On Impulsive Delay Integrodifferential Equations
with Integral Impulses

Kishor D. Kucche 1

<EMAIL_ADDRESS>
Pallavi U. Shikhare 2

<EMAIL_ADDRESS>
1,2 Department of Mathematics, Shivaji University, Kolhapur-416 004,
Maharashtra, India.

###### Abstract

The present research paper is devoted to investigate the existence, uniqueness
of mild solutions for impulsive delay integrodifferential equations with
integral impulses in Banach spaces. We also investigate the dependence of
solutions on initial conditions, parameters and the functions involved in the
equations. Our analysis is based on semigroup theory, fixed point technique
and an application of Pachpatte’s type integral inequality with integral
impulses. An example is provided in support of existence result.

Key words: Impulsive integrodifferential equation; Impulsive integral
inequality; Integral jump condition; Fixed point, Dependence of solutions.
2010 Mathematics Subject Classification: 37L05, 47D60, 34A37, 34G20, 35B30,
35A23.

##  1 Introduction

Numerous real world physical phenomena can’t be modeled as differential
equations with classical initial conditions, specially the issues in the part
of biology and medical sciences, for instances, the dynamics of populations
subject to sudden changes (e.g. diseases, harvesting and so forth.) and few
problems in the field of physics, chemical technology, population dynamics,
medicine, mechanics, biotechnology and economics. But, such physical phenomena
can be modeled by means of differential equations with impulsive conditions.
It very well may be noticed that the impulsive conditions are the blends of
classical initial conditions and the short-term perturbations whose span can
be insignificant in correlation with the span of the procedure. Therefore the
differential and integrodifferential equations with impulse effect have
importance in the modeling many physical phenomena in which sudden changes
occurs. The details relating to the theory and applications of impulsive
differential equations (IDEs) and its impact in further development can be
found in the monograph by Bainov and Simeonov [1] , Lakshmikantham et al. [2],
and Samoilenko and Perestyuk [3] and in the interesting papers [4, 5, 6, 7, 8,
9, 10, 11, 12, 13].

In the begin, Frigon and O’regan [5], using the fixed point approach
determined variety of existence principles for the IDE

$\begin{cases}w^{\prime}(t)=f\left(t,w(t)\right),\,0<t<b,\,~{}t\not=t_{k},\\\
w(t_{k}^{+})=I_{k}\left(w(t_{k}^{-})\right),\,~{}k=1,2,\cdots,m,\end{cases}$
(1.1)

with the initial condition $w(0)=w_{0}$, where
$x(t^{+}_{k})=\underset{\epsilon\to 0^{+}}{\lim}x(t_{k}+\epsilon)$ and
$x(t^{-}_{k})=\underset{\epsilon\to 0^{-}}{\lim}x(t_{k}+\epsilon)$. In
addition, utilizing the idea of upper and lower solutions, authors have
derived existence results for the IDE (1.1) with boundary condition
$w(0)=w(b)$. First, Wang et al. [4] presented the idea of Ulam’s type
stabilities for ordinary IDE and derived these stability results by means of
integral inequality of Gronwall type for piecewise continuous functions. Liu
[14] have extended the theory to nonlinear impulsive evolution equation

$\begin{cases}w^{\prime}(t)=\mathscr{A}w(t)+f\left(t,w(t)\right),\,0<t<b,\,~{}t\not=t_{k},\\\
\Delta w(t_{k})=I_{k}\left(w(t_{k})\right),\,~{},\,~{}k=1,2,\cdots,m,\\\
w(0)=w_{0},\end{cases}$ (1.2)

in a Banach space, where $\mathscr{A}:D(A)\subseteq X\to X$ is the
infinitesimal generator of a strongly continuous semigroup of bounded linear
operators and applying the semigroup theory explored the existence and
uniqueness of the mild solutions by utilizing fixed point argument. Further,
the author have demonstrated that if f is continuously differentiable then the
mild solutions give rise to classical solution. Hernández, D. O’regan [9]
analyzed a class of abstract impulsive differential equations of the form
(1.2) considering extraordinary sorts of condition on the function $I_{k}$, so
as it would allow to think about the impulsive partial differential equations.

Utilizing the Lerey-Schauder nonlinear alternative, Arjunan and Anguraj [15]
derived existence of mild solutions for the impulsive delay differential
equations.

$\begin{cases}w^{\prime}(t)=\mathscr{A}w(t)+f\left(t,w_{t}\right),\,t\in
I=[0,b],~{}t\not=t_{k},~{}k=1,2,\cdots,m,\\\
w(t)=\phi(t),\,t\in[-r,0],\,0<r<\infty,\\\ \Delta
w(t_{k})=I_{k}\left(w(t_{k})\right),\,~{}k=1,2,\cdots,m\end{cases}$ (1.3)

Benchora et al. [16] have extended the above study to first and second order
impulsive semilinear neutral functional differential equations.

Thiramanus et al. [17] built up the impulsive differential and impulsive
integral inequalities with integral jump conditions and gave its applications
to examine the boundedness and other properties of IDEs with integral jump
conditions. Thaiprayoon et al. [18, 19] exhibited the maximum principles for
second-order impulsive integrodifferential equations with integral jump
conditions and also developed the monotone iterative technique for a periodic
boundary value problem. More subtleties relating to integral jump conditions
can be found in [17, 18, 19].

It is seen that the theory of impulsive differential equations have been
significant progress. However, very few work have been accounted relating to
the theory of impulsive delay differential equations with integral impulses.
Motivated by the previously mentioned work and the applicability of impulsive
differential equation in the scientific community it is imperative to
examining the existence, uniqueness, stability and data dependence of solution
for broad class of impulsive integrodifferential equations.

Motivated by the work of [17, 18, 19], in the present paper, we consider the
impulsive integrodifferential equation with integral impulses and finite delay
described in the form:

$\displaystyle
w^{\prime}(t)=\mathscr{A}w(t)+V\left(t,w_{t},\int_{0}^{t}U(t,s,w_{s})ds\right),~{}t\in
I=[0,b],~{}t\not=t_{k},~{}k=1,2,\cdots,m,$ (1.4) $\displaystyle
w(t)=\varsigma(t),~{}t\in[-r,0],$ (1.5) $\displaystyle\Delta
w(t_{k})=I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,w_{s})ds\right),~{}k=1,2,\cdots,m.$
(1.6)

where $\mathscr{A}:D(A)\subseteq X\to X$ is the infinitesimal generator of a
strongly continuous semigroup of bounded linear operators
$\\{\mathscr{T}(t)\\}_{t\geq 0}$ in the Banach
$\left(X,\left\|\cdot\right\|\right)$; $V:I\times C([-r,0],X)\times X\to
X,~{}U:I\times I\times C([-r,0],X)\to X$ and $G:J\times X\to X$ are given
functions, $\varsigma\in
C([-r,0],X),(0<r<\infty),0=t_{0}<t_{1}<\cdots<t_{m}<t_{m+1}=b,\,I_{k}\in
C(X,X)$ are bounded functions, $0\leq\theta_{k}\leq\tau_{k}\leq t_{k}-t_{k-1}$
for $k=1,2,\cdots,m$, $\Delta w(t_{k})=w(t^{+})-w(t^{-}_{k})$,
$w(t^{+}_{k})=\underset{\epsilon\to 0^{+}}{\lim}w(t_{k}+\epsilon)$ and
$w(t^{-}_{k})=\underset{\epsilon\to 0^{-}}{\lim}w(t_{k}+\epsilon)$. For any
continuous function $w$ defined on
$[-r,b]-\left\\{t_{1},\cdots,t_{m}\right\\}$ and $t\in[0,b]$, denote by
$w_{t}$ the element of $C([-r,0],X)$ defined by
$w_{t}(\theta)=w(t+\theta),~{}\theta\in[-r,0]$.

Our fundamental point is to research the existence, uniqueness of mild
solutions for the problem (1.4)–(1.6) and examine the dependence of mild
solution on initial conditions, parameters and on the functions involved in
the right hand side of the equations (1.4)–(1.6). The innovative idea in the
present paper is the utilization of the Pachpatte’s type integral inequality
with integral impulses. The innovative thought to be noted here that all the
qualitative properties mentioned above of the mild solutions for the problem
(1.4)–(1.6) have been acquired by means of Pachpatte’s type integral
inequality with integral impulses for broad class of impulsive Volterra
integrodifferential equations just with Lipschitz conditions on the functions
$V$and $U$.

For more uses of the Pachpatte’s integral inequalities in the investigation of
qualitative properties of solutions and different kind of Ulam’s types
stabilities of general form of integrodifferential equations we refer to the
work of Pachpatte [20, 21] and Kucche et al. [22, 23].

The paper is organized as follows. In section $2$, we introduce notations and
state theorems which are useful in further analysis. Section $3$ deals with
existence and uniqueness results pertaining to the problem (1.4)–(1.6). In
section $4$ we establish the dependence of solution on different data. In
section 5, an example is given to illustrate the existence and uniqueness
results.

##  2 Preliminaries

In this section, we introduce notations and few theorems that are used
throughout this paper. Let $(X,\left\|\cdot\right\|)$ be a Banach space. Then

$\mathscr{C}=C([-r,0],X)=\left\\{\varsigma~{}:~{}\varsigma:[-r,0]\to
X~{}\text{is continuous}\right\\}$

is a Banach space with respect to the norm

$\left\|\varsigma\right\|=\sup\left\\{\left\|\varsigma(\theta)\right\|:-r<\theta<0\right\\}.$

Further, $C(I,X)=\left\\{w~{}:~{}w:I\to X~{}\text{is continuous}\right\\}$ is
a Banach space with respect to the norm

$\left\|w\right\|=\sup\left\\{\left\|w(t)\right\|:0<t<b\right\\}.$

######  Theorem 2.1 ([24])

Let $\\{\mathscr{T}(t)\\}_{t\geq 0}$ is a $C_{0}$-semigroup. There exists
constants $\omega\geq 0$ and $M\geq 1$ such that

$\|\mathscr{T}(t)\|\leq Me^{\omega t},~{}0\leq t<\infty.$

We utilize the following fixed point theorem due to Burton and Kirk [25].

######  Theorem 2.2 ([25])

Let $\mathcal{X}$ be a Banach space and
$\mathcal{A,\,B}:\mathcal{X}\to\mathcal{X}$ two operators satisfying:

  * (i)

$\mathcal{A}$ is a contraction, and

  * (ii)

$\mathcal{B}$ is completely continuous.

Then either

  * (a)

the operator equation $w=\mathcal{A}(w)+\mathcal{B}(w)$ has a solution, or

  * (b)

the set

$\mathcal{Z}=\left\\{w\in\mathcal{X}:w=\lambda\mathcal{A}\left(\frac{w}{\lambda}\right)+\lambda\mathcal{B}(w)\right\\}$

is unbounded for $\lambda\in(0,1)$.

The qualitative analysis of the solution of equations (1.4)–(1.6) has been
done by utilizing the following Pachpatte’s type integral inequality.

######  Theorem 2.3 ([26])

Let the sequence $\left\\{t_{k}\right\\}^{\infty}_{k=1}$ satisfies $0\leq
t_{0}<t_{1}<\cdots~{}\text{and}~{}\underset{t\rightarrow\infty}{\lim}~{}t_{k}=\infty.$
Let $u\in PC^{1}\left(\mathbb{R}_{+},\mathbb{R}_{+}\right)$ and $u(t)$ is
left-continuous at $t_{k},\,k=1,2,\cdots$. Assume that $f,~{}g$ are
nonnegative continuous functions defined on $\mathbb{R}_{+}$ and $n(t)$ is a
positive and nondecreasing continuous function defined on $\mathbb{R}_{+}$. In
addition, assume that constants $\beta_{k}\geq
0,~{}0\leq\theta_{k}\leq\tau_{k}\leq t_{k}-t_{k-1}$ for $k=1,2,\cdots$.
If

$\displaystyle u(t)$ $\displaystyle\leq
n(t)+\int_{0}^{t}f(s)u(s)ds+\int_{0}^{t}f(s)\left(\int_{0}^{s}g(\sigma)u(\sigma)d\sigma\right)ds+\underset{0<t_{k}<t}{\sum}~{}\beta_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}u(s)ds,\quad
t\geq 0,$

then

$\displaystyle u(t)$ $\displaystyle\leq
n(t)\underset{0<t_{k}<t}{\prod}~{}C_{k}\exp\left(\int_{t_{\alpha}}^{t}f(s)\left[1+\int_{0}^{s}g(\sigma)d\sigma\right]ds\right),\quad
t\geq 0,$

where

$\displaystyle\alpha$ $\displaystyle=\max\left\\{k~{}:~{}t\geq
t_{k},~{}k=1,2,\cdots\right\\}$

and

$\displaystyle C_{k}$
$\displaystyle=\exp\left(\int_{t_{k-1}}^{t_{k}}f(s)\left[1+\int_{0}^{s}g(\sigma)d\sigma\right]ds\right)+\beta_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\exp\left(\int_{t_{k-1}}^{s}f(\sigma)\left[1+\int_{0}^{\sigma}g(\xi)d\xi\right]d\sigma\right)ds.$

##  3 Existence and Uniqueness Results

To define the concept of mild solution of (1.4)–(1.6), we consider the
following space

$\displaystyle\varSigma=\left\\{w:[-r,b]\to X:w_{k}\in
C(I_{k},X),~{}k=0,\cdots,m~{}\text{and there
exist}~{}w(t^{-}_{k})~{}\text{and}\right.$
$\displaystyle\left.w(t^{+}_{k}),~{}\text{with}~{}w(t^{-}_{k})=w(t_{k}),~{}k=1,\cdots,m,~{}w(t)=\varsigma(t),~{}\forall~{}t\in[-r,0]\right\\}.$

Define $\left\|\cdot\right\|_{\varSigma}:\varSigma\to[0,\infty)$ by

$\left\|w\right\|_{\varSigma}=\max\left\\{\left\|w_{k}\right\|_{I_{k}}:~{}k=0\cdots
m\right\\},$

where $w_{k}$ is the restriction of $w$ to
$I_{k}=[t_{k},~{}t_{k+1}],~{}k=0,\cdots,m$. Then
$(\varSigma,\left\|\cdot\right\|_{\varSigma})$ is a Banach space.

