# Half-integer and full-integer topological defects in polar active matter:
Emergence, crossover, and coexistence

Aboutaleb Amiri  Max Planck Institute for the Physics of complex systems,
Dresden, Germany  Romain Mueller  Rudolf Peierls Centre for Theoretical
Physics, University of Oxford, UK  Amin Doostmohammadi
<EMAIL_ADDRESS>The Niels Bohr Institute, University of Copenhagen,
Copenhagen, Denmark

###### Abstract

The presence and significance of active topological defects is increasingly
realised in diverse biological and biomimetic systems. We introduce a
continuum model of polar active matter, based on conservation laws and
symmetry arguments, that recapitulates both polar and apolar (nematic)
features of topological defects in active turbulence. Using numerical
simulations of the continuum model, we demonstrate the emergence of both half-
and full-integer topological defects in polar active matter. Interestingly, we
find that crossover from active turbulence with half- to full-integer defects
can emerge with the coexistence region characterized by both defect types.
These results put forward a minimal, generic framework for studying
topological defect patterns in active matter which is capable of explaining
the emergence of half-integer defects in polar systems such as bacteria and
cell monolayers, as well as predicting the emergence of coexisting defect
states in active matter.

Topological defects denote singularities in the order parameter field, marking
the regions where the order breaks down Kosterlitz (2016); Michel (1980);
Bowick and Giomi (2009). They are topological in the sense that no smooth
local variation in the order parameter space can remove them De Gennes and
Prost (1993); Chaikin and Lubensky (1995) and they are prevalent in various
physical systems ranging from cosmic strings in particle physics model of
early universe Durrer  _et al._ (2002) to vortices in superfluid helium films
Kosterlitz (2016) and flux tubes in superconductors Tinkham (2004), to
disclination lines in liquid crystals Poulin  _et al._ (1997). More recently,
topological defects are being increasingly identified in biological systems,
where, in analogy to liquid crystals, they mark singularities in the
orientation field associated with the alignment of the constituents of
biological systems. These range from topological defects in subcellular
filaments such as actin Schaller and Bausch (2013); Kumar  _et al._ (2018) or
microtubules Nédélec  _et al._ (1997); Roostalu  _et al._ (2018); Duclos  _et
al._ (2020), to defects in bacterial alignment Dell’Arciprete  _et al._
(2018); Meacock  _et al._ (2021), and topological defects in the orientation
field associated with the elongation of fibroblasts Duclos  _et al._ (2017),
epithelial Saw  _et al._ (2017a), and stem cells Kawaguchi  _et al._ (2017).
Remarkably, not only these topological defects are found within numerous
biological systems and across subcellular to multicellular scales, they appear
to play an important role in various biological processes such as cell death
and extrusion in epithelia Saw  _et al._ (2017a), accumulation sites for
bacteria and stem cells Meacock  _et al._ (2021); Copenhagen  _et al._ (2020);
Kawaguchi  _et al._ (2017), and determinants of morphological features in
bacterial colonies Doostmohammadi  _et al._ (2016) and developing animal Hydra
Maroudas-Sacks  _et al._ (2021). The distinguishing feature of these
realizations of topological defects, compared to their counterparts in non-
living systems, is that they are characterized by active flows continuously
being generated due to the activity of the constituents of the living material
(see Doostmohammadi  _et al._ (2018); Shankar  _et al._ (2020) for recent
reviews of active topological defects).

Despite abundant and growing identifications of topological defects in various
living biological systems, fundamental questions regarding their nature remain
unanswered. One particularly puzzling observation is the abundance of half-
integer defects (defects with nematic, head-tail, symmetry) in systems with a
clear polar symmetry such as motile bacteria Copenhagen  _et al._ (2020);
Meacock  _et al._ (2021) and eukaryotic cells Saw  _et al._ (2017a); Kawaguchi
_et al._ (2017); Blanch-Mercader  _et al._ (2018). Polarity in this context
determines the direction of motion of the self-propelled cell and yet when
multicellular collections of these polar entities are formed, such as in
biofilm layers or epithelial monolayers, the emerging topological defects show
half-integer charges - in systems with polar symmetry full-integer defects are
expected Meyer (1973); Kruse  _et al._ (2004).

Indeed, continuum theories of polar active matter predict topological defects
in the form of asters, vortices and spirals that have a full-integer charge
Kruse  _et al._ (2004); Elgeti  _et al._ (2011) as has been reported for
microtubule-motor protein mixtures Nédélec  _et al._ (1997); Surrey  _et al._
(2001). On the other hand, based on the emergence of half-integer defects in
several cellular systems, most of existing continuum models treat these as
active nematics, describing the coarse-grained alignment of constituent
particles through nematic tensor, modeling activity through a stress term
proportional to this nematic tensor, and neglecting polarity altogether. It is
not clear how these distinct, yet relevant, symmetries compete in a real
biological system, nor why half-integer defects emerge in active polar
systems. While recent experiments and agent-based models have reported
coexisting polar waves and nematic bands in actomyosin motility assay Huber
_et al._ (2018), the study of topological defects in systems with mixed polar-
nematic symmetry is non-existent and a generic theoretical framework for
describing defect dynamics in such systems is lacking.

Here we address this problem by introducing a continuum formulation for polar
active entities that shows the emergence of both half-integer and full-integer
defects within one framework. Moreover, compared to the current models of
active nematics that are based on tensorial equations for the nematic
orientation, we introduce a vector-based model that holds all the essential
features and allows for recovering both polar and nematic topological defects.

Figure 1: (a) Snapshots of active nematic turbulence state from continuum
equations of polar active matter with the elasticity ratio $K_{p}/K=0.0$.
Shown here are the vorticity of the full computational domain (left) and a
selected zoomed-in region marked by dashed lines (right). In the zoomed-in
regions black solid lines (top) and black arrows (bottom) represent
streamlines and the polar director field, respectively. In the bottom zoomed-
in panel the half-integer $+1/2$, $-1/2$ topological defects are marked by red
comets and blue triangles, respectively. (b) Corresponding averaged flow field
and isotropic stress around $+1/2$ topological defects from continuum
equations of polar active matter with the elasticity ratio $K_{p}/K=0.0$ (top)
and are compared to the experimental measurements for epithelial monolayers of
MDCK cells (bottom). In the flow fields colormaps indicate velocity magnitude
and black arrows indicate velocity vectors, while in the stress maps, the
colormaps indicate isotropic stresses. All colormaps are normalized by maximum
values such that the velocity magnitudes range between $0$ (blue) to $+1$
(red) and isotropic stress maps range between $-1$ (blue) to $+1$ (red).
Experimental measurements courtesy of Lakshmi Balasubramaniam from Benoit
Ladoux’s lab.

We begin by describing a force balance equation governing both self-propulsion
and active stress generation in the same active system. We describe the
direction of self-propulsion by a local polarity vector $\vec{p}$ such that
each active particle generates a polar force
$\vec{f}^{\text{pol.}}=\alpha\vec{p}$, where $\alpha$ controls the force
strength. In addition to this polar force, each particle generates a dipolar
contribution to the active stress that can be described at leading order by
the stresslet
${\bm{\sigma}}^{\text{active}}=-\zeta\left(\vec{p}\vec{p}^{\text{T}}-p^{2}{\bf
I}/2\right)$ Prost  _et al._ (2015); Jülicher  _et al._ (2018), where ${\bf
I}$ is the identity tensor, and $\zeta$ is the activity coefficient. The
active contributions, from the polar force and active stress, are balanced by
viscous and elastic passive stresses and any existing friction with the
underlying substrate, through the momentum equation

$\rho\left(\partial_{t}\vec{u}+\vec{u}\cdot\vec{\nabla}\vec{u}\right)=\vec{f}^{\text{pol.}}+\vec{\nabla}\cdot{\bm{\sigma}}^{\text{active}}+\vec{\nabla}\cdot{\bm{\sigma}}^{\text{passive}}-\xi\vec{u},$
(1)

where $\rho$ is the fluid density, $\vec{u}$ the velocity field that satisfies
incompressibiliy condition ($\vec{\nabla}\cdot\vec{u}=0$), and $\xi$ the
friction coefficient. The passive stress ${\bm{\sigma}}^{\text{passive}}$
comprises pressure, elastic stress ${\bm{\sigma}}^{\text{elastic}}$, and
viscous stress ${\bm{\sigma}}^{\text{viscous}}=2\eta{\bf
E}=2\eta(\vec{\nabla}\vec{u})^{\text{S}}$, where $\eta$ is the shear viscosity
and ${\bf E}$ is the rate of strain tensor characterizing the symmetric part
of the velocity gradient. The elastic stress
${\bm{\sigma}}^{\text{elastic}}=\frac{\lambda+1}{2}\vec{p}\vec{h}+\frac{\lambda-1}{2}\vec{h}\vec{p}-\frac{\lambda}{2}\vec{p}\cdot\vec{h}$,
is described in terms of the polarity field and its conjugate field $\vec{h}$
known as the molecular field (defined below), with $\lambda$ denoting the flow
alignment parameter De Gennes and Prost (1993). For completeness we have
retained all inertial terms and passive stresses in Eq. (1). In many
biological realizations of active matter, including microscopic active
particles, the inertia is negligible compared to strong viscous dissipation
and as such the lhs of (1) drops out. Furthermore, elastic stresses are
normally dominated by active contributions and are commonly neglected in
studies of active systems Giomi  _et al._ (2014); Blanch-Mercader and
Casademunt (2017). Under these conditions the momentum equation simplifies to
a force balance

$\xi\vec{u}-\alpha\vec{p}=\eta\nabla^{2}\vec{u}-\zeta\vec{\nabla}\cdot\left(\vec{p}\vec{p}^{\text{T}}-p^{2}\frac{{\bf
I}}{2}\right).$ (2)

The above force balance equation is only represented for simplicity. The
overall framework and all the conclusions presented in this paper hold in the
presence of inertial terms and elastic stresses as well.

