# On a non local non-homogeneous fractional Timoshenko system with frictional
and viscoelastic damping terms

Said Mesloub, Eman Alhazzani, Gadain Hassan Eltayeb  Mathematics Department,
College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi
Arabia<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We are devoted to the study of a nonhomogeneous time fractional timoshenko
system with frictional and viscoelastic damping terms. We are concerned with
the well-posedness of the given problem. The approach relies on some
fucntional analysis tools, operator theory, a prori estimates and density
arguments.

###### Key words and phrases:

Fractional Timoshenko system; memory term damping; frictional damping; non
local constraint; a priori estimate; Well posedness.
2010 Mathematics Subject Classification: 35B40, 35B37, 35L55

##  1\. Introduction

Vibrations of beams are not always safe and welcomed because of their great
and irreparable damages effects. In this situation, researchers try to
introduce some damping mechanisms (viscous damping, thermoelastic damping,
modal damping, frictional damping) in such a way that these damaging and
destructive vibrations are perfectly reduced. In other words, an intensive
investigation has been carried out to impose minimal conditions to provide and
guarantee stability of Timoshenko systems using several types of dissipative
mechanisms.

As a classical and a simple model [1], Timoshenko studied the following
coupled hyperbolic system

$\left\\{\begin{array}[]{c}\begin{array}[]{c}\rho_{1}\theta_{tt}-\kappa(\theta_{x}-\mathcal{\phi})_{x}=0,\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ }(x,t)\in\text{ }(0,L)\times(0,\infty)\\\
\rho_{2}\mathcal{\phi}_{tt}=\kappa^{\ast}\mathcal{\phi}_{xx}+\kappa(\theta_{x}-\mathcal{\phi})\text{
\ \ \ }(x,t)\in\text{ }(0,L)\times(0,\infty),\end{array}\\\
(\theta_{x}-\mathcal{\phi})\mid_{x=0}^{x=L}=0,\text{ \
}\mathcal{\phi}_{x}\mid_{x=0}^{x=L}=0,\end{array}\right.$ (1.1)

describing the transverse vibration of a beam. where $L$ is the length of the
beam in its equilibrium configuration. The function $\theta$ models the
transverse displacement of the beam and $\mathcal{\phi}$ models the rotation
angle of its filament. The coefficients $\rho_{1},$ $\rho_{2},$ $\kappa$ and
$\kappa^{\ast}$ are respectively the density, the polar moment of inertia of a
cross section, the shear modulus and the Young’s modulus of elasticity.
Timoshenko system (1.1) was generalized and studied by many authors. As
mentioned at the beginning of the introduction, different types of dampings
were added to the Timoshenko system for the purpose of its stabilization. For
example, in [4], researchers investigated the exponential stability for a
Timoshenko system having two weak dampings

$\left\\{\begin{array}[]{c}\rho_{1}\theta_{tt}=\kappa(\theta_{x}-\mathcal{\phi})_{x}-\theta_{t},\text{
\ \ in }(0,L)\times(0,\infty),\\\
\rho_{2}\mathcal{\phi}_{tt}=\kappa^{\ast}\mathcal{\phi}_{xx}-\kappa(\theta_{x}-\mathcal{\phi})_{x}-\mathcal{\phi}_{t},\text{
\ \ in }(0,L)\times(0,\infty),\\\
\theta(0,t)=\theta(L,t)=\mathcal{\phi}(0,t)=\mathcal{\phi}(L,t)=0,\text{ \
}t>0.\end{array}\right.$ (1.2)

In [2], authors proved some exponential decay results for a Timoshenko system
with a memory damping term

$\left\\{\begin{array}[]{c}\rho_{1}\theta_{tt}-\kappa_{1}(\theta_{x}+\mathcal{\phi})_{x}=0,\text{
\ \ \ in }(0,L)\times(0,\infty)\\\
\rho_{2}\mathcal{\phi}_{tt}-\kappa_{2}\mathcal{\phi}_{xx}+\kappa_{1}(\theta_{x}+\mathcal{\phi})+h\ast\mathcal{\phi}_{xx}(x,t)=0,\text{
in }(0,L)\times(0,\infty)\\\
\theta(0,t)=\theta(L,t)=\mathcal{\phi}(0,t)=\mathcal{\phi}(L,t)=0,\\\
\theta(x,0)=\theta_{0},\text{ }\theta_{t}(x,0)=\theta_{1},\text{
}\mathcal{\phi}(x,0)=\mathcal{\phi}_{0},\text{
}\mathcal{\phi}_{t}(x,0)=\mathcal{\phi}_{1}.\end{array}\right.$ (1.3)

Authors considered and studied in [5] the effect of frictional and
viscoelastic dampings, and proved some exponential and polynomial decay
results for the system

$\left\\{\begin{array}[]{c}\theta_{tt}-(\theta_{x}+\mathcal{\phi})_{x}=0,\\\
\mathcal{\phi}_{tt}-\mathcal{\phi}_{xx}+\theta_{x}+\mathcal{\phi}+\int\limits_{0}^{t}g(t-s)(a(x)\mathcal{\phi}_{x}(x,s))_{x}ds+b(x)h(\mathcal{\phi}_{t})=0,\\\
\theta(0,t)=\theta(1,t)=\mathcal{\phi}(0,t)=\mathcal{\phi}(1,t)=0,\text{ \
}t>0.\end{array}\right.$ (1.4)

We also mention that in [$44$], the authors investigated the exponential
stabilization of a Timoshenko system by a thermal effect damping.