Let $w\in\varSigma$ is a solution of problem the (1.4)–(1.6). Then comparing
with the abstract Cauchy problem [24], we have

$w(t)=\mathscr{T}(t)\varsigma(0)+\int_{0}^{t}\mathscr{T}(t-s)V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)ds,~{}t\in[0,t_{1}).$

This gives

$\displaystyle
w(t^{-}_{1})=\mathscr{T}(t_{1})\varsigma(0)+\int_{0}^{t_{1}}\mathscr{T}(t_{1}-s)V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)ds.$
(3.1)

Now, by condition (1.6), we have

$w(t^{+}_{1})=w(t^{-}_{1})+I_{1}\left(\int_{t_{1}-\tau_{1}}^{t_{1}-\theta_{1}}G(s,w_{s})ds\right).$

Thus again by comparing with abstract Cauchy problem [24], and using semigroup
property for $t\in(t_{1},t_{2})$ we have

$\displaystyle w(t)$
$\displaystyle=\mathscr{T}(t-t_{1})w(t^{+}_{1})+\int_{t_{1}}^{t}\mathscr{T}(t-s)V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)ds$
$\displaystyle=\mathscr{T}(t-t_{1})\left[w(t^{-}_{1})+I_{1}\left(\int_{t_{1}-\tau_{1}}^{t_{1}-\theta_{1}}G(s,w_{s})ds\right)\right]$
$\displaystyle\qquad+\int_{t_{1}}^{t}\mathscr{T}(t-s)V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)ds$
$\displaystyle=\mathscr{T}(t-t_{1})\mathscr{T}(t_{1})\varsigma(0)+\int_{0}^{t_{1}}\mathscr{T}(t-t_{1})\mathscr{T}(t_{1}-s)V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)ds$
$\displaystyle\qquad+\mathscr{T}(t-t_{1})I_{1}\left(\int_{t_{1}-\tau_{1}}^{t_{1}-\theta_{1}}G(s,w_{s})ds\right)+\int_{t_{1}}^{t}\mathscr{T}(t-s)V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)ds$
$\displaystyle=\mathscr{T}(t)\varsigma(0)+\int_{0}^{t}\mathscr{T}(t-s)V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)ds+\mathscr{T}(t-t_{1})I_{1}\left(\int_{t_{1}-\tau_{1}}^{t_{1}-\theta_{1}}G(s,w_{s})ds\right).$

Reiterating these procedures, one can prove that

$\displaystyle w(t)$
$\displaystyle=\mathscr{T}(t)\varsigma(0)+\int_{0}^{t}\mathscr{T}(t-s)V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)ds$
$\displaystyle\qquad+\underset{0<t_{k}<t}{\sum}\mathscr{T}(t-t_{k})I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,w_{s})ds\right),~{}t\in
I.$

###### Definition 3.1

A function $w:[-r,b]\to X$ is called a mild solution of (1.4)–(1.6) if
$~{}w(t)=\varsigma(t),\text{on}\,[-r,0],$ and

$\displaystyle w(t)$
$\displaystyle=\mathscr{T}(t)\varsigma(0)+\int_{0}^{t}\mathscr{T}(t-s)V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)ds$
$\displaystyle\qquad+\underset{0<t_{k}<t}{\sum}\mathscr{T}(t-t_{k})I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,w_{s})ds\right),~{}t\in
I.$

To obtain existence and uniqueness results pertaining to the problem
(1.4)–(1.6), we need the following conditions:

  * (H1)

There exists functions $N_{V},N_{U}\in C(I,\mathbb{R}_{+})$ and the constant
$L_{G}>0$ such that

    * (i)

$\|V(t,\vartheta_{1},w_{1})-V(t,\vartheta_{2},w_{2})\|\leq
N_{V}(t)\left(\|\vartheta_{1}-\vartheta_{2}\|_{\mathscr{C}}+\|w_{1}-w_{2}\|\right);$

    * (ii)

$\|U(t,s,\vartheta_{1})-U(t,s,\vartheta_{2})\|\leq
N_{U}(t)\,\|\vartheta_{1}-\vartheta_{2}\|_{\mathscr{C}};$

    * (iii)

$\|G(t,\vartheta_{1})-G(t,\vartheta_{2})\|\leq
L_{G}\,\|\vartheta_{1}-\vartheta_{2}\|_{\mathscr{C}};$

for all $t,s\in I,~{}\vartheta_{1},\vartheta_{2}\in\mathscr{C}$ and
$w_{1},w_{2}\in X.$

  * (H2)

There exist constants $\mathscr{D}_{k}(k=1,\cdots,m)$ such that
$\left\|I_{k}(w_{1})-I_{k}(w_{2})\right\|\leq\mathscr{D}_{k}\left\|w_{1}-w_{2}\right\|;~{}k=1,\cdots,m$
for $w_{1},w_{2}\in X$.

###### Theorem 3.1

(Existence) Assume that the hypotheses (H1)–(H2) are satisfied and let

$\displaystyle\sum_{k=1}^{m}2b\,\mathscr{M}L_{G}\,\mathscr{D}_{k}<1.$ (3.2)

Then the problem (1.4)–(1.6) has at least one solution in $\varSigma$.

Proof: Transform the problem (1.4)–(1.6) into a fixed point problem consider
the operators $\mathcal{A},\mathcal{B}:\varSigma\to\varSigma$ defined by

$\mathcal{A}(w)(t)=\begin{cases}0&\text{if $t\in[-r,0],$}\\\
\underset{0<t_{k}<t}{\sum}\mathscr{T}(t-t_{k})I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,w_{s})ds\right)&\text{if
$t\in[0,b]$}.\end{cases}$

and

$\mathcal{B}(w)(t)=\begin{cases}\varsigma(t)&\text{if $t\in[-r,0],$}\\\
\mathscr{T}(t)\varsigma(0)+\int_{0}^{t}\mathscr{T}(t-s)V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)ds&\text{if
$t\in[0,b]$}.\end{cases}$

In the view of the Theorem 2.2 to prove the operator equation
$w=\mathcal{A}(w)+\mathcal{B}(w)$ have fixed point, we prove the following
statements.

  * (i)

$\mathcal{B}:\varSigma\to\varSigma$ is completely continuous.

  * (ii)

$\mathcal{A}:\varSigma\to\varSigma$ is contraction.

  * (iii)

the set

$\mathcal{Z}=\left\\{w\in\varSigma:w=\lambda\mathcal{A}\left(\frac{w}{\lambda}\right)+\lambda\mathcal{B}(w)\right\\}$

is bounded for some $\lambda\in(0,1)$.

The proof of the operator $\mathcal{B}$ is completely continuous is given in
the following steps.
Step 1: $\mathcal{B}:\varSigma\to\varSigma$ is continuous .

Let any $w\in\varSigma$ and $\left\\{w_{n}\right\\}^{\infty}_{n=1}$ be any
sequence in $\varSigma$ such that $w_{n}\to w$. Then
$\left\|w_{n}\right\|_{\varSigma}\leq
q^{{}^{\prime}},\,\forall\,n\in\mathbb{N}$ and $q^{{}^{\prime}}>0$ and hence
$\left\|w\right\|_{\varSigma}\leq q^{{}^{\prime}}$. Thus we have
$\left\\{w_{n}\right\\}\subseteq B_{q^{{}^{\prime}}}$ and
$w\,\epsilon\,B_{q^{{}^{\prime}}}$. Now by Lebesgue dominated convergence
theorem,

$\displaystyle\left\|\mathcal{B}(w_{n})-\mathcal{B}(w)\right\|_{\varSigma}$
$\displaystyle\leq\underset{[-r,b]}{\sup}\left\\{\int_{0}^{t}\left\|\mathscr{T}(t-s)\right\|_{B(X)}\left\|V\left(s,w_{n_{s}},\int_{0}^{s}U\left(s,\sigma,w_{n_{\sigma}}\right)d\sigma\right)\right.\right.$
$\displaystyle\qquad\left.\left.-V\left(s,w_{s},\int_{0}^{s}U\left(s,\sigma,w_{\sigma}\right)d\sigma\right)\right\|ds\right\\}\to
0\,\text{as}\,n\to\infty.$

Thus $\mathcal{B}$ is continuous.

Step 2: $\mathcal{B}$ maps bounded sets into bounded sets in $\varSigma$.

Let $w\in B_{q}=\left\\{w\in\varSigma:\left\|w\right\|_{\varSigma}\leq
q\right\\}$. It is sufficient to prove that there exist $l>0$ such that
$\left\|\mathcal{B}(w)\right\|_{\varSigma}\leq l,\,\forall\,w\in B_{q}$. Now
for any $t\in[0,b]$,

$\displaystyle\left\|\mathcal{B}(w)(t)\right\|$
$\displaystyle\quad\leq\left\|\mathscr{T}(t)\right\|_{B(X)}\left\|\varsigma(0)\right\|+\int_{0}^{t}\left\|\mathscr{T}(t-s)\right\|_{B(X)}\left\|V\left(s,w_{s},\int_{0}^{s}U\left(s,\sigma,w_{\sigma}\right)d\sigma\right)\right.$
$\displaystyle\qquad\left.-V\left(s,0,\int_{0}^{s}U\left(s,\sigma,0\right)d\sigma\right)\right\|ds+\int_{0}^{t}\left\|\mathscr{T}(t-s)\right\|_{B(X)}\left\|V\left(s,0,\int_{0}^{s}U\left(s,\sigma,0\right)d\sigma\right)\right\|ds.$
(3.3)

By the Theorem 2.1 there exist constant $\mathscr{M}\geq 1$ such that

$\displaystyle\left\|\mathscr{T}(t)\right\|_{B(X)}\leq\mathscr{M},\,t\geq 0.$
(3.4)

Thus from inequality (3) and by hypothesis (H1)(i) and (H1)(ii) we have

$\displaystyle\left\|\mathcal{B}(w)(t)\right\|$
$\displaystyle\quad\leq\mathscr{M}\left\|\varsigma\right\|_{\mathscr{C}}+\int_{0}^{t}\mathscr{M}N_{V}(s)\left\\{\left\|w_{s}\right\|_{\mathscr{C}}+\int_{0}^{s}N_{U}(\sigma)\left\|w_{\sigma}\right\|_{\mathscr{C}}d\sigma\right\\}ds$
$\displaystyle\qquad+\int_{0}^{t}\mathscr{M}\left\|V\left(s,0,\int_{0}^{s}U\left(s,\sigma,0\right)d\sigma\right)\right\|ds$
$\displaystyle\quad\leq\mathscr{M}\left\|\varsigma\right\|_{\mathscr{C}}+\int_{0}^{b}\mathscr{M}N_{V}(s)\left\\{\left\|w_{s}\right\|_{\mathscr{C}}+\int_{0}^{s}N_{U}(\sigma)\left\|w_{\sigma}\right\|_{\mathscr{C}}d\sigma\right\\}ds$
$\displaystyle\qquad+\int_{0}^{b}\mathscr{M}\left\|V\left(s,0,\int_{0}^{s}U\left(s,\sigma,0\right)d\sigma\right)\right\|ds$
$\displaystyle\quad\leq\mathscr{M}\left\|\varsigma\right\|_{\mathscr{C}}+\int_{0}^{b}q\mathscr{M}N_{V}(s)\left\\{1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right\\}ds+\int_{0}^{b}\mathscr{M}\left\|V\left(s,0,\int_{0}^{s}U\left(s,\sigma,0\right)d\sigma\right)\right\|ds\,=l^{*}.$

Further, for any $t\in[-r,0]$ we have,
$\left\|\mathcal{B}(w)(t)\right\|=\left\|\varsigma(t)\right\|\leq\left\|\varsigma\right\|_{\mathscr{C}}$.
Thus

$\left\|\mathcal{B}\,w\right\|_{\varSigma}=\underset{[-r,b]}{\sup}\left\|\mathcal{B}(w)(t)\right\|\leq\max\left\\{\left\|\varsigma\right\|_{\mathscr{C}},\,l^{*}\right\\}=l.$

Step 3: $\mathcal{B}$ maps bounded sets into equicontinuous sets of
$\varSigma$.

