# Mathematical modelling of heat transfer in closed electrical contacts and
electrical potential field dynamics with Thomson effect

Targyn A. Nauryz1,2, Stanislav N. Kharin1,2 Adriana C. Briozzo3,4, Julieta
Bollati3,4 1International School of Economics, Kazakh-British Technical
University, Tole Bi 59, Almaty, Kazakhstan
2Institute of Mathematics and Mathematical Modeling, Pushkina 125, Almaty,
Kazakhstan
3CONICET, Argentina
4Depto de Matemática, F.C.E.,Universidad Austral,
Paraguay 1950, S2000FZF Rosario, Argentina

###### Abstract

In this study we develop a mathematical model that describe the behavior of
electromagnetic fields and heat transfer in closed electrical contacts that
arises when instantaneous explosion of the micro-asperity which involves
vaporization zone and liquid, solid zones where temperature is defined by a
generalized heat equation with Thomson effect. This model account for the
nonlinear nature of the thermal coefficients and electrical conductivity
depended on temperature. Our proposed solutions are based on similarity
transformation which allows us to reduce a Stefan-type problem to a system of
nonlinear integral equations whose existence of solution is proved by the
fixed point theory in Banach spaces.

###### keywords:

Stefan problem , Generalized heat equation , Thomson effect , Similarity
solution, Nonlinear integral equations , Nonlinear thermal coefficient, Fixed
point theorem

††journal: Journal

##  1 Introduction

Stefan problems are fundamental in understanding phase transition phenomena,
particularly in situations involving heat transfer and solidification
processes. They were first introduced by Stefan J. in his seminal work in Ste
(1889). These problems concern the determination of the moving boundary
between phases during the process of solidification or melting.

The classical Stefan problem arises in scenarios where a material undergoes a
phase change, such as freezing or melting, subject to certain boundary
conditions and physical constraints. One of the key aspects of Stefan problems
is the existence of a sharp interface, known as the Stefan interface, which
separates the regions of different phases.

Caffarelli and Souganidis in CaSo (1988) proposed a rate equation approach to
tackle Stefan problems, providing a mathematical framework to describe the
evolution of the phase boundary. This approach has since been widely utilized
in the analysis and numerical simulation of phase-change problems.

Theoretical investigations into Stefan problems have also contributed
significantly to the field. Rubinstein’s comprehensive work Rub (1971)
provides a mathematical foundation for understanding the Stefan problem from a
theoretical perspective. In Alex (1993) Alexiades and Solomon delved into the
mathematical intricacies of these processes, shedding light on the underlying
principles governing phase transitions.

Furthermore, the study of free and moving boundary problems, including Stefan
problems, has garnered considerable attention. In Cra (1984) Crank provides a
comprehensive treatment of such problems, offering valuable insights into
their mathematical formulation and solution techniques.

In recent years, researchers have explored various aspects of Stefan problems,
including the investigation of similarity solutions. Lipton and Witelski LipWi
(2007) analyze similarity solutions of the one-phase Stefan problem,
elucidating important characteristics of the phase-change process.

Stefan problems, which traditionally deal with phase-change phenomena under
classical heat conduction assumptions, have seen extensions to encompass more
complex physical scenarios. These extensions, often referred to as non-
classical Stefan problems, involve variations in thermal coefficients,
boundary conditions, or latent heat dependencies, among other factors. The
investigation of non-classical Stefan problems has significant implications in
various fields, including materials science, engineering, and mathematical
physics.

One avenue of research in non-classical Stefan problems involves the
consideration of thermal coefficients that vary with temperature or position.
In BoBr (2021), BrNatTar (2007) were explored Stefan problems for diffusion-
convection equations with temperature-dependent thermal coefficients,
providing insights into the behavior of phase-change processes under such
conditions. Similarly, in Ku2Raj (2019), Ku2Raj (2020) Kumar et al.
investigated Stefan problems with variable thermal coefficients, highlighting
the impact of these variations on the phase-change dynamics.

Another aspect of non-classical Stefan problems involves incorporating
convective boundary conditions or heat flux conditions on fixed faces. The
study BrNatTar (2007) examined the existence of exact solutions for one-phase
Stefan problems with nonlinear thermal coefficients, incorporating Tirskii’s
method to handle such complexities. Additionally, the paper BrTaz (2006)
addressed a one-phase Stefan problem for a non-classical heat equation with a
heat flux condition on the fixed face, contributing to the understanding of
phase-change phenomena under non-standard boundary conditions.

Moreover, the consideration of latent heat dependencies that vary with
position adds another layer of complexity to non-classical Stefan problems.
The work BoTa (2021) provided an explicit solution for a Stefan problem with
latent heat depending on the position, utilizing Kummer functions to analyze
the phase-change dynamics under such conditions.

Thermal phenomena in electrical apparatus, such as welding, arcing, and
bridging, contribute to their failure and are highly complex. These phenomena
depend on various factors including current, voltage, contact force, contact
material properties, and arc duration Holm (1984), Slade (1999). Experimental
investigations usually focus on cumulative probability representations of
resulting values Koren (1980)-Yamasaki (1984), as direct experimental
observation of these processes is often challenging or even impossible due to
their extremely short duration.

Hence, mathematical modeling plays a crucial role in understanding the
dynamics of such processes, improving the endurance and reliability of contact
systems, and predicting and preventing failures in electrical apparatus.
However, many existing mathematical models for describing processes like
electrical contact welding have limitations. They often make simplifying
assumptions, such as adiabatic heating conditions, neglecting phase
transformations and dynamic changes in contact radius Islaf (1982)-Zhenbiao
(1996). Additionally, important factors like bounce length and duration, time
and spatial variations of welding characteristics, and properties of contact
materials are often overlooked Zhenbiao (1996), Zhienbia (1992).

Efforts have been made in Kharin (1991)-SarErNauH (2017) to address these
aspects and the study of electrical contacts involves intricate thermal
dynamics influenced by non-linearities in material properties and heat
generation mechanisms. Non-linear Stefan problems offer a valuable
mathematical framework to model and analyze these complex phenomena, providing
insights into heat transfer processes during phase transitions within
electrical contacts NauBri (2023)-NauKas (2023). This paper aims to further
develop the existing models while also incorporating the Thomson effect.

The Thomson effect refers to the phenomenon where a temperature difference is
created across an electrical conductor when an electric current flows through
it. This effect occurs due to the interaction between the current-carrying
electrons and the lattice structure of the conductor.

In the context of a closure electrical contact after the instantaneous
explosion of a micro-asperity, it’s important to note that micro-asperities
are tiny protrusions or irregularities on the surface of a material. An
explosion or sudden release of energy can cause these micro-asperities to
rupture or deform.

After such an explosion, the closure electrical contact may be affected in
several ways. The intense energy release can lead to the melting or
vaporization of the micro-asperities, altering the surface characteristics of
the contact. This can potentially disrupt the normal flow of electric current
and create temperature variations due to the Thomson effect.

The Thomson effect in this scenario could result in localized heating or
cooling at the contact points, depending on the direction of the current flow.
This temperature difference might affect the electrical conductivity and
overall performance of the closure electrical contact.

It’s worth noting that the specific details and consequences of the Thomson
effect on a closure electrical contact after an instantaneous explosion of
micro-asperity would depend on various factors such as the material
properties, energy released, and contact configuration. Therefore, a detailed
analysis considering these factors would be necessary to fully understand the
implications of the Thomson effect in such a situation.

Figure 1: Contact zones: $Z_{0}:(0<z<s(t))$-vaporization zone,
$Z_{1}:(s(t)<z<r(t))$-liquid zone, $Z_{2}:(r(t)<z)$-solid zone

In the initial phase of a closed electrical contact, when a micro-asperity
undergoes sudden ignition, the contact region comprises both a metallic
vaporization zone and a liquid domain, see Figure 1. Modeling the metallic
vapor zone, denoted as $Z_{0}$ with a height range of $0<z<s(t)$, is a complex
undertaking. We propose that the temperature within this region decreases
linearly from the ionization temperature of the metallic vapor, denoted as
$T_{ion}$, which occurs after the explosion at the fixed face $z=0$, to the
boiling temperature $T_{b}$ at the free boundary that separates the vapor and
liquid phases. The temperature field within the vapor zone $Z_{0}$ exhibits a
gradual and linear decrease

$T_{V}(z,t)=\frac{z}{s(t)}(T_{b}-T_{ion})+T_{ion},\quad 0\leq z\leq s(t),$ (1)

where the following boundary conditions hold

$T_{V}(0,t)=T_{ion},$ (2) $T_{V}(s(t),t)=T_{b}.$ (3)

Temperature distribution and electrical potential field of the zones $Z_{1}$
and $Z_{2}$ are defined by

$c(T_{1})\gamma(T_{1})\dfrac{\partial T_{1}}{\partial
t}=\dfrac{1}{z^{\nu}}\dfrac{\partial}{\partial
z}\left[\lambda(T_{1})z^{\nu}\dfrac{\partial T_{1}}{\partial
z}\right]+\sigma_{T_{1}}\dfrac{\partial T_{1}}{\partial
z}\dfrac{\partial\varphi_{1}}{\partial
z}+\dfrac{1}{\rho(T_{1})}\left(\dfrac{\partial\varphi_{1}}{\partial
z}\right)^{2},$ (4) $\dfrac{1}{z^{\nu}}\dfrac{\partial}{\partial
z}\left[\dfrac{1}{\rho(T_{1})}z^{\nu}\dfrac{\partial\varphi_{1}}{\partial
z}\right]=0,\quad s(t)<z<r(t),\quad t>0,\quad 0<\nu<1,$ (5)
$c(T_{2})\gamma(T_{2})\dfrac{\partial T_{2}}{\partial
t}=\dfrac{1}{z^{\nu}}\dfrac{\partial}{\partial
z}\left[\lambda(T_{2})z^{\nu}\dfrac{\partial T_{2}}{\partial
z}\right]+\sigma_{T_{2}}\dfrac{\partial T_{2}}{\partial
z}\dfrac{\partial\varphi_{2}}{\partial
z}+\dfrac{1}{\rho(T_{2})}\left(\dfrac{\partial\varphi_{2}}{\partial
z}\right)^{2},$ (6) $\dfrac{1}{z^{\nu}}\dfrac{\partial}{\partial
z}\left[\dfrac{1}{\rho(T_{2})}z^{\nu}\dfrac{\partial\varphi_{2}}{\partial
z}\right]=0,\quad r(t)<z,\quad t>0,\quad 0<\nu<1,$ (7)
$T_{1}(s(t),t)=T_{b},\quad t>0,$ (8)
$-\lambda\left(T_{1}(s(t),t)\right)\dfrac{\partial T_{1}}{\partial
z}\Bigg{|}_{z=s(t)}=\dfrac{Q_{0}e^{-s_{0}^{2}}}{2a\sqrt{\pi t}},\quad t>0,$
(9) $\varphi_{1}(s(t),t)=0,\quad t>0,$ (10)
$T_{1}(r(t),t)=T_{2}(r(t),t)=T_{m}>0,\quad t>0,$ (11)
$\varphi_{1}(r(t),t)=\varphi_{2}(r(t),t),\quad t>0,$ (12)
$-\lambda\left(T_{1}(r(t),t)\right)\dfrac{\partial T_{1}}{\partial
z}\Bigg{|}_{z=r(t)}+\lambda\left(T_{2}(r(t),t)\right)\dfrac{\partial
T_{2}}{\partial z}\Bigg{|}_{z=r(t)}=l_{m}\gamma_{m}\dfrac{dr}{dt},\quad t>0,$
(13) $\dfrac{1}{\rho(T_{1}(r(t),t))}\dfrac{\partial\varphi_{1}}{\partial
z}\Bigg{|}_{z=r(t)}=\dfrac{1}{\rho(T_{2}(r(t),t))}\dfrac{\partial\varphi_{2}}{\partial
z}\Bigg{|}_{z=r(t)},\quad t>0,$ (14) $T_{2}(+\infty,t)=0,\quad t>0,$ (15)
$\varphi_{2}(+\infty,t)=\dfrac{U_{c}}{2},\quad t>0,$ (16)
$T_{2}(z,0)=\varphi_{2}(z,0)=0,\quad z>0,\qquad s(0)=r(0)=0,\quad$ (17)

where $T_{1},\;T_{2}$ and $\varphi_{1},\;\varphi_{2}$ are temperatures and
electrical potential fields for liquid and solid zones,
$c(T_{i})$,$\gamma(T_{i})$ and $\lambda(T_{i})$ are specific heat, density and
thermal conductivity which depend on the temperature, $\sigma_{T_{i}}$ is the
Thomson coefficient, $\rho(T_{i})$ is the electrical resistivity, $Q_{0}>0$ is
the power of the heat flux, $T_{m}$ is the melting temperature, $U_{c}$ is the
contact voltage, $s(t)$ and $r(t)$ are locations of the boiling and melting
interfaces.