$\left\\{\begin{array}[]{c}\rho_{1}\theta_{tt}-\kappa_{1}(\theta_{x}+\mathcal{\phi})_{x}=0,\text{
\ \ \ in }(0,L)\times(0,\infty)\\\
\rho_{2}\mathcal{\phi}_{tt}-\kappa_{2}\mathcal{\phi}_{xx}+\kappa_{1}(\theta_{x}+\mathcal{\phi})+\gamma\mathcal{\omega}_{x},\text{
in }(0,L)\times(0,\infty)\\\
\rho_{2}\mathcal{\omega}_{tt}-\kappa_{3}\mathcal{\omega}_{xx}+\beta\int\limits_{0}^{t}g(t-s)\mathcal{\omega}_{xx}(x,s)ds+\gamma\mathcal{\phi}_{ttx},\text{
in }(0,L)\times(0,\infty).\end{array}\right.$ (1.5)

In [3], the author considered a Timoshenko linear thermoelastic system with
linear frictional damping and a distributed delay. He proved the well-
posedness, and proved that the system is exponentially stable regardless of
the speeds of wave propagation.There are many other papers in the literature
dealing with the stabilization of different version of Timoshenko systems. For
more results concerning the stabilization and controllability of Timoshenko
systems, we refer the reader to [6, 7, 8, 9), 10, 11, 12, 13, 14, 15, 31].

Recently, a generalization of the Timoshenko system (1.1) into fractional
setting is studied in [45] by using a fractional version of resolvents. The
author established the well posedness of a fractional Timoshenko system, and
proved that lower order fractional terms can stabilize the system in a Mittag-
Leffler fashion. More precisely, the author considered the initial boundary
value problem

$\left\\{\begin{array}[]{c}\rho_{1}\partial_{t}^{\alpha}(\partial_{t}^{\alpha}\theta)-\kappa_{1}(\theta_{x}+\mathcal{\phi})_{x}=0,\text{
\ \ \ in }(0,1)\times(0,\infty)\\\
\rho_{2}\partial_{t}^{\alpha}(\partial_{t}^{\alpha}\mathcal{\phi+}a\mathcal{\phi})-\kappa_{2}\mathcal{\phi}_{xx}+\kappa_{1}(\theta_{x}+\mathcal{\phi}),\text{
in }(0,1)\times(0,\infty)\\\ \theta(0,t)=\theta(1,t)=0,\text{
}\mathcal{\phi}(0,t)=\mathcal{\phi}(1,t)=0,\text{ \ }t>0\\\
\theta(x,0)=\theta_{0}(x)\ ,\ \mathcal{\phi(}x,0)=\psi(x).\end{array}\right.$

Motivated by the above results on Timoshenko systems, we consider a non local
initial boundary value problem for a non-homogeneous fractional Timoshenko
system with a frictional damping in the first equation and a viscoelastic
memory damping term in the second equation. The system is complemented with
initial conditions and non local purely boundary integral conditions. At the
beginning of the year 1963, Cannon [20] was the first researcher to
investigate a non local problem with a non local constraint (energy
specification) of the form $\int\limits_{0}^{l}\chi(x)U(x,t)dt=\tau(t)$, where
$\chi(x),$ and $\tau(t)$ are given functions. More precisely, he used the
potential method to investigate the well posedness of the heat equation
subject to the specification of energy. This type of conditions arise mainly
when the data cannot be measured directly on the boundary, but only their
averages (weighted averages) are known. Due to their importance, physical
significance (mean, total flux, total energy,..) and numerous applications in
different fields of science and engineering, such as underground water flow,
vibration problems, heat conduction, medical science, nuclear reactor
dynamics, thermoelasticity, and plasma physics and control theory, several
authors extensively studied this type of problems. We can cite for example
[16, 21, 22, 23, 24, 25, 26, $28,29,30,31$]. Note that theoretical study of
non local problems is connected with great difficulties, since the presence of
integral terms in the boundary conditions can greatly complicate the
application of classical methods of functional analysis method, especially
when it comes to the fractional case . A functional analysis method based on
some a priori bounds and on the density of the range of the unbounded operator
corresponding to the abstract formulation of the given problem is used to
prove the well posedness of the posed problem. This is shown through the
introduction of some multiplier operators, some classical and fractional
inequalities, and the establishment of some properties, involving fractional
derivatives.

To the best of our knowledge, the treated fractional system problem
(2.1)-(2.4) has never been studied and explored in the literature. This work
can be considered as a contribution in the development of the traditional
functional analysis method, the so called energy inequality method used to
prove the well posedness of mixed problems with integral boundary conditions.
For some classical cases, the reader can refeer for example to example [16,
17, 18, 19, 27], and for some fractional cases, the reader should refeer to
$[32,33,34,36,37,38,39,40,41].$ We should also mention here that there are
some important papers dealing with numerical aspects for Timoshenko systems,
and having many applications, for which the reader can refer to [47 48, 49,
50]. There are some papers dealing with Timoshenko system with fractional
operator in the memory [42, 43].