The velocity field is coupled to the spatio-temporal evolution of the polarity
vector through:

$\partial_{t}\vec{p}+\vec{u}\cdot\vec{\nabla}\vec{p}+\lambda{\bf
E}\cdot\vec{p}+{\bm{\Omega}}\cdot\vec{p}=\frac{1}{\gamma}\vec{h},$ (3)

where the flow alignment parameter $\lambda$ characterizes the response of the
polar alignment to the symmetric and anti-symmetric parts of the velocity
gradient tensor (denoted by the rate of strain ${\bf E}$ and vorticity
${\bm{\Omega}}$ tensors, respectively), and $\gamma$ is the rotational
viscosity that controls the relaxation of the polar alignment to the minimum
of the free energy $\mathcal{F}$ through the molecular field
$\vec{h}=-\delta\mathcal{F}/\delta\vec{p}$. The free energy is described as

$\displaystyle\mathcal{F}=\int d\vec{x}$
$\displaystyle\Bigg{\\{}A\left(-\frac{p^{2}}{2}+\frac{p^{4}}{4}\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle+\frac{K_{p}}{2}(\vec{\nabla}\vec{p})^{2}+\frac{K}{2}\left(\vec{\nabla}(\vec{p}\vec{p}^{\text{T}}-p^{2}\frac{{\bf
I}}{2})\right)^{2}\Bigg{\\}},$

where the first term under the integral controls the isotropic-polar
transition favoring the emergence of finite polarity at $|\vec{p}|=1$. The
second term penalizes gradients in polarity and is controlled by the
elasticity coefficient $K_{p}$. However, by virtue of the symmetry additional
term is allowed that penalizes gradients in nematic (apolar) alignment of the
particles with its strength controlled by the coefficient $K$. Various
additional contributions to the free energy have been considered in coarse-
grained models of self-propelled particles introducing separate equations for
polarity vector and nematic tensor fields Baskaran and Marchetti (2008, 2010,
2012); Chaté (2020); Vafa (2020). As we show here, the competition between the
additional nematic elasticity, controlled by $K$, and the regular polar
elasticity, controlled by $K_{p}$ allows for continuous transition between
polar and nematic topological defects. As such the model presented here
provides, generic, minimal continuum formulation of active self-propelled
particles, with emergent polar and nematic properties.

We simulate equations (1),(3) using a hybrid lattice-Boltzmann method,
combining finite-difference method for the evolution of polarity vector Eq.
(3), and the lattice-Boltzmann method for solving the Navier-Stokes equation
Eq. (1) with $\rho=40$ and $\eta=2/3$ in lattice Boltzmann units, ensuring
that the Reynolds number in the simulations is negligible $Re\ll 1$ Thampi
_et al._ (2014); Doostmohammadi  _et al._ (2017). Simulations are initialized
by setting quiescent velocity field and random polar alignments and periodic
boundary conditions are used on the domain of the size $L_{x}\times
L_{y}=512\times 512$. The dimensionless ratio of the viscosities is fixed
$\eta/\gamma=2/3$, the alignment parameter is $\lambda=0.1$, and the
dimensionless activity $\bar{\zeta}=\zeta/A$ and elasticities $K_{p}/K$ are
varied throughout this study.

We begin by investigating the case of pure apolar elasticity $K_{p}/K=0$,
setting the polar elasticity to zero ($K_{p}=0$). At first, we also set the
polar and friction to zero $\alpha=0$, $\xi=0$, which reduces the number of
parameters while still allowing to retain the minimal physics that show
topological defect formation and the decisive role of the new apolar elastic
term. As shown in Fig. 1a, at statistical steady-state, the system evolves
into active turbulence characterized by chaotic patterns of polar ordering and
flow vortices Wensink  _et al._ (2012); Alert  _et al._ (2020). Remarkably,
the orientation field associated with the polarity vectors demonstrates the
emergence of half-integer topological defects, in the form of comet-shaped
$+1/2$ and trefoil shaped $-1/2$ defects, interleaving the vorticity patterns
in the velocity field. This is striking, since the emergence of such half-
integer defects has so far been only associated to continuum active nematics
where a nematic tensor is described to account for coarse-grained orientation
of active particles Simha and Ramaswamy (2002); Ramaswamy (2010); Marchetti
_et al._ (2013); Doostmohammadi  _et al._ (2018). More importantly, measuring
the average velocity field around the emergent half-integer defects confirms
their self-propulsive feature and produces flow patterns in agreement with
experimentally measured flow fields for twitching bacteria Meacock  _et al._
(2021) and epithelial and progenitor stem cell layers Saw  _et al._ (2017b);
Kawaguchi  _et al._ (2017); Balasubramaniam  _et al._ (2020) (see Fig. 1b).
Additionally, the isotropic stress patterns characterizing the compressive and
tensile stresses around the self-propelled defect are in agreement with the
experimentally measured stresses for epithelial monolayers of Madine Darby
Canine Kidney (MDCK) cells (Fig. 1b). Together, these results show that, at
the limit of zero polar elasticity $K_{p}=0$, our proposed minimal model
reproduces the emergence and active dynamics of half-integer topological
defects, from spatio-temporal evolution of the polarity vectors and without
revoking any dynamic evolution for an additional nematic tensor.

Figure 2: Emergence of active polar turbulence and active turbulence state
with mixed symmetry. The colormaps are the same as in Fig. 1 and only a
zoomed-in section of the entire simulation domain is shown. The full-integer
$+1$, $-1$ topological defects are marked by orange asters and green squares,
while the half-integer $+1/2$, $-1/2$ topological defects are marked by red
comets and blue triangles, respectively.

We find that at the other limit of zero apolar elasticity $K=0$ and finite
polar elasticity $K_{p}/K=\infty$, the minimal model recovers the emergence of
active turbulence interleaved by full-integer topological defects in the form
of spiral vortex $+1$ and anti-vortex $-1$ defects (Fig. 2a). The existence of
such full-integer defects in active polar systems has been predicted
theoretically Kruse  _et al._ (2004) and shown experimentally in motility
assays of actin or microtubule filaments where the motion is generated by
motor proteins Surrey  _et al._ (2001); Schaller and Bausch (2013); Ross  _et
al._ (2019). Numerical studies have explored the hydrodynamics of interactions
between pairs of full-integer defects in active polar systems Elgeti  _et al._
(2011), however - surprisingly - the emergence and dynamics of active
turbulence interleaved with full-integer defects is not well-understood. In
particular, earlier studies of the active polar systems did not find any full-
integer defects in the active turbulence phase Giomi and Marchetti (2012) and
only recently it has been shown numerically that such defects can mark the
transition from active polar turbulence to phase turbulence upon increasing
polarity Chatterjee  _et al._ (2019).

Figure 3: Stability-diagram in the activity-relative elasticity phase space.
Colormap indicates the average charge of the system with $0$ corresponding to
no topological defect, $0.5$ to half-integer, and $1.0$ to full-integer
topological defects, with values between $0.5-1.0$ marking the co-existence
region. (lower panel) A cut in the phase space marked by dashed black line
corresponding to activity value $\bar{\zeta}=3.5$.

Importantly, when both polar and apolar elasticities are finite and non-zero,
the minimal model predicts the emergence of an active topological state with a
mixed symmetry. Here, the active turbulence, characterized by the emergence of
vortices and jets in the flow profile, is accompanied by the emergence of both
half-integer and full-integer topological defects in the orientation of polar
director field. Therefore, varying the ratio of polar and apolar elasticities
$K_{p}/K$ from a purely apolar elasticity ($K_{p}=0$) to a purely polar
elasticity ($K=0$) results in a cross-over region where half-integer comet-
like and trefoil-like defects coexist with their full-integer counterparts
(Fig. 2b). This is a hallmark of a state with mixed symmetry and indicates
that the minimal model unifies both polar and nematic active turbulence.