This paper is structured as follows. In Section 2, we use the similarity
transformation to obtain an equivalent system of coupled integral equations
for the problem (4)-(17). In Section 3, we define proper spaces in order to
apply the fixed point Banach theorem to prove the existence of solution to the
system of coupled integral equations.

The contribution of the problem addressed in our paper holds significant
implications for electrical engineering. By developing a mathematical model
that captures the behavior of electromagnetic fields and heat transfer in
closed electrical contacts, particularly during instantaneous micro-asperity
explosions, we offer valuable insights into the complex dynamics of these
systems.

Our model accounts for the non-linear nature of thermal coefficients and
temperature-dependent electrical conductivity, factors that are crucial in
accurately representing real-world scenarios. By considering vaporization,
liquid, and solid zones within the contact, we provide a comprehensive
framework for analyzing the thermal and electromagnetic effects associated
with such phenomena.

Furthermore, our approach, which utilizes similarity transformations to reduce
the Stefan-type problem to a system of nonlinear integral equations, offers
practical methodologies for analyzing and predicting the behavior of
electrical contacts under extreme conditions. The rigorous establishment of
the validity of this approach through discussions and proofs supported by
fixed point theory in Banach space enhances the reliability and applicability
of our proposed solutions.

Overall, our contribution bridges theoretical insights with practical
applications, providing electrical engineers with tools and methodologies to
better understand, design, and optimize the performance and reliability of
electrical contacts in various operating conditions.

##  2 Integral formulation

In this section, taking into account that the problem (4)-(17) can be thought
as a Stefan-type problem, we look for similarity type solutions that depend on
the similarity variable

$\eta=\dfrac{z}{2a\sqrt{t}},$

with $a=\sqrt{\tfrac{\lambda_{0}}{\rho_{0}c_{0}}}$ where $\lambda_{0}$,
$\rho_{0}$ and $c_{0}$ are reference thermal coefficients.

We propose the following transformation

$f_{i}(\eta)=\frac{T_{i}(z,t)-T_{m}}{T_{m}},\qquad\phi_{i}(\eta)=\varphi_{i}(z,t),\quad,\quad
i=1,2.$ (18)

According to this transformation, the location of the boiling and melting
fronts are given by

$s(t)=2as_{0}\sqrt{t},\qquad\qquad r(t)=2ar_{0}\sqrt{t},$ (19)

where $s_{0}$ and $r_{0}$ must be determined as part of the solution.

Therefore, the problem (4)-(17) can be rewritten as the following form

$\left[L(f_{i})\eta^{\nu}f_{i}^{\prime}\right]^{\prime}+2a\eta^{\nu+1}N(f_{i})f_{i}^{\prime}+\dfrac{\sigma_{f_{i}}}{c_{0}\gamma_{0}a}\eta^{\nu}f_{i}^{\prime}\phi_{i}^{\prime}+\dfrac{\eta^{\nu}}{c_{0}\gamma_{0}T_{m}aK(f_{i})}\left(\phi_{i}^{\prime}\right)^{2}=0,$
(20) $\left[\dfrac{1}{K(f_{i})}\eta^{\nu}\phi_{i}^{\prime}\right]^{\prime}=0,$
(21) $i=1:\quad s_{0}<\eta<r_{0},\qquad\quad i=2:\quad\eta>r_{0},$
$f_{1}(s_{0})=B,$ (22)
$L(f_{1}(s_{0}))f_{1}^{\prime}(s_{0})=-Qe^{-s_{0}^{2}},$ (23)
$\phi_{1}(s_{0})=0,$ (24) $f_{1}(r_{0})=f_{2}(r_{0})=0,$ (25)
$\phi_{1}(r_{0})=\phi_{2}(r_{0}),$ (26)
$-L(f_{1}(r_{0}))f_{1}^{\prime}(r_{0})=-L(f_{2}(r_{0}))f_{2}^{\prime}(r_{0})+Mr_{0},$
(27)
$\dfrac{1}{K(f_{1}(r_{0}))}\phi_{1}^{\prime}(r_{0})=\dfrac{1}{K(f_{2}(r_{0}))}\phi_{2}^{\prime}(r_{0}),$
(28) $f_{2}(+\infty)=-1,$ (29) $\phi_{2}(+\infty)=\dfrac{U_{c}}{2},$ (30)

where

$B=\frac{T_{b}-T_{m}}{T_{m}},\qquad
Q=\dfrac{Q_{0}}{\lambda_{0}T_{m}\sqrt{\pi}}>0,\quad
M=\dfrac{2l_{m}\gamma_{m}a^{2}}{\lambda_{0}T_{m}}>0$ (31)

and for $i=1,2$:

$N(f_{i})=\frac{c(f_{i}T_{m}+T_{m})\gamma(f_{i}T_{m}+T_{m})}{c_{0}\gamma_{0}},$
(32) $L(f_{i})=\frac{\lambda(f_{i}T_{m}+T_{m})}{\lambda_{0}},$ (33)
$K(f_{i})=\rho(f_{i}T_{m}+T_{m}),$ (34) $\sigma_{f_{i}}=\sigma_{T_{i}},$ (35)

From (21), (24), (26), (28) and (30), we obtain the solution for electrical
potential field for liquid and solid zones explicitly in function of
$f_{1},f_{2},s_{0}$ and $r_{0}$ as

$\phi_{1}(\eta,s_{0},r_{0},f_{1},f_{2})=\dfrac{U_{c}F_{1}(\eta,s_{0},f_{1})}{2H(r_{0},s_{0},f_{1},f_{2})},\quad
s_{0}\leq\eta\leq r_{0},$ (36)
$\phi_{2}(\eta,s_{0},r_{0},f_{1},f_{2})=\dfrac{U_{c}\left(F_{1}(r_{0},s_{0},f_{1})+F_{2}(\eta,r_{0},f_{2})\right)}{2H(r_{0},s_{0},f_{1},f_{2})},\quad\eta\geq
r_{0},$ (37)

where

$F_{1}(\eta,s_{0},f_{1})=\int\limits_{s_{0}}^{\eta}\dfrac{K(f_{1}(v))}{v^{\nu}}dv,\quad
s_{0}\leq\eta\leq r_{0},$ (38)
$F_{2}(\eta,r_{0},f_{2})=\displaystyle\int\limits_{r_{0}}^{\eta}\dfrac{K(f_{2}(v))}{v^{\nu}}dv,\quad\eta\geq
r_{0},$ (39)

and

$H(r_{0},s_{0},f_{1},f_{2})=F_{1}(r_{0},s_{0},f_{1})+F_{2}(+\infty,r_{0},f_{2}).$
(40)

In addition, from $\eqref{eq26}$, $\eqref{eq28}$ and $\eqref{eq30}$, we get

$f_{1}(\eta)=s_{0}^{\nu}Q\exp(-s_{0}^{2})\left[\Phi_{1}(r_{0},s_{0},f_{1},f_{2})-\Phi_{1}(\eta,s_{0},f_{1},f_{2})\right]$
$+\frac{D_{1}^{*}}{H^{2}(r_{0},s_{0},f_{1},f_{2})}\left[{G_{1}(r_{0},s_{0},f_{1},f_{2})}-G_{1}(\eta,s_{0},f_{1},f_{2})\right],\quad
s_{0}\leq\eta\leq r_{0},$ (41)

and from $\eqref{eq26}$, $\eqref{eq30}$ and (29) we get

$f_{2}(\eta)=\left[\frac{D_{2}^{*}}{H^{2}(r_{0},s_{0},f_{1},f_{2})}G_{2}(+\infty,r_{0},f_{1},f_{2})-1\right]\frac{\Phi_{2}(\eta,r_{0},f_{1},f_{2})}{\Phi_{2}(+\infty,r_{0},f_{1},f_{2})}$
$-\frac{D_{2}^{*}}{H^{2}(r_{0},s_{0},f_{1},f_{2})}G_{2}(\eta,r_{0},f_{1},f_{2}),\quad\eta\geq
r_{0}.$ (42)

Moreover, from conditions (22) and (27) we obtain the following equations

$s_{0}^{\nu}Q\exp(-s_{0}^{2})\Phi_{1}(r_{0},s_{0},f_{1},f_{2})+\frac{D_{1}^{*}}{H^{2}(r_{0},s_{0},f_{1},f_{2})}{G_{1}(r_{0},s_{0},f_{1},f_{2})}=B,$
(43)

and

$E_{1}(r_{0},s_{0},f_{1},f_{2})\left[Q\exp(-s_{0}^{2})s_{0}^{\nu}+\frac{D_{1}^{*}}{H^{2}(r_{0},s_{0},f_{1},f_{2})}H_{1}(r_{0},s_{0},f_{1},f_{2})\right]$
$-\frac{1}{\Phi_{2}(+\infty,r_{0},f_{1},f_{2})}\left[1-\frac{D_{2}^{*}}{H^{2}(r_{0},s_{0},f_{1},f_{2})}G_{2}(+\infty,r_{0},f_{1},f_{2})\right]=Mr_{0}^{\nu+1}$
(44)

where

$\Phi_{1}(\eta,s_{0},f_{1},f_{2})=\displaystyle\int\limits_{s_{0}}^{\eta}\dfrac{E_{1}(v,s_{0},f_{1},f_{2})}{L(f_{1}(v))v^{\nu}}dv,\quad
s_{0}\leq\eta\leq r_{0},$ (45)
$\Phi_{2}(\eta,r_{0},f_{1},f_{2})=\displaystyle\int\limits_{r_{0}}^{\eta}\dfrac{E_{2}(v,r_{0},f_{1},f_{2})}{L(f_{2}(v))v^{\nu}}dv,\quad\eta\geq
r_{0},$ (46)
$G_{1}(\eta,s_{0},f_{1},f_{2})=\displaystyle\int\limits_{s_{0}}^{\eta}\dfrac{E_{1}(v,s_{0},f_{1},f_{2})}{L(f_{1}(v))v^{\nu}}H_{1}(v,r_{0},f_{1},f_{2})dv,\quad
s_{0}\leq\eta\leq r_{0},$ (47)
$G_{2}(\eta,r_{0},f_{1},f_{2})=\displaystyle\int\limits_{r_{0}}^{\eta}\dfrac{E_{2}(v,s_{0},f_{1},f_{2})}{L(f_{2}(v))v^{\nu}}H_{2}(v,r_{0},f_{1},f_{2})dv\quad\eta\geq
r_{0}$ (48)
$H_{1}(\eta,s_{0},f_{1},f_{2})=\displaystyle\int\limits_{s_{0}}^{\eta}\dfrac{K(f_{1}(v))}{v^{\nu}E_{1}(v,s_{0},f_{1},f_{2})}dv,\quad
s_{0}\leq\eta\leq r_{0},$ (49)
$H_{2}(\eta,r_{0},f_{1},f_{2})=\displaystyle\int\limits_{r_{0}}^{\eta}\dfrac{K(f_{2}(v))}{v^{\nu}E_{2}(v,r_{0},f_{1},f_{2})}dv\quad\eta\geq
r_{0}$ (50)
$E_{1}(\eta,s_{0},f_{1},f_{2})=\exp\left(-\displaystyle\int\limits_{s_{0}}^{\eta}\left[2av\tfrac{N(f_{1}(v))}{L(f_{1}(v))}+\tfrac{D_{1}}{H(r_{0},s_{0},f_{1},f_{2})}\tfrac{K(f_{1}(v))}{L(f_{1}(v))v^{\nu}}\right]dv\right),\quad
s_{0}\leq\eta\leq r_{0},$ (51)
$E_{2}(\eta,r_{0},f_{1},f_{2})=\exp\left(-\displaystyle\int\limits_{r_{0}}^{\eta}\left[2av\tfrac{N(f_{2}(v))}{L(f_{2}(v))}+\tfrac{D_{2}}{H(r_{0},s_{0},f_{1},f_{2})}\tfrac{K(f_{2}(v))}{L(f_{2}(v))v^{\nu}}\right]dv\right),\quad\eta\geq
r_{0},$ (52)

and the coefficients $D_{i}$ and $D_{i}^{*}$ for $i=1,2$ are given by

$D_{i}=\dfrac{\sigma_{f_{i}}U_{c}}{2c_{0}\gamma_{0}a},\qquad\qquad
D_{i}^{*}=\frac{U_{c}D_{i}}{2}.$ (53)

In conclusion, to find a similarity solution to the problem (4)-(17) is
equivalent to obtain $f_{1}$, $f_{2}$, $s_{0}$ and $r_{0}$ such that (41),
(42), (43) and (44) hold. Notice that the electric potential fields $\phi_{1}$
and $\phi_{2}$ are explicitly given by (36) and (37) in function of $f_{1}$,
$f_{2}$, $s_{0}$ and $r_{0}$.