Let $B_{q}=\left\\{w\in\varSigma:\left\|w\right\|_{\varSigma}\leq q\right\\}$
be any bounded set of $\varSigma$. Then utilizing hypothesis (H1)(i) and
(H1)(ii) for any $\xi_{1},\xi_{2}\in[0,b]$ with $\xi_{1}<\xi_{2}$ we have,

$\displaystyle\left\|\mathcal{B}(w)(\xi_{2})-\mathcal{B}(w)(\xi_{1})\right\|$
$\displaystyle\leq\left\|\mathscr{T}(\xi_{1})-\mathscr{T}(\xi_{2})\right\|_{B(X)}\left\|\varsigma(0)\right\|+\int_{0}^{\xi_{2}}\left\|\mathscr{T}(\xi_{2}-s)\right\|_{B(X)}\left\|V\left(s,w_{s},\int_{0}^{s}U\left(s,\sigma,w_{\sigma}\right)d\sigma\right)\right\|ds$
$\displaystyle\qquad-\int_{0}^{\xi_{1}}\left\|\mathscr{T}(\xi_{1}-s)\right\|_{B(X)}\left\|V\left(s,w_{s},\int_{0}^{s}U\left(s,\sigma,w_{\sigma}\right)d\sigma\right)\right\|ds$
$\displaystyle=\left\|\mathscr{T}(\xi_{1})-\mathscr{T}(\xi_{2})\right\|_{B(X)}\left\|\varsigma(0)\right\|+\int_{0}^{\xi_{1}}\left\|\mathscr{T}(\xi_{2}-s)-\mathscr{T}(\xi_{1}-s)\right\|_{B(X)}\left\|V\left(s,w_{s},\int_{0}^{s}U\left(s,\sigma,w_{\sigma}\right)d\sigma\right)\right\|ds$
$\displaystyle\qquad+\int_{\xi_{1}}^{\xi_{2}}\left\|\mathscr{T}(\xi_{2}-s)\right\|_{B(X)}\left\|V\left(s,w_{s},\int_{0}^{s}U\left(s,\sigma,w_{\sigma}\right)d\sigma\right)\right\|ds$
$\displaystyle\leq\left\|\mathscr{T}(\xi_{1})-\mathscr{T}(\xi_{2})\right\|_{B(X)}\left\|\varsigma(0)\right\|+\int_{0}^{\xi_{1}}\left\|\mathscr{T}(\xi_{2}-s)-\mathscr{T}(\xi_{1}-s)\right\|_{B(X)}$
$\displaystyle\quad\left\|V\left(s,w_{s},\int_{0}^{s}U\left(s,\sigma,w_{\sigma}\right)d\sigma\right)-V\left(s,0,\int_{0}^{s}U\left(s,\sigma,0\right)d\sigma\right)\right\|ds$
$\displaystyle\quad+\int_{0}^{\xi_{1}}\left\|\mathscr{T}(\xi_{1})-\mathscr{T}(\xi_{2})\right\|_{B(X)}\left\|V\left(s,0,\int_{0}^{s}U\left(s,\sigma,0\right)d\sigma\right)\right\|ds$
$\displaystyle\qquad+\int_{\xi_{1}}^{\xi_{2}}\left\|\mathscr{T}(\xi_{2}-s)\right\|_{B(X)}\left\|V\left(s,w_{s},\int_{0}^{s}U\left(s,\sigma,w_{\sigma}\right)d\sigma\right)-V\left(s,0,\int_{0}^{s}U\left(s,\sigma,0\right)d\sigma\right)\right\|ds$
$\displaystyle\quad\qquad+\int_{\xi_{1}}^{\xi_{2}}\left\|\mathscr{T}(\xi_{2}-s)\right\|_{B(X)}\left\|V\left(s,0,\int_{0}^{s}U\left(s,\sigma,0\right)d\sigma\right)\right\|ds$
$\displaystyle\leq\left\|\mathscr{T}(\xi_{1})-\mathscr{T}(\xi_{2})\right\|_{B(X)}\left\|\varsigma\right\|_{\mathscr{C}}+\int_{0}^{\xi_{1}}\left\|\mathscr{T}(\xi_{2}-s)-\mathscr{T}(\xi_{1}-s)\right\|_{B(X)}N_{V}(s)\left\\{\left\|w_{s}\right\|_{\mathscr{C}}\right.$
$\displaystyle\quad\left.+\int_{0}^{s}N_{U}(\sigma)\left\|w_{\sigma}\right\|_{\mathscr{C}}d\sigma\right\\}ds+\int_{0}^{\xi_{1}}\left\|\mathscr{T}(\xi_{1})-\mathscr{T}(\xi_{2})\right\|_{B(X)}\left\|V\left(s,0,\int_{0}^{s}U\left(s,\sigma,0\right)d\sigma\right)\right\|ds$
$\displaystyle\qquad+\int_{\xi_{1}}^{\xi_{2}}\left\|\mathscr{T}(\xi_{2}-s)\right\|_{B(X)}N_{V}(s)\left\\{\left\|w_{s}\right\|_{\mathscr{C}}+\int_{0}^{s}N_{U}(\sigma)\left\|w_{\sigma}\right\|_{\mathscr{C}}d\sigma\right\\}ds$
$\displaystyle\quad\qquad+\int_{\xi_{1}}^{\xi_{2}}\left\|\mathscr{T}(\xi_{2}-s)\right\|_{B(X)}\left\|V\left(s,0,\int_{0}^{s}U\left(s,\sigma,0\right)d\sigma\right)\right\|ds$
$\displaystyle\leq\left\|\mathscr{T}(\xi_{1})-\mathscr{T}(\xi_{2})\right\|_{B(X)}\left\|\varsigma\right\|_{\mathscr{C}}+\int_{0}^{\xi_{1}}\left\|\mathscr{T}(\xi_{2}-s)-\mathscr{T}(\xi_{1}-s)\right\|_{B(X)}q\,N_{V}(s)\left\\{1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right\\}ds$
$\displaystyle\quad+\int_{0}^{\xi_{1}}\left\|\mathscr{T}(\xi_{1})-\mathscr{T}(\xi_{2})\right\|_{B(X)}\left\|V\left(s,0,\int_{0}^{s}U\left(s,\sigma,0\right)d\sigma\right)\right\|ds$
$\displaystyle\qquad+\int_{\xi_{1}}^{\xi_{2}}\left\|\mathscr{T}(\xi_{2}-s)\right\|_{B(X)}\,q\,N_{V}(s)\left\\{1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right\\}ds$
$\displaystyle\quad\qquad+\int_{\xi_{1}}^{\xi_{2}}\left\|\mathscr{T}(\xi_{2}-s)\right\|_{B(X)}\left\|V\left(s,0,\int_{0}^{s}U\left(s,\sigma,0\right)d\sigma\right)\right\|ds$

As $\xi_{1}\to\xi_{2}$ the right hand side of the above inequality tends to
zero. For the cases $-r<\xi_{1}<\xi_{2}\leq 0$ and $-r<\xi_{1}<0<\xi_{2}<b\leq
0$ the equicontinuity one can be obtain easily via uniform continuity of the
function $\zeta$ and above case. Thus,
$\left\|\mathcal{B}(w)(t_{1})-\mathcal{B}(w)(t_{2})\right\|\leq\aleph|t_{1}-t_{2}|;$
with constant $\aleph>0,\forall\,t\in[-r,b]$ and we conclude that
$\mathcal{B}_{B_{q}}$ is an equicontinuous family of functions.

The equicontinuity and uniform boundedness of the set
$\mathcal{B}_{B_{q}}=\left\\{(\mathcal{B}w)(t)~{}:\left\|w\right\|_{\varSigma}\leq
q,\,-r\leq r\leq b\right\\}$ is as of now illustrated. In the perspective of
Arzela-Ascoli’s, if we prove

$\left\\{(\mathcal{B}w)(t)~{}:\left\|w\right\|_{\varSigma}\leq q,\,-r\leq
r\leq b\right\\}$

is precompact in $\varSigma$ for every $t\in[-r,b]$ then $\mathcal{B}_{B_{q}}$
is precompact in $\varSigma$.

Let $0<t<b$ be fixed and let $\epsilon$ be a real number satisfying
$0<\epsilon<t$. For $w\in B_{q},$ we define

$(\mathcal{B}_{\epsilon}w)(t)=T(t)\varsigma(0)+T(\epsilon)\int_{0}^{(t-\epsilon)}T(t-\epsilon-s)V\left(s,w_{s},\int_{0}^{s}U\left(s,\sigma,w_{\sigma}\right)d\sigma\right)ds.$

Since $T(t)$ is a compact operator, the set

$\mathcal{W}_{\epsilon}(t)=\left\\{(\mathcal{B}_{\epsilon}w)(t)~{}:w\in
B_{q}\right\\}$

is compact in $\varSigma$ for every $\epsilon,\,0<\epsilon<t$. Moreover, for
every $w\in B_{q}$ we have

$\left\|\mathcal{B}(w)(t)-\mathcal{B}_{\epsilon}(w)(t)\right\|\leq\int_{(t-\epsilon)}^{t}\left\|T(t-s)\right\|_{B(X)}\left\|V\left(s,w_{s},\int_{0}^{s}U\left(s,\sigma,w_{\sigma}\right)d\sigma\right)\right\|ds.$

See that the precompact sets are arbitrarily close to the set
$\left\\{(\mathcal{B}w)(t)~{}:w\in B_{q}\right\\}$ and hence
$\left\\{(\mathcal{B}w)(t)~{}:w\in B_{q}\right\\}$ is precompact in
$\varSigma$. We have proved that the operator $\mathcal{B}$ is completely
continuous.

Next, we prove that the operator $\mathcal{A}$ is a contraction. Let any
$w_{1},w_{2}\in\varSigma$. Then by using (H1)(iii) and (H2) then for any
$t\in[0,b]$, we get

$\displaystyle\left\|\mathcal{A}(w_{1})(t)-\mathcal{A}(w_{2})(t)\right\|$
$\displaystyle\quad\leq\underset{0<t_{k}<t}{\sum}\left\|\mathscr{T}(t-t_{k})\right\|_{B(X)}\left[\left\|I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,(w_{1})_{s})ds\right)-I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,(w_{2})_{s})ds\right)\right\|\right]$
$\displaystyle\quad\leq\sum_{k=1}^{m}\mathscr{M}L_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\left\|(w_{1})_{s}-(w_{2})_{s}\right\|_{\mathscr{C}}ds$
$\displaystyle\quad\leq\sum_{k=1}^{m}\mathscr{M}L_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\left\|w_{1}-w_{2}\right\|_{\varSigma}ds$
$\displaystyle\quad\leq\sum_{k=1}^{m}2b\,\mathscr{M}L_{G}\,\mathscr{D}_{k}\left\|w_{1}-w_{2}\right\|_{\varSigma}.$

Finally, we prove that the set

$\mathcal{Z}=\left\\{w\in\varSigma:w=\lambda\mathcal{A}\left(\frac{w}{\lambda}\right)+\lambda\mathcal{B}(w)\,\text{for
some}\,0<\lambda<1\right\\}$

is bounded. Consider the operator equation,

$w=\lambda\mathcal{A}\left(\frac{w}{\lambda}\right)+\lambda\mathcal{B}(w),\,w\in\mathcal{Z}\,\text{\,for
some}\,0<\lambda<1.$

Then

$\displaystyle w(t)$ $\displaystyle=\lambda
T(t)\varsigma(0)+\lambda\int_{0}^{t}\mathscr{T}(t-s)V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)ds$
$\displaystyle\qquad+\lambda\underset{0<t_{k}<t}{\sum}\mathscr{T}(t-t_{k})I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\frac{G(s,w_{s})}{\lambda}ds\right),\,t\in[0,b].$

By using (H1)–(H2), for $\lambda\in(0,1)$ and $t\in[0,b]$, we have

$\displaystyle\left\|w(t)\right\|$
$\displaystyle\leq|\lambda|\left\|T(t)\right\|_{B(X)}\left\|\varsigma\right\|_{\mathscr{C}}+|\lambda|\int_{0}^{t}\left\|\mathscr{T}(t-s)\right\|_{B(X)}\left\|V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)\right.$
$\displaystyle\qquad\left.-V\left(s,0,\int_{0}^{s}U(s,\sigma,0)d\sigma\right)\right\|ds+|\lambda|\int_{0}^{t}\left\|\mathscr{T}(t-s)\right\|_{B(X)}\left\|V\left(s,0,\int_{0}^{s}U(s,\sigma,0)d\sigma\right)\right\|ds$
$\displaystyle\quad\qquad+|\lambda|\underset{0<t_{k}<t}{\sum}\left\|\mathscr{T}(t-t_{k})\right\|_{B(X)}\left\|I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\frac{G(s,w_{s})}{\lambda}ds\right)-I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\frac{G(s,0)}{\lambda}ds\right)\right\|$
$\displaystyle\qquad\qquad+|\lambda|\sum_{k=1}^{m}\left\|\mathscr{T}(t-t_{k})\right\|_{B(X)}\left\|I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\frac{G(s,0)}{\lambda}ds\right)\right\|$
$\displaystyle\leq\mathscr{M}\left\|\varsigma\right\|_{\mathscr{C}}+\int_{0}^{t}MN_{v}(s)\left\\{\left\|w_{s}\right\|_{\mathscr{C}}+\int_{0}^{s}\left\|w_{\sigma}\right\|_{\mathscr{C}}d\sigma\right\\}ds+\int_{0}^{b}\mathscr{M}\left\|V\left(s,0,\int_{0}^{s}U(s,\sigma,0)d\sigma\right)\right\|ds$
$\displaystyle\qquad+\underset{0<t_{k}<t}{\sum}\mathscr{M}L_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\left\|w_{s}\right\|_{\mathscr{C}}ds+\sum_{k=1}^{m}\mathscr{M}\left\|I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\frac{G(s,0)}{\lambda}ds\right)\right\|.$