To quantify the emergence of mixed symmetry and transition from states with
half-integer to full-integer topological defects, we measure the absolute
value of the topological charge of the system $\langle|q|\rangle_{x,t}$,
averaged over space and time, for varying dimensionless relative elasticity
$K_{r}=(K_{p}-K)/(K_{p}+K)$. As such, if all defects are half-integer
$\langle|q|\rangle_{x,t}=1/2$ and if all defects are polar
$\langle|q|\rangle_{x,t}=1$. As evident from Fig. 3 upon increasing the
relative elasticity $K_{r}$ the active system continuously crossovers from
active nematic turbulence, characterized by apolar half-integer topological
defects, to an active turbulence state with mixed polar and apolar symmetry,
where half- and full-integer defects coexist, and to active polar turbulence,
characterized by only polar full-integer topological defects. While, recent
studies have suggested existence of universal scaling behaviors in active
turbulence Alert  _et al._ (2020), the precise role of topological defects in
setting turbulent flow characteristics in active fluids remains largely
unexplored. We further characterize the flow properties of active turbulence
states by measuring their associated kinetic energy spectrum and vorticity-
vorticity correlation functions (Fig. 4). Interestingly, moving from active
nematic turbulence to active polar turbulence is accompanied by an enhanced
decay of the kinetic energy towards smaller scales and leads to lower
vorticity correlation length, indicating that the nature of topological
defects can have a marked impact on active turbulent flows.

Figure 4: Turbulent flow characteristics. (a) Kinetic energy spectrum and (b)
vorticity-vorticity correlation function for the activity value
$\bar{\zeta}=3.5$. The wave number $\kappa$ and length $r$ are normalised by
the integral length $\lfloor=\int
d\kappa~{}\left[E(\kappa)/\kappa\right]~{}/\int d\kappa~{}E(\kappa)$.

The minimal model presented herein is the first to show how apolar, half-
integer, topological defects can emerge in a continuum representation of polar
active matter. Not only this presents a significant reduction in the
complexity compared to the current active nematic formulation which is based
on tensorial representation of the orientation, but also overcomes the
limitation of active nematics by capturing the impact of the polarity of the
particles. This is important because the majority of biological living systems
such as bacterial suspensions or cellular monolayers for which half-integer
topological defects have been identified Meacock  _et al._ (2021); Copenhagen
_et al._ (2020); Saw  _et al._ (2017b); Duclos  _et al._ (2017); Kawaguchi
_et al._ (2017), are composed of polar entities that continuously self-propel
in the direction of polarity. Without the polar self-propelled forces, the
current active nematic framework is basically modeling shakers that do not
move but actively generate flows around themselves. Additionally, our findings
provide a first characterization of the active turbulence with coexisting
half- and full-integer defects. As such our approach unifies active polar and
active nematics systems in one framework and provides testable predictions for
observing states with mixed symmetry in the experiments. Indeed, recent
experiments on motility-assays have shown how modifying the interaction
between the filaments can result in the coexistence of phases with polar and
nematic symmetry in the absence of topological defects Huber  _et al._ (2018).
Future experiment could investigate the active turbulence generation in a
dense system of polar filaments-motor protein mixtures and probe the co-
existence of half- and full-integer topological defects within the active
turbulence. Moreover, although recent experiments on epithelial monolayers
have identified half-integer defects in the orientation field corresponding to
the deformable shape of the cells Saw  _et al._ (2017b); Blanch-Mercader  _et
al._ (2018); Balasubramaniam  _et al._ (2020), earlier studies on epithelial
monolayers have consistently reported the emergence of ‘rosette’ structures,
which closely resemble full-integer topological defects Blankenship  _et al._
(2006); Razzell  _et al._ (2014), though their associated orientation field is
yet to be fully characterized. We conjecture that epithelial monolayers could
realize the active turbulence with mixed half- and full-integer topological
defects and hope that our study triggers further experiments analyzing
biological states with such mixed symmetry.

## Acknowledgement and Funding

AD acknowledges support from the Novo Nordisk Foundation (grant no.
NNF18SA0035142), Villum Fonden (grant no. 29476), and funding from the
European Union’s Horizon 2020 research and innovation program under the Marie
Sklodowska-Curie grant agreement no. 847523 (INTERACTIONS).

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Analysis of Volterra Integrodifferential Equations with Nonlocal and Boundary
Conditions via Picard Operator

Pallavi U. Shikhare 1

<EMAIL_ADDRESS>
Kishor D. Kucche 2

<EMAIL_ADDRESS>
J. Vanterler da C. Sousa 3

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

3 Department of Applied Mathematics, Imecc-Unicamp, 13083-859, Campinas, SP,
Brazil.

###### Abstract

This article investigates the existence and uniqueness of solutions to the
second order Volterra integrodifferential equations with nonlocal and boundary
conditions through its integral equivalent equations and fixed point of
Banach. Further, utilising the Picard operator theory we obtain the dependency
of solutions on the initial nonlocal data and on functions involved on the
right hand side of the equations.

Key words: Nonlocal conditions; Integrodifferential equations; Fixed point
theorem; Picard operator; Data dependency.
2010 Mathematics Subject Classification: 45J05, 34G20, 47H10, 34B15.

##  1 Introduction

Over the years, the study of solutions of differential equations has been the
subject of research, and continues for a number of reasons, from theoretical
comfort, which is about existence, uniqueness, controllability, among others,
and in the practical sense, involving the stability and continuous dependence
on data [1, 2, 3, 4, 5, 6]. So there are several attractive points to
investigate the solutions of the various types of differential equations. On
the other hand, we can also highlight the integrodifferential equations that
have gained prominence in the scientific community [5, 7, 8, 9, 10, 11, 12,
13, 14].

The nonlocal condition is a generalization of the classical initial condition.
Studies with the nonlocal conditions is driven by theoretical premium, yet
additionally there are several events happened in engineering, physics and
life sciences that can be described by means of differential equations subject
to nonlocal conditions [15]–[20]. Therefore differential equations with
nonlocal condition has turned into an active zone of research .

In 1991, Byszewski [1] introduced the nonlocal Cauchy problem in abstract
spaces. The following years, Byszewski [2], carried out another important
work, also addressing the existence and uniqueness of classical solutions to a
nonlocal Cauchy problem of the abstract type in a Banach space. In the
literature many researcher has been commented on nonlocal conditions and
investigated various class of differential and integrodifferential equations
for existences, uniqueness and dependency of solutions [21, 22, 23, 24].

Wang et al. [25], utilizing the Picard and weakly Picard operator theory
combined with Bielecki norms analysed the nonlocal Cauchy problem in Banach
spaces of the form:

$w^{\prime}(t)=f\left(t,w(t)\right),\,t\in[0,b],~{}w(0)=w_{0}+g(w),$

for existence, uniqueness and dependency of solutions. On the other hand,
Otrocol and Ilea [26, 27], using the technique of Picard weak operators,
examined the existence and qualitative properties of solutions for
differential and integrodifferential equations with abstract Volterra
operators.

There are many other important and interesting works related to this theme
that will contribute to the growth of the theory and make possible new
research, so we suggest some work for a more in-depth reading [4, 8, 24, 28,
29, 30].

Motivated by [2, 25, 26, 28], the main objective of this paper is to discuss
some basic problems, such as, existence and uniqueness and dependency of
solutions of the following second order Volterra integrodifferential equations
with nonlocal and boundary conditions (VIDNBC)

$\displaystyle
w^{{}^{\prime\prime}}(t)=\mathscr{F}\left(t,w(t),w^{\prime}(t),\int_{0}^{t}\mathscr{G}\left(t,s,w(s),w^{\prime}(s)\right)ds\right),\,t\in
J=[0,T],\,T>0,$ (1.1) $\displaystyle w(0)+\sum_{k=1}^{p}c_{k}w(t_{k})=w_{0},$
(1.2) $\displaystyle w^{\prime}(T)=\beta\,w^{\prime}(0),\,\,1<\beta<+\infty,$
(1.3)

by utilizing the fixed point theorem and Picard operator theory, where
$0<t_{1}<t_{2}<\cdots<t_{p}\leq T$;
$\mathscr{F}:J\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R},\,\mathscr{G}:J\times
J\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ are given functions satisfying
some assumptions that will be specified later and $c_{1},c_{2}\cdots
c_{p},\,(p\in\mathbb{N})$ are the constants such that
$\sum\limits_{k=1}^{p}c_{k}\not=-1$.