In the next section, to address the existence and uniqueness of solutions, we
employ a rigorous analytical approach. We leverage similarity transformations
to reduce the problem to a set of ordinary differential equations,
facilitating a more tractable analysis. Additionally, we draw upon fixed point
theory in Banach space to establish the validity of our proposed solutions.

Through thorough discussions and rigorous proofs, we demonstrate the soundness
of our approach and the robustness of our solutions. This section not only
provides theoretical insights into the existence and uniqueness of solutions
within the context of electrical contact phenomena but also offers practical
methodologies for analyzing and predicting the behavior of such systems under
extreme conditions.

Overall, our analysis in this section contributes to a deeper understanding of
the mathematical underpinnings of electromagnetic fields and heat transfer in
closed electrical contacts. By establishing the existence and uniqueness of
solutions, we lay a solid foundation for further exploration and optimization
of electrical contact systems in various engineering applications.

##  3 Existence of solution

In order to prove the existence and uniqueness of solution $f_{1},f_{2}$ to
equations (41) and (42), we assume fixed positive constants $0<s_{0}<r_{0}$
and consider the Banach space

$\mathcal{C}=C[s_{0},r_{0}]\times C_{b}[r_{0},+\infty)$ (54)

endowed with the norm

$||\vec{f}||=\|(f_{1},f_{2})\|=\max\left\\{||f_{1}||_{C[s_{0},r_{0}]},||f_{2}||_{C_{b}[r_{0},+\infty)}\right\\}$

where $C[s_{0},r_{0}]$ denotes the space of continuous functions defined on
the interval $[s_{0},r_{0}]$ and $C_{b}[r_{0},+\infty)$ represents the space
of continuous and bounded functions on the interval $[r_{0},+\infty).$ We
define the closed subset $\mathcal{M}$ of $C_{b}[r_{0},+\infty)$ by

$\mathcal{M}=\\{f_{2}\in
C_{b}[r_{0},+\infty):f_{2}(r_{0})=0,f_{2}(+\infty)=-1\\}.$

We consider the operator $\Psi$ on
$\mathcal{K}=C[s_{0},r_{0}]\times\mathcal{M}$ given by

$\Psi(\vec{f})=(V_{1}(\vec{f}),V_{2}(\vec{f})),$ (55)

where $V_{1}(\vec{f})$, $V_{2}(\vec{f})$ are defined by

$\begin{array}[]{llll}&V_{1}(\vec{f})(\eta)=s_{0}^{\nu}Q\exp(-s_{0}^{2})\left[\Phi_{1}(r_{0},s_{0},f_{1},f_{2})-\Phi_{1}(\eta,s_{0},f_{1},f_{2})\right]\\\
\\\
&+\frac{D_{1}^{*}}{H^{2}(r_{0},s_{0},f_{1},f_{2})}\Big{[}{G_{1}(r_{0},s_{0},f_{1},f_{2})}-G_{1}(\eta,s_{0},f_{1},f_{2})\Big{]},\quad
s_{0}\leq\eta\leq r_{0},\end{array}$ (56)

$\begin{array}[]{llll}&V_{2}(\vec{f})(\eta)=\Big{[}\frac{D_{2}^{*}}{H^{2}(r_{0},s_{0},f_{1},f_{2})}G_{2}(+\infty,r_{0},f_{1},f_{2})-1\Big{]}\frac{\Phi_{2}(\eta,r_{0},f_{1},f_{2})}{\Phi_{2}(+\infty,r_{0},f_{1},f_{2})}\\\
\\\
&-\frac{D_{2}^{*}}{H^{2}(r_{0},s_{0},f_{1},f_{2})}G_{2}(\eta,r_{0},f_{1},f_{2}),\quad\eta\geq
r_{0}.\end{array}$ (57)

Notice that solving the system of equations (41) and (42) is equivalent to
obtain a fixed point to the operator $\Psi$.

Taking into account that $\mathcal{K}$ is a closed subset of $\mathcal{C}$ we
will prove that $\Psi(\mathcal{K})\subset\mathcal{K}$ and $\Psi$ is a
contraction mapping in order to apply the fixed point Banach theorem.

For this purpose we will assume that there exists positive coefficients $\mu$,
$L_{im},\;L_{iM},\;N_{im}$ and $N_{iM}$, $\widetilde{L}_{i}$ ,
$\widetilde{N}_{i}$ and $\widetilde{K}_{i}$ for $i=1,2$ such that

  1. (A1)

for each $f_{1}\in C[s_{0},r_{0}]:s_{0}\leq v\leq r_{0}$

$L_{1m}\eta^{\mu}\leq L(f_{1})(\eta)\leq L_{1M}\eta^{\mu},$ (58)
$N_{1m}\eta^{-\mu}\leq N(f_{1})(\eta)\leq N_{1M}\eta^{-\mu},$ (59)
$K_{1m}\eta^{-\mu}\leq K(f_{1})(\eta)\leq K_{1M}\eta^{-\mu},$ (60)

  2. (A2)

for each $f_{2}\in\mathcal{M},\;\eta\geq r_{0}:$

$L_{2m}\eta^{\mu}\leq L(f_{2})(\eta)\leq L_{2M}\eta^{\mu},$ (61)
$N_{2m}\eta^{-\mu}\leq N(f_{2})(\eta)\leq N_{2M}\eta^{-\mu},$ (62)
$K_{2m}\eta^{-\mu}\leq K(f_{2})(\eta)\leq K_{2M}\eta^{-\mu},$ (63)

  3. (A3)

for each $f_{1},g_{1}\in C[s_{0},r_{0}],s_{0}\leq\eta\leq r_{0}$:

$|L(f_{1}(\eta))-L(g_{1}(\eta))|\leq\widetilde{L}_{1}||f_{1}-g_{1}||,$ (64)
$|N(f_{1}(\eta))-N(g_{1}(\eta))|\leq\widetilde{N}_{1}||f_{1}-g_{1}||,$ (65)
$|K(f_{1}(\eta))-K(g_{1}(\eta))|\leq\widetilde{K}_{1}\eta^{-\mu}||f_{1}-g_{1}||,$
(66)

  4. (A4)

for each $f_{2},g_{2}\in\mathcal{M},\eta\geq r_{0}$ :

$|L(f_{2}(\eta))-L(g_{2}(\eta))|\leq\widetilde{L}_{2}||f_{2}-g_{2}||,$ (67)
$|N(f_{2}(\eta))-N(g_{2}(\eta))|\leq\widetilde{N}_{2}||f_{2}-g_{2}||,$ (68)
$|K(f_{2}(\eta))-K(g_{2}(\eta))|\leq\widetilde{K}_{2}\eta^{-\mu}||f_{2}-g_{2}||,$
(69)

  5. (A5)

$\mu>2$.

From now on, hypothesis (A1)-(A5) will be assumed throughout the paper.

We will present preliminary results that will be useful to prove the existence
and uniqueness of a fixed point to the operator $\Psi$.

######  Lemma 1.

For every $\vec{f}=(f_{1},f_{2})$, $\vec{g}=(g_{1},g_{2})\in\mathcal{K}$, the
following inequalities hold:

$H(r_{0},s_{0},f_{1},f_{2})\geq H_{inf}(r_{0},s_{0}),$ (70)
$H(r_{0},s_{0},f_{1},f_{2})\leq H_{sup}(r_{0},s_{0}),$ (71)
$|H(r_{0},s_{0},f_{1},f_{2})-H(r_{0},s_{0},g_{1},g_{2})|\leq\widetilde{H}(r_{0},s_{0})||\vec{f}-\vec{g}||,$
(72)

where

$H_{inf}(r_{0},s_{0}):=\tfrac{K_{1m}}{\mu+\nu-1}\left(\tfrac{1}{s_{0}^{\mu+\nu-1}}-\tfrac{1}{r_{0}^{\mu+\nu-1}}\right)$
(73)
$H_{sup}(r_{0},s_{0}):=\tfrac{1}{\mu+\nu-1}\left(\tfrac{K_{1M}}{s_{0}^{\mu+\nu-1}}+\tfrac{K_{2M}}{r_{0}^{\mu+\nu-1}}\right)$
(74)
$\widetilde{H}(r_{0},s_{0}):=\tfrac{1}{\mu+\nu-1}\left(\tfrac{\widetilde{K}_{1}}{s_{0}^{\mu+\nu-1}}+\tfrac{\widetilde{K}_{2}}{r_{0}^{\mu+\nu-1}}\right)$
(75)

###### Proof.

Taking into account the definition of $H$ given by (40) and assumptions (60)
and (63) we have

$\begin{array}[]{ll}H(r_{0},s_{0},f_{1},f_{2})\geq\tfrac{K_{1m}}{\mu+\nu-1}\left(\tfrac{1}{s_{0}^{\mu+\nu-1}}-\tfrac{1}{r_{0}^{\mu+\nu-1}}\right)+\tfrac{K_{2m}}{\mu+\nu-1}\tfrac{1}{r_{0}^{\mu+\nu-1}}\\\
\\\
\geq\tfrac{K_{1m}}{\mu+\nu-1}\left(\tfrac{1}{s_{0}^{\mu+\nu-1}}-\tfrac{1}{r_{0}^{\mu+\nu-1}}\right),\end{array}$

and then we get (70). Inequality (71) follows analogously. In addition, taking
into account assumptions (66) and (69) we get

$\begin{array}[]{ll}&|H(r_{0},s_{0},f_{1},f_{2})-H(r_{0},s_{0},g_{1},g_{2})|\\\\[9.95863pt]
&\leq\left(\widetilde{K}_{1}\displaystyle\int_{s_{0}}^{r_{0}}\tfrac{1}{v^{\mu+\nu}}dv+\widetilde{K}_{2}\displaystyle\int_{r_{0}}^{+\infty}\tfrac{1}{v^{\mu+\nu}}dv\right)||\vec{f}-\vec{g}||\\\
\\\
&\leq\tfrac{1}{\mu+\nu-1}\left(\widetilde{K}_{1}\left(\tfrac{1}{s_{0}^{\mu+\nu-1}}-\tfrac{1}{r_{0}^{\mu+\nu-1}}\right)+\tfrac{\widetilde{K}_{2}}{r_{0}^{\mu+\nu-1}}\right)||\vec{f}-\vec{g}||\\\
\\\
&\leq\tfrac{1}{\mu+\nu-1}\left(\tfrac{\widetilde{K}_{1}}{s_{0}^{\mu+\nu-1}}+\tfrac{\widetilde{K}_{2}}{r_{0}^{\mu+\nu-1}}\right)||\vec{f}-\vec{g}||,\end{array}$

as a consequence, inequality (72) holds. ∎

######  Lemma 2.

For every $\vec{f}=(f_{1},f_{2})$, $\vec{g}=(g_{1},g_{2})\in\mathcal{K}$, the
following inequalities hold:

  1. 1.

$E_{1}(\eta,s_{0},f_{1},f_{2})\geq E_{1inf}(r_{0},s_{0})$ (76)
$E_{1}(\eta,s_{0},f_{1},f_{2})\leq 1$ (77)
$|E_{1}(\eta,s_{0},f_{1},f_{2})-E_{1}(\eta,s_{0},g_{1},g_{2})|\leq\widetilde{E}_{1}(r_{0},s_{0})||\vec{f}-\vec{g}||$
(78)

where

$E_{1inf}(r_{0},s_{0}):=\exp\left(-\left[a\tfrac{N_{1M}}{L_{1m}(\mu-1)}\tfrac{1}{s_{0}^{2\mu-2}}+\tfrac{D_{1}K_{1M}}{H_{inf}(r_{0},s_{0})L_{1m}(2\mu+\nu-1)}\tfrac{1}{s_{0}^{2\mu+\nu-1}}\right]\right)$
(79)
$\begin{array}[]{l}\widetilde{E}_{1}(r_{0},s_{0}):=2a\left[\tfrac{\widetilde{N}_{1}}{L_{1m}(\mu-2)}\tfrac{1}{s_{0}^{\mu-2}}+\tfrac{N_{1M}\widetilde{L}_{1}}{L_{1m}^{2}(3\mu-2)}\tfrac{1}{s_{0}^{3\mu-2}}\right]\\\
\\\
+D_{1}\Big{(}\tfrac{\widetilde{K}_{1}}{H_{inf}(r_{0},s_{0})L_{1m}(2\mu+\nu-1)}\tfrac{1}{s_{0}^{2\mu+\nu-1}}\\\
\\\
+\tfrac{K_{1M}}{H_{inf}(r_{0},s_{0})L_{1m}}\Big{(}\tfrac{\widetilde{H}(r_{0},s_{0})}{H_{inf}(r_{0},s_{0})(2\mu+\nu-1)}\tfrac{1}{s_{0}^{2\mu+\nu-1}}+\tfrac{\widetilde{L}_{1}}{L_{1m}(3\mu+\nu-1)}\tfrac{1}{s_{0}^{3\mu+\nu-1}}\Big{)}\Big{)}\end{array}$
(80)

  2. 2.