Denote

$\mathcal{H}=\int_{0}^{b}\mathscr{M}\,\left\|V\left(s,0,\int_{0}^{s}U(s,\sigma,0)d\sigma\right)\right\|ds\,\text{\,and\,}\,\mathcal{Q}=\sum_{k=1}^{m}\mathscr{M}\left\|I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\frac{G(s,0)}{\lambda}ds\right)\right\|.$

Set ${\mathfrak{P}}(t)=\sup\\{\|w(s)\|:s\in[-r,t]\\},~{}t\in[0,b]$. Observe
that $\|w_{t}\|_{\mathscr{C}}\leq{\mathfrak{P}}(t)$ for all $t\in[0,b]$ and
there is $\eta\in[-r,t]$ such that ${\mathfrak{P}}(t)=\|w(\eta)\|$. Hence for
$\eta\in[0,t]$ we have

$\displaystyle{\mathfrak{P}}(t)$ $\displaystyle=\|w(\eta)\|$
$\displaystyle\leq\mathscr{M}\left\|\varsigma\right\|_{\mathscr{C}}+\mathcal{H}+\mathcal{Q}+\int_{0}^{\eta}\mathscr{M}N_{v}(s)\left\\{\left\|w_{s}\right\|_{\mathscr{C}}+\int_{0}^{s}N_{U}(\sigma)\left\|w_{\sigma}\right\|_{\mathscr{C}}d\sigma\right\\}ds$
$\displaystyle\qquad+\underset{0<t_{k}<t}{\sum}\mathscr{M}L_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\left\|w_{s}\right\|_{\mathscr{C}}ds$
$\displaystyle\leq\mathscr{M}\left\|\varsigma\right\|_{\mathscr{C}}+\mathcal{H}+\mathcal{Q}+\int_{0}^{t}\mathscr{M}N_{v}(s)\left\\{{\mathfrak{P}}(s)+\int_{0}^{s}N_{U}(\sigma){\mathfrak{P}}(\sigma)d\sigma\right\\}ds$
$\displaystyle\qquad+\underset{0<t_{k}<t}{\sum}\mathscr{M}L_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}{\mathfrak{P}}(s)ds.$
(3.5)

If $\eta\in[-r.0]$ then
${\mathfrak{P}}(s)=\left\|\varsigma(t)\right\|\leq\left\|\varsigma\right\|_{\mathscr{C}}$.
In this case the inequality (3) holds clearly since $\mathscr{M}\geq 1$.
Applying the Pachpatte’s type integral inequality with integral impulses given
in Theorem 2.3 to (3) with

$~{}u(t)={\mathfrak{P}}(t),\,\,n(t)=\mathscr{M}\left\|\varsigma\right\|_{\mathscr{C}}+\mathcal{H}+\mathcal{Q},~{}f(t)=\mathscr{M}N_{V}(t),~{}g(t)=N_{U}(t),~{}\text{and}\,~{}\beta_{k}=\mathscr{M}L_{G}\,\mathscr{D}_{k}$

we obtain

$\displaystyle{\mathfrak{P}}(t)$
$\displaystyle\leq\left(\mathscr{M}\left\|\varsigma\right\|_{\mathscr{C}}+\mathcal{H}+\mathcal{Q}\right)\underset{0<t_{k}<t}{\prod}~{}C_{k}\exp\left(\int_{t_{\alpha}}^{t}\mathscr{M}N_{V}(s)\left[1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right]ds\right)$
$\displaystyle\leq\left(\mathscr{M}\left\|\varsigma\right\|_{\mathscr{C}}+\mathcal{H}+\mathcal{Q}\right)\underset{0<t_{k}<t}{\prod}~{}C_{k}\exp\left(\int_{t_{\alpha}}^{b}\mathscr{M}N_{V}(s)\left[1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right]ds\right)$
$\displaystyle:=\mathcal{K},\,t\in[0,b],$ (3.6)

where

$\displaystyle C_{k}$
$\displaystyle=\exp\left(\int_{t_{k-1}}^{t_{k}}\mathscr{M}N_{V}(s)\left[1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right]ds\right)$
$\displaystyle\qquad+\mathscr{M}L_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\exp\left(\int_{t_{k-1}}^{s}\mathscr{M}N_{V}(\sigma)\left[1+\int_{0}^{\sigma}N_{U}(\xi)d\xi\right]d\sigma\right)ds.$
(3.7)

This gives $\left\|w\right\|_{\varSigma}\leq\mathcal{K}$. We have proved that
the set $\mathcal{Z}$ is bounded. In the view of the Theorem 2.2, the operator
equation $w=\mathcal{A}(w)+\mathcal{B}(w),\,w\in\varSigma$ has a fixed point
which is solution of problem (1.4)–(1.6). $\Box$

###### Theorem 3.2

(Uniqueness) If the functions $V,~{}U$ and $G$ satisfies the conditions
(H1)–(H2) then the problem (1.4)–(1.6) has unique solution.

Proof: Let $w_{1}$ and $w_{2}$ be the mild solutions of (1.4)–(1.6) then we
have

$\displaystyle w_{1}(t)-w_{2}(t)$
$\displaystyle=\int_{0}^{t}\mathscr{T}(t-s)\left[V\left(s,(w_{1})_{s},\int_{0}^{s}U(s,\sigma,(w_{1})_{\sigma})d\sigma\right)-V\left(s,(w_{2})_{s},\int_{0}^{s}U(s,\sigma,(w_{2})_{\sigma})d\sigma\right)\right]ds$
$\displaystyle\qquad+\underset{0<t_{k}<t}{\sum}\mathscr{T}(t-t_{k})\left[I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,(w_{1})_{s})\right)ds-
I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,(w_{2})_{s})ds\right)\right],~{}t\in[0,b]$
(3.8)

and

$\displaystyle w_{1}(t)-w_{2}(t)=0,\quad t\in[-r,0].$ (3.9)

Then using the hypotheses (H1) and (H2),

$\displaystyle\left\|w_{1}(t)-w_{2}(t)\right\|$
$\displaystyle\leq\int_{0}^{t}\left\|\mathscr{T}(t-s)\right\|_{B(X)}N_{V}(s)\left\\{\left\|(w_{1}-w_{2})(s)\right\|_{\mathscr{C}}+\int_{0}^{s}N_{U}(\sigma)\left\|(w_{1}-w_{2})(\sigma)\right\|_{\mathscr{C}}d\sigma\right\\}ds$
$\displaystyle\qquad+\underset{0<t_{k}<t}{\sum}\left\|\mathscr{T}(t-t_{k})\right\|_{B(X)}L_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\left\|(w_{1}-w_{2})(s)\right\|_{\mathscr{C}}ds,\,t\in[0,b].$

Set

$\displaystyle\tilde{{\mathfrak{P}}}(t)=\sup\\{\|(w-v)(s)\|:s\in[-r,t]\\},~{}t\in[0,b].$
(3.10)

Observe that $\|(w-v)_{t}\|_{\mathscr{C}}\leq\tilde{{\mathfrak{P}}}(t)$ for
all $t\in[0,b]$ and there is $\eta\in[-r,t]$ such that
$\tilde{{\mathfrak{P}}}(t)=\|(w-v)(\tilde{\eta})\|$. Hence for
$\tilde{\eta}\in[0,t]$ we have

$\displaystyle\tilde{{\mathfrak{P}}}(t)$
$\displaystyle\leq\int_{0}^{\tilde{\eta}}\mathscr{M}N_{V}(s)\left\\{\tilde{{\mathfrak{P}}}(s)+\int_{0}^{s}N_{U}(\sigma)\tilde{{\mathfrak{P}}}(\sigma)d\sigma\right\\}ds+\underset{0<t_{k}<t}{\sum}\mathscr{M}L_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\tilde{{\mathfrak{P}}}(s)ds$
$\displaystyle\leq\int_{0}^{t}\mathscr{M}N_{V}(s)\left\\{\tilde{{\mathfrak{P}}}(s)+\int_{0}^{s}N_{U}(\sigma)\tilde{{\mathfrak{P}}}(\sigma)d\sigma\right\\}ds+\underset{0<t_{k}<t}{\sum}\mathscr{M}L_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\tilde{{\mathfrak{P}}}(s)ds.$
(3.11)

If $\tilde{\eta}\in[-r.0]$ then $\tilde{{\mathfrak{P}}}(s)=0$ and the
inequality (3) holds obviously since $\mathscr{M}\geq 1$. Applying the
Pachpatte’s type integral inequality given in Theorem 2.3 to (3) with

$~{}u(t)=\tilde{{\mathfrak{P}}}(t),\,\,n(t)=0,~{}f(t)=\mathscr{M}N_{V}(t),~{}g(t)=N_{U}(t),~{}\text{and}\,~{}\beta_{k}=\mathscr{M}L_{G}\,\mathscr{D}_{k}$

gives that $\tilde{{\mathfrak{P}}}(t)\leq 0$ and hence
$\left\|w_{1}-w_{2}\right\|_{\varSigma}=0$ . This proves uniqueness. $\Box$

##  4 Dependence of Solutions

######  Theorem 4.1 (Dependence on initial conditions)

Assume that the functions $V,~{}U$ and $G$ satisfies the conditions (H1)–(H2).
Let $w_{i}\,(i=1,2)$ be the mild solutions of the following problem
corresponding to $\varsigma_{i}(t),\,(i=1,2)$,

$\displaystyle w^{\prime}(t)$
$\displaystyle=\mathscr{A}w(t)+V\left(t,w_{t},\int_{0}^{t}U(t,s,w_{s})ds\right),~{}t\in
I=[0,b],~{}t\not=t_{k},~{}k=1,2,\cdots,m$ (4.1) $\displaystyle w(t)$
$\displaystyle=\varsigma_{i}(t),\,(i=1,2),~{}t\in[-r,0],$ (4.2)
$\displaystyle\Delta w(t_{k})$
$\displaystyle=I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,w_{s})ds\right),~{}k=1,2,\cdots,m.$
(4.3)

Then,

$\displaystyle\left\|w_{1}-w_{2}\right\|_{\varSigma}\leq\mathscr{M}\left\|\varsigma_{1}-\varsigma_{2}\right\|_{\mathscr{C}}\underset{0<t_{k}<t}{\prod}~{}C_{k}\exp\left(\int_{t_{\alpha}}^{b}\mathscr{M}N_{V}(s)\left[1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right]ds\right)$
(4.4)

where $C_{k}\,(k=1,2,\cdots,m)$ are given in (3).