##  2\. Formulation of the problem and function spaces

Given the interval $I=(0,L),$ we consider the non-homogeneous fractional
viscoelastic beam model with frictional damping of Timoshenko type

$\left\\{\begin{array}[]{c}\mathcal{L}_{1}(\theta,\mathcal{\phi})=\rho_{1}\partial_{t}^{\alpha+1}\theta-\kappa_{1}(\theta_{x}+\mathcal{\phi})_{x}+\theta_{t}=F(x,t)\\\
\mathcal{L}_{2}(\theta,\mathcal{\phi})=\rho_{2}\partial_{t}^{\alpha+1}\mathcal{\phi}-\kappa_{2}\mathcal{\phi}_{xx}+\kappa_{1}(\theta_{x}+\mathcal{\phi})+\int\limits_{0}^{t}m(t-s)\mathcal{\phi}_{xx}(x,s)ds=G(x,t),\end{array}\right.$
(2.1)

in the unknowns $(\theta,\phi):(x,t)$ $\in I\times[0,T]\rightarrow\mathbb{R},$
the strictly positive constants $\rho_{1},\rho_{2},\kappa_{1}$ and
$\kappa_{2}$ satisfy the relation

$\frac{\rho_{1}}{\kappa_{1}}=\frac{\rho_{2}}{\kappa_{2}},$

and $\ f\ ,\ g\ ,\ \varphi\ ,\ \psi,$ $F,$ and $G\ $are given functions, and
$m:$ $\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is a twice differentiable
function such that

$\kappa_{2}-\int\limits_{0}^{T}m(t)dt=l>0,\text{ \ }m^{\prime}(t)<0,\text{
}\forall t\geq 0.$ (2.2)

The system (2.1) is complemented with the initial conditions

$\left\\{\begin{array}[]{c}\Gamma_{1}\theta=\theta(x,0)=\varphi(x)\ ,\
\Gamma_{2}\theta=\theta_{t}(x,0)=\psi(x)\ ,\\\
\Gamma_{1}\mathcal{\phi}=\mathcal{\phi}(x,0)=f(x)\ ,\
\Gamma_{2}\mathcal{\phi}=\mathcal{\phi}_{t}(x,0)=g(x),\end{array}\right.$
(2.3)

and the non local boundary integral conditions

$\int\limits_{0}^{L}\theta dx=0\ ,\ \int\limits_{0}^{L}x\theta
dx=0,\int\limits_{0}^{L}\mathcal{\phi}dx=0\ ,\
\int\limits_{0}^{L}x\mathcal{\phi}dx=0.$ (2.4)

This system of coupled hyperbolic equations represents a Timoshenko model for
a thick beam of length $L,$where $\theta$ is the transverse displacement of
the beam and $\mathcal{\phi}$ is the rotation angle of the filament of the
beam. The coefficients $\rho_{1},\rho_{2},\kappa_{1}$ and $\kappa_{2}$ are
respectively the density, the polar moment of inertia of a cross section, the
shear modulus and the Young’s modulus of elasticity. The integral conditions
represent the averages (weighted averages) of the total transverse
displacement of the beam and the rotation angle of the filament of the beam.

Our aim is to study the well posedness of the solution of problem (2.1),
(2.4). That is on the basis of some a priori bounds and on the density of the
range of the operator generated by the problem under consideration, we prove
the existence, uniqueness and continuous dependence of the solution on the
given data of problem (2.1), (2.4). We now introduce some function spaces
needed throughout the sequel. Let $L^{2}(Q^{T})$ be the Hilbert space of
square integrable functions on $Q^{T}=(0,1)\times(0,T),$ $T<\infty,$ with
scalar product and norm respectively

$(Z,S)_{L^{2}(Q^{T})}=\int_{Q^{T}}ZSdxdt,\text{ \ \ \
}\left\|Z\right\|_{L^{2}(Q^{T})}^{2}=\int_{Q^{T}}Z^{2}dxdt.$ (2.5)

We also use the space $L^{2}((0,1))$ on the interval $(0,1)$, whose definition
is analogous to the space on $Q.$ Let $B_{2}^{1}(0,L)$ be the space obtained
by completion of the space $C^{0}(0,L)$ of real continuous functions with
compact support in the interval $(0,L)$ with respect to the inner product

$(\gamma,\gamma^{\ast})_{B_{2}^{1}(0,L)}=\int\limits_{0}^{L}\mathcal{I}_{x}\gamma.\Im_{x}\gamma^{\ast}dx,$

where $\mathcal{I}_{x}\gamma=\int\limits_{0}^{x}\gamma(\zeta)d\zeta$ for every
fixed $x\in(0,L).$ The associated norm is
$\left\|\gamma\right\|_{B_{2}^{1}(0,L)}^{2}=\sqrt{(\gamma,\gamma)_{B_{2}^{1}(0,L)}}=\int\limits_{0}^{L}\left(\mathcal{I}_{x}\gamma\right)^{2}dx.$
We denote by $C(\overline{J};L^{2}(0,L))$ with $J=(0,T)$ the set of all
continuous functions $\gamma(.,t):J\rightarrow L^{2}(0,L)$ with norm