$|\Phi_{1}(\eta,s_{0},f_{1},f_{2})-\Phi_{1}(\eta,s_{0},g_{1},g_{2})|\leq\widetilde{\Phi}_{1}(r_{0},s_{0})||\vec{f}-\vec{g}||$
(81)

where

$\widetilde{\Phi}_{1}(r_{0},s_{0}):=\tfrac{\widetilde{E}_{1}(r_{0},s_{0})}{L_{1m}(\mu+\nu-1)}\frac{1}{s_{0}^{\nu+\mu-1}}+\tfrac{\widetilde{L}_{1}}{L_{1m}^{2}(2\mu+\nu-1)}\frac{1}{s_{0}^{\nu+2\mu-1}}$
(82)

  3. 3.

$H_{1}(\eta,s_{0},f_{1},f_{2})\geq H_{1inf}(\eta,s_{0})$ (83)
$H_{1}(\eta,s_{0},f_{1},f_{2})\leq H_{1sup}(r_{0},s_{0})$ (84)
$|H_{1}(\eta,s_{0},f_{1},f_{2})-H_{1}(\eta,s_{0},g_{1},g_{2})|\leq\widetilde{H}_{1}(r_{0},s_{0})||\vec{f}-\vec{g}||$
(85)

where

$H_{1inf}(\eta,s_{0}):=\tfrac{K_{1m}}{(\mu+\nu-1)}\left(\tfrac{1}{s_{0}^{\mu+\nu-1}}-\tfrac{1}{\eta^{\mu+\nu-1}}\right)$
(86)
$H_{1sup}(r_{0},s_{0}):=\tfrac{K_{1M}}{E_{1inf}(r_{0},s_{0})}\tfrac{1}{(\mu+\nu-1)}\tfrac{1}{s_{0}^{\mu+\nu-1}}$
(87)
$\widetilde{H}_{1}(r_{0},s_{0}):=\left(\widetilde{K}_{1}+\tfrac{K_{1M}\widetilde{E}_{1}(r_{0},s_{0})}{E_{1inf}(r_{0},s_{0})}\right)\tfrac{1}{E_{1inf}(r_{0},s_{0})(\mu+\nu-1)}\tfrac{1}{s_{0}^{\mu+\nu-1}}$
(88)

  4. 4.

$G_{1}(\eta,s_{0},f_{1},f_{2})\geq G_{1inf}(\eta,r_{0},s_{0})$ (89)
$G_{1}(\eta,s_{0},f_{1},f_{2})\leq G_{1sup}(r_{0},s_{0})$ (90)
$|G_{1}(\eta,s_{0},f_{1},f_{2})-G_{1}(\eta,s_{0},g_{1},g_{2})|\leq\widetilde{G}_{1}(r_{0},s_{0})||\vec{f}-\vec{g}||$
(91)

where

$G_{1inf}(\eta,r_{0},s_{0}):=\tfrac{K_{1m}E_{1inf}(r_{0},s_{0})}{2L_{1M}(\mu+\nu-1)^{2}}\left(\tfrac{1}{s_{0}^{\mu+\nu-1}}-\tfrac{1}{\eta^{\mu+\nu-1}}\right)^{2},$
(92)
$G_{1sup}(r_{0},s_{0}):=\tfrac{H_{1sup}(r_{0},s_{0})}{L_{1m}}\tfrac{1}{(\mu+\nu-1)}\tfrac{1}{s_{0}^{\mu+\nu-1}},$
(93)
$\widetilde{G}_{1}(r_{0},s_{0}):=H_{1sup}(r_{0},s_{0})\widetilde{\Phi}_{1}(r_{0},s_{0})+\tfrac{\widetilde{H}_{1}(r_{0},s_{0})}{L_{1m}}\tfrac{1}{(\mu+\nu-1)}\tfrac{1}{s_{0}^{\mu+\nu-1}}.$
(94)

###### Proof.

From the definition of $E_{1}$ given by (51), assumptions (58)-(60) and
inequality (70) we obtain that

$\begin{array}[]{l}\displaystyle\int_{s_{0}}^{\eta}2av\tfrac{N(f_{1}(v))}{L(f_{1}(v)}+\tfrac{D_{1}}{H(r_{0},s_{0},f_{1},f_{2})}\tfrac{K(f_{1}(v))}{L(f_{1}(v))v^{\nu}}dv\\\
\\\
\leq\displaystyle\int_{s_{0}}^{\eta}2a\tfrac{N_{1M}}{L_{1m}}\tfrac{1}{v^{2\mu-1}}+\tfrac{D_{1}K_{1M}}{H_{inf}(r_{0},s_{0})L_{1m}}\tfrac{1}{v^{2\mu+\nu}}dv\\\
\\\ \leq
2a\tfrac{N_{1M}}{L_{1m}(2\mu-2)}\tfrac{1}{s_{0}^{2\mu-2}}+\tfrac{D_{1}K_{1M}}{H_{inf}(r_{0},s_{0})L_{1m}(2\mu+\nu-1)}\tfrac{1}{s_{0}^{2\mu+\nu-1}}.\\\
\\\ \end{array}$

As a consequence it results that $E_{1}(\eta,s_{0},f_{1},f_{2})\geq
E_{1inf}(r_{0},s_{0})$ with $E_{1inf}(r_{0},s_{0})$ given by (79).

In addition, as $E_{1}$ is a negative exponential function it follows that
$E_{1}(\eta,s_{0},f_{1},f_{2})\leq 1$.

Let us analyse the difference of $E_{1}$. From inequality
$|\exp(-x)-\exp(-y)|\leq|x-y|$ we have

$\begin{array}[]{l}\left|E_{1}(\eta,s_{0},f_{1},f_{2})-E_{1}(\eta,s_{0},g_{1},g_{2})\right|\leq\displaystyle\int_{s_{0}}^{\eta}2av\left|\tfrac{N(f_{1}(v))}{L(f_{1}(v))}-\tfrac{N(g_{1}(v))}{L(g_{1}(v))}\right|dv\\\
\\\
+D_{1}\displaystyle\int_{s_{0}}^{\eta}\left|\tfrac{K(f_{1}(v))}{H(r_{0},s_{0},f_{1},f_{2})L(f_{1}(v))v^{\nu}}-\tfrac{K(g_{1}(v))}{H(r_{0},s_{0},g_{1},g_{2})L(g_{1}(v))v^{\nu}}\right|dv.\end{array}$
(95)

On one hand,

$\begin{array}[]{l}\displaystyle\int_{s_{0}}^{\eta}2av\left|\tfrac{N(f_{1}(v))}{L(f_{1}(v))}-\tfrac{N(g_{1}(v))}{L(g_{1}(v))}\right|dv\\\
\\\ \leq 2a\displaystyle\int_{s_{0}}^{\eta}\left\lvert
v\tfrac{N(f_{1}(v))L(g_{1}(v))-N(g_{1}(v))L(g_{1}(v))+N(f_{1}(v))L(g_{1}(v))-N(g_{1}(v))L(f_{1}(v))}{L(f_{1}(v))L(g_{1}(v))}\right\rvert
dv\\\ \\\ \leq
2a\left[\displaystyle\int_{s_{0}}^{\eta}\tfrac{v\widetilde{N}_{1}||\vec{f}-\vec{g}||}{L_{1m}v^{\mu}}dv+\displaystyle\int_{s_{0}}^{\eta}\tfrac{N_{1M}\widetilde{L}_{1}v^{1-\mu}||\vec{f}-\vec{g}||}{L_{1m}^{2}v^{2\mu}}dv\right]\\\
\\\ \leq
2a\left[\tfrac{\widetilde{N}_{1}}{L_{1m}(\mu-2)}\tfrac{1}{s_{0}^{\mu-2}}+\tfrac{N_{1M}\widetilde{L}_{1}}{L_{1m}^{2}(3\mu-2)}\tfrac{1}{s_{0}^{3\mu-2}}\right]||\vec{f}-\vec{g}||\end{array}$
(96)

On the other hand,

$\begin{array}[]{l}D_{1}\displaystyle\int_{s_{0}}^{\eta}\left|\tfrac{K(f_{1}(v))}{H(r_{0},s_{0},f_{1},f_{2})L(f_{1}(v))v^{\nu}}-\tfrac{K(g_{1}(v))}{H(r_{0},s_{0},g_{1},g_{2})L(g_{1}(v))v^{\nu}}\right|dv\\\
\\\ \leq
D_{1}\Bigg{(}\displaystyle\int_{s_{0}}^{\eta}\left|\tfrac{K(f_{1}(v))}{H(r_{0},s_{0},f_{1},f_{2})L(f_{1}(v))v^{\nu}}-\tfrac{K(g_{1}(v))}{H(r_{0},s_{0},f_{1},f_{2})L(f_{1}(v))v^{\nu}}\right|dv\\\
\\\
+\displaystyle\int_{s_{0}}^{\eta}\left|\tfrac{K(g_{1}(v))}{H(r_{0},s_{0},f_{1},f_{2})L(f_{1}(v))v^{\nu}}-\tfrac{K(g_{1}(v))}{H(r_{0},s_{0},g_{1},g_{2})L(g_{1}(v))v^{\nu}}\right|dv\Bigg{)}\\\
\\\ \leq
D_{1}\Bigg{(}\displaystyle\int_{s_{0}}^{\eta}\tfrac{|K(f_{1}(v))-K(g_{1}(v))|}{H(r_{0},s_{0},f_{1},f_{2})L(f_{1}(v))v^{\nu}}dv\\\
\\\
+\displaystyle\int_{s_{0}}^{\eta}|K(g_{1}(v))|\left|\tfrac{1}{H(r_{0},s_{0},f_{1},f_{2})L(f_{1}(v))v^{\nu}}-\tfrac{1}{H(r_{0},s_{0},g_{1},g_{2})L(g_{1}(v))v^{\nu}}\right|dv\Bigg{)}.\\\
\\\ \end{array}$ (97)

From assumptions (58), (60), (66) and inequalities (70), (72) we get that

$\displaystyle\int_{s_{0}}^{\eta}\tfrac{|K(f_{1}(v))-K(g_{1}(v))|}{H(r_{0},s_{0},f_{1},f_{2})L(f_{1}(v))v^{\nu}}dv\leq\tfrac{\widetilde{K}_{1}}{H_{inf}(r_{0},s_{0})L_{1m}(2\mu+\nu-1)}\tfrac{1}{s_{0}^{2\mu+\nu-1}}||\vec{f}-\vec{g}||$
(98)

and

$\begin{array}[]{l}\displaystyle\int_{s_{0}}^{\eta}|K(g_{1}(v))|\left|\tfrac{1}{H(r_{0},s_{0},f_{1},f_{2})L(f_{1}(v))v^{\nu}}-\tfrac{1}{H(r_{0},s_{0},g_{1},g_{2})L(g_{1}(v))v^{\nu}}\right|dv\\\
\\\ \leq
K_{1M}\Bigg{(}\displaystyle\int_{s_{0}}^{\eta}\tfrac{|H(r_{0},s_{0},g_{1},g_{2})-H(r_{0},s_{0},f_{1},f_{2})|}{H(r_{0},s_{0},f_{1},f_{2})L(f_{1}(v))H(r_{0},s_{0},g_{1},g_{2})}\tfrac{dv}{v^{\mu+\nu}}\\\
\\\
+\displaystyle\int_{s_{0}}^{\eta}\tfrac{|L(g_{1}(v))-L(f_{1}(v))|}{L(f_{1}(v))H(r_{0},s_{0},g_{1},g_{2})L(g_{1}(v))}\tfrac{dv}{v^{\mu+\nu}}\Bigg{)}\\\
\\\
\leq\tfrac{K_{1M}}{H_{inf}(r_{0},s_{0})L_{1m}}\Bigg{(}\tfrac{\widetilde{H}(r_{0},s_{0})}{H_{inf}(r_{0},s_{0})(2\mu+\nu-1)}\tfrac{1}{s_{0}^{2\mu+\nu-1}}+\tfrac{\widetilde{L}_{1}}{L_{1m}(3\mu+\nu-1)}\tfrac{1}{s_{0}^{3\mu+\nu-1}}\Bigg{)}||\vec{f}-\vec{g}||.\end{array}$
(99)

Then inequalities (95)-(99) yields to

$\left|E_{1}(\eta,s_{0},f_{1},f_{2})-E_{1}(\eta,s_{0},g_{1},g_{2})\right|\leq\widetilde{E}_{1}(r_{0},s_{0})||\vec{f}-\vec{g}||$

with $\widetilde{E}_{1}$ given by (80).