Proof: Let any $\varsigma_{i}\in\mathscr{C}\,(i=1,2)$ and $w_{i}\,(i=1,2)$ be
the corresponding mild solutions problems (4.1)–(4.3). Then, by hypotheses
(H1) and (H2),

$\displaystyle\left\|w_{1}(t)-w_{2}(t)\right\|$
$\displaystyle\leq\left\|\mathscr{T}(t)\right\|_{B(X)}\left\|\varsigma_{1}-\varsigma_{2}\right\|_{\mathscr{C}}+\int_{0}^{t}\left\|\mathscr{T}(t-s)\right\|_{B(X)}N_{V}(s)\left\\{\left\|(w_{1}-w_{2})(s)\right\|_{\mathscr{C}}\right.$
$\displaystyle\left.+\int_{0}^{s}N_{U}(\sigma)\left\|(w_{1}-w_{2})(\sigma)\right\|_{\mathscr{C}}d\sigma\right\\}ds+\underset{0<t_{k}<t}{\sum}\left\|\mathscr{T}(t-t_{k})\right\|_{B(X)}L_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\left\|(w_{1}-w_{2})(s)\right\|_{\mathscr{C}}ds,\,t\in[0,b].$

Consider the function $\tilde{{\mathfrak{P}}}$ defined by (3.10). Then ,

$\displaystyle\tilde{{\mathfrak{P}}}(t)$
$\displaystyle\leq\mathscr{M}\left\|\varsigma_{1}-\varsigma_{2}\right\|_{\mathscr{C}}+\int_{0}^{t}\mathscr{M}N_{V}(s)\left\\{\tilde{{\mathfrak{P}}}(s)+\int_{0}^{s}N_{U}(\sigma)\tilde{{\mathfrak{P}}}(\sigma)d\sigma\right\\}ds$
$\displaystyle\qquad+\underset{0<t_{k}<t}{\sum}\mathscr{M}L_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\tilde{{\mathfrak{P}}}(s)ds.$
(4.5)

Applying the Pachpatte’s type integral inequality to (4) with

$~{}u(t)=\tilde{{\mathfrak{P}}}(t),\,\,n(t)=\mathscr{M}\left\|\varsigma_{1}-\varsigma_{2}\right\|_{\mathscr{C}},~{}f(t)=\mathscr{M}N_{V}(t),~{}g(t)=N_{U}(t),~{}\text{and}\,~{}\beta_{k}=\mathscr{M}L_{G}\,\mathscr{D}_{k}$

we get

$\displaystyle\tilde{{\mathfrak{P}}}(t)$
$\displaystyle\leq\mathscr{M}\left\|\varsigma_{1}-\varsigma_{2}\right\|_{\mathscr{C}}\underset{0<t_{k}<t}{\prod}~{}C_{k}\exp\left(\int_{t_{\alpha}}^{t}\mathscr{M}N_{V}(s)\left[1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right]ds\right)$
$\displaystyle\leq\mathscr{M}\left\|\varsigma_{1}-\varsigma_{2}\right\|_{\mathscr{C}}\underset{0<t_{k}<t}{\prod}~{}C_{k}\exp\left(\int_{t_{\alpha}}^{b}\mathscr{M}N_{V}(s)\left[1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right]ds\right).$

Which gives the inequality (4.4). $\Box$

Consider the functional impulsive integrodifferential equation involving the
parameter of the form:

$\displaystyle w^{\prime}(t)$
$\displaystyle=\mathscr{A}w(t)+V\left(t,\varrho_{1},w_{t},\int_{0}^{t}U(t,s,w_{s})ds\right),~{}t\in
I=[0,b],~{}t\not=t_{k},~{}k=1,2,\cdots,m,$ (4.6) $\displaystyle\Delta
w(t_{k})$
$\displaystyle=I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,\mu_{1},w_{s})ds\right),~{}k=1,2,\cdots,m$
(4.7)

and

$\displaystyle w^{\prime}(t)$
$\displaystyle=\mathscr{A}w(t)+V\left(t,\varrho_{2},w_{t},\int_{0}^{t}U(t,s,w_{s})ds\right),~{}t\in
I=[0,b],~{}t\not=t_{k},~{}k=1,2,\cdots,m,$ (4.8) $\displaystyle\Delta
w(t_{k})$
$\displaystyle=I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,\mu_{2},w_{s})ds\right),~{}k=1,2,\cdots,m,$
(4.9)

subject to

$\displaystyle w(t)$ $\displaystyle=\varsigma(t),~{}t\in[-r,0],$ (4.10)

where $\varrho_{1},\,~{}\varrho_{2},\,\mu_{1},\,\mu_{2}\in\mathbb{R}$,
$V:I\times\mathbb{R}\times\mathscr{C}\times X\to X$ and
$G:I\times\mathbb{R}\times\mathscr{C}$ are given functions,
$U\,\text{and}\,~{}\varsigma$ are as in (1.4)–(1.6).
Then we have following theorem showing dependence of solution on parameters.

######  Theorem 4.2 (Dependence on initial parameters)

Let the hypotheses (H1)(iii) and (H2) are holds. Suppose that

  * (H3)

there exist $\tilde{N_{V}}\in C([0,b],\mathbb{R})$ and
$\varOmega_{\varrho},\varOmega_{1}>0$ such that

    * (i)

$\|V(t,\varrho,\vartheta_{1},w_{1})-V(t,\varrho,\vartheta_{2},w_{2})\|\leq\varOmega_{\varrho}\tilde{N_{V}}(t)\left(\|\vartheta_{1}-\vartheta_{2}\|_{\mathscr{C}}+\|w_{1}-w_{2}\|\right);$

    * (ii)

$\|V(t,\varrho_{1},\vartheta_{1},w_{1})-V(t,\varrho_{2},\vartheta_{1},w_{1})\|\leq\varOmega_{1}\left|\varrho_{1}-\varrho_{2}\right|,$

for all
$t\in[0,b],~{}\varrho,\varrho_{1},\varrho_{2}\in\mathbb{R},~{}\vartheta_{1},\vartheta_{2}\in\mathscr{C}$
and $w_{1},w_{2}\in X$ and

  * (H4)

there exist $\tilde{L_{G}},\varOmega_{\mu},\varOmega_{2}>0$ such that

    * (i)

$\|G(t,\mu,\vartheta_{1})-G(t,\mu,\vartheta_{2})\|\leq\varOmega_{\mu}\tilde{L_{G}}\,\|\vartheta_{1}-\vartheta_{2}\|_{\mathscr{C}};$

    * (ii)

$\|G(t,\mu_{1},\vartheta_{1})-G(t,\mu_{2},\vartheta_{1})\|\leq\varOmega_{2}\left|\mu_{1}-\mu_{2}\right|,$

for all $t\in[0,b],~{}\mu,\mu_{1},\mu_{2}\in\mathbb{R}$ and
$\vartheta_{1},\vartheta_{2}\in\mathscr{C}$. Then the mild solution $w_{1}$ of
(4.6)–(4.7) subject to (4.10) and $w_{2}$ of (4.8)–(4.9) subject to (4.10)
satisfy the

$\displaystyle\left\|w_{1}-w_{2}\right\|_{\varSigma}$
$\displaystyle\leq\left(b\mathscr{M}\varOmega_{1}\left|\varrho_{1}-\varrho_{2}\right|+\sum_{k=1}^{m}2b\mathscr{M}\varOmega_{2}\mathscr{D}_{k}\left|\mu_{1}-\mu_{2}\right|\right)\underset{0<t_{k}<t}{\prod}~{}C_{k}\exp\left(\int_{t_{\alpha}}^{b}\mathscr{M}\tilde{N_{V}}(s)\left[1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right]ds\right),$
(4.11)

where

$\displaystyle C_{k}$
$\displaystyle=\exp\left(\int_{t_{k-1}}^{t_{k}}\mathscr{M}\tilde{N_{V}}(s)\left[1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right]ds\right)$
$\displaystyle\qquad+\mathscr{M}\tilde{L_{G}}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\exp\left(\int_{t_{k-1}}^{s}\mathscr{M}\tilde{N_{V}}(\sigma)\left[1+\int_{0}^{\sigma}N_{U}(\xi)d\xi\right]d\sigma\right)ds.$

Proof: Let $w_{1}$ be the mild solution of (4.6)–(4.7) subject to (4.10) and
let $w_{2}$ be the mild solution of (4.8)–(4.9) subject to (4.10). Then by
using hypotheses (H1)(iii), (H2), (H3) and (H4) we have

$\displaystyle\left\|w_{1}(t)-w_{2}(t)\right\|$
$\displaystyle\leq\int_{0}^{t}\left\|\mathscr{T}(t-s)\right\|\left\|V\left(s,\varrho_{1},(w_{1})_{s},\int_{0}^{s}U(s,\sigma,(w_{1})_{\sigma})d\sigma\right)-V\left(s,\varrho_{2},(w_{2})_{s},\int_{0}^{s}U(s,\sigma,(w_{2})_{\sigma})d\sigma\right)\right\|ds$
$\displaystyle\qquad+\underset{0<t_{k}<t}{\sum}\left\|\mathscr{T}(t-t_{k})\right\|\left\|I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,\mu_{1},(w_{1})_{s})\right)-I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,\mu_{2},(w_{2})_{s})ds\right)\right\|$
$\displaystyle\leq\int_{0}^{t}\left\|\mathscr{T}(t-s)\right\|\left\|V\left(s,\varrho_{1},(w_{1})_{s},\int_{0}^{s}U(s,\sigma,(w_{1})_{\sigma})d\sigma\right)-V\left(s,\varrho_{1},(w_{2})_{s},\int_{0}^{s}U(s,\sigma,(w_{2})_{\sigma})d\sigma\right)\right\|ds$
$\displaystyle+\int_{0}^{t}\left\|\mathscr{T}(t-s)\right\|\left\|V\left(s,\varrho_{1},(w_{2})_{s},\int_{0}^{s}U(s,\sigma,(w_{2})_{\sigma})d\sigma\right)-V\left(s,\varrho_{2},(w_{2})_{s},\int_{0}^{s}U(s,\sigma,(w_{2})_{\sigma})d\sigma\right)\right\|ds$
$\displaystyle\qquad+\underset{0<t_{k}<t}{\sum}\left\|\mathscr{T}(t-t_{k})\right\|\left\|I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,\mu_{1},(w_{1})_{s})\right)-I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,\mu_{1},(w_{2})_{s})ds\right)\right\|$
$\displaystyle\qquad\qquad+\underset{0<t_{k}<t}{\sum}\left\|\mathscr{T}(t-t_{k})\right\|\left\|I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,\mu_{1},(w_{2})_{s})\right)-I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,\mu_{2},(w_{2})_{s})ds\right)\right\|$
$\displaystyle\leq
b\,\mathscr{M}\varOmega_{1}\left|\varrho_{1}-\varrho_{2}\right|+\sum_{k=1}^{m}2\,b\,\mathscr{M}\,\varOmega_{2}\,\mathscr{D}_{k}\left|\mu_{1}-\mu_{2}\right|+\int_{0}^{t}\mathscr{M}\tilde{N_{V}}(s)\left\\{\left\|(w_{1}-w_{2})(s)\right\|_{\mathscr{C}}\right.$
$\displaystyle\qquad\left.+\int_{0}^{s}N_{U}(\sigma)\left\|(w_{1}-w_{2})(\sigma)\right\|_{\mathscr{C}}d\sigma\right\\}ds+\underset{0<t_{k}<t}{\sum}\mathscr{M}\tilde{L_{G}}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\left\|(w_{1}-w_{2})(s)\right\|_{\mathscr{C}}ds.$

Then with the function $\tilde{{\mathfrak{P}}}$ given in (3.10), we have,

$\displaystyle\tilde{{\mathfrak{P}}}(t)$ $\displaystyle\leq
b\,\mathscr{M}\varOmega_{1}\left|\varrho_{1}-\varrho_{2}\right|+\sum_{k=1}^{m}2\,b\,\mathscr{M}\,\varOmega_{2}\,\mathscr{D}_{k}\left|\mu_{1}-\mu_{2}\right|+\int_{0}^{t}\mathscr{M}\tilde{N_{V}}(s)\left\\{\tilde{{\mathfrak{P}}}(t)+\int_{0}^{s}N_{U}(\sigma)\tilde{{\mathfrak{P}}}(\sigma)d\sigma\right\\}ds$
$\displaystyle\qquad+\underset{0<t_{k}<t}{\sum}\mathscr{M}\tilde{L}_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\tilde{{\mathfrak{P}}}(s)ds.$
(4.12)

Again, application of Pachpatte’s type integral inequality with

$~{}u(t)=\tilde{{\mathfrak{P}}}(t),\,\,n(t)=b\,\mathscr{M}\varOmega_{1}\left|\varrho_{1}-\varrho_{2}\right|+\sum_{k=1}^{m}2\,b\,\mathscr{M}\,\varOmega_{2}\,\mathscr{D}_{k}\left|\mu_{1}-\mu_{2}\right|,~{}f(t)=\mathscr{M}\tilde{N_{V}}(t),~{}g(t)=N_{U}(t)$

$~{}\text{and}\,~{}\beta_{k}=\mathscr{M}\tilde{L_{G}}\,\mathscr{D}_{k}$ gives

$\displaystyle{\mathfrak{P}}(t)\leq\left(b\mathscr{M}\varOmega_{1}\left|\varrho_{1}-\varrho_{2}\right|+\sum_{k=1}^{m}2b\mathscr{M}\varOmega_{2}\mathscr{D}_{k}\left|\mu_{1}-\mu_{2}\right|\right)\underset{0<t_{k}<t}{\prod}~{}C_{k}\exp\left(\int_{t_{\alpha}}^{b}\mathscr{M}\tilde{N_{V}}(s)\left[1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right]ds\right).$

Which gives the inequality (4.2). $\Box$

Consider the following problem

$\displaystyle v^{\prime}(t)$
$\displaystyle=\mathscr{A}v(t)+\widehat{V}\left(t,v_{t},\int_{0}^{t}U(t,s,v_{s})ds\right),~{}t\in
I=[0,b],~{}t\not=t_{k},~{}k=1,2,\cdots,m$ (4.13) $\displaystyle v(t)$
$\displaystyle=\widehat{\varsigma}(t),~{}t\in[-r,0]$ (4.14)
$\displaystyle\Delta v(t_{k})$
$\displaystyle=\widehat{I_{k}}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,v_{s})ds\right),~{}k=1,2,\cdots,m,$
(4.15)

where $\widehat{V}:I\times\times\mathscr{C}\times X\to
X,\,\widehat{\varsigma}(t)\in\mathscr{C},\,\widehat{I_{k}}:X\to X,\,k=1,\cdots
m$ are given functions. The functions $U\,\text{and}\,~{}G$ as in (1.4)–(1.6).

The next theorem gives dependance of the solutions on the functions involved
in the equations .