The growth and branching of the field of differential and integrodifferential
equations with nonlocal and boundary conditions provide the scientific and the
academic community the main reasons for the elaboration of the present paper.
A systematic and detailed study under taken on the existence, uniqueness and
dependencies of the data, provides a wider range of tools and work that may
prove to be useful and important for future research.

This paper is organized as follows. Find out preliminarily facts in section 2.
In section 3, we will investigate the first main result of the article, that
is, the existence and uniqueness of solutions of second order Volterra
integrodifferential equations with nonlocal and boundary conditions, through
Banach’s contraction principle. Sections 4, identified with dependency of
solutions through Picard operators theory. In section 5, we illustrate an
example to verify the results. Concluding remakes closed the paper.

##  2 Preliminaries

######  Definition 2.1 ([25],[31],[32])

. Let $\left(X,d\right)$ be a metric space. An operator
$\mathcal{A}:X\rightarrow X$ is a Picard operator (PO), if there exists
$w^{\ast}\in X$ satisfying the following conditions:

(a) |  $\mathbf{F}_{\mathcal{A}}=\\{w^{\ast}\\}$, where $\mathbf{F}_{\mathcal{A}}:=\\{w\in X:\mathcal{A}\left(w\right)=w\\}$.
---|---
(b) |  the sequence $\left(\mathcal{A}^{n}(w_{0})\right)_{n\in\mathbb{N}}$ converges to $w^{\ast}$ for all $w_{0}\in X$.

######  Theorem 2.1 ([25],[31],[32])

. Let $(Y,d)$ be a complete metric space and $\mathcal{A},\mathcal{B}:Y\to Y$
two operators. We suppose the following:

(a) |  $\mathcal{A}$ is a contraction with contraction constant $\alpha$ and $\mathbf{F}_{\mathcal{A}}=\\{w^{\ast}_{\mathcal{A}}\\}$ ;
---|---
(b) |  $\mathcal{B}$ has fixed point and $w^{\ast}_{\mathcal{B}}\in\mathbf{F}_{\mathcal{B}}$;
(c) |  there exist $\rho>0$ such that $d\left(\mathcal{A}(w),\mathcal{B}(w)\right)\leq\rho$ for all $w\in Y.$

Then

$d\left(w^{\ast}_{\mathcal{A}},w^{\ast}_{\mathcal{B}}\right)\leq\frac{\rho}{1-\alpha}.$

##  3 Existence and uniqueness:

###### Definition 3.1

The function $w\in C^{2}(J,\mathbb{R})$ is said to be solution of the problem
(1.1)–(1.3) if $w$ satisfied (1.1) and nonlocal and boundary conditions (1.2)
and (1.3) respectively.

###### Lemma 3.1

A function $w\in C^{2}(J,\mathbb{R})$ is a solution of VIDNBC (1.1)–(1.3) if
and only if $w\in C^{1}(J,\mathbb{R})$ is a solution of the following integral
equations:

$\displaystyle
w(t)=\left(w_{0}-\sum_{k=1}^{p}c_{k}\left[\frac{t_{k}}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right.\right.$
$\displaystyle\qquad\left.\left.+\int_{0}^{t_{k}}(t_{k}-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right]\right)\bigg{/}\left(1+\sum_{k=1}^{p}c_{k}\right)$
$\displaystyle\qquad\quad+\frac{t}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds$
$\displaystyle\qquad\qquad+\int_{0}^{t}(t-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$
(3.1)

Proof: Let $w\in C^{2}(J,\mathbb{R})$ is a solution of problem (1.1)–(1.3).
Integrating (1.1) from $0$ from $t$, we get

$\displaystyle
w^{\prime}(t)=w^{\prime}(0)+\int_{0}^{t}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$
(3.2)

Again integrate above equation from $0$ from $t$, we have

$\displaystyle
w(t)=w(0)+w^{\prime}(0)t+\int_{0}^{t}\mathscr{(}t-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$
(3.3)

From (3.2), we have

$\displaystyle
w^{\prime}(T)=w^{\prime}(0)+\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$
(3.4)

From (3.4) and (1.3), we have

$\displaystyle\beta\,w^{\prime}(0)=w^{\prime}(0)+\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$

This gives

$\displaystyle
w^{\prime}(0)=\frac{1}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$
(3.5)

Putting the value of $w^{\prime}(0)$ from (3.5) in equation (3.3), we obtain

$\displaystyle w(t)$
$\displaystyle=w(0)+\frac{t}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds$
$\displaystyle\qquad+\int_{0}^{t}\mathscr{(}t-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$
(3.6)

Therefore for any $k\,(k=1,2,\dots,p)$

$\displaystyle w(t_{k})$
$\displaystyle=w(0)+\frac{t_{k}}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds$
$\displaystyle\qquad+\int_{0}^{t_{k}}\mathscr{(}t_{k}-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$
(3.7)

Using (3) in the nonlocal conditions (1.2) we obtain

$\displaystyle
w(0)=\left(w_{0}-\sum_{k=1}^{p}c_{k}\left[\frac{t_{k}}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right.\right.$
$\displaystyle\qquad\left.\left.+\int_{0}^{t_{k}}(t_{k}-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right]\right)\bigg{/}\left(1+\sum_{k=1}^{p}c_{k}\right).$

Putting this value of $w(0)$ in (3), we get

$\displaystyle
w(t)=\left(w_{0}-\sum_{k=1}^{p}c_{k}\left[\frac{t_{k}}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right.\right.$
$\displaystyle\qquad\left.\left.+\int_{0}^{t_{k}}(t_{k}-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right]\right)\bigg{/}\left(1+\sum_{k=1}^{p}c_{k}\right)$
$\displaystyle\qquad\quad+\frac{t}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds$
$\displaystyle\qquad\qquad+\int_{0}^{t}(t-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$

Which is equation (3.1). Conversely let $w\in C^{1}(J,\mathbb{R})$ is solution
of (3.1) we prove that $w$ satisfied (1.1)–(1.3). Differentiating (3.1) with
respect to $t$ we get

$\displaystyle
w^{\prime}(t)=\frac{1}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds$
$\displaystyle\qquad\qquad+\int_{0}^{t}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$
(3.8)

On differentiating (3), we obtain

$w^{{}^{\prime\prime}}(t)=\mathscr{F}\left(t,w(t),w^{\prime}(t),\int_{0}^{t}\mathscr{G}\left(t,s,w(s),w^{\prime}(s)\right)ds\right),\,t\in
J,$

which is (1.1). From (3), we have

$w^{\prime}(0)=\frac{1}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds$

and

$\displaystyle w^{\prime}(T)$
$\displaystyle=\frac{1}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds$
$\displaystyle\qquad\qquad+\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$
$\displaystyle=\left(\frac{\beta}{\beta-1}\right)\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds$
$\displaystyle=\beta\,w^{\prime}(0).$

Further, from (3), we have

$\displaystyle w(0)$
$\displaystyle=\left(w_{0}-\sum_{k=1}^{p}c_{k}\left[\frac{t_{k}}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right.\right.$
$\displaystyle\quad\left.\left.+\int_{0}^{t_{k}}(t_{k}-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right]\right)\bigg{/}\left(1+\sum_{k=1}^{p}c_{k}\right)$

and for each $i\,(i=1,2,\dots p)$,

$\displaystyle
w(t_{i})=\left(w_{0}-\sum_{k=1}^{p}c_{k}\left[\frac{t_{k}}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right.\right.$
$\displaystyle\qquad\left.\left.+\int_{0}^{t_{k}}(t_{k}-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right]\right)\bigg{/}\left(1+\sum_{k=1}^{p}c_{k}\right)$
$\displaystyle\qquad\quad+\frac{t_{i}}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds$
$\displaystyle\qquad\qquad+\int_{0}^{t_{i}}(t_{i}-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$

Therefore

$\displaystyle w(0)+\sum_{i=1}^{p}c_{i}w(t_{i})$
$\displaystyle=\left(w_{0}-\sum_{k=1}^{p}c_{k}\left[\frac{t_{k}}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right.\right.$
$\displaystyle\qquad\left.\left.+\int_{0}^{t_{k}}(t_{k}-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right]\right)\bigg{/}\left(1+\sum_{k=1}^{p}c_{k}\right)$
$\displaystyle\qquad+\sum_{i=1}^{p}c_{i}\left\\{\left(w_{0}-\sum_{k=1}^{p}c_{k}\left[\frac{t_{k}}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right.\right.\right.$
$\displaystyle\qquad\left.\left.\left.+\int_{0}^{t_{k}}(t_{k}-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right]\right)\bigg{/}\left(1+\sum_{k=1}^{p}c_{k}\right)\right.$
$\displaystyle\left.\qquad+\frac{t_{i}}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right.$
$\displaystyle\left.\qquad+\int_{0}^{t_{i}}(t_{i}-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right\\}$
$\displaystyle=w_{0}.$

Which is conditions (1.3). This complete the proof. $\Box$

###### Theorem 3.2

Assume that:

  * (H1)

Let
$\mathscr{F}\in\left(J\times\mathbb{R}\times\mathbb{R}\times\mathbb{R},\mathbb{R}\right),\,\mathscr{G}\in\left(J\times
J\times\mathbb{R}\times\mathbb{R},\mathbb{R}\right)$ and there exist constants
$L_{\mathscr{F}},L_{\mathscr{G}}>0$ such that

$\left|\mathscr{F}(t,w_{1},w_{2},w_{3})-\mathscr{F}(t,v_{1},v_{2},v_{3})\right|\leq
L_{\mathscr{F}}\left(\sum_{j=1}^{3}\left|w_{j}-v_{j}\right|\right)$

and

$\left|\mathscr{G}(t,s,w_{1},w_{2})-\mathscr{G}(t,s,v_{1},v_{2})\right|\leq
L_{\mathscr{G}}\left(\sum_{j=1}^{2}\left|w_{j}-v_{j}\right|\right)$

for all $t,s\in J$ and $w_{j},v_{j}\in\mathbb{R}\,(j=1,2,3)$.