From the definition of $\Phi_{1}$ given by (36) we have that

$\begin{array}[]{ll}|\Phi_{1}(\eta,s_{0},f_{1},f_{2})-\Phi_{1}(\eta,s_{0},g_{1},g_{2})|\leq\displaystyle\int_{s_{0}}^{\eta}\left|\tfrac{E_{1}(v,s_{0},f_{1},f_{2})}{L(f_{1}(v))}-\tfrac{E_{1}(v,s_{0},g_{1},g_{2})}{L(g_{1}(v))}\right|\tfrac{dv}{v^{\nu}}\\\
\\\
\leq\displaystyle\int_{s_{0}}^{\eta}\tfrac{|E_{1}(v,s_{0},f_{1},f_{2})-E_{1}(v,s_{0},g_{1},g_{2})|}{L(f_{1}(v))}\tfrac{dv}{v^{\nu}}+\displaystyle\int_{s_{0}}^{\eta}\tfrac{E_{1}(v,s_{0},g_{1},g_{2})|L(f_{1}(v))-L(g_{1}(v))|}{L(f_{1}(v))L(g_{1}(v))}\tfrac{dv}{v^{\nu}}\\\
\\\
\leq\left(\tfrac{\widetilde{E}_{1}(r_{0},s_{0})}{L_{1m}}\displaystyle\int_{s_{0}}^{\eta}\tfrac{1}{v^{\mu+\nu}}dv+\tfrac{\widetilde{L}_{1}}{L_{1m}^{2}}\displaystyle\int_{s_{0}}^{\eta}\tfrac{1}{v^{2\mu+\nu}}dv\right)||\vec{f}-\vec{g}||\leq\widetilde{\Phi}_{1}(r_{0},s_{0})||\vec{f}-\vec{g}||\end{array}$

where $\widetilde{\Phi}_{1}(r_{0},s_{0})$ is given by (82).

Taking into account the definition of $H_{1}$ given by (49) we esasily obtain
that

$H_{1}(\eta,s_{0},f_{1},f_{2})\geq
K_{1m}\displaystyle\int_{s_{0}}^{\eta}\tfrac{1}{v^{\mu+\nu}}dv\geq\tfrac{K_{1m}}{(\mu+\nu-1)}\left(\tfrac{1}{s_{0}^{\mu+\nu-1}}-\tfrac{1}{\eta^{\mu+\nu-1}}\right),$

$|H_{1}(\eta,s_{0},f_{1},f_{2})|\leq\tfrac{K_{1M}}{E_{1inf}(r_{0},s_{0})}\displaystyle\int_{s_{0}}^{\eta}\tfrac{1}{v^{\mu+\nu}}dv\leq\tfrac{K_{1M}}{E_{1inf}(r_{0},s_{0})}\tfrac{1}{(\mu+\nu-1)}\tfrac{1}{s_{0}^{\mu+\nu-1}},$

then (83) and (84) hold. In addition

$\begin{array}[]{ll}|H_{1}(\eta,s_{0},f_{1},f_{2})-H_{1}(\eta,s_{0},g_{1},g_{2})|\leq\displaystyle\int_{s_{0}}^{\eta}\left|\tfrac{K(f_{1}(v))}{E_{1}(v,s_{0},f_{1},f_{2})}-\tfrac{K(g_{1}(v))}{E_{1}(v,s_{0},g_{1},g_{2})}\right|\tfrac{dv}{v^{\nu}}\\\
\\\
\leq\displaystyle\int_{s_{0}}^{\eta}\tfrac{|K(f_{1}(v))-K(g_{1}(v))|}{E_{1}(v,s_{0},f_{1},f_{2})}\tfrac{dv}{v^{\nu}}+\displaystyle\int_{s_{0}}^{\eta}\tfrac{K(g_{1}(v))|E_{1}(v,s_{0},f_{1},f_{2})-E_{1}(v,s_{0},g_{1},g_{2})|}{E_{1}(v,s_{0},f_{1},f_{2})E_{1}(v,s_{0},g_{1},g_{2})}\tfrac{dv}{v^{\nu}}\\\
\\\
\leq\tfrac{1}{E_{1inf}(r_{0},s_{0})}\left(\widetilde{K}_{1}+\tfrac{K_{1M}\widetilde{E}_{1}(r_{0},s_{0})}{E_{1inf}(r_{0},s_{0})}\right)\displaystyle\int_{s_{0}}^{\eta}\tfrac{1}{v^{\mu+\nu}}dv\;\;||\vec{f}-\vec{g}||\leq\widetilde{H}_{1}(r_{0},s_{0})||\vec{f}-\vec{g}||\end{array}$

where $\widetilde{H}_{1}$ is given by (88).

From the definition of $G_{1}$ it follows that

$\begin{array}[]{ll}G_{1}(\eta,s_{0},f_{1},f_{2})|\geq\tfrac{E_{1inf}(r_{0},s_{0})}{L_{1M}}\displaystyle\int_{s_{0}}^{\eta}\tfrac{H_{1inf}(v,s_{0})}{v^{\mu+\nu}}dv\\\
\geq\tfrac{K_{1m}E_{1inf}(r_{0},s_{0})}{L_{1M}(\mu+\nu-1)}\displaystyle\int_{s_{0}}^{\eta}\tfrac{1}{v^{\mu+\nu}}\left(\tfrac{1}{s_{0}^{\mu+\nu-1}}-\tfrac{1}{v^{\mu+\nu-1}}\right)dv\geq
G_{1inf}(\eta,r_{0},s_{0}),\\\ \end{array}$

where $G_{1inf}$ is given by (92) and

$\begin{array}[]{ll}|G_{1}(\eta,s_{0},f_{1},f_{2})|\leq\displaystyle\int\limits_{s_{0}}^{\eta}\left|\tfrac{E_{1}(v,s_{0},f_{1},f_{2})H_{1}(v,s_{0},f_{1},f_{2})}{L(f_{1}(v))}\right|\tfrac{dv}{v^{\nu}}\\\
\\\
\leq\tfrac{H_{1sup}(r_{0},s_{0})}{L_{1m}}\displaystyle\int_{s_{0}}^{\eta}\tfrac{dv}{v^{\mu+\nu}}\leq
G_{1sup}(r_{0},s_{0}),\end{array}$

where $G_{1sup}$ is given by (93). Moreover

$\begin{array}[]{ll}|G_{1}(\eta,s_{0},f_{1},f_{2})-G_{1}(\eta,s_{0},g_{1},g_{2})|\\\
\\\
\leq\displaystyle\int_{s_{0}}^{\eta}|H_{1}(v,s_{0},f_{1},f_{2})|\left|\tfrac{E_{1}(v,s_{0},g_{1},g_{2})}{L(g_{1}(v))}-\tfrac{E_{1}(v,s_{0},f_{1},f_{2})}{L(f_{1}(v))}\right|\tfrac{dv}{v^{\nu}}\\\
\\\
+\displaystyle\int_{s_{0}}^{\eta}\tfrac{E_{1}(v,s_{0},g_{1},g_{2})|H_{1}(v,s_{0},f_{1},f_{2})-H_{1}(v,s_{0},g_{1},g_{2})|}{L(g_{1}(v))}\tfrac{dv}{v^{\nu}}\\\
\\\ \leq\widetilde{G}_{1}(r_{0},s_{0})||\vec{f}-\vec{g}||,\end{array}$

with $\widetilde{G}_{1}$ defined by (94).

∎

######  Lemma 3.

For every $\vec{f}=(f_{1},f_{2})$, $\vec{g}=(g_{1},g_{2})\in\mathcal{K}$, the
following inequalities hold:

  1. 1.

$E_{2}(\eta,r_{0},f_{1},f_{2})\geq E_{2inf}(r_{0},s_{0}),$ (100)
$E_{2}(\eta,r_{0},f_{1},f_{2})\leq 1,$ (101)
$|E_{2}(\eta,r_{0},f_{1},f_{2})-E_{2}(\eta,r_{0},g_{1},g_{2})|\leq\widetilde{E}_{2}(r_{0},s_{0})||\vec{f}-\vec{g}||,$
(102)

where

$E_{2inf}(r_{0},s_{0}):=\exp\left(-\left[a\tfrac{N_{2M}}{L_{2m}(\mu-1)}\tfrac{1}{r_{0}^{2\mu-2}}+\tfrac{D_{2}K_{2M}}{H_{inf}(r_{0},s_{0})L_{2m}(2\mu+\nu-1)}\tfrac{1}{r_{0}^{2\mu+\nu-1}}\right]\right),$
(103)
$\begin{array}[]{l}\widetilde{E}_{2}(r_{0},s_{0}):=2a\left[\tfrac{\widetilde{N}_{2}}{L_{2m}(\mu-2)}\tfrac{1}{r_{0}^{\mu-2}}+\tfrac{N_{2M}\widetilde{L}_{2}}{L_{2m}^{2}(3\mu-2)}\tfrac{1}{r_{0}^{3\mu-2}}\right]\\\
\\\
+D_{2}\Big{(}\tfrac{\widetilde{K}_{2}}{H_{inf}(r_{0},s_{0})L_{2m}(2\mu+\nu-1)}\tfrac{1}{r_{0}^{2\mu+\nu-1}}\\\
\\\
+\tfrac{K_{2M}}{H_{inf}(r_{0},s_{0})L_{2m}}\Big{(}\tfrac{\widetilde{H}(r_{0})}{H_{inf}(r_{0},s_{0})(2\mu+\nu-1)}\tfrac{1}{r_{0}^{2\mu+\nu-1}}+\tfrac{\widetilde{L}_{2}}{L_{2m}(3\mu+\nu-1)}\tfrac{1}{r_{0}^{3\mu+\nu-1}}\Big{)}\Big{)}.\end{array}$
(104)

  2. 2.

$\Phi_{2}(\eta,r_{0},f_{1},f_{2})\geq\Phi_{2inf}(\eta,r_{0},s_{0}),$ (105)
$\Phi_{2}(\eta,r_{0},f_{1},f_{2})\leq\Phi_{2sup}(r_{0}),$ (106)
$|\Phi_{2}(\eta,r_{0},f_{1},f_{2})-\Phi_{2}(\eta,r_{0},g_{1},g_{2})|\leq\widetilde{\Phi}_{2}(r_{0},s_{0})||\vec{f}-\vec{g}||,$
(107)

where

$\Phi_{2inf}(\eta,r_{0},s_{0}):=\tfrac{E_{2inf}(r_{0},s_{0})}{L_{2M}}\tfrac{1}{(\mu+\nu-1)}\left(\tfrac{1}{r_{0}^{\mu+\nu-1}}-\tfrac{1}{\eta^{\mu+\nu-1}}\right),$
(108)
$\Phi_{2sup}(r_{0}):=\tfrac{1}{L_{2m}}\tfrac{1}{(\mu+\nu-1)}\tfrac{1}{r_{0}^{\mu+\nu-1}},$
(109)
$\widetilde{\Phi}_{2}(r_{0},s_{0}):=\tfrac{\widetilde{E}_{2}(r_{0},s_{0})}{L_{2m}}\tfrac{1}{(\mu+\nu-1)}\tfrac{1}{r_{0}^{\mu+\nu-1}}+\tfrac{\widetilde{L}_{2}}{L_{2m}^{2}}\tfrac{1}{(2\mu+\nu-1)}\tfrac{1}{r_{0}^{2\mu+\nu-1}}.$
(110)

  3. 3.

$H_{2}(\eta,r_{0},f_{1},f_{2})\leq H_{2inf}(\eta,r_{0},s_{0}),$ (111)
$H_{2}(\eta,r_{0},f_{1},f_{2})\leq H_{2sup}(r_{0},s_{0}),$ (112)
$|H_{2}(\eta,r_{0},f_{1},f_{2})-H_{2}(\eta,r_{0},g_{1},g_{2})|\leq\widetilde{H}_{2}(r_{0},s_{0})||\vec{f}-\vec{g}||,$
(113)

where

$H_{2inf}(\eta,r_{0}):=\tfrac{K_{2m}}{(\mu+\nu-1)}\left(\tfrac{1}{r_{0}^{\mu+\nu-1}}-\tfrac{1}{\eta^{\mu+\nu-1}}\right),$
(114)
$H_{2sup}(r_{0},s_{0}):=\tfrac{K_{2M}}{E_{2inf}(r_{0},s_{0})}\tfrac{1}{(\mu+\nu-1)}\tfrac{1}{r_{0}^{\mu+\nu-1}},$
(115)
$\widetilde{H}_{2}(r_{0},s_{0}):=\left(\widetilde{K}_{2}+\tfrac{K_{2M}\widetilde{E}_{2}(r_{0},s_{0})}{E_{2inf}(r_{0},s_{0})}\right)\tfrac{1}{E_{2inf}(r_{0},s_{0})(\mu+\nu-1)}\tfrac{1}{r_{0}^{\mu+\nu-1}}.$
(116)

  4. 4.