$\left\|\gamma\right\|_{C(J;L^{2}(0,L))}^{2}=\underset{0\leq t\leq
T}{\sup}\left\|\gamma(.,t)\right\|_{L^{2}(0,L)}^{2}<\infty,$ (2.6)

and $C(\overline{J};B_{2}^{1}(0,L))$ the set of functions
$\gamma(.,t):\overline{J}\rightarrow B_{2}^{1}(0,L)$ with norm

$\left\|\gamma\right\|_{C(\overline{J};B_{2}^{1}(0,L))}^{2}=\underset{0\leq
t\leq
T}{\sup}\left\|\mathcal{I}_{x}\gamma(.,t)\right\|_{L^{2}(0,L)}^{2}=\underset{0\leq
t\leq T}{\sup}\left\|\gamma(.,t)\right\|_{B_{2}^{1}(0,L)}^{2}<\infty.$ (2.7)

To obtain a priori estimate for the solution, we write down our problem (2.1),
(2.4) in its operator form: $\mathcal{GZ}=H$ with
$\mathcal{Z}=(\mathcal{\theta},\phi)$,
$\mathcal{GZ}=(\mathcal{S}_{1}(\mathcal{\theta},\phi),\mathcal{S}_{2}(\mathcal{\theta},\phi))$
and $H=(H_{1},H_{2})$ where

$\left\\{\begin{array}[]{c}\begin{array}[]{c}\mathcal{S}_{1}(\mathcal{\theta},\phi)=\\{\mathcal{L}_{1}(\mathcal{\theta},\phi),\Gamma_{1}\mathcal{\theta},\Gamma_{2}\mathcal{\theta}\\}\\\
\mathcal{S}_{2}(\mathcal{\theta},\phi)=\\{\mathcal{L}_{2}(\mathcal{\theta},\phi),\Gamma_{1}\phi,\Gamma_{2}\phi\\}\end{array}\\\
H_{1}=\\{F,\varphi,\psi\\},\text{ }H_{2}=\\{G,f,g\\}.\end{array}\right.,\text{
}$ (2.8)

The operator $\mathcal{G}$ is an unbounded operator of domain of definition
$D(\mathcal{G})$ consisting of elements
$(\mathcal{\theta},\phi)\in\left(L^{2}(\overline{J};L^{2}(0,L))\right)^{2}$
such that
$\mathcal{\theta}_{x},\phi_{x},\mathcal{\theta}_{t},\phi_{t},\mathcal{\theta}_{tt},\phi_{tt},\mathcal{\theta}_{xx},\phi_{xx}$
belonging to $L^{2}(\overline{J};L^{2}(0,L))$ verifying initial and boundary
conditions (2.3) and (2.4). The operator $\mathcal{G}$ is acting from the
Banach space $\mathcal{B}$ into the Hilbert space $\mathcal{E}$, where
$\mathcal{B}$ is the Banach space obtained by completing $D(\mathcal{G})$ with
respect to the norm

$\|\mathcal{Z}\|_{\mathcal{B}}^{2}=\|\mathcal{\theta}(.,t)\|_{C(\overline{J};L^{2}(0,L))}^{2}+\|\phi(.,t)\|_{C(\overline{J};L^{2}(0,L))}^{2}.$
(2.9)

And
$\mathcal{E}=\left[L^{2}(Q^{T})\times(L^{2}(0,L))^{2}\right]\times\left[L^{2}(Q^{T})\times(L^{2}(0,L))^{2}\right]$
is the Hilbert space consisting of vector-valued functions
$H=\left(\\{F,\varphi,\psi\\},\\{G,f,g\\}\right)$ for which the norm

$\displaystyle\|H\|_{\mathcal{E}}^{2}$ $\displaystyle=$
$\displaystyle\|F\|_{L^{2}(Q^{T})}^{2}+\|\varphi\|_{L^{2}(0,L)}^{2}+\|\psi\|_{L^{2}(0,L)}^{2}+\|G\|_{L^{2}(Q^{T})}^{2}$
(2.10) $\displaystyle+\|f\|_{L^{2}(0,L)}^{2}+\|g\|_{L^{2}(0,L)}^{2}.$

is finite.

##  3\. Preliminaries (Definitis and lemmas)

In this section, we provide some definitions and lemmas needed for
establishing different proves in the sequel.

Definition 1. [51] The time fractional derivative of order $\beta,$ with
$\beta\in(1,2)$ for a function $V$ is defined by

${}^{C}\partial_{t}^{\beta}V(x,t)=\frac{1}{\Gamma(2-\beta)}\int\limits_{0}^{t}\frac{V_{\tau\tau}(x,\tau)}{(t-\tau)^{\beta-1}}d\tau,$
(3.1)

and for $\beta\in(0,1)$ it is defined by

${}^{C}\partial_{t}^{\beta}V(x,t)=\frac{1}{\Gamma(1-\beta)}\int\limits_{0}^{t}\frac{V_{\tau}(x,\tau)}{(t-\tau)^{\beta}}d\tau$

where $\Gamma(1-\beta)$ is the Gamma function.