  * (H2)

There exist a constant $\gamma>0$ such that

$q=\frac{L_{\mathscr{F}}}{\gamma}\left(1+\frac{L_{\mathscr{G}}}{\gamma}\right)\left(1+\left[1+\left\\{T\beta+T\beta\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right\\}\right]\frac{e^{\gamma\,T}}{\beta-1}\right)<1.$

Then the VIDNBC (1.1)–(1.3) has a unique solution in $C^{2}(J,\mathbb{R})$.

Proof: Consider the space $C^{1}(J,\mathbb{R})$ with the norm

$\left\|w\right\|_{1}=\underset{t\in
J}{\max}\left\\{\frac{\left|w(t)\right|+\left|w^{\prime}(t)\right|}{e^{\gamma\,t}}\right\\},\,\gamma>0,\,w\in
C^{1}(J,\mathbb{R}).$

Then $\left(C^{1}(J,\mathbb{R}),\left\|\cdot\right\|_{1}\right)$ is a Banach
space. Define the operator
$\mathcal{P}:(C^{1}(J,\mathbb{R}),\left\|\cdot\right\|_{1})\to(C^{1}(J,\mathbb{R}),\left\|\cdot\right\|_{1})$

$\displaystyle\mathcal{P}(w)(t)=\left(w_{0}-\sum_{k=1}^{p}c_{k}\left[\frac{t_{k}}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right.\right.$
$\displaystyle\qquad\qquad\left.\left.+\int_{0}^{t_{k}}(t_{k}-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right]\right)\bigg{/}\left(1+\sum_{k=1}^{p}c_{k}\right)$
$\displaystyle\qquad\qquad\quad+\frac{t}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds$
$\displaystyle\qquad\qquad\qquad+\int_{0}^{t}(t-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$

Then the fixed point of
$w=\mathcal{P}w,\,\,w\in\left(C^{1}(J,\mathbb{R}),\left\|\cdot\right\|_{1}\right)$
is the solution of (1.1)–(1.3). Let any $w,v\in C^{1}(J,\mathbb{R})$ and $t\in
J$. Then

$\displaystyle\left|(\mathcal{P}w)(t)-(\mathcal{P}v)(t)\right|$
$\displaystyle\leq\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\left|\frac{t_{k}}{\beta-1}\int_{0}^{T}\left\\{\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-F\left(s,v(s),v^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,v(\sigma),v^{\prime}(\sigma)\right)d\sigma\right)\right\\}ds\right.$
$\displaystyle\quad\left.+\int_{0}^{t_{k}}(t_{k}-s)\left\\{\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-\mathscr{F}\left(s,v(s),v^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,v(\sigma),v^{\prime}(\sigma)\right)d\sigma\right)\right\\}ds\right|$
$\displaystyle+\left|\frac{t}{\beta-1}\int_{0}^{T}\left\\{\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-F\left(s,v(s),v^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,v(\sigma),v^{\prime}(\sigma)\right)d\sigma\right)\right\\}ds\right.$
$\displaystyle\quad\left.+\int_{0}^{t}(t-s)\left\\{\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-\mathscr{F}\left(s,v(s),v^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,v(\sigma),v^{\prime}(\sigma)\right)d\sigma\right)\right\\}ds\right|$
$\displaystyle\leq\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\left|\frac{T}{\beta-1}\int_{0}^{T}\left\\{\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-F\left(s,v(s),v^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,v(\sigma),v^{\prime}(\sigma)\right)d\sigma\right)\right\\}ds\right.$
$\displaystyle\quad\left.+\int_{0}^{T}T\left\\{\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-\mathscr{F}\left(s,v(s),v^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,v(\sigma),v^{\prime}(\sigma)\right)d\sigma\right)\right\\}ds\right|$
$\displaystyle+\left|\frac{T}{\beta-1}\int_{0}^{T}\left\\{\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-F\left(s,v(s),v^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,v(\sigma),v^{\prime}(\sigma)\right)d\sigma\right)\right\\}ds\right.$
$\displaystyle\quad\left.+\int_{0}^{T}T\left\\{\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-\mathscr{F}\left(s,v(s),v^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,v(\sigma),v^{\prime}(\sigma)\right)d\sigma\right)\right\\}ds\right|$
$\displaystyle=\left(1+\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right)\left|\frac{T}{\beta-1}\int_{0}^{T}\left\\{\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-\mathscr{F}\left(s,v(s),v^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,v(\sigma),v^{\prime}(\sigma)\right)d\sigma\right)\right\\}ds\right.$
$\displaystyle\quad\left.+\int_{0}^{T}T\left\\{\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-\mathscr{F}\left(s,v(s),v^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,v(\sigma),v^{\prime}(\sigma)\right)d\sigma\right)\right\\}ds\right|$
$\displaystyle\leq\left(1+\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right)\left[\frac{T}{\beta-1}\int_{0}^{T}L_{\mathscr{F}}\,e^{\gamma
s}e^{-\gamma
s}\left\\{\left|w(s)-v(s)\right|+\left|w^{\prime}(s)-v^{\prime}(s)\right|\right\\}ds\right.$
$\displaystyle\quad\left.+\frac{T}{\beta-1}\int_{0}^{T}\int_{0}^{s}L_{\mathscr{F}}\,L_{\mathscr{G}}\,e^{\gamma\sigma}e^{-\gamma\sigma}\left\\{\left|w(\sigma)-v(\sigma)\right|+\left|w^{\prime}(\sigma)-v^{\prime}(\sigma)\right|\right\\}d\sigma\,ds\right.$
$\displaystyle\quad\left.+T\int_{0}^{T}L_{\mathscr{F}}\,e^{\gamma s}e^{-\gamma
s}\left\\{\left|w(s)-v(s)\right|+\left|w^{\prime}(s)-v^{\prime}(s)\right|\right\\}ds\right.$
$\displaystyle\quad\left.+T\int_{0}^{T}\int_{0}^{s}L_{\mathscr{F}}\,L_{\mathscr{G}}\,e^{\gamma\sigma}e^{-\gamma\sigma}\left\\{\left|w(\sigma)-v(\sigma)\right|+\left|w^{\prime}(\sigma)-v^{\prime}(\sigma)\right|\right\\}d\sigma\,ds\right]$
$\displaystyle\leq\left(1+\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right)\left\\{\frac{T}{\beta-1}L_{\mathscr{F}}\left(\frac{e^{\gamma
T}}{\gamma}-\frac{1}{\gamma}\right)\left\|w-v\right\|_{1}+\frac{T}{\beta-1}L_{\mathscr{F}}\,L_{\mathscr{G}}\left(\frac{e^{\gamma
T}}{\gamma^{2}}-\frac{1}{\gamma^{2}}-\frac{T}{\gamma}\right)\left\|w-v\right\|_{1}\right.$
$\displaystyle\quad\quad\left.+T\,L_{\mathscr{F}}\left(\frac{e^{\gamma
T}}{\gamma}-\frac{1}{\gamma}\right)\left\|w-v\right\|_{1}+T\,L_{\mathscr{F}}\,L_{\mathscr{G}}\left(\frac{e^{\gamma
T}}{\gamma^{2}}-\frac{1}{\gamma^{2}}-\frac{T}{\gamma}\right)\left\|w-v\right\|_{1}\right\\}$
$\displaystyle\leq\left(1+\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right)\left\\{\frac{T}{\beta-1}L_{\mathscr{F}}\frac{e^{\gamma
T}}{\gamma}+\frac{T}{\beta-1}L_{\mathscr{F}}\,L_{\mathscr{G}}\frac{e^{\gamma
T}}{\gamma^{2}}+T\,L_{\mathscr{F}}\frac{e^{\gamma
T}}{\gamma}+T\,L_{\mathscr{F}}\,L_{\mathscr{G}}\frac{e^{\gamma
T}}{\gamma^{2}}\right\\}\left\|w-v\right\|_{1}$
$\displaystyle=\frac{L_{\mathscr{F}}}{\gamma}\left(1+\frac{L_{\mathscr{G}}}{\gamma}\right)\left\\{T\beta+T\beta\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right\\}\frac{e^{\gamma\,T}}{\beta-1}\left\|w-v\right\|_{1}.$
(3.9)