$G_{2}(\eta,r_{0},f_{1},f_{2})\leq G_{2inf}(\eta,r_{0},s_{0}),$ (117)
$G_{2}(\eta,r_{0},f_{1},f_{2})\leq G_{2sup}(r_{0},s_{0}),$ (118)
$|G_{2}(\eta,r_{0},f_{1},f_{2})-G_{2}(\eta,r_{0},g_{1},g_{2})|\leq\widetilde{G}_{2}(r_{0},s_{0})||\vec{f}-\vec{g}||,$
(119)

where

$G_{2inf}(\eta,r_{0},s_{0}):=\tfrac{K_{2m}E_{2inf}(r_{0},s_{0})}{2L_{2M}(\mu+\nu-1)^{2}}\left(\tfrac{1}{r_{0}^{\mu+\nu-1}}-\tfrac{1}{\eta^{\mu+\nu-1}}\right)^{2},$
(120)
$G_{2sup}(r_{0},s_{0}):=\tfrac{H_{2sup}(r_{0},s_{0})}{L_{2m}}\tfrac{1}{(\mu+\nu-1)}\tfrac{1}{r_{0}^{\mu+\nu-1}},$
(121)
$\widetilde{G}_{2}(r_{0},s_{0}):=H_{2sup}(r_{0},s_{0})\widetilde{\Phi}_{2}(r_{0},s_{0})+\tfrac{\widetilde{H}_{2}(r_{0},s_{0})}{L_{2m}}\tfrac{1}{(\mu+\nu-1)}\tfrac{1}{r_{0}^{\mu+\nu-1}}.$
(122)

###### Proof.

The proof follows analogously to the previous lemma. ∎

######  Lemma 4.

For every $\vec{f}=(f_{1},f_{2}),\vec{g}=(g_{1},g_{2})\in\mathcal{K}$ it
follows that

$||V_{1}(\vec{f})-V_{1}(\vec{g})||_{C[s_{o},r_{0}]}\leq\varepsilon_{1}(r_{0},s_{0})||\vec{f}-\vec{g}||$

where

$\varepsilon_{1}(r_{0},s_{0})=2s_{0}^{\nu}Q\exp(-s_{0}^{2})\widetilde{\Phi}_{1}(r_{0},s_{0})+2D_{1}^{*}\bigg{(}\tfrac{G_{1sup}(r_{0},s_{0})2H_{sup}(r_{0},s_{0})\widetilde{H}(r_{0},s_{0})}{H^{4}_{inf}(r_{0},s_{0})}+\tfrac{\widetilde{G}_{1}(r_{0},s_{0})}{H^{2}_{inf}(r_{0},s_{0})}\bigg{)}$
(123)

###### Proof.

Taking into account that

$\begin{array}[]{l}\bigg{|}\frac{G_{1}(\eta,s_{0},f_{1},f_{2})}{H^{2}(r_{0},s_{0},f_{1},f_{2})}-\frac{G_{1}(\eta,s_{0},g_{1},g_{2})}{H^{2}(r_{0},s_{0},g_{1},g_{2})}\bigg{|}\leq\par\frac{|G_{1}(\eta,s_{0},f_{1},f_{2})||H^{2}(r_{0},s_{0},f_{1},f_{2})-H^{2}(r_{0},s_{0},g_{1},g_{2})|}{H^{2}(r_{0},s_{0},f_{1},f_{2})H^{2}(g_{1},g_{2})}\\\\[11.38092pt]
+\frac{|G_{1}(\eta,s_{0},f_{1},f_{2})-G_{1}(\eta,s_{0},g_{1},g_{2})|}{H^{2}(r_{0},s_{0},g_{1},g_{2})}\leq\bigg{[}\frac{G_{1sup}(r_{0},s_{0})2H_{sup}(r_{0},s_{0})\widetilde{H}(r_{0},s_{0})}{H^{4}_{inf}(r_{0},s_{0})}+\frac{\widetilde{G}_{1}(r_{0},s_{0})}{H^{2}_{inf}(r_{0},s_{0}}\bigg{]}||\vec{f}-\vec{g}||\\\
\end{array}$ (124)

then, for each $\eta\in[s_{0},r_{0}]$ it follows that

$\begin{array}[]{l}|V_{1}(\vec{f})(\eta)-V_{1}(\vec{g})(\eta)|\leq\\\\[7.11317pt]
\leq
s_{0}^{\nu}Q\exp(-s_{0}^{2})\big{[}|\Phi_{1}(r_{0},s_{0},f_{1},f_{2})-\Phi_{1}(r_{0},s_{0},g_{1},g_{2})|\\\
\\\
+|\Phi_{1}(\eta,s_{0},f_{1},f_{2})-\Phi_{1}(\eta,s_{0},g_{1},g_{2})|\big{]}\\\
\\\
+\big{|}\frac{D_{1}^{*}G_{1}(r_{0},s_{0},f_{1},f_{2})}{H^{2}(r_{0},s_{0},f_{1},f_{2})}-\frac{D_{1}^{*}G_{1}(r_{0},s_{0},g_{1},g_{2})}{H^{2}(r_{0},s_{0},g_{1},g_{2})}\bigg{|}+\bigg{|}\frac{D_{1}^{*}G_{1}(\eta,s_{0},f_{1},f_{2})}{H^{2}(r_{0},s_{0},f_{1},f_{2})}-\frac{D_{1}^{*}G_{1}(\eta,s_{0},g_{1},g_{2})}{H^{2}(r_{0},s_{0},g_{1},g_{2})}\big{|},\\\
\\\
\leq\Big{[}2s_{0}^{\nu}Q\exp(-s_{0}^{2})\widetilde{\Phi}_{1}(r_{0},s_{0})+2D_{1}^{*}\big{(}\frac{G_{1sup}(r_{0},s_{0})2H_{sup}(r_{0},s_{0})\widetilde{H}(r_{0},s_{0})}{H^{4}_{inf}(r_{0},s_{0})}+\frac{\widetilde{G}_{1}(r_{0},s_{0})}{H^{2}_{inf}(r_{0},s_{0})}\big{)}\Big{]}||\vec{f}-\vec{g}||\\\
\\\ =\varepsilon_{1}(r_{0},s_{0})||\vec{f}-\vec{g}||.\end{array}$ (125)

∎

######  Lemma 5.

For every $\vec{f}=(f_{1},f_{2}),\vec{g}=(g_{1},g_{2})\in\mathcal{K}$ it
follows that

$||V_{2}(\vec{f})-V_{2}(\vec{g})||_{C_{b}[r_{0},+\infty)}\leq\varepsilon_{2}(r_{0},s_{0})||\vec{f}-\vec{g}||$

where

$\varepsilon_{2}(r_{0},s_{0})=\varepsilon_{21}(r_{0},s_{0})+\varepsilon_{22}(r_{0},s_{0})+\varepsilon_{23}(r_{0},s_{0})$
(126)

with

$\begin{array}[]{ll}\varepsilon_{21}(r_{0},s_{0})=\frac{2\widetilde{\Phi}_{2}(r_{0},s_{0})}{\Phi_{2inf}(+\infty,r_{0},s_{0})},\\\
\\\
\varepsilon_{22}(r_{0},s_{0})=\frac{\widetilde{G}_{2}(r_{0},s_{0})}{H^{2}_{inf}(r_{0},s_{0})}+\frac{2G_{2sup}(r_{0},s_{0})H_{sup}(r_{0},s_{0})\widetilde{H}(r_{0},s_{0})}{H^{4}_{inf}(r_{0},s_{0})},\\\
\\\
\varepsilon_{23}(r_{0},s_{0})=\frac{\Phi_{2sup}(r_{0},s_{0})}{\Phi_{2inf}(+\infty,r_{0},s_{0})}\varepsilon_{22}(r_{0},s_{0})+\frac{G_{2sup}(r_{0},s_{0})}{H^{2}_{inf}(r_{0},s_{0})}\varepsilon_{21}(r_{0},s_{0}).\end{array}$

###### Proof.

On one hand we have that

$\begin{array}[]{ll}\bigg{|}\frac{\Phi_{2}(\eta,r_{0},f_{1},f_{2})}{\Phi_{2}(+\infty,r_{0},f_{1},f_{2})}-\frac{\Phi_{2}(\eta,r_{0},g_{1},g_{2})}{\Phi_{2}(+\infty,r_{0},g_{1},g_{2})}\bigg{|}\leq\frac{|\Phi_{2}(\eta,r_{0},f_{1},f_{2})-\Phi_{2}(\eta,r_{0},g_{1},g_{2})|}{\Phi_{2}(+\infty,r_{0},f_{1},f_{2})}\\\\[11.38092pt]
+\frac{\Phi_{2}(\eta,r_{0},g_{1},g_{2})}{\Phi_{2}(+\infty,r_{0},g_{1},g_{2})}\frac{|\Phi_{2}(+\infty,r_{0},f_{1},f_{2})-\Phi_{2}(+\infty,r_{0},g_{1},g_{2})|}{\Phi_{2}(+\infty,r_{0},f_{1},f_{2})}\\\\[11.38092pt]
\leq\frac{|\Phi_{2}(\eta,r_{0},f_{1},f_{2})-\Phi_{2}(\eta,r_{0},g_{1},g_{2})|}{\Phi_{2}(+\infty,r_{0},f_{1},f_{2})}+\frac{|\Phi_{2}(+\infty,r_{0},f_{1},f_{2})-\Phi_{2}(+\infty,r_{0},g_{1},g_{2})|}{\Phi_{2}(+\infty,r_{0},f_{1},f_{2})}\\\\[11.38092pt]
\leq\frac{2\widetilde{\Phi}_{2}(r_{0},s_{0})}{\Phi_{2inf}(+\infty,r_{0},s_{0})}||\vec{f}-\vec{g}||=\varepsilon_{21}(r_{0},s_{0})||\vec{f}-\vec{g}||.\end{array}$
(127)

On the other hand we obtain that

$\begin{array}[]{ll}\bigg{|}\frac{G_{2}(\eta,r_{0},f_{1},f_{2})}{H^{2}(s_{0},r_{0},f_{1},f_{2})}-\frac{G_{2}(\eta,r_{0},g_{1},g_{2})}{H^{2}(s_{0},r_{0},g_{1},g_{2})}\bigg{|}\\\
\\\
\leq\frac{|G_{2}(\eta,r_{0},f_{1},f_{2})-G_{2}(\eta,r_{0},g_{1},g_{2})|}{H^{2}(s_{0},r_{0},f_{1},f_{2})}+\frac{|G_{2}(\eta,r_{0},g_{1},g_{2})||H^{2}(s_{0},r_{0},g_{1},g_{2})-H^{2}(s_{0},r_{0},f_{1},f_{2})|}{H^{2}(s_{0},r_{0},f_{1},f_{2})H^{2}(s_{0},r_{0},g_{1},g_{2})}\\\
\\\
\leq\left(\frac{\widetilde{G}_{2}(r_{0},s_{0})}{H^{2}_{inf}(r_{0},s_{0})}+\frac{2G_{2sup}(r_{0},s_{0})H_{sup}(r_{0},s_{0})\widetilde{H}(r_{0},s_{0})}{H^{4}_{inf}(r_{0},s_{0})}\right)||\vec{f}-\vec{g}||\\\
\\\ =\varepsilon_{22}(r_{0},s_{0})||\vec{f}-\vec{g}||.\end{array}$ (128)