Definition 2. [51]. The fractional Riemann-Liouville integral of order
$0<\beta<1$ [$52$] which is given by

$D_{t}^{-\beta}\upsilon(x,t)=\frac{1}{\Gamma(\beta)}\int\limits_{0}^{t}\frac{\upsilon(x,\tau)}{(t-\tau)^{1-\beta}}d\tau.$
(2.6)

Lemma 2.1 $[35]$. Let $E(s)$ be nonnegative and absolutely continuous on
$[0,T]$, and suppose that for almost all $s\in[0,T]$, $R$ satisfies the
inequality

$\frac{dE}{ds}\leq A(s)E(s)+B(s),$ (3.2)

where the functions $A(s)$ and $B(s)$ are summable and nonnegative on $[0,T].$
Then

$E(s)\leq\exp\left\\{\int\limits_{0}^{s}A(t)dt\right\\}\left(E(0)+\int\limits_{0}^{s}B(t)dt\right).$
(3.3)

Lemma 2.3. $[34]$ Let a nonnegative absolutely continuous function
$\mathcal{Q}(t)$ satisfy the inequality

${}^{C}\partial_{t}^{\beta}\mathcal{Q}(t)\leq
b_{1}\mathcal{Q}(t)+b_{2}(t),\text{ \ \ \ }0<\beta<1,$ (3.4)

for almost all $t\in[0,T],$ where $b_{1}$ is a positive constant and
$b_{2}(t)$ is an integrable nonnegative function on $[0,T].$ Then

$\mathcal{Q}(t)\leq\mathcal{Q}(0)E_{\beta}(b_{1}t^{\beta})+\Gamma(\beta)E_{\beta,\beta}(b_{1}t^{\beta})D_{t}^{-\beta}b_{2}(t),$
(3.5)

where

$E_{\beta}(x)=\sum_{n=0}^{\infty}\frac{x^{n}}{\Gamma(\beta n+1)}\text{ and
}E_{\beta,\mu}(x)=\sum_{n=0}^{\infty}\frac{x^{n}}{\Gamma(\beta n+\mu)},$

are the MIttag-Leffler functions

##  4\. A priori estimate and its Consequences

In this section, we establish an energy inequality from which we deduce the
uniqueness and continuous dependence of solution of problem (2.1)-(2.4) on the
given data.

Theorem 3.1. For any function $\mathcal{Z}=(\mathcal{\theta},\phi)\in
D(\mathcal{G})$ the following a priori estimates holds

$\displaystyle\|\mathcal{\theta}(.,t)\|_{C(\overline{J};L^{2}(0,L))}^{2}+\|\phi(.,t)\|_{C(\overline{J};L^{2}(0,L))}^{2}$
(4.1) $\displaystyle\leq$
$\displaystyle\mathcal{F}^{\ast}\left(\|\varphi\|_{L^{2}(0,L)}^{2}+\|\psi\|_{L^{2}(0,L)}^{2}+\|g\|_{L^{2}(0,L)}^{2}+\|f\|_{L^{2}(0,L)}^{2}\right.$
$\displaystyle\left.+\|F\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|G\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right),$

and

$\displaystyle
D^{\alpha-1}\left(\left\|\mathcal{\theta}_{t}\right\|_{B_{2}^{1}(0,L)}+\left\|\phi_{t}\right\|_{B_{2}^{1}(0,L)}\right)$
(4.2) $\displaystyle\leq$
$\displaystyle\mathcal{F}^{\ast}\left(\|\varphi\|_{L^{2}(0,L)}^{2}+\|\psi\|_{L^{2}(0,L)}^{2}+\|g\|_{L^{2}(0,L)}^{2}+\|f\|_{L^{2}(0,L)}^{2}\right.$
$\displaystyle\left.+\|F\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|G\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right).$

where $\mathcal{F}^{\ast}$ is a positive constant independent of
$\mathcal{Z=}(\mathcal{\theta},\phi)$ given by

$\mathcal{F}^{\ast}=\mathcal{M}\omega\max\left\\{1,\frac{T^{\alpha}}{\alpha\Gamma(\alpha)}\right\\},$

with

$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\Gamma(\alpha)E_{\alpha,\alpha}(\omega
t^{\alpha})\left(\max\left\\{1,\frac{T^{\alpha}}{\alpha\Gamma(\alpha)}\right\\}\right)$
$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\mathcal{W}^{\ast}(\mathcal{W}^{\ast}e^{\mathcal{W}^{\ast}T}+1),$

and $\mathcal{W}^{\ast}$ is given by (4.24).