######  Theorem 4.3 (Dependence on functions)

Let the hypotheses (H1)–(H2) are holds. Suppose that

  * (H3)

there exists $\mathscr{P},\,\mathscr{J},\mathscr{N}_{k}>0$ such that

    * (i)

$\|V(t,\vartheta_{1},w_{1})-\widehat{V}(t,\vartheta_{1},w_{1})\|\leq\mathscr{P}$;

    * (ii)

$\left\|\varsigma(t)-\widehat{\varsigma}(t)\right\|\leq\mathscr{J}$;

    * (iii)

$\left\|I_{k}(w_{1})-\widehat{I_{k}}(w_{1})\right\|\leq\mathscr{N}_{k};\,k=1,\cdots
m$,

for all $t\in I,~{}\vartheta_{1},\in\mathscr{C}$ and $w_{1}\in X$. If $w$ and
$v$ are the mild solutions of (1.4)–(1.6) and (4.13)–(4.15) respectively then,

$\displaystyle\left\|w-v\right\|_{\varSigma}\leq\left(\mathscr{M}\mathscr{J}+b\,\mathscr{M}\,\mathscr{P}+\sum_{K=1}^{m}\mathscr{M}\mathscr{N}_{k}\right)\underset{0<t_{k}<t}{\prod}~{}C_{k}\exp\left(\int_{t_{\alpha}}^{b}\mathscr{M}N_{V}(s)\left[1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right]ds\right),$
(4.16)

where $C_{k}\,(k=1,2,\cdots,m)$ is given in (3).

Proof: Let $w$ is the mild solution of (1.4)–(1.6) and $v$ is the mild
solution (4.13)–(4.15). Then, for $t\in[0,b]$,

$\displaystyle\left\|w(t)-v(t)\right\|$
$\displaystyle\leq\left\|\mathscr{T}(t)\right\|_{B(X)}\left\|\varsigma(0)-\widehat{\varsigma}(0)\right\|+\int_{0}^{t}\left\|\mathscr{T}(t-s)\right\|_{B(X)}\left\|V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)\right.$
$\displaystyle\quad\left.-\widehat{V}\left(s,v_{s},\int_{0}^{s}U(s,\sigma,v_{\sigma})d\sigma\right)\right\|ds$
$\displaystyle\qquad+\underset{0<t_{k}<t}{\sum}\left\|\mathscr{T}(t-t_{k})\right\|_{B(X)}\left\|I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,w_{s})ds\right)-\widehat{I_{k}}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,v_{s})ds\right)\right\|$
$\displaystyle\leq\left\|\mathscr{T}(t)\right\|_{B(X)}\left\|\varsigma(t)-\widehat{\varsigma}(t)\right\|+\int_{0}^{t}\left\|\mathscr{T}(t-s)\right\|_{B(X)}\left\|V\left(s,w_{s},\int_{0}^{s}U(s,\sigma,w_{\sigma})d\sigma\right)\right.$
$\displaystyle\quad\left.-V\left(s,v_{s},\int_{0}^{s}U(s,\sigma,v_{\sigma})d\sigma\right)\right\|ds+\int_{0}^{t}\left\|\mathscr{T}(t-s)\right\|_{B(X)}\left\|V\left(s,v_{s},\int_{0}^{s}U(s,\sigma,v_{\sigma})d\sigma\right)\right.$
$\displaystyle\qquad\quad\left.-\widehat{V}\left(s,v_{s},\int_{0}^{s}U(s,\sigma,v_{\sigma})d\sigma\right)\right\|ds$
$\displaystyle\qquad\qquad+\underset{0<t_{k}<t}{\sum}\left\|\mathscr{T}(t-t_{k})\right\|_{B(X)}\left\|I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,w_{s})ds\right)-I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,v_{s})ds\right)\right\|$
$\displaystyle\qquad\qquad\quad+\underset{0<t_{k}<t}{\sum}\left\|\mathscr{T}(t-t_{k})\right\|_{B(X)}\left\|I_{k}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,v_{s})ds\right)-\widehat{I_{k}}\left(\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}G(s,v_{s})ds\right)\right\|$
$\displaystyle\leq\mathscr{M}\mathscr{J}+\int_{0}^{t}\mathscr{M}N_{V}(s)\left\\{\left\|w_{s}-v_{s}\right\|_{\mathscr{C}}+\int_{0}^{s}N_{U}(\sigma)\left\|w_{\sigma}-v_{\sigma}\right\|_{\mathscr{C}}d\sigma\right\\}ds+b\mathscr{M}\mathscr{P}$
$\displaystyle\qquad+\underset{0<t_{k}<t}{\sum}\mathscr{M}L_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\left\|w_{s}-v_{s}\right\|_{\mathscr{C}}ds+\sum_{K=1}^{m}\mathscr{M}\mathscr{N}_{k}.$

With the function $\tilde{{\mathfrak{P}}}$ in (3.10), we have

$\displaystyle\tilde{{\mathfrak{P}}}(t)\leq\mathscr{M}\mathscr{J}+b\mathscr{M}\mathscr{P}+\sum_{K=1}^{m}\mathscr{M}\mathscr{N}_{k}+\int_{0}^{t}\mathscr{M}N_{V}(s)\left\\{\tilde{{\mathfrak{P}}}(s)+\int_{0}^{s}N_{U}(\sigma)\tilde{{\mathfrak{P}}}(\sigma)d\sigma\right\\}ds$
$\displaystyle\qquad\quad+\underset{0<t_{k}<t}{\sum}\mathscr{M}L_{G}\,\mathscr{D}_{k}\int_{t_{k}-\tau_{k}}^{t_{k}-\theta_{k}}\tilde{{\mathfrak{P}}}(s)ds.$
(4.17)

By applying the Pachpatte’s type integral inequalities we obtain

$\displaystyle\tilde{{\mathfrak{P}}}(t)\leq\left(\mathscr{M}\mathscr{J}+b\,\mathscr{M}\,\mathscr{P}+\sum_{K=1}^{m}\mathscr{M}\mathscr{N}_{k}\right)\underset{0<t_{k}<t}{\prod}~{}C_{k}\exp\left(\int_{t_{\alpha}}^{b}\mathscr{M}N_{V}(s)\left[1+\int_{0}^{s}N_{U}(\sigma)d\sigma\right]ds\right).$

Which gives the desired inequality (4.16). $\Box$

##  5 Examples

Consider the nonliner Volterra delay integrodifferntial equations with
integral impluses

$\displaystyle w^{\prime}(t)-w(t)$
$\displaystyle=1-\sin(5)-\sin(w(t-5))+\int_{0}^{t}[1+\cos(w(s-5))]ds,~{}t\in
I=[0,2],~{}t\not=t_{1}=1,$ (5.1) $\displaystyle w(t)$
$\displaystyle=t,~{}t\in[-1,0],$ (5.2) $\displaystyle\Delta w(t_{1})$
$\displaystyle=\sin\left(\int_{t_{1}-\tau_{1}}^{t_{1}-\theta_{1}}G(s,w(t-5))ds\right),$
(5.3)

where $G$ is any Lipschitz continuous function with Lipschitz constant
$L_{G}$.

Consider the real Banach space $(\mathbb{R},\left|\cdot\right|)$ and define
$\mathscr{T}(t):[0,\infty)\to\mathcal{B}(\mathbb{R})$ by
$\mathscr{T}(t)x=e^{t}x,\,~{}x\in\mathbb{R}.$ Then
$\\{\mathscr{T}(t)\\}_{t\geq 0}$ is semigroup of bounded linear operators in
$\mathbb{R}.$ with infinitesimal generator $\mathscr{A}=I=\text{Indentity
operator on}\,\mathbb{R}$.

For the choices $\theta_{1}=\tau_{1}=\frac{1}{2}$ and the function
$I_{1}:\mathbb{R}\to\mathbb{R}$ defined by $I_{1}(x)=\sin x,x\in\mathbb{R}$,
the problem (5.1)–(5.3) can be written as

$\displaystyle
w^{\prime}(t)=\mathscr{A}w(t)+V\left(t,w(t-5),\int_{0}^{t}U(t,s,w(s-5))ds\right)$
(5.4) $\displaystyle w(t)=t,~{}t\in[-1,0],$ (5.5) $\displaystyle\Delta
w(t_{1})=0,$ (5.6)

where $U:[0,2]\times[0,2]\times\mathbb{R}\to\mathbb{R}$ is defined by

$U\left(t,s,w\right)=-1+\cos w$

and $V:[0,2]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is defined by

$V\left(t,v,w\right)=1-\sin(5)-\sin v+w.$

Note that

$\left|U(t,s,w_{1})-U(t,s,v_{1})\right|\leq\ \left|\cos w_{1}-\cos
v_{1}\right|\leq\ \left|w_{1}-v_{1}\right|,$

and

$\left|V(t,w_{1},w_{2})-V(t,v_{1},v_{2})\right|\leq\ \left|\sin w_{1}-\sin
v_{1}\right|+|w_{2}-v_{2}|\leq\ |w_{1}-v_{1}|+|w_{2}-v_{2}|,$

for any $t,s\in[0,2]$ and $w_{1},w_{2},v_{1},v_{2}\in\mathbb{R}$.

In this case, the condition (3.2) takes the form

$2\,b\mathscr{M}L_{G}\,\mathscr{D}_{1}=4\mathscr{M}L_{G}<1$

If we choose $L_{G}$ such that $0<L_{G}<\frac{1}{4\mathscr{M}}$ then by
applying Theorem 3.1 the problem (5.1)–(5.3) has at least one solution.

## Acknowledgement

The second author is financially supported by UGC, New Delhi, India. Ref:
F1-17.1/2017-18/RGNF-2017-18-SC-MAH-43083.

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# Molecular Shattering

Ryan J. Farber,1 and Max Gronke,1
1Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-857481
Garching, Germany
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Recent observations suggest galaxies may ubiquitously host a molecular
component to their multiphase circumgalactic medium. However, the structure
and kinematics of the molecular CGM remains understudied theoretically and
largely unconstrained observationally. Recent work suggests molecular gas
clouds with efficient cooling survive acceleration in hot winds similar to
atomic clouds. Yet the pressure-driven fragmentation of molecular clouds when
subjected to external shocks remains unstudied. We perform radiative, inviscid
hydrodynamics simulations of clouds perturbed out of pressure equilibrium to
explore the process of pressure-driven fragmentation to molecular
temperatures, finding significant departures from the atomic case. We find
molecular clouds can shatter into a mist above a critical size corresponding
to the timescale for pressure equilibration exceeding the maximum cooling time
despite all these clouds having a large final overdensity. Our results may
have implications for the formation of the first molecules and for resolving
the molecular phase of the CGM in observations and cosmological simulations.

###### keywords:

galaxies:evolution – ISM: clouds – hydrodynamics – Galaxy: halo – ISM:
molecules

††pubyear: 2022††pagerange: Molecular Shattering–A

##  1 Introduction

The recent development of state-of-the-art submillimeter arrays (e.g., ALMA)
and integral field spectrograph instruments (e.g., MUSE) is beginning to shed
light on the evolution of the molecular phase in the galactic baryon cycle
(tumlinson2017circumgalactic; Veilleux2020). While the hot ionized phase of
the galactic winds has been well studied since ROSAT, (strickland1996rosat;
Strickland2007) a similar picture is only beginning to emerge regarding the
cold molecular phase. It is becoming clear the cold molecular phase is a
ubiquitous phase, detected in galactic outflows and the circumgalactic medium
(CGM) both from within the Local Group (Di2019; Di2020; su2021molecular),
nearby starbursts (M82 strickland2009supernova , NGC 253
strickland2000chandra; strickland2002chandra), close AGN (e.g., Cen A;
Charmandis2000) and at high redshifts (weiss2012variations;
vayner2017galactic). Theoretically, the preponderance of molecular gas is
shocking as supposedly there is sufficient mass at temperatures above 10$4$ K
in the circumgalactic medium to account for all missing baryons (werk2014cos).
Yet the molecular phase tends to dominate the gas mass at least in the
interstellar medium and observations detect massive reservoirs of outflowing
molecular gas (Cicone2014). Whether this gas survives the acceleration process
or is restricted to small heights beyond the galactic midplane depends
crucially on the size distribution of molecular clouds embedded in those hot
winds (farber2022survival).

Previous work has considered the resulting cloud size distribution of warm
$\sim$104 K clouds subjected to strong shocks (Gronke2020Misty) finding these
clouds can shatter into a ‘mist’ of tiny droplets if the shock is strong
enough and the initial overdensity is large enough. However, the
generalization of these results to the molecular regime remains unstudied.

##  2 Methods

We performed our simulations with a modified version of FLASH4.2.2
(Fryxell2000; Dubey2008; Farber2018; farber2022survival), utilizing the
unsplit staggered mesh solver (Lee2009; Lee2013) to solve the compressible,
Eulerian fluid equations with radiative cooling included as a sink term. We
utilize the second-order MUSCL-Hancock predictor-corrector method with the
HLLC Riemann solver and mc slope limiter.