Similarly we have

$\displaystyle\left|(\mathcal{P}w)^{{}^{\prime}}(t)-(\mathcal{P}v)^{{}^{\prime}}(t)\right|$
$\displaystyle\leq\left|\frac{1}{\beta-1}\int_{0}^{T}\left[\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-F\left(s,v(s),v^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,v(\sigma),v^{\prime}(\sigma)\right)d\sigma\right)\right]ds\right|$
$\displaystyle+\left|\int_{0}^{t}\left[\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-\mathscr{F}\left(s,v(s),v^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,v(\sigma),v^{\prime}(\sigma)\right)d\sigma\right)\right]ds\right|$
$\displaystyle\leq\frac{L_{\mathscr{F}}}{\gamma}\left(1+\frac{L_{\mathscr{G}}}{\gamma}\right)\left(\frac{e^{\gamma\,T}}{\beta-1}+e^{\gamma\,t}\right)\left\|w-v\right\|_{1}.$
(3.10)

Thus from (3) and (3) we have

$\displaystyle\left|(\mathcal{P}w)(t)-(\mathcal{P}v)(t)\right|+\left|(\mathcal{P}w)^{{}^{\prime}}(t)-(\mathcal{P}v)^{{}^{\prime}}(t)\right|$
$\displaystyle\leq\frac{L_{\mathscr{F}}}{\gamma}\left(1+\frac{L_{\mathscr{G}}}{\gamma}\right)\left\\{\left(T\beta+T\beta\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right)\frac{e^{\gamma\,T}}{\beta-1}+\frac{e^{\gamma\,T}}{\beta-1}+e^{\gamma\,t}\right\\}\left\|w-v\right\|_{1},\,t\in
J.$ (3.11)

Therefore

$\displaystyle\left\|\mathcal{P}w-\mathcal{P}v\right\|_{1}$
$\displaystyle=\underset{t\in
J}{\max}\,\,\frac{1}{e^{\gamma\,t}}\left\\{\left|(\mathcal{P}w)(t)-(\mathcal{S}w)(t)\right|+\left|(\mathcal{P}w)^{{}^{\prime}}(t)-(\mathcal{S}w)(t)\right|\right\\}$
$\displaystyle\leq\frac{L_{\mathscr{F}}}{\gamma}\left(1+\frac{L_{\mathscr{G}}}{\gamma}\right)\left(1+\left[1+\left\\{T\beta+T\beta\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right\\}\right]\frac{e^{\gamma\,T}}{\beta-1}\right)\left\|w-v\right\|_{1}.$

Choose $\gamma>0$ such that

$\frac{L_{\mathscr{F}}}{\gamma}\left(1+\frac{L_{\mathscr{G}}}{\gamma}\right)\left(1+\left[1+\left\\{T\beta+T\beta\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right\\}\right]\frac{e^{\gamma\,T}}{\beta-1}\right)<1.$

Hence $\mathcal{P}$ is contraction and by Banach contraction principle
$\mathcal{P}$ has a fixed point $\bar{w}$ in
$\left(C^{1}(J,\mathbb{R}),\left\|\cdot\right\|_{1}\right)$ which is solution
of VIDNBC (1.1)–(1.3). $\Box$

##  4 Dependency of solutions via PO:

In this section we give the dependence of solution of problem (1.1)–(1.3) on
the functions involved in the right hand side of equations (1.1)–(1.2) and on
the initial nonlocal data via Picard operator theory. Note that [25] every
contraction operator is a Picard operator.

$\displaystyle
w^{{}^{\prime\prime}}(t)=\tilde{\mathscr{F}}\left(t,w(t),w^{\prime}(t),\int_{0}^{t}\tilde{\mathscr{G}}\left(t,s,w(s),w^{\prime}(s)\right)ds\right),\,t\in
J=[0,T],\,T>0,$ (4.1) $\displaystyle
w(0)+\sum_{k=1}^{p}c_{k}w(t_{k})=\tilde{w}_{0},$ (4.2) $\displaystyle
w^{\prime}(T)=\beta\,w^{\prime}(0),\,\,1<\beta<+\infty.$ (4.3)

Then its equivalent Volterra integral equation is

$\displaystyle
w(t)=\left(\tilde{w}_{0}-\sum_{k=1}^{p}c_{k}\left[\frac{t_{k}}{\beta-1}\int_{0}^{T}\tilde{\mathscr{F}}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\tilde{\mathscr{G}}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right.\right.$
$\displaystyle\qquad\left.\left.+\int_{0}^{t_{k}}(t_{k}-s)\tilde{\mathscr{F}}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\tilde{\mathscr{G}}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right]\right)\bigg{/}\left(1+\sum_{k=1}^{p}c_{k}\right)$
$\displaystyle\qquad\quad+\frac{t}{\beta-1}\int_{0}^{T}\tilde{\mathscr{F}}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\tilde{\mathscr{G}}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds$
$\displaystyle\qquad\qquad+\int_{0}^{t}(t-s)\tilde{\mathscr{F}}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\tilde{\mathscr{G}}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$
(4.4)

###### Theorem 4.1

Suppose the following:

  * $(H1)^{{}^{\prime}}$

All the conditions in Theorem 3.2 are satisfied and $w^{\ast}\in
C^{1}\left(J,\mathbb{R}\right)$ is the unique solution of the integral
equation (3.1).

  * $(H2)^{{}^{\prime}}$

There exists $L_{\tilde{\mathscr{F}}},L_{\tilde{\mathscr{G}}}>0$ such that

$\left|\tilde{\mathscr{F}}(t,w_{1},w_{2},w_{3})-\tilde{\mathscr{F}}(t,v_{1},v_{2},v_{3})\right|\leq
L_{\tilde{\mathscr{F}}}\left(\sum_{j=1}^{3}\left|w_{j}-v_{j}\right|\right)$

and

$\left|\tilde{\mathscr{G}}(t,s,w_{1},w_{2})-\tilde{\mathscr{G}}(t,s,v_{1},v_{2})\right|\leq
L_{\tilde{\mathscr{G}}}\left(\sum_{j=1}^{2}\left|w_{j}-v_{j}\right|\right)$

for all $t,s\in J$ and $w_{j},v_{j}\in\mathbb{R}\,(j=1,2,3)$.

  * $(H3)^{{}^{\prime}}$

There exists a function $\mu(\cdot)\in L^{1}\left(J,\mathbb{R}_{+}\right)\cap
C\left(J,\mathbb{R}_{+}\right)$ such that

$\left|\mathscr{F}(t,u,v,w)-\tilde{\mathscr{F}}(t,u,v,\tilde{w})\right|\leq\mu(t)$

for all $t\in J$ and $u,\,v,\,w,\,\tilde{w}\in\mathbb{R}$.

Then if $v^{\ast}$ is the solution of integral equations (4) then

$\left\|w^{\ast}-v^{\ast}\right\|_{1}\leq\frac{\left|\frac{1}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\left|w_{0}-\tilde{w}_{0}\right|+\frac{\beta
L_{\mu}}{\beta-1}\left[1+\left(1+\left|\frac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right)T\right]}{1-q},$

where $L_{\mu}=\int_{0}^{T}\mu(s)\,ds$.