Proof. Define the integro-differential operators
$\mathcal{M}_{1}\mathcal{\theta}=-\mathcal{I}_{x}^{2}\mathcal{\theta}_{t}$ and
$\mathcal{M}_{2}\phi=-\mathcal{I}_{x}^{2}\phi_{t},$ where

$\mathcal{I}_{x}^{2}\mathcal{\theta}(x,t)=\int\limits_{0}^{x}\int\limits_{0}^{\xi}\mathcal{\theta}(\eta,t)d\eta
d\xi,\text{
}\mathcal{I}_{x}^{2}\phi(x,t)=\int\limits_{0}^{x}\int\limits_{0}^{\xi}\phi(\eta,t)d\eta
d\xi,$

and consider the identity

$\displaystyle\left(\rho_{1}\partial_{t}^{\alpha+1}\mathcal{\theta},\mathcal{M}_{1}\mathcal{\theta}\right)_{L^{2}(0,L)}-\kappa_{1}((\mathcal{\theta}_{x}+\phi)_{x},\mathcal{M}_{1}\mathcal{\theta})_{L^{2}(0,L)}+\left(\mathcal{\theta}_{t},\mathcal{M}_{1}\mathcal{\theta}\right)_{L^{2}(0,L)}$
$\displaystyle+(\rho_{2}\partial_{t}^{\alpha+1}\phi,\mathcal{M}_{2}\phi)_{L^{2}(0,L)}-\kappa_{2}((\phi_{xx},\mathcal{M}_{2}\phi)_{L^{2}(0,L)}+\kappa_{1}((\mathcal{\theta}_{x}+\phi),\mathcal{M}_{2}\phi)_{L^{2}(0,L)}$
$\displaystyle+$
$\displaystyle\left(\int\limits_{0}^{t}m(t-s)\phi_{xx}(x,s)ds,\mathcal{M}_{2}\phi\right)_{L^{2}(0,L)}$
(4.3) $\displaystyle=$
$\displaystyle(F(x,t),\mathcal{M}_{1}\mathcal{\theta})_{L^{2}(0,L)}+(G(x,t),\mathcal{M}_{2}\phi)_{L^{2}(0,L)}.$

The standard integration by parts of each term in (4.3) and conditions (2.3),
(2.4) give

$\displaystyle\left(\rho_{1}\partial_{t}^{\alpha+1}\mathcal{\theta},\mathcal{M}_{1}\mathcal{\theta}\right)_{L^{2}(0,L)}$
$\displaystyle=$
$\displaystyle\frac{\rho_{1}}{2}(\partial_{t}^{\alpha}(\mathcal{I}_{x}\mathcal{\theta}_{t}),\mathcal{I}_{x}\mathcal{\theta}_{t})_{L^{2}(0,L)}$
(4.4) $\displaystyle\geq$
$\displaystyle\frac{\rho_{1}}{2}\partial_{t}^{\alpha}\left\|\mathcal{I}_{x}\mathcal{\theta}_{t}\right\|_{L^{2}(0,L)},$
$\displaystyle\left(\rho_{2}\partial_{t}^{\alpha+1}\phi,\mathcal{M}_{2}\phi\right)_{L^{2}(0,L)}$
$\displaystyle=$
$\displaystyle\frac{\rho_{2}}{2}(\partial_{t}^{\alpha}(\mathcal{I}_{x}\phi_{t}),\mathcal{I}_{x}\phi_{t})_{L^{2}(0,L)}$
(4.5) $\displaystyle\geq$
$\displaystyle\frac{\rho_{2}}{2}\partial_{t}^{\alpha}\left\|\mathcal{I}_{x}\phi_{t}\right\|_{L^{2}(0,L)},$
$-\left(\mathcal{\theta}_{t},\mathcal{M}_{1}\mathcal{\theta}\right)_{L^{2}(0,L)}=\|\mathcal{I}_{x}\mathcal{\theta}_{t}\|_{L^{2}(0,L)}^{2},$
(4.6)
$\displaystyle\kappa_{1}(\mathcal{\theta}_{xx},\mathcal{I}_{x}^{2}\mathcal{\theta}_{t})_{L^{2}(0,L)}$
$\displaystyle=$
$\displaystyle\kappa_{1}\mathcal{I}_{x}^{2}\mathcal{\theta}_{t}.\mathcal{\theta}_{x}]_{0}^{L}-\kappa_{1}\int\limits_{0}^{L}\mathcal{I}_{x}\mathcal{\theta}_{t}.\mathcal{\theta}_{x}dx=\kappa_{1}\int\limits_{0}^{L}\mathcal{\theta}_{t}\mathcal{\theta}dx$
(4.7) $\displaystyle=$
$\displaystyle\frac{\kappa_{1}}{2}\frac{\partial}{\partial
t}\|\mathcal{\theta}\|_{L^{2}(0,L)}^{2},$