In addition

$\begin{array}[]{ll}\bigg{|}\frac{G_{2}(+\infty,r_{0},f_{1},f_{2})}{H^{2}(s_{0},r_{0},f_{1},f_{2})}\frac{\Phi_{2}(\eta,r_{0},f_{1},f_{2})}{\Phi_{2}(+\infty,r_{0},f_{1},f_{2})}-\frac{G_{2}(+\infty,r_{0},g_{1},g_{2})}{H^{2}(s_{0},r_{0},g_{1},g_{2})}\frac{\Phi_{2}(\eta,r_{0},g_{1},g_{2})}{\Phi_{2}(+\infty,r_{0},g_{1},g_{2})}\bigg{|}\\\
\\\
\leq\frac{\Phi_{2}(\eta,r_{0},f_{1},f_{2})}{\Phi_{2}(+\infty,r_{0},f_{1},f_{2})}\bigg{|}\frac{G_{2}(+\infty,r_{0},f_{1},f_{2})}{H^{2}(s_{0},r_{0},f_{1},f_{2})}-\frac{G_{2}(+\infty,r_{0},g_{1},g_{2})}{H^{2}(s_{0},r_{0},g_{1},g_{2})}\bigg{|}\\\
\\\
+\frac{G_{2}(+\infty,r_{0},g_{1},g_{2})}{H^{2}(s_{0},r_{0},g_{1},g_{2})}\bigg{|}\frac{\Phi_{2}(\eta,r_{0},f_{1},f_{2})}{\Phi_{2}(+\infty,r_{0},f_{1},f_{2})}-\frac{\Phi_{2}(\eta,r_{0},g_{1},g_{2})}{\Phi_{2}(+\infty,r_{0},g_{1},g_{2})}\bigg{|}\\\
\\\
\leq\left(\frac{\Phi_{2sup}(r_{0},s_{0})}{\Phi_{2inf}(+\infty,r_{0},s_{0})}\varepsilon_{22}(r_{0},s_{0})+\frac{G_{2sup}(r_{0},s_{0})}{H^{2}_{inf}(r_{0},s_{0})}\varepsilon_{21}(r_{0},s_{0})\right)||\vec{f}-\vec{g}||\\\
\\\ =\varepsilon_{23}(r_{0},s_{0})||\vec{f}-\vec{g}||.\end{array}$ (129)

Taking into account the previous inequalities, for each $\eta\geq r_{0}$, it
follows that

$\begin{array}[]{l}|V_{2}(\vec{f})(\eta)-V_{2}(\vec{g})(\eta)|\\\\[7.11317pt]
\leq\bigg{|}\frac{D_{2}^{*}G_{2}(+\infty,r_{0},f_{1},f_{2})}{H^{2}(r_{0},s_{0},f_{1},f_{2})}\frac{\Phi_{2}(\eta,r_{0},f_{1},f_{2})}{\Phi_{2}(+\infty,r_{0},f_{1},f_{2})}-\frac{D_{2}^{*}G_{2}(+\infty,r_{0},g_{1},g_{2})}{H^{2}(r_{0},s_{0},g_{1},g_{2})}\frac{\Phi_{2}(\eta,r_{0},g_{1},g_{2})}{\Phi_{2}(+\infty,r_{0},g_{1},g_{2})}\bigg{|}\\\\[7.11317pt]
+\bigg{|}\frac{\Phi_{2}(\eta,r_{0},f_{1},f_{2})}{\Phi_{2}(+\infty,r_{0},f_{1},f_{2})}-\frac{\Phi_{2}(\eta,r_{0},g_{1},g_{2})}{\Phi_{2}(+\infty,r_{0},g_{1},g_{2})}\bigg{|}+\bigg{|}\frac{D_{2}^{*}G_{2}(\eta,r_{0},f_{1},f_{2})}{H^{2}(r_{0},s_{0},f_{1},f_{2})}-\frac{D_{2}^{*}G_{2}(\eta,r_{0},g_{1},g_{2})}{H^{2}(r_{0},s_{0},g_{1},g_{2})}\bigg{|}\\\\[7.11317pt]
\leq\varepsilon_{2}(r_{0},s_{0})||\vec{f}-\vec{g}||.\end{array}$ (130)

∎

######  Theorem 1.

For every $\vec{f}=(f_{1},f_{2})$, $\vec{g}=(g_{1},g_{2})\in\mathcal{K}$ it
follows that

$||\Psi(\vec{f})-\Psi(\vec{g})||\leq\varepsilon(r_{0},s_{0})||\vec{f}-\vec{g}||$

with

$\varepsilon(r_{0},s_{0})=\max\left\\{\varepsilon_{1}(r_{0},s_{0}),\varepsilon_{2}(r_{0},s_{0})\right\\}$
(131)

where $\varepsilon_{1}(r_{0},s_{0})$ and $\varepsilon_{2}(r_{0},s_{0})$ are
given by (123) and (126), respectively.

###### Proof.

From the previous lemmas we have that

$\begin{array}[]{ll}||\Psi(\vec{f})-\Psi(\vec{g})||=\max\left\\{||V_{1}(\vec{f})-V_{1}(\vec{g})||_{C[s_{0},r_{0}]},||V_{2}(\vec{f})-V_{2}(\vec{g})||_{C_{b}[r_{0},+\infty)}\right\\}\\\
\\\
=\max\left\\{\varepsilon_{1}(r_{0},s_{0})||\vec{f}-\vec{g}||,\varepsilon_{2}(r_{0},s_{0})||\vec{f}-\vec{g}||\right\\}=\varepsilon(r_{0},s_{0})||\vec{f}-\vec{g}||.\end{array}$

∎

Now we will look for conditions that guarantee that $\Psi$ is a contraction
mapping.

For each $s_{0}>0$ fixed, we define the functions
$\varepsilon_{1,s_{0}}(r_{0})=\varepsilon_{1}(r_{0},s_{0})$ and
$\varepsilon_{2,s_{0}}(r_{0})=\varepsilon_{2}(r_{0},s_{0})$, for all
$r_{0}>s_{0}$ where $\varepsilon_{1}$, $\varepsilon_{2}$ are given by (123)
and (126), respectively. The following results hold:

######  Lemma 6.

  1. a)

The function $\varepsilon_{1,s_{0}}$ is a decreasing function that satisfies
$\varepsilon_{1,s_{0}}(s_{0})=+\infty$ and
$\varepsilon_{1,s_{0}}(+\infty)=j_{1}(s_{0})$ where

$\begin{array}[]{ll}j_{1}(s_{0})=2s_{0}^{\nu}Q\exp(-s_{0}^{2})\widetilde{\Phi}_{1}(+\infty,s_{0})\\\
\\\
+2D_{1}^{*}\bigg{(}\tfrac{G_{1sup}(+\infty,s_{0})2H_{sup}(+\infty,s_{0})\widetilde{H}(+\infty,s_{0})}{H^{4}_{inf}(+\infty,s_{0})}+\tfrac{\widetilde{G}_{1}(+\infty,s_{0})}{H^{2}_{inf}+\infty,s_{0})}\bigg{)}\end{array}$
(132)

  2. b)

If

$\tfrac{2D_{1}^{*}\widetilde{K}_{1}}{L_{1m}K_{1m}^{2}}\left(\tfrac{2K_{1M}}{K_{1m}^{2}}+1\right)<1$
(133)

then there exists a unique $s_{1}>0$ such that $j_{1}(s_{0})<1$ for all
$s_{0}>s_{1}$. Moreover, for each $s_{0}>s_{1}$ there exists
$r_{1}=r_{1}(s_{0})>s_{0}$ such that $\varepsilon_{1,s_{0}}(r_{1})=1$ and
$\varepsilon_{1,s_{0}}(r_{0})<1$ for all $r_{0}>r_{1}$.

###### Proof.

  1. a)

According to the definition of $\varepsilon_{1}$ given by (123), the proof
follows straightforwardly from Lemmas 1 and 2.

  2. b)

From the definition of $j_{1}$ given by (132), we have that it is a decreasing
function that satisfies $j_{1}(0)=+\infty$ and
$j_{1}(+\infty)=\tfrac{2D_{1}^{*}\widetilde{K}_{1}}{L_{1m}K_{1m}^{2}}\left(\tfrac{2K_{1M}}{K_{1m}^{2}}+1\right)$.
Then, assuming (133) it follows that there exists a unique $s_{1}>s_{0}$ such
that $j_{1}(s_{1})=1$ and $j_{1}(s_{0})<1$ for all $s_{0}>s_{1}$. Moreover,
from item a), for each $s_{0}>s_{1}$ there exists $r_{1}=r_{1}(s_{0})>s_{0}$
such that $\varepsilon_{1,s_{0}}(r_{1})=1$ and
$\varepsilon_{1,s_{0}}(r_{0})<1$ for all $r_{0}>r_{1}$.

∎

######  Lemma 7.

  1. a)

The function $\varepsilon_{2,s_{0}}$ is a decreasing function that satisfies
$\varepsilon_{2,s_{0}}(s_{0})=+\infty$ and $\varepsilon_{2,s_{0}}(+\infty)=0$.

  2. b)

For each $s_{0}>0$ there exists $r_{2}=r_{2}(s_{0})>s_{0}$ such that
$\varepsilon_{2,s_{0}}(r_{2})=1$ and $\varepsilon_{2,s_{0}}(r_{0})<1$ for all
$r_{0}>r_{2}$.

###### Proof.

  1. a)

It follows from Lemmas 1 and 3 taking into account that $\varepsilon_{2}$ is
defined by (126).

  2. b)

It is clear from item a)

∎

######  Theorem 2.

If inequality (133) holds, then for each $(r_{0},s_{0})\in\Sigma$ with

$\Sigma=\\{(r_{0},s_{0}):\;s_{0}>s_{1},\;r_{0}>\overline{r}_{0}(s_{0})\\}$
(134)

we have that $\varepsilon(r_{0},s_{0})<1$ where $\varepsilon$ is given by
(131) and

$\overline{r}_{0}(s_{0})=\max\\{r_{1}(s_{0}),r_{2}(s_{0})\\}$ (135)

with $s_{1},r_{1}$ and $r_{2}$ defined in Lemmas 6 and 7, respectively.

###### Proof.

The proof follows immediately by Lemmas 6 and 7. ∎

######  Corollary 1.

Under the assumption (133), for each $(r_{0},s_{0})\in\Sigma$, the operator
$\Psi$ defined by (55) is a contraction mapping.

######  Theorem 3.

Under the assumption (133), for each $(r_{0},s_{0})\in\Sigma$, there exists a
unique fixed point $(f_{1}^{*},f_{2}^{*})\in\mathcal{K}$ to the operator
$\Psi$.

###### Proof.

First, notice that $\mathcal{K}$ is a closed subset of the Banach space
$\mathcal{C}$ given by (54). In addition, it is easy to see that
$\Psi(\vec{f})\in\mathcal{K}$ given that $V_{1}(\vec{f})\in C[s_{0},r_{0}]$,
$V_{2}(\vec{f})\in C_{b}[r_{0},+\infty)$, $V_{2}(\vec{f})(r_{0})=0$ and
$V_{2}(\vec{f})(+\infty)=0$. Finally, according to Corollary 1, under the
assumption (133), for each $(r_{0},s_{0})\in\Sigma$ it results that $\Psi$ is
a contraction mapping. As a consequence, applying the fixed point Banach
theorem, there exists a unique fixed point
$(f_{1}^{*},f_{2}^{*})\in\mathcal{K}$ to the operator $\Psi$ for each
$(r_{0},s_{0})\in\Sigma$.

∎

######  Corollary 2.

If (133) holds, for each $(r_{0},s_{0})\in\Sigma$, there exists a unique
solution $(f_{1}^{*},f_{2}^{*})$ to the system of equations (41)-(42).

It remains to prove the existence of solution $(r_{0},s_{0})\in\Sigma$ to the
system of equations given by (43) and (44) where $f_{1}=f_{1}^{*}$ and
$f_{2}=f_{2}^{*}$ are the unique solutions to the equations (41)-(42). For
that purpose we will need some preliminary results.

Let us notice that equation (43) can be rewritten as

$X(r_{0},s_{0})=Y(r_{0},s_{0})$ (136)

where

$X(r_{0},s_{0})=Z(r_{0},s_{0})-B,\quad
Z(r_{0},s_{0})=\tfrac{D_{1}^{*}G_{1}(r_{0},s_{0},f_{1}^{*},f_{2}^{*})}{H^{2}(r_{0},s_{0},f_{1}^{*},f_{2}^{*})}$
(137)

and

$\begin{array}[]{ll}Y(r_{0},s_{0})=-Qs_{0}^{\nu}\exp(-s_{0}^{2})\Phi_{1}(r_{0},s_{0},f_{1}^{*},f_{2}^{*}).\end{array}$
(138)

######  Lemma 8.

The following properties hold:

  * a)

$Y(r_{0},s_{0})<0$ for each $(r_{0},s_{0})\in\Sigma$.

  * b)

$Z(r_{0},s_{0})>Z_{inf}(r_{0},s_{0})$ for each $(r_{0},s_{0})\in\Sigma$, where

$Z_{inf}(r_{0},s_{0})=\tfrac{D_{1}^{*}E_{1inf}(r_{0},s_{0})K_{1m}}{2L_{1M}}\left(\tfrac{r_{0}^{\mu+\nu-1}-s_{0}^{\mu+\nu-1}}{K_{1M}r_{0}^{\mu+\nu-1}+K_{2M}s_{0}^{\mu+\nu-1}}\right)^{2}.$

  * c)

For a fixed $s_{0}>s_{1}$, if we assume

$X(\overline{r}_{0}(s_{0}),s_{0})<Y(\overline{r}_{0}(s_{0}),s_{0})$ (139)

then $Z_{inf}(\cdot,s_{0})$ is an increasing function that satisfies

$Z_{inf}(\overline{r_{0}}(s_{0}),s_{0})<B,\qquad
Z_{inf}(+\infty,s_{0})=j_{2}(s_{0}),$ (140)

where

$j_{2}(s_{0})=\tfrac{D_{1}^{*}E_{1inf}(+\infty,s_{0})K_{1m}}{L_{1M}K_{1M}^{2}},$
(141)

is an increasing function that satisfies
$j_{2}(+\infty)=\tfrac{D_{1}^{*}K_{1m}}{2L_{1M}K_{1M}^{2}}$.