To model optically thin radiative cooling we use the townsend2009 exact
integration scheme to improve the stability of our solutions and to model the
cooling process to machine precision accuracy.111Rather, the principal source
of error is relegated to interpolation from fits to the cooling curve. As in
farber2022survival we apply a piecewise power law fit to the Sutherland1993
cooling curve down to 104 K, extended down to 300 K with the cooling curve of
Dalgarno1972.

In each of our simulations we initialize four overlapping spherical clouds at
a thermally unstable mean temperature ($T_{\mathrm{cl}}$) of 4 $\times 10^{5}$
K and mean density ($n_{\mathrm{cl}}$) of 0.1 cm-3 in hydrostatic equilibrium
with a hot ambient medium $T_{\mathrm{hot}}$= $4\times 10^{7}$ and an initial
density contrast ($\chi_{\mathrm{I}}$) of 100. One cloud was placed at domain
center with the other three offset at most one cloud radius
($r_{\mathrm{cl}}$) per dimension, as in Gronke2020Misty to introduce
asymmetry into our initial conditions, more representative of observed
filamentary clouds (e.g., young2005spitzer). To roughly model a turbulent
power spectrum, we set the density for cloud material by drawing from a
Gaussian with ($\mu$, $\sigma$) = (1, 0.01) $n_{\mathrm{cl}}$, truncated at
3$\sigma$. We set the temperature to maintain a fixed initial pressure of 4
$\times 10^{4}$ cm-3 K.

Note that we turn off radiative cooling above 0.6 $T_{\mathrm{hot}}$to roughly
model the effect of radiative heating of the tenuous ambient medium (e.g., due
to the metagalactic radiation field, faucher2009new; haardt2012radiative;
faucher2020cosmic). However, since we fix $T_{\mathrm{hot}}$ = $4\times
10^{7}$ which has a cooling time already of $\sim$100s Myr we do not expect
this choice to affect our results. Moreover, fixing $\chi_{\mathrm{I}}$ = 100
unlikely affects our results since in general $\chi_{\mathrm{F}}$ is $\gg$
300, the threshold found by Gronke2020Misty for atomic gas shattering.

We utilize static mesh refinement in our simulations to ensure uniform
coverage at the center of the domain with an extended box at lower resolution
to prevent complete outflow of material and mitigate the effect of boundary
conditions on our simulation results. Specifically, we perform our simulations
in a box with resolution 16 cells per $r_{\mathrm{cl}}$ in the innermost (8
$r_{\mathrm{cl}}$)3, outside of which the resolution degrades at most a factor
of four to a full box size of(24 $r_{\mathrm{cl}}$)3.

We performed a suite of simulations varying the imposed temperature floor
($T_{\mathrm{fl}}$) from 400 K to 4 $\times 10^{5}$ K and physical scale of
$r_{\mathrm{cl}}$ from 0.04 pc to 0.5 Mpc to cover uniformly (in log-space)
five decades in temperature from and seven decades in cloud size. We find
these two parameters play a key role in the dynamical evolution of clouds
cooling to molecular temperatures as we now show.

##  3 Results

We begin our presentation of the results by considering the morphological
evolution of two particular clouds from our simulation suite. The only
parameter differentiating these two clouds is their initial radius
($r_{\mathrm{cl}}$, 10 pc & 103 pc at P = 4 $\times 10^{6}$cm-3K) which plays
a critical role in their evolution as we now show. Each cloud is initialized
with a temperature $T_{\mathrm{cl}}$ = 4 $\times 10^{5}$ K and density
$n_{\mathrm{cl}}$ = 0.1 cm-3 in initial hydrostatic equilibrium with a hot
ambient medium $T_{\mathrm{hot}}$ = 4 $\times 10^{7}$ K with an initial
density contrast $\chi_{\mathrm{I}}$ = 100. Note that we initialize four
overlapping spherical clouds resulting in the asymmetric shapes seen in the
density projections of Fig. 1, top panel.

Since $T_{\mathrm{cl}}$ is set at roughly the peak of the cooling curve
(Sutherland1993) the clouds rapidly cool. In the first $t_{\mathrm{sc}}$ the
resulting pressure differential between the ambient medium and the
underpressured clouds drives a contraction of the cloud material. Due to mass
conservation the density in the resulting ‘cores’ increase as they contract.
At the same time the cores are accelerated apparently towards the center of
the initial cloud. In the second $t_{\mathrm{sc}}$ the cores appear to mix and
partially merge, attaining larger central densities. A subset of the core gas
appears to encounter a ‘nozzle’ that forms as the cores are accelerated
towards the center and merge. Such gas is shot out of the nozzle in a stream
which condenses into a large clump that rapidly leaves the domain (see the
second row of Fig 1).

Two other condensations from the stream form at roughly the same time the
merged core structure fragments into roughly (half a) dozen clumps for the
(smaller) larger cloud (see the third row of Fig 1). The fragments are
evidently rotating,222A movie is available at https://cloudflare-
ipfs.com/ipfs/bafybeidvnhg2bc24ruv25xaj76htwaxzirrthh4yudhwqgorngirb2cvuq as
can be seen more clearly for the smaller clumps. Presumably some vorticity was
generated during the contraction and merger phase and the slight rotational
velocity of the larger cores is amplified in the fragmentation process due to
conservation of angular momentum. The rapidly rotating small clumps fragment
into smaller, more rapidly rotating clumps. The smaller cloud appears to
cascade this fragmentation process down to the grid scale and the resulting
incredibly small fragments which are rapidly rotating are seen to evaporate
into the hot ambient medium when they are ejected at large differential
velocity. Meanwhile the larger cores are only slowly rotating and have not
fragmented much in the past 10 $t_{\mathrm{sc}}$ (see the 4th row, left column
of Fig. 1). In contrast, the larger cloud’s fragments do not rotate as rapidly
and one can observe simultaneous coaggulation and fragmentation occurring. The
net effect appears to be an increasing number of clumps but coaggulation
appears to truncate the minimum fragment size and regulate the rotational
velocity. Note that while the smallest clumps are relatively larger and
consequently less rapidly rotating in the larger cloud case, the cores
fragment earlier and are rotating (albeit still slowly) compared to the small
cloud’s cores.

For the large cloud case, it appears coaggulation will triumph in the end, as
a new core can be seen to be forming at domain center from the merging of
smaller clumps. Additionally, some material which was ejected from the central
regions is now returning to the plotted sub-region. Note however that the
number of clumps was increasing until the end of the time we simulated as
driven by the cascading shattering of the small rotating clumps (see the 4th
row, right column of Fig. 1). Meanwhile the smaller cloud maintains two rather
starkly differing populations of clumps: large slowly rotating cores and
$\sim$grid-scale rapidly rotating (and in some cases evaporating) clumps. Only
towards 30 $t_{\mathrm{sc}}$ (4th row, left column of Fig. 1) do the cores
begin to fragment, rotate more rapidly, and generate a more substantial number
of clumps. The lack of a strong coaggulation force in this case leads to the
cores continuing to disperse and it is only towards 40 $t_{\mathrm{sc}}$ that
a large number of clumps are observed (5th row, left column of Fig. 1)compared
to 33 $t_{\mathrm{sc}}$, the final snapshot of the larger cloud case. In
summary we find the initial cloud radius (or mass) plays a key role through
the coaggulation force in setting the shattering timescale.

Figure 1: Mass-weighted density projections illustrating the dynamical
evolution of two characteristic clouds. The left and right columns
respectively show the evolution of characteristic ‘small’ and ‘large’ clouds
with time evolution from top to bottom. The smaller cloud (10 pc, left column)
undergoes shattering at much later times than the larger cloud (103 pc, right
column).

Given that the density of the clumps $\sim$ 1 cm-3 is two orders of magnitude
smaller than the density of the cores ($\sim$100 cm-3) one may wonder as to
the multiphase evolution of the cloud temperature. In Figure 2 we present
minimum temperature projections. Note that the evolution is largely three-
dimensional so many clumps are missed when looking at slice plots. Moreover,
mass-weighted projections still favor the substantial shells of warm 104 K gas
at the expense of embedded cold 103 K gas. We choose to display minimum
temperature projections to specifically determine the evolution of the coldest
phase. Note that in Fig. 1 we project along the x-axis while in Fig. 2 we
project along the z-axis.

In the first row of Fig. 2 note mixing is occurring, made evident particularly
with the evaporation of the smallest fragments for the small cloud (left
column), via the wispy red-white material. One can likewise observe mixing is
occurring at the interface of the cold and hot gas for the clumps in the
larger cloud case by the white $\sim 10^{5}$ K boundary between the cold and
hot gas. The small cloud (left column) only exhibits very cold gas in its
large cores which also contain a $10^{4}$ K warm phase, while the smallest
fragments are rather warmer. On the other hand the larger cloud case (right
column) exhibits a very cold $<10^{3}$ K phase in almost all clumps, including
the smallest ones.

In the middle row of Fig. 2 for the small cloud (left column) we still observe
only the large cores possess $10^{3}$ K gas. For the large cloud (right
column) some small clouds are now vigorously mixing with the ambient medium
(amber material towards the right edge) possibly being ablated, but again
essentially all clumps possess a $10^{3}$ K phase, often embedded within a
$10^{4}$ K phase. In the bottom row we show the same final snapshot as in Fig.
1.

It would seem a large number of clumps are being ablated for the small cloud
case. Although the majority of clumps do not possess $10^{3}$ K gas, some of
the small clumps do. Possibly they fragmented from the larger cores that
originally had $10^{3}$ K gas. For the larger cloud we can now see clearly the
smallest clumps on the right side are not accreting material but indeed are
undergoing ablation. This process occurs more at the perimeter with large
relative velocity between clumps and ambient medium compared to the more
tranquil gas at domain center. Across 20 $t_{\mathrm{sc}}$ in time we observe
a persistent difference where the large cloud’s fragments essentially all have
a very cold $10^{3}$ K phase whereas only a fraction of clumps have $10^{3}$ K
gas for the small cloud case.

Figure 2: Temperature projections showing the minimum value along the line of
sight. While most of the clumps in the large cloud case have gas close to the
floor temperature, clumps in the small cloud case are largely at or above 104
K.

We next examine more closely the dependence on the relative number of clumps
for different gas phases as a function of initial cloud radius. In Fig. 3 we
have plotted the number of clumps vs. time. We depict cloud radii as different
colors from blue (small) to pink (large). We indicate two temperature cuts: T
$<$ 2 $T_{\mathrm{fl}}$ as solid curves and T $<$ 20 $T_{\mathrm{fl}}$ as
dashed curves. Note there can be more clumps in the cold phase than in the
warm phase depending on the clump connectivity and substructure. We plot
simulations with $T_{\mathrm{fl}}$ $\sim$ 8000 K in the top panel and
$T_{\mathrm{fl}}$ $\sim$ 4000 K in the bottom panel to show the separation in
clump number with phase clearly relates to the peak in the cooling time at
8000 K (see Fig. 4). That is, the top panels all have roughly the same number
of clumps with the two temperature cuts with gas at $\sim$$T_{\mathrm{fl}}$
(not presented here).

In the bottom panel clearly there exist two populations of clumps: a larger
number of warm clumps and smaller number of cold clumps. Note that the
separation between the number of clumps diminishes with increasing
$r_{\mathrm{cl}}$. Also note that the larger clouds shatter earlier, which we
conjecture is related to vorticity driven by the coaggulation force, which we
plan to study in more depth in future work. The smallest cloud
$r_{\mathrm{cl}}$ $\sim$ 0.5 pc forming only $\sim$10 clumps at 80
$t_{\mathrm{sc}}$ can hardly be said to have shattered. To determine what
separates i) small clouds which barely fragment, ii) larger clouds with two
distinct populations, iii) and even larger clouds with roughly all clumps very
cold, we compare the sound crossing time to the cooling time in Fig. 4.

Figure 3: Time evolution of the number of clumps formed in a batch of our
simulations. The top panel shows the $\chi_{\mathrm{F}}$= $5\times 10^{3}$
case and the bottom panel shows the $\chi_{\mathrm{F}}$= $10^{4}$ case with
curves colored to indicate initial cloud radius from $r_{\mathrm{cl}}$ = 0.5
pc (blue) to $r_{\mathrm{cl}}$ = 104 pc (pink).

In Fig. 4 the green band indicates the range in cooling times depending on
whether cloud material cools isobarically (lower bound) or isochorically
(upper bound). We plot the sound crossing times of for clouds with
$r_{\mathrm{cl}}$ = 20, 103, and 105 pc as black, red, and blue curves
respectively. Evidently for the largest cloud case all material has a shorter
cooling time than sound crossing time and therefore we expect all clumps can
cool uninhibited to $T_{\mathrm{fl}}$. Smaller clouds are in sonic contact to
lower temperatures and re-attain sonic contact at higher temperatures, at some
point preventing material from reaching the true $T_{\mathrm{fl}}$. At an
intermediate regime, the densest/largest clumps can cool rapidly enough to
reach the floor temperature while more tenuous/smaller clumps hit the ‘wall’
in cooling time at 8000 K, explaining the two populations we observed in Figs.
2 & 3.

Figure 4: Maximum cooling time and sound crossing times vs. temperature. We
plot three curves for the sound crossing time corresponding to clouds with
$r_{\mathrm{cl}}$of 20 pc (black), 103 pc (red), and 105 pc (blue) for clouds
of three initial radii: prediction.