and in the same manner, we have

$\kappa_{2}(\phi_{xx},\mathcal{I}_{x}^{2}\phi_{t})_{L^{2}(0,L)}=\frac{\kappa_{2}}{2}\frac{\partial}{\partial
t}\|\phi\|_{L^{2}(0,L)}^{2},$ (4.8)
$\displaystyle\kappa_{1}(\phi_{x},\mathcal{I}_{x}^{2}\mathcal{\theta}_{t})_{L^{2}((0,L))}$
$\displaystyle=$
$\displaystyle\kappa_{1}\mathcal{I}_{x}^{2}\mathcal{\theta}_{t}.\phi]_{0}^{L}dt-\kappa_{1}\int\limits_{0}^{L}\mathcal{I}_{x}\mathcal{\theta}_{t}.\phi
dx$ (4.9) $\displaystyle=$
$\displaystyle-\kappa_{1}\int\limits_{0}^{L}\mathcal{I}_{x}\mathcal{\theta}_{t}.\phi
dx,$
$-\kappa_{1}(\mathcal{\theta}_{x},\mathcal{I}_{x}^{2}\phi_{t})_{L^{2}(Q^{\tau})}=\kappa_{1}\int\limits_{0}^{L}\mathcal{I}_{x}\phi_{t}.\mathcal{\theta}dx,$
(4.10)
$\displaystyle-\kappa_{1}(\phi,\mathcal{I}_{x}^{2}\phi_{t})_{L^{2}(0,L)}$
(4.11) $\displaystyle=$
$\displaystyle-\kappa_{1}\mathcal{I}_{x}^{2}\phi_{t}.\Im_{x}\phi]_{0}^{L}+\kappa_{1}\int\limits_{0}^{\tau}\int\limits_{0}^{L}\mathcal{I}_{x}\phi_{t}.\mathcal{I}_{x}\phi
dxdt$ $\displaystyle=$ $\displaystyle\frac{1}{2}\frac{\partial}{\partial
t}\|\mathcal{I}_{x}\phi\|_{L^{2}(0,L)}^{2},$
$\displaystyle-\left(\int\limits_{0}^{t}m(t-s).\phi_{xx}(x,s)ds,\mathcal{I}_{x}^{2}\phi_{t}\right)_{L^{2}(0,L)}$
(4.12) $\displaystyle=$
$\displaystyle-\int\limits_{0}^{L}\left(\int\limits_{0}^{t}m(t-s).\phi_{xx}(x,s)ds\right)\mathcal{I}_{x}^{2}\phi_{t}dx$
$\displaystyle=$
$\displaystyle-\int\limits_{0}^{t}m(t-s).\phi_{x}(x,s)ds).\mathcal{I}_{x}^{2}\phi_{t}]_{0}^{L}dx+\int\limits_{0}^{L}\left(\int\limits_{0}^{t}m(t-s).\phi_{x}(x,s)ds\right)\mathcal{I}_{x}\phi_{t}dx$
$\displaystyle=$
$\displaystyle\left(\int\limits_{0}^{t}m(t-s).\phi(x,s)ds)\right)\mathcal{I}_{x}\phi_{t}]_{0}^{L}dx-\int\limits_{0}^{L}\left(\int\limits_{0}^{t}m(t-s).\phi(x,s)ds\right)\phi_{t}dx$
$\displaystyle=$
$\displaystyle-\int\limits_{0}^{L}\left(\int\limits_{0}^{t}m(t-s).\phi(x,s)ds\right)\phi_{t}dx.$

Substituting equalities (4.4)-(4.12) into ((4.3)),we obtain

$\displaystyle\frac{\rho_{1}}{2}\partial_{t}^{\alpha}\left\|\mathcal{I}_{x}\mathcal{\theta}_{t}\right\|_{L^{2}(0,L)}+\frac{\rho_{2}}{2}\partial_{t}^{\alpha}\left\|\mathcal{I}_{x}\phi_{t}\right\|_{L^{2}(0,L)}+\frac{\kappa_{1}}{2}\frac{\partial}{\partial
t}\|\mathcal{\theta}\|_{L^{2}(0,L)}^{2}$ (4.13)
$\displaystyle+\frac{\kappa_{2}}{2}\frac{\partial}{\partial
t}\|\phi\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{x}\mathcal{\theta}_{t}\|_{L^{2}(0,L)}^{2}+\frac{1}{2}\frac{\partial}{\partial
t}\|\mathcal{I}_{x}\phi\|_{L^{2}(0,L)}^{2}$ $\displaystyle=$
$\displaystyle\kappa_{1}\int\limits_{0}^{L}\phi\mathcal{I}_{x}\mathcal{\theta}_{t}dx-\kappa_{1}\int\limits_{0}^{L}\mathcal{\theta
I}_{x}\phi_{t}dx-\int\limits_{0}^{L}\left(\int\limits_{0}^{t}m(t-s).\phi(x,s)ds\right)\phi_{t}dx$
$\displaystyle-\int\limits_{0}^{L}F\mathcal{I}_{x}^{2}\mathcal{\theta}_{t}dx-\int\limits_{0}^{L}G\mathcal{I}_{x}^{2}\phi_{t}dx.$

Replacing $t$ by $\tau$, integrating with respect to $\tau$ from zero to $t$
and using the given conditions, we obtain