Proof: Consider the operators
$\mathcal{P},\mathcal{S}:(C^{1}(J,\mathbb{R}),\left\|\cdot\right\|_{1})\to(C^{1}(J,\mathbb{R}),\left\|\cdot\right\|_{1})$
defined by

$\displaystyle\mathcal{P}(w)(t)$
$\displaystyle=\left(w_{0}-\sum_{k=1}^{p}c_{k}\left[\frac{t_{k}}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right.\right.$
$\displaystyle\qquad\left.\left.+\int_{0}^{t_{k}}(t_{k}-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right]\right)\bigg{/}\left(1+\sum_{k=1}^{p}c_{k}\right)$
$\displaystyle\qquad\quad+\frac{t}{\beta-1}\int_{0}^{T}\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds$
$\displaystyle\qquad\qquad+\int_{0}^{t}(t-s)\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$

and

$\displaystyle\mathcal{S}(w)(t)$
$\displaystyle=\left(\tilde{w}_{0}-\sum_{k=1}^{p}c_{k}\left[\frac{t_{k}}{\beta-1}\int_{0}^{T}\tilde{\mathscr{F}}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\tilde{\mathscr{G}}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right.\right.$
$\displaystyle\qquad\left.\left.+\int_{0}^{t_{k}}(t_{k}-s)\tilde{\mathscr{F}}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\tilde{\mathscr{G}}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds\right]\right)\bigg{/}\left(1+\sum_{k=1}^{p}c_{k}\right)$
$\displaystyle\qquad\quad+\frac{t}{\beta-1}\int_{0}^{T}\tilde{\mathscr{F}}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\tilde{\mathscr{G}}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds$
$\displaystyle\qquad\qquad+\int_{0}^{t}(t-s)\tilde{\mathscr{F}}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\tilde{\mathscr{G}}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)ds.$

By condition $(H1)^{{}^{\prime}}\,\,\,\mathcal{P}$ is a contraction with
contraction constant $q$. Let $\mathbf{F}_{\mathcal{P}}=\\{w^{\ast}\\}$.
Following same steps in the proof of the Theorem 3.2, the operator that
$\mathcal{S}$ is contraction with contraction constant

$\tilde{q}=\frac{L_{\tilde{\mathscr{F}}}}{\gamma}\left(1+\frac{L_{\tilde{\mathscr{G}}}}{\gamma}\right)\left(1+\left[1+\left\\{T\beta+T\beta\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right\\}\right]\frac{e^{\gamma\,T}}{\beta-1}\right)<1.$

Hence the VIDNBC (4.1)–(4.3) has a unique solution. Let
$\mathbf{F}_{\mathcal{S}}=\\{v^{\ast}\\}$. For any
$w\in\left(C^{1}(J,\mathbb{R}),\left\|\cdot\right\|_{1}\right)$. Then any
$t\in J$, we have

$\displaystyle\left|(\mathcal{P}w)(t)-(\mathcal{S}w)(t)\right|$
$\displaystyle\leq\left|\dfrac{1}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\left|w_{0}-\tilde{w}_{0}\right|+\left(1+\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right)\left|\frac{T}{\beta-1}\int_{0}^{T}\left\\{\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-\tilde{\mathscr{F}}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\tilde{\mathscr{G}}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right\\}ds\right.$
$\displaystyle\quad\left.+\int_{0}^{T}T\left\\{\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-\tilde{\mathscr{F}}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\tilde{\mathscr{G}}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right\\}ds\right|$
$\displaystyle\leq\left|\dfrac{1}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\left|w_{0}-\tilde{w}_{0}\right|+\left(1+\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right)\left\\{\frac{T}{\beta-1}\int_{0}^{T}\mu(s)ds+T\int_{0}^{T}\mu(s)ds\right\\}$
(4.5)
$\displaystyle\leq\left|\dfrac{1}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\left|w_{0}-\tilde{w}_{0}\right|+\left(1+\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right)\frac{\beta
TL_{\mu}}{\beta-1}.$ (4.6)

Similarly we have

$\displaystyle\left|(\mathcal{P}w)^{{}^{\prime}}(t)-(\mathcal{S}w)^{{}^{\prime}}(t)\right|$
$\displaystyle\leq\left|\frac{1}{\beta-1}\int_{0}^{T}\left[\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-\tilde{\mathscr{F}}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\tilde{\mathscr{G}}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right]ds\right|$
$\displaystyle+\left|\int_{0}^{t}\left[\mathscr{F}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\mathscr{G}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right.\right.$
$\displaystyle\quad\left.\left.-\tilde{\mathscr{F}}\left(s,w(s),w^{\prime}(s),\int_{0}^{s}\tilde{\mathscr{G}}\left(s,\sigma,w(\sigma),w^{\prime}(\sigma)\right)d\sigma\right)\right]ds\right|$
$\displaystyle\leq\frac{1}{\beta-1}\int_{0}^{T}\mu(s)ds+\int_{0}^{T}\mu(s)ds$
(4.7) $\displaystyle\leq\frac{\beta L_{\mu}}{\beta-1}.$ (4.8)

From (4) and (4)

$\displaystyle\left\|\mathcal{P}w-\mathcal{S}w\right\|_{1}$
$\displaystyle=\underset{t\in
J}{\max}\,\,\frac{1}{e^{\gamma\,t}}\left\\{\left|(\mathcal{P}w)(t)-(\mathcal{S}w)(t)\right|+\left|(\mathcal{P}w)^{{}^{\prime}}(t)-(\mathcal{S}w)^{{}^{\prime}}(t)\right|\right\\}$
$\displaystyle\leq\left|\frac{1}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\left|w_{0}-\tilde{w}_{0}\right|+\frac{\beta
L_{\mu}}{\beta-1}\left[1+\left(1+\left|\frac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right)T\right].$

Using the Theorem 2.1 we get following inequality

$\displaystyle\left\|w^{\ast}-v^{\ast}\right\|_{1}\leq\frac{\left|\frac{1}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\left|w_{0}-\tilde{w}_{0}\right|+\frac{\beta
L_{\mu}}{\beta-1}\left[1+\left(1+\left|\frac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right)T\right]}{1-q}.$
(4.9)

$\Box$

###### Remark 4.2

  * (i)

From inequality (4.9) it follows that the solutions of the VIDNBC (1.1)–(1.3)
depends on the initial nonlocal data and functions involved on the right hand
side of equation.

  * (ii)

If $\mu(t)=0$ then $L_{\mu}=0$. In this case inequality (4.9) gives dependency
of solutions on initial nonlocal data.

  * (iii)

If $w_{0}=\tilde{w}_{0}$ then inequality (4.9) gives dependency of solutions
on functions involved on the right hand side of equation.

  * (iv)

If $\mu(t)=0$ and $w_{0}=\tilde{w}_{0}$ then $L_{\mu}=0$, in this case
inequality (4.9) gives uniqueness of solution.

##  5 Examples:

###### Example 5.1

Consider the second order Volterra integrodifferential equations with nonlocal
and boundary conditions:

$\displaystyle
w^{{}^{\prime\prime}}(t)=0.010540+\frac{1}{10}\sin\left(\frac{1}{10}\right)-\frac{\cos(w(t))}{1000}-\frac{\sin(w^{\prime}(t))}{100}$
$\displaystyle\qquad\qquad\qquad+\frac{1}{100}\int_{0}^{t}\frac{1}{10}\left[w(s)-\frac{e^{\frac{s}{10}}}{10}\sin(w(s))+\frac{e^{\frac{s}{10}}}{10}\cos(w^{\prime}(s))\right]\,ds,\,~{}t\in
J=[0,1],$ (5.1) $\displaystyle
w(0)+w(t_{1})+w(t_{2})-w(t_{2})+(0)w(t_{4})+w(t_{5})=3.10,\,\,$ (5.2)
$\displaystyle w^{\prime}(1)=e^{\frac{1}{10}}\,w^{\prime}(0).\,$ (5.3)

Comparing with the equation (1.1)–(1.3) we have
$T=1,\,\beta=e^{\frac{1}{10}},\,t_{1}=0.2,\,t_{2}=0.4,\,t_{3}=0.6,\,t_{4}=0.8,\,t_{5}=1$
and $c_{1}=c_{2}=1,\,c_{3}=-1,\,c_{4}=0,\,c_{5}=1$ such that
$\sum\limits_{k=1}^{5}c_{k}=2\not=-1$.

  1. (i)

Define
$\mathscr{G}:[0,1]\times[0,1]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ by

$\mathscr{G}\left(t,s,w(s),w^{\prime}(s)\right)=\frac{1}{10}\left[w(s)-\frac{e^{\frac{s}{10}}}{10}\sin(w(s))+\frac{e^{\frac{s}{10}}}{10}\cos(w^{\prime}(s))\right].$

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

$\displaystyle\left|\mathscr{G}(t,s,w_{1},w_{2})-\mathscr{G}(t,s,v_{1},v_{2})\right|$
$\displaystyle\leq\frac{1}{10}\left|w_{1}-v_{1}\right|+\frac{e^{\frac{s}{10}}}{10}\left|\sin
w_{1}-\sin v_{1}\right|+\frac{e^{\frac{s}{10}}}{10}\left|\cos w_{2}-\cos
v_{2}\right|$
$\displaystyle\leq\frac{1}{10}\left|w_{1}-v_{1}\right|+\frac{e^{\frac{1}{10}}}{10}\left|w_{1}-v_{1}\right|+\frac{e^{\frac{1}{10}}}{10}\left|w_{2}-v_{2}\right|$
$\displaystyle\leq\frac{1+e^{\frac{1}{10}}}{10}\left\\{\left|w_{1}-v_{1}\right|+\left|w_{2}-v_{2}\right|\right\\}.$

  2. (ii)