  * e)

If we assume

$\tfrac{D_{1}^{*}K_{1m}}{2L_{1M}K_{1M}^{2}}>B$ (142)

then there exists a unique $s_{2}=\min\\{s_{0}\geq s_{1}:j_{2}(s_{0})\geq
B\\}$. Moreover, for each $s_{0}>s_{2}$, we have that $j_{2}(s_{0})>B$.

  * f)

If (139) and (142) holds for each $s_{0}>s_{2}$ there exists a unique
$r_{B}(s_{0})>s_{0}$ such that $Z_{inf}(r_{0},s_{0})>B$ for all
$r_{0}>r_{B}(s_{0})$.

###### Proof.

It is clear from the definition of the function $Y$ given by (138).

It follows from the inequalities obtained in Lemmas 1 and 2.

From the definition of $Z_{inf}$ it it easy to see that $Z_{inf}(\cdot,s_{0})$
is an increasing function for each fixed $s_{0}>s_{1}$. In addition,
assumption (139) and item a) leads to

$Z_{inf}(\overline{r}_{0}(s_{0}),s_{0})-B<Z(\overline{r}_{0}(s_{0}),s_{0})-B<Y(\overline{r}_{0}(s_{0}),s_{0})<0,$

Then it follows that $Z_{inf}(\overline{r}_{0}(s_{0}),s_{0})<B$. Finally,
taking a limit gives that $Z_{inf}(+\infty,s_{0})=j_{2}(s_{0})$ for each
$s_{0}>s_{1}$.

First, notice that hypothesis (142) can be rewritten as $j_{2}(+\infty)>B$.
From the fact that $j_{2}$ is an increasing function, we can conclude that
there exists a unique $s_{2}=\min\left\\{s_{0}\geq s_{1}:j_{2}(s_{0})\geq
B\right\\}$. Notice that $s_{2}=s_{1}$ in case that $j_{2}(s_{1})>B$. As a
consequence for each $s_{0}>s_{2}$, we get that $j_{2}(s_{0})>B$.

For each fixed $s_{0}>s_{2}$, we have that
$Z_{inf}(\overline{r}_{0}(s_{0}),s_{0})<B$ from item c) and
$Z_{inf}(+\infty,s_{0})>B$ from item e). Then there exists a unique
$r_{B}=r_{B}(s_{0})>\overline{r}_{0}(s_{0})$ such that
$Z_{inf}(r_{B}(s_{0}),s_{0})=B$ and $Z_{inf}(r_{0},s_{0})>B$ for all
$r_{0}>r_{B}(s_{0})$. ∎

######  Lemma 9.

For each $s_{0}>s_{2}$, if we assume (139) and (142) there exists at least one
solution
$r_{0}^{*}=r_{0}^{*}(s_{0},f_{1}^{*},f_{2}^{*})\in\left(\overline{r}_{0}(s_{0}),r_{B}(s_{0})\right)$
to the equation (43).

###### Proof.

For each $s_{0}>s_{2}$, taking into account assumption (139) and the fact that
from item f) of Lemma 8 the following inequality holds

$X(r_{B}(s_{0}),s_{0})\geq
Z_{inf}(r_{B}(s_{0}),s_{0})-B=0>Y(r_{B}(s_{0}),s_{0})$

we obtain that there exists at least one solution
$r_{0}^{*}\in\left(\overline{r}_{0}(s_{0}),r_{B}(s_{0})\right)$ to the
equation (43). ∎

Now we will analyze the equation (44). If we replace $r_{0}$ by
$r_{0}^{*}(s_{0})$ and $(f_{1},f_{2})$ by $(f_{1}^{*},f_{2}^{*})$, it results
equivalent to

$W(r_{0}^{*}(s_{0}),s_{0})=M$ (143)

where

$\begin{array}[]{l}W(r_{0}^{*}(s_{0}),s_{0})=\tfrac{E_{1}(r_{0}^{*}(s_{0}),s_{0},f_{1}^{*},f_{2}^{*})}{r_{0}^{\nu+1}}\Bigg{[}Q\exp(-s_{0}^{2})s_{0}^{\nu}+\frac{D_{1}^{*}}{H^{2}(r_{0}^{*}(s_{0}),s_{0},f_{1}^{*},f_{2}^{*})}H_{1}(r_{0}^{*}(s_{0}),s_{0},f_{1}^{*},f_{2}^{*})\Bigg{]}\\\
\\\
-\frac{1}{r_{0}^{*}(s_{0})^{\nu+1}\Phi_{2}(+\infty,r_{0}^{*}(s_{0}),f_{1}^{*},f_{2}^{*})}\Bigg{[}1-\frac{D_{2}^{*}}{H^{2}(r_{0}^{*}(s_{0}),s_{0},f_{1}^{*},f_{2}^{*})}G_{2}(+\infty,r_{0}^{*}(s_{0}),f_{1}^{*},f_{2}^{*})\Bigg{]}.\end{array}$
(144)

######  Lemma 10.

If any of the following system of inequalities hold

$\left\\{\begin{array}[]{ll}W_{inf}(s_{2})>M\\\ \\\
W_{sup}(+\infty)<M\end{array}\right.\qquad\text{or}\qquad\left\\{\begin{array}[]{ll}W_{sup}(s_{2})<M\\\
\\\ W_{inf}(+\infty)>M\end{array}\right.$ (145)

then there exists at least one solution $\widehat{s_{0}}>s_{2}$ to the
equation (143), where

$\begin{array}[]{l}W_{inf}(s_{0})=\tfrac{E_{1inf}(r_{0}^{*}(s_{0}),s_{0})}{r_{B}^{\nu+1}(s_{0})}\Bigg{[}Q\exp(-s_{0}^{2})s_{0}^{\nu}+\frac{D_{1}^{*}}{H_{sup}^{2}(r_{0}^{*}(s_{0}),s_{0})}H_{1inf}(r_{0}^{*}(s_{0}),s_{0})\Bigg{]}\\\
\\\
-\frac{1}{\overline{r_{0}}^{\nu+1}(s_{0})\Phi_{2inf}(+\infty,r_{0}^{*}(s_{0}),s_{0})}+\frac{1}{r_{B}^{\nu+1}(s_{0})\Phi_{2sup}(r_{0}^{*}(s_{0}))}\frac{D_{2}^{*}}{H_{sup}^{2}(r_{0}^{*}(s_{0}),s_{0})}G_{2inf}(+\infty,r_{0}^{*}(s_{0}),s_{0}).\end{array}$
(146)
$\begin{array}[]{l}W_{sup}(s_{0})=\tfrac{1}{\overline{r_{0}}^{\nu+1}(s_{0})}\Bigg{[}Q\exp(-s_{0}^{2})s_{0}^{\nu}+\frac{D_{1}^{*}}{H_{inf}^{2}(r_{0}^{*}(s_{0}),s_{0})}H_{1sup}(r_{0}^{*}(s_{0}),s_{0})\\\
\\\
+\frac{1}{\Phi_{2inf}(+\infty,r_{0}^{*}(s_{0}),s_{0})}\frac{D_{2}^{*}}{H_{inf}^{2}(r_{0}^{*}(s_{0}),s_{0})}G_{2sup}(r_{0}^{*}(s_{0}),s_{0}).\Bigg{]}.\end{array}$
(147)

The prior analysis allows to establish the following existence theorem.

######  Theorem 4.

Assuming (A1)-(A5), (133), (139), (142) and (145) there exists at least one
solution $(\widehat{s_{0}},r_{0}^{*}(\widehat{s_{0}}),f_{1}^{*},f_{2}^{*})$
where $(f_{1}^{*},f_{2}^{*})$ is the unique fixed point of the operator $\Psi$
corresponding to $(\widehat{s_{0}},r_{0}^{*}(\widehat{s_{0}}))\in\Sigma$.

######  Corollary 3.

If (A1)-(A5), (133), (139), (142) and (145) hold there exists at least one
solution to the problem (4)-(17) where

$\left\\{\begin{array}[]{lll}T_{1}(z,t)=T_{m}f_{1}^{*}\left(\tfrac{z}{2a\sqrt{t}}\right)+T_{m},&s(t)\leq
z\leq r(t),\;t>0\\\ \\\
T_{2}(z,t)=T_{m}f_{2}^{*}\left(\tfrac{z}{2a\sqrt{t}}\right)+T_{m},&z\geq
r(t),\;t>0\\\ \\\
\varphi_{1}(z,t)=\dfrac{U_{c}}{2}\cdot\dfrac{F_{1}\left(\tfrac{z}{2a\sqrt{t}},\widehat{s_{0}},f_{1}^{*}\right)}{H(r_{0}^{*}(\widehat{s_{0}}),\widehat{s_{0}},f_{1}^{*},f_{2}^{*})}&s(t)\leq
z\leq r(t),\;t>0\\\ \\\
\varphi_{2}(z,t)=\dfrac{U_{c}}{2}\cdot\dfrac{F_{1}(r_{0}^{*}(\widehat{s_{0}}),\widehat{s_{0}},f_{1}^{*})+F_{2}\left(\tfrac{z}{2a\sqrt{t}},r_{0}^{*}(\widehat{s_{0}}),f_{2}^{*}\right)}{H(r_{0}^{*}(\widehat{s_{0}}),\widehat{s_{0}},f_{1}^{*},f_{2}^{*})},&z\geq
r(t),\;t>0\\\ \\\ \end{array}\right.$

with $s(t)=2a\widehat{s_{0}}\sqrt{t}$ and
$r(t)=2ar_{0}^{*}(\widehat{s_{0}})\sqrt{t}$.

The contribution of the section on existence and uniqueness in our paper lies
in the development of a comprehensive mathematical model that captures the
intricate behavior of electromagnetic fields and heat transfer in closed
electrical contacts. We address the complex phenomenon of instantaneous micro-
asperity explosion, incorporating vaporization, liquid, and solid zones, while
accounting for non-linear thermal coefficients and temperature-dependent
electrical conductivity.

Our proposed solutions, based on similarity transformations, facilitate the
reduction of the problem to ordinary differential equations, streamlining the
analysis process. We rigorously establish the validity of this approach
through discussions and proofs supported by fixed point theory in Banach
space. This section not only offers theoretical insights into the existence
and uniqueness of solutions but also provides practical methodologies for
tackling the intricate dynamics of electrical contacts under extreme
conditions.

##  4 Conclusions

In conclusion, this study has presented novel mathematical models to
characterize the intricate dynamics of electromagnetic fields and heat
transfer within closed electrical contacts, particularly focusing on the
instantaneous explosion of micro-asperities. These explosions give rise to
complex phenomena involving vaporization zones, as well as liquid and solid
phases, where temperatures are governed by a generalized heat equation
incorporating Thomson effect considerations.

One of the key strengths of our models lies in their ability to capture the
nonlinear behavior exhibited by thermal coefficients and electrical
conductivity, both of which are temperature-dependent. By employing similarity
transformations, we have effectively reduced the problem to a set of ordinary
differential equations, thereby facilitating tractable analysis and solution.

The validity and utility of our approach have been rigorously demonstrated
through discussions and proofs grounded in fixed point theory within the
framework of Banach spaces. This theoretical underpinning not only enhances
our confidence in the proposed solutions but also provides a solid foundation
for future research endeavors in related domains.

Furthermore, the insights gained from this study hold significant implications
for various practical applications involving electrical contacts, such as in
the design and optimization of electronic devices, electrical connectors, and
power transmission systems. By elucidating the intricate interplay between
electromagnetic fields and heat transfer phenomena, our work contributes to
advancing the understanding and engineering of such systems in both industrial
and academic contexts.

In essence, this research underscores the importance of integrating
mathematical modeling with theoretical analysis to unravel the complexities
inherent in multifaceted physical processes. Moving forward, we envision
further refinement and extension of our models to address additional
complexities and pave the way for enhanced performance and reliability of
electrical contact systems in diverse real-world scenarios.

##  5 Acknowledgement

The study was sponsored by AP19675480 ”Problems of heat conduction with a free
boundary arising in modeling of switching processes in electrical devices”
from Ministry of Sciences and Higher Education of the Republic of Kazakhstan
and projects O06-24CI1901 and O06-24CI1903 from Austral University, Rosario

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