To test this prediction of two critical radii: one for atomic gas shattering
and one for molecular shattering, we plot the number of clumps (from zero as
red to $>100$ as blue) in the two temperature cuts (T $<$ 20 $T_{\mathrm{fl}}$
as triangles and T $<$ 2 $T_{\mathrm{fl}}$ as circles/squares) for all of the
simulations we performed in Fig. 5 where we plot the initial $r_{\mathrm{cl}}$
vs $T_{\mathrm{fl}}$. As would be expected, for $T_{\mathrm{cl}}$ =
$T_{\mathrm{fl}}$ no shattering occurs. As $T_{\mathrm{fl}}$ drops more clouds
atomically shatter, in agreement with (Gronke2020Misty). Note that at a fixed
$T_{\mathrm{fl}}$ sufficiently large clouds fail to shatter as coaggulation
dominates over fragmentation.

We find remarkably good agreement between our simulations and the
(Gronke2020Misty) coaggulation size threshold $\sim$($r_{\mathrm{cl}}$ /
$l_{\mathrm{sh}}$)1/6 plotted as the grey curve. For $T_{\mathrm{fl}}$ $<$
8000 K we find that sufficiently small clouds fail to molecularly shatter (red
squares) while they typically still atomically shatter (blue right-side-up
triangles). The critical size for molecular shattering indicated by the green
curve is $R_{\mathrm{crit}}$ = ($t_{\mathrm{cool}}$
$c_{s}$|${}_{T_{\mathrm{loc,min}}}$ where $t_{\mathrm{cool}}$ $c_{s}$ are
evaulated at $T_{\mathrm{loc,min}}$ the temperature at which the cooling time
is at a local minimum. That is, $T_{\mathrm{loc,min}}=2\times 10^{4}$ K for
8000 K $<$ T $<$ 2 $\times 10^{4}$ K else $T_{\mathrm{fl}}$. We find
remarkably good agreement between our molecular shattering criterion and the
results of our simulations.

Figure 5: Comparison of simulation results with critical conditions for
shattering. Right-side-up (up-side-down) triangles indicate greater than
(fewer than) 25 clumps with temperature $<$ 20 $T_{\mathrm{fl}}$(with
characteristic temperatures of $\sim$104 K which we refer to as atomic
shattering) formed in a simulation. For simulations with $T_{\mathrm{fl}}$$<$
8000 K we additionally mark circles (squares) horizontally offset from the
triangles to indicate molecular shattering (lack thereof) for greater than
(less than) 25 clumps with temperature $<$ 2 $T_{\mathrm{fl}}$. We indicate
the number of clumps by coloring each symbol from dark red for one clump to
dark blue for 100 clumps. The coaggulation size criterion of Gronke2020Misty
is included as the grey dashed curve and our predicted critical size for
molecular shattering is drawn as the green curve.

##  4 Discussion

In this work, we extended the ‘shattering’ idea of McCourt2018 into the
colder, molecular regime. We found that $<10^{4}\,$K gas fragmentation does
occur if333for 300 K $<$ T $<$ 2000 K; for 2000 K $<$ T $<$ 8000 K
$r_{\mathrm{cl}}$$>$ 300 pc ($T_{\mathrm{fl}}$/2000K)0( nfl/0.1cm-3)-1

$r_{\rm cl}>\text{min}(c_{\rm s}t_{\rm cool})\big{|}_{T<8000\,K}\sim
3\text{pc}({T_{\mathrm{fl}}}/300K)^{-0.5}({n_{\mathrm{fl}}}/100cm^{-3})^{-1}.$
(1)

This condition is akin to the criterion $r>{\rm min}(c_{\rm s}t_{\rm cool})$
for molecular shattering with the difference here the minimum is the global
minimum (i.e., at $T\sim 10^{4}\,$K) whereas the minimum in Eq. (1) is the
local minimum beyond the ‘bump’ at $T\sim 8000\,$K (cf. Fig. 4). In other
words, the proposed criterion for molecular shattering compares the sound
crossing time to the minimum cooling time $\lesssim 10^{4}\,K$, i.e., the
timescales when the cloud can regain and loose pressure equilibrium,
respectively. It follows, hence, the logic of the original ‘shattering’ idea
by McCourt2018.

However, in § XXX we also discussed the droplet dynamics following the
‘launching’ phase – either due to strong pulsations of the cloud or due to
rotation and centrifugal forces. During their trajectory in the hot medium,
the droplets experience shear forces which can strip off and mix the cold gas.
In farber2022survival, we discussed the survival of molecular gas in such as
‘cloud crushing’ setup and found that the coldest gas can survive if $t_{\rm
cool,max}/t_{\rm cc}<1$ where $t_{\rm cool,max}$ is the maximum cooling time
$\lesssim 10^{4}\,K$ (i.e., for our employed cooling curve at $\sim 8000\,$K)
and $t_{\rm cc}\sim\chi^{1/2}r_{\rm cl}/v_{\rm wind}$ is the ‘cloud crushing’
timescale it takes a cold cloud of size $r_{\rm cl}$ to be mixed into the hot
wind of velocity $v_{\rm wind}$ (in the adiabatic case).

Comparing the criteria for ‘molecular shattering’ put forward in this paper
and the survival criterion for cold gas from farber2022survival adapted to the
droplets launched, we find:

$\displaystyle\frac{t_{\rm cool,min}}{t_{\rm sc}}\frac{t_{\rm cc}}{t_{\rm
cool,max}}=$ $\displaystyle\frac{t_{\rm cool,min}}{t_{\rm
cool,max}}\chi^{1/2}\frac{r_{\rm d}}{r_{\rm cl}}\frac{c_{\rm s}}{v_{\rm d}}$
(2) $\displaystyle\sim$
$\displaystyle\left(\frac{\chi}{10^{4}}\right)^{-3/2}\left(\frac{T}{T_{0.3}}\right)^{\alpha}\frac{c_{\rm
s}}{v_{\rm d}}\frac{r_{\rm d}}{r_{\rm cl}}$ (3)

where we normalized the quantities to $T_{\rm floor}=T_{0.3}\equiv 300\,$K,
and $\Lambda(T)/\Lambda(T_{0.3})\sim(T/T_{0.3})^{\alpha}$ describes the
cooling curve at this low temperatures. Note that above since $t_{\rm
cool}(T_{0.3})/t_{\rm cool,max}\sim 10^{-2}$ the prefactors cancel with the
$\chi/10^{4}$ term. Furthermore, since the cloud pulsated on a timescale of
$\sim t_{\rm sc}$, the ‘launching’ velocity of $v_{\rm d}\lesssim c_{\rm s}$.
While the ‘typical’ droplet radius $r_{\rm d}$ is difficult to assess, it is
reasonable to assume that the cloud breaks up into several fragments, thus,
$r_{\rm cl}/r_{\rm d}\sim$a few. This implies that for $T_{\rm floor}\sim
300\,$K, $\chi\sim 10^{4}$, for $\chi\sim 10^{4}$ the two criteria actually
agree! Furthermore, the cooling curve employed her has a slope of $\alpha\sim
1.5$ at $T\lesssim 8000\,$K. Hence, for the temperature range considered we
have $t_{\rm cool,min}/t_{\rm sc}\sim(\chi/10^{4})^{-1}t_{\rm cool,max}/t_{\rm
cc}$. – which means that the two criteria actually agree for the entire
temperature range considered.

While the ‘shattering’ criteria are as described by McCourt2018, it is
noteworthy that the dynamics of fragmentation is vastly different. Both in the
atomic case (cf. Gronke2020Misty) as well as in the molecular case focused on
here, the could seems to contract and then fragment upon expansion.
McCourt2018 envisioned a ‘Jeans like’ instability in which hierarchical
fragmentation occurs while the cloud cools, i.e., on a timescale of the
(initial) cooling time. The fragmentation we observe occurs after expansion,
that is, after the (much longer) sound crossing time of the cloud. This is
closer to the picture conjectured by Waters2019 who studied thermal
instability in 1D and found pulsations in the cold medium (see also Das2021)
and extrapolated their findings to a regime where the pulsations are stronger
than the restoring force leading to what they call ‘splattering’. In addition
to this pulsational fragmentation, we see in our simulations fast rotations of
the clumps – induced by the contraction phase and the non-symmetric initial
conditions – which are the dominant process of further fragmentation. Because
of of the different nature and the importance of rotation we dub this process
‘rattering’ MG: OK, the last sentence is a joke! ;) But you can think about it
a bit…. Interestingly, coagulation of the droplets leads to a averaging out of
the spin, and thus a reduced further fragmentation rate. We see this in … MG:
Maybe some words here? Otherwise we can maybe leave the coagulation part out?

Even at large distances from galaxies, studies have shown that apart from the
$\sim 10^{4}\,$K ‘cold’ phase even colder, molecular gas exists both in the
CGM (citation) and ICM (citation). While in the CGM it is yet uncertain what
the morphology of this molecular gar is, the existance of molecular filaments
in cluster centers has been long known (citation). However, there – as well as
in the CGM – the origin of this molecular gas is unknown . In the context of
the ICM, the $\sim 10^{4}\,$K gas is thought to prediminantly originate from
thermal instability, i.e., to form in-situ (Sharma2012; McCourt2012) and
potentially ‘shatter’ during the cooling process (McCourt2018). Our results
suggest that this a similar process might be responsible to form a colder
phase leading to a ‘fog’ of $\lesssim 1000\,$K gas. For typical cluster
conditions ($T\sim 10^{7}-10^{8}\,$K, $n\sim 0.1\ccm$ in the central regions
to $n\sim 10^{-}4\ccm$ in the outskirts), the molecular shattering length-
scale (cf. Eq. XXX) is $\sim 0.01-100\,$pc. This would imply that fragments
formed in such manner are very small, with a low volume filling fraction
before they potentially coagulate to form the molecular streams we observe. A
similar ‘molecular fog’ picture might apply to the CGM where our molecular
length scale is (using $T_{\rm hot}\sim 10^{6}\,K$, $n_{\rm hot}\sim
10^{-2}\ccm$) is $\sim 10\,$pc. That such a colder phase will form and
molecules will form in it does naturally depend on the form of heating – which
is unkown in the ICM and CGM – as well as on the dust abundance. Dust has been
detected in both the CGM and ICM but its origin and survival is theoretically
not yet well understood (citation).

Another

  * •

MG: done reiterate criteri. discussion on different criteria (see below).

  * •

compare to physics / applications (ICM centers, ISM? CGM (check pressure?),
galactic winds (close to starbust; AGN)

  * •

MG: done fragmentation process: shattering vs splattering vs shedding (Mikes
vs PG / GO vs new torque induced). role of coagulation –> averaging out
rotating velocity due to merging

  * •

shedding details: cooling, contraction, conservation of angular momentum –>
spinup, more shedding. some dependence on IC (non symmetric) [with connection
to extrinsic turbulence –> ICs but also startup of spin]

some words about caveats

  * •

B fields, CR, …

  * •

No chemisty –> first exploration of physics effect, chemistry later; thermal
history important for dust

###  4.1 Notes about observations

  * •

Potentially: molecular (H2) detection in BLR ($\lesssim 3$pc,
https://www.eso.org/sci/meetings/2020/UVES2020/Presentations/Noterdaeme_H2.pdf

  * •

MG: done ICM TI –> see noted iPad

  * •

molecular winds

  * •

maybe other: galactic center? ISM?

##  5 Conclusions

We study the nonlinear evolution of rapidly cooling clumps of cold gas to
molecular temperatures to determine the structure of cold gas in the
interstellar/circumgalactic/intracluster media. We find significant deviations
in our results from previous work studying cooling down to neutral
temperatures. Our specific conclusions are as follows:

  1. 1.

Size-Dependent Shattering Temperature. For simulations that permit cooling
below $10^{4}$ K, small clumps fail to cool below $10^{4}$ K, while large
clumps cool down to the temperature floor and sufficiently small clouds
fragment into a few clumps rather than shatter.

  2. 2.

Clumpy vs. Foggy. The number of clumps formed increases with the size of the
progenitor cloud.

  3. 3.

Schattering Criteria. In addition to the Gronke2020Misty shattering criterion
for $T_{\mathrm{fl}}$$>$ 104 K, we find the analytical condition for clouds to
shatter to molecular temperatures, requiring the sound crossing time to exceed
the maximum cooling time.

These results help to explain the cold gas structure in astrophysical
conditions. However, the inclusion of additional physical effects such as
magnetic fields, cosmic rays, and chemical networks may introduce deviations
from the picture outlined in this paper. We hope to explore the impact of
these processes in future work.

## Acknowledgements

We thank the referee for a helpful review. RJF gratefully acknowledges X, Y, Z
for helpful discussions. RJF thanks X. MG was supported by X and acknowledges
support from X and Y. Freya acknowledgement. This project utilized the
visualization and data analysis package yt (Turk2011); we are grateful to the
yt community for their support. Could also cite numpy and matplotlib here.

## Data Availability

Data related to this work will be shared on reasonable request to the
corresponding author.

##  Appendix A Resolution Convergence