Define
$\mathscr{F}:[0,1]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$
by

$\displaystyle\mathscr{F}\left(t,w(t),w^{\prime}(t),\int_{0}^{t}\mathscr{G}\left(t,s,w(s),w^{\prime}(s)\right)ds\right)$
$\displaystyle=0.010540+\frac{1}{10}\sin\left(\frac{1}{10}\right)-\frac{\cos(w(t))}{1000}-\frac{\sin(w^{\prime}(t))}{100}$
$\displaystyle\qquad+\frac{1}{100}\int_{0}^{t}\frac{1}{10}\left[w(s)-\frac{e^{\frac{s}{10}}}{10}\sin(w(s))+\frac{e^{\frac{s}{10}}}{10}\cos(w^{\prime}(s))\right]\,ds.$
$\displaystyle=0.010540+\frac{1}{10}\sin\left(\frac{1}{10}\right)-\frac{\cos(w(t))}{1000}-\frac{\sin(w^{\prime}(t))}{100}+\int_{0}^{t}G\left(t,s,w(s),w^{\prime}(s)\right)ds.$

Then for any $t\in[0,1]$ and
$w_{1},w_{2},w_{3},v_{1},v_{2},v_{3}\in\mathbb{R}$, we have

$\displaystyle\left|\mathscr{F}(t,w_{1},w_{2},w_{3})-\mathscr{F}(t,v_{1},v_{2},v_{3})\right|$
$\displaystyle\leq\frac{1}{1000}\left|\cos w_{1}-\cos
v_{1}\right|+\frac{1}{100}\left|\sin w_{2}-\sin
v_{2}\right|+\frac{1}{100}\left|w_{3}-v_{3}\right|$
$\displaystyle\leq\frac{1}{100}\left\\{\left|w_{1}-v_{1}\right|+\left|w_{2}-v_{2}\right|+\left|w_{3}-v_{3}\right|\right\\}.$

We have proved that $L_{\mathscr{F}}$ and $L_{\mathscr{G}}$ satisfied the
conditions (H1) with
$L_{\mathscr{F}}=\frac{1}{100},\,\text{and}\,\,L_{\mathscr{G}}=\frac{1+e^{\frac{1}{10}}}{10}$.

$\displaystyle q$
$\displaystyle=\frac{L_{\mathscr{F}}}{\gamma}\left(1+\frac{L_{\mathscr{G}}}{\gamma}\right)\left(1+\left[1+\left\\{T\beta+T\beta\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right\\}\right]\frac{e^{\gamma\,T}}{\beta-1}\right)$
$\displaystyle=\frac{\frac{1}{100}}{\gamma}\left(1+\frac{\frac{1+e^{\frac{1}{10}}}{10}}{\gamma}\right)\left(1+\left[1+\left\\{1\,e^{\frac{1}{10}}+1\,e^{\frac{1}{10}}\left|\frac{2}{1+2}\right|\right\\}\right]\frac{e^{\gamma\,1}}{e^{\frac{1}{10}}-1}\right).$

Note for $\gamma=1,$ we have

$q=\frac{1}{100}\left(1+\frac{1+e^{\frac{1}{10}}}{10}\right)\left(1+\left[1+\left\\{e^{\frac{1}{10}}+e^{\frac{1}{10}}\left|\frac{2}{3}\right|\right\\}\right]\frac{e}{e^{\frac{1}{10}}-1}\right)=0.901278<1.$

Thus all the assumptions of the Theorem 3.2 are satisfied. Applying the
Theorem 3.2, the problem (5.1)–(5.3) has unique solution on $[0,1]$. One can
verify that $w(t)=e^{\frac{t}{10}},\,t\in[0,1]$ is the unique solution
(5.1)–(5.3).

###### Example 5.2

Consider the following second order Volterra integrodifferential equations:

$\displaystyle
w^{{}^{\prime\prime}}(t)=\frac{2}{10}-\frac{t^{2}}{1000}-\frac{(9-t)}{1000}+\frac{\cos(w(t))}{100}-\frac{w^{\prime}(t)}{100}$
$\displaystyle\qquad\qquad\qquad+\frac{1}{100}\int_{0}^{t}\left[\left(\frac{1+2s}{10}\right)\sin(w(s))+w^{\prime}(s)\right]\,ds,\,t\in
J=[0,2],$ (5.4)

subject to

$\displaystyle w(0)+w(t_{1})+w(t_{2})+w(t_{3})+w(t_{4})=1.35,\,\,$ (5.5)
$\displaystyle w^{\prime}(2)=5\,w^{\prime}(0),\,$ (5.6)

Comparing with the equation (1.1)–(1.3) we have
$T=2,\,\beta=5,\,t_{1}=0.5,\,t_{2}=1,\,t_{3}=1.5,\,t_{4}=2$ and
$c_{1}=c_{2}=c_{3}=c_{4}=1$ such that $\sum\limits_{k=1}^{4}c_{k}=4\not=-1$.
Define
$\mathscr{G}:[0,2]\times[0,2]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ by

$\mathscr{G}\left(t,s,w(s),w^{\prime}(s)\right)=\left[\left(\frac{1+2s}{10}\right)\sin(w(s))+w^{\prime}(s)\right].$

Then

$\left|\mathscr{G}(t,s,w_{1},w_{2})-\mathscr{G}(t,s,v_{1},v_{2})\right|\leq\left\\{\left|w_{1}-v_{1}\right|+\left|w_{2}-v_{2}\right|\right\\}\,\,t,s\in[0,2]\,\,\text{and}\,\,w_{1},w_{2},v_{1},v_{2}\in\mathbb{R}.$

Further, define
$\mathscr{F}:[0,2]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$
by

$\displaystyle\mathscr{F}\left(t,w(t),w^{\prime}(t),\int_{0}^{t}\mathscr{G}\left(t,s,w(s),w^{\prime}(s)\right)ds\right)$
$\displaystyle=\frac{2}{10}-\frac{t^{2}}{1000}-\frac{(9-t)}{1000}+\frac{\cos(w(t))}{100}-\frac{w^{\prime}(t)}{100}+\frac{1}{100}\int_{0}^{t}\mathscr{G}\left(t,s,w(s),w^{\prime}(s)\right)ds.$

Then

$\displaystyle\left|\mathscr{F}(t,w_{1},w_{2},w_{3})-\mathscr{F}(t,v_{1},v_{2},v_{3})\right|$
$\displaystyle\leq\frac{1}{100}\left\\{\left|w_{1}-v_{1}\right|+\left|w_{2}-v_{2}\right|+\left|w_{3}-v_{3}\right|\right\\},\,t\in[0,2]\,\,\text{and}\,\,w_{1},w_{2},w_{3},v_{1},v_{2},v_{3}\in\mathbb{R}.$

All the conditions of the Theorem 3.2 are satisfied with
$L_{\mathscr{F}}=\frac{1}{100}$ and $L_{\mathscr{G}}=1$. Take $\gamma=2$ then
we have

$\displaystyle
q=\frac{L_{\mathscr{F}}}{\gamma}\left(1+\frac{L_{\mathscr{G}}}{\gamma}\right)\left(1+\left[1+\left\\{T\beta+T\beta\left|\dfrac{\sum\limits_{k=1}^{p}c_{k}}{1+\sum\limits_{k=1}^{p}c_{k}}\right|\right\\}\right]\frac{e^{\gamma\,T}}{\beta-1}\right)=0.8395<1.$

Thus the Theorem 3.2 guarantee uniqueness solutions of (5.4)–(5.6). By actual
substitution one can verify that $w(t)=\frac{(t+t^{2})}{10},\,t\in[0,2]$ is
the unique solution (5.4)–(5.6).

##  6 Concluding Remarks

We conclude the paper with the objectives achieved: the existence, uniqueness
and dependency of solution for a second order Volterra integrodifferential
equations with nonlocal and boundary conditions through the Picard operators
theory. Note that the fractional calculus is the branch of mathematical
analysis, which has been on the rise over the decade, for its well-established
theory and its important results that have resulted to the point it has
attained today [33, 34]. Although many papers have been published and the
theory has been expanded, there are many avenues to be covered. In this sense,
an interesting idea is to adapt some results obtained here for the field of
fractional calculus, especially fractional differential equations, to put
other results, and to present the improvements that can be obtained, in
particular, making some comparisons through examples, in order to elucidate
the results obtained and consequently, the differences between the whole case
and the fractional one [35, 36].

## Acknowledgement

The first author is financially supported by UGC, New Delhi, India (Ref:
F1-17.1/2017-18/RGNF-2017-18-SC-MAH-43083). The third author JVCS acknowledges
the financial support of a PNPD-CAPES (process number nº88882.305834/2018-01)
scholarship of the Postgraduate Program in Applied Mathematics of IMECC-
Unicamp.

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