϵ∈(1,2). Fractal Fract.
**2023**, 7, 37. [Google Scholar] [CrossRef] - Zhang, X.; Zhang, X.; Zhang, M. On the concept of general solution for impulsive differential equations of fractional order q∈(0,1). Appl. Math. Comp.
**2014**, 247, 72–89. [Google Scholar] [CrossRef] - Zhou, Y.; Jiao, F. Existence of mild solutions for fractional neutral evolution equations. Comp. Math. Appl.
**2010**, 59, 1063–1077. [Google Scholar] [CrossRef] [Green Version] - Haloi, R. Some existence results on impulsive differential equations. In Proceedings of the Sixth International Conference on Mathematics and Computing; Giri, D., Buyya, R., Ponnusamy, S., De, D., Adamatzky, A., Abawajy, J.H., Eds.; Advances in Intelligent Systems and Computing; Springer: Singapore, 2021; Volume 1262. [Google Scholar]
- Ranjinia, M.C.; Angurajb, A. Nonlocal impulsive fractional semilinear differential equations with almost sectorial operators. Malaya J. Math.
**2013**, 2, 43–53. [Google Scholar] - Benedetti, I.; Obukhovskii, V.; Taddei, V. On solvability of the impulsive Cauchy problem for integro-differential inclusions with non-densely defined operators. Phil. Trans. R. Soc. A
**2021**, 379, 20190384. [Google Scholar] [CrossRef] [PubMed] - Ke, T.D.; Kinh, C.T. Generalized Cauchy problem involving a class of degenerate fractional differential equations. Dyn. Contin. Discret. Impuls. Syst. Ser. A Math. Anal.
**2014**, 21, 449–472. [Google Scholar] - Xiao, T.-J.; Liang, J. Abstract degenerate Cauchy problems in locally convex spaces. J. Math. Anal. Appl.
**2001**, 259, 398–412. [Google Scholar] - Favini, A.; Yagi, A. Degenerate Differential Equations in Banach Spaces; Chapman and Hall/CRC Pure and Applied Mathematics: New York, NY, USA, 1998. [Google Scholar]
- Fečkan, M.; Zhou, Y.; Wang, J.R. On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simulat.
**2012**, 17, 3050–3060. [Google Scholar] [CrossRef] - Fečkan, M.; Zhou, Y.; Wang, J.R. Response to "Comments on the concept of existence of solution for impulsive fractional differential equations [Commun Nonlinear Sci Numer Simul 2014;19:401??.]". Commun. Nonlinear Sci. Numer. Simulat.
**2014**, 19, 4213–4215. [Google Scholar] [CrossRef] - Wang, J.R.; Fečkan, M.; Zhou, Y. On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equations
**2011**, 8, 345–361. [Google Scholar] - Wang, J.R.; Li, X.; Wei, W. On the natural solution of an impulsive fractional differential equation of order q∈(1,2). Commun. Nonlinear Sci. Numer. Simulat.
**2012**, 17, 4384–4394. [Google Scholar] [CrossRef] - Kostić, M. Abstract Degenerate Volterra Integro-Differential Equations; Mathematical Institute SANU: Belgrade, Serbia, 2020. [Google Scholar]
- Kostić, M. Generalized Semigroups and Cosine Functions; Mathematical Institute SANU: Belgrade, Serbia, 2011. [Google Scholar]
- Kostić, M. Almost Periodic and Almost Automorphic Type Solutions of Abstract Volterra Integro-Differential Equations; W. de Gruyter: Berlin, Germany, 2019. [Google Scholar]
- Xiao, T.-J.; Liang, J. The Cauchy Problem for Higher Order Abstract Differential Equations; Springer: Berlin, Germany, 1998. [Google Scholar]
- Prüss, J. Evolutionary Integral Equations and Applications; Birkhäuser-Verlag: Basel, Switzerland, 1993. [Google Scholar]
- Plekhanova, M.V.; Fedorov, V.E. Optimal Control of Degenerate Evolution Systems; Publication Center IIGU Univeristy: Chelyabinsk, Russia, 2013. (In Russian) [Google Scholar]
- Du, W.-S.; Kostić, M.; Velinov, D. Almost Periodic Type Solutions of the Abstract Impulsive Volterra Integro-Differential Inclusions. Preprint. Available online: https://www.researchgate.net/publication/366547982 (accessed on 1 December 2022).

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Du, W.-S.; Kostić, M.; Velinov, D.
Abstract Impulsive Volterra Integro-Differential Inclusions. *Fractal Fract.* **2023**, *7*, 73.
https://doi.org/10.3390/fractalfract7010073

**AMA Style**

Du W-S, Kostić M, Velinov D.
Abstract Impulsive Volterra Integro-Differential Inclusions. *Fractal and Fractional*. 2023; 7(1):73.
https://doi.org/10.3390/fractalfract7010073

**Chicago/Turabian Style**

Du, Wei-Shih, Marko Kostić, and Daniel Velinov.
2023. "Abstract Impulsive Volterra Integro-Differential Inclusions" *Fractal and Fractional* 7, no. 1: 73.
https://doi.org/10.3390/fractalfract7010073