$\displaystyle\frac{\rho_{1}}{2}D^{\alpha-1}\left\|\mathcal{I}_{x}\mathcal{\theta}_{t}\right\|_{L^{2}(0,L)}+\frac{\rho_{2}}{2}D^{\alpha-1}\left\|\mathcal{I}_{x}\phi_{t}\right\|_{L^{2}(0,L)}+\frac{\kappa_{1}}{2}\|\mathcal{\theta}(.,t)\|_{L^{2}(0,L)}^{2}$
(4.14)
$\displaystyle+\frac{\kappa_{2}}{2}\|\phi(.,t)\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{x}\mathcal{\theta}_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\frac{1}{2}\|\mathcal{I}_{x}\phi(.,t)\|_{L^{2}(0,L)}^{2}$
$\displaystyle=$
$\displaystyle\kappa_{1}\left(\phi,\mathcal{I}_{x}\mathcal{\theta}_{\tau}\right)_{L^{2}(0,t;L^{2}(0,L))}-\kappa_{1}\left(\mathcal{\theta},\mathcal{I}_{x}\phi_{\tau}\right)_{L^{2}(0,t;L^{2}(0,L))}$
$\displaystyle-\left(F,\mathcal{I}_{x}^{2}\mathcal{\theta}_{\tau}\right)_{L^{2}(0,t;L^{2}(0,L))}-\left(G,\mathcal{I}_{x}^{2}\phi_{\tau}\right)_{L^{2}(0,t;L^{2}(0,L))}$
$\displaystyle+\frac{\rho_{1}t^{1-\alpha}}{2(\Gamma(1-\alpha)(1-\alpha))}\left\|\mathcal{I}_{x}\psi\right\|_{L^{2}(0,L)}^{2}+\frac{\kappa_{1}}{2}\|\varphi\|_{L^{2}(0,L)}^{2}$
$\displaystyle+\frac{\rho_{2}t^{1-\alpha}}{2(\Gamma(1-\alpha)(1-\alpha))}\left\|\mathcal{I}_{x}g\right\|_{L^{2}(0,L)}^{2}+\frac{\kappa_{2}}{2}\|f\|_{L^{2}(0,L)}^{2}$
$\displaystyle+\frac{1}{2}\|\mathcal{I}_{x}f\|_{L^{2}(0,L)}^{2}-\int\limits_{0}^{t}\int\limits_{0}^{L}\left(\int\limits_{0}^{\tau}m(\tau-s).\phi(x,s)ds\right)\phi_{\tau}dxd\tau.$

The last term on the right-hand side needs to be evaluated as follows

$\displaystyle-\int\limits_{0}^{t}\int\limits_{0}^{L}\left(\int\limits_{0}^{\tau}m(\tau-s).\phi(x,s)ds\right)\phi_{\tau}dxd\tau$
(4.15) $\displaystyle=$
$\displaystyle-\int\limits_{0}^{L}\left(\int\limits_{0}^{\tau}m(\tau-s).\phi(x,s)ds\right)\phi
dx]_{0}^{t}+\int\limits_{0}^{\tau}\int\limits_{0}^{L}m(0)\phi^{2}dxd\tau$
$\displaystyle+\int\limits_{0}^{t}\int\limits_{0}^{L}\left(\int\limits_{0}^{\tau}m^{\prime}(\tau-s).\phi(x,s)ds\right)\phi(x,\tau)dxd\tau$
$\displaystyle=$
$\displaystyle-\int\limits_{0}^{L}\left(\int\limits_{0}^{t}m(t-s).\phi(x,s)ds\right)\phi(x,t)dx+m(0)\|\phi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}$
$\displaystyle+\int\limits_{0}^{t}\int\limits_{0}^{L}\left(\int\limits_{0}^{\tau}m^{\prime}(\tau-s).\phi(x,s)ds\right)\phi
dxd\tau.$

By replacing (4.15) into (4.14), and estimating different terms on the right-
hand side of (by using Cauchy $\epsilon$ inequality, a Poincare type
inequality) (4.14) as follows

$\displaystyle\kappa_{1}\left(\phi,\mathcal{I}_{x}\mathcal{\theta}_{t}\right)_{L^{2}(0,t;L^{2}(0,L))}$
(4.16) $\displaystyle\leq$
$\displaystyle\frac{\kappa_{1}\epsilon_{1}}{2}\|\phi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\frac{\kappa_{1}}{2\epsilon_{1}}\|\mathcal{I}_{x}\mathcal{\theta}_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2},\
\ \ \ \ \ \ \ \ \ \ $
$\displaystyle-\kappa_{1}\left(\mathcal{\theta},\mathcal{I}_{x}\phi_{\tau}\right)_{L^{2}(0,t;L^{2}(0,L))}$
(4.17) $\displaystyle\leq$
$\displaystyle\frac{\kappa_{1}\epsilon_{2}}{2}\|\mathcal{\theta}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\frac{\kappa_{1}}{2\epsilon_{2}}\|\mathcal{I}_{x}\phi_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2},\
\ \ \ \ \ \ \ \ $
$\displaystyle-\int\limits_{0}^{L}\left(\int\limits_{0}^{t}m(t-s).\phi(x,s)ds\right)\phi(x,t)dx$
(4.18) $\displaystyle\leq$
$\displaystyle\frac{\epsilon_{3}}{2}\int\limits_{0}^{L}\phi^{2}(x,t)dx+\frac{1}{2\epsilon_{3}}\int\limits_{0}^{L}\left(\int\limits_{0}^{t}m(t-s)\phi(x,s)ds\right)^{2}dx$
$\displaystyle\leq$
$\displaystyle\frac{\epsilon_{3}}{2}\int\limits_{0}^{L}\phi^{2}(x,t)dx+\frac{1}{2\epsilon_{3}}\int\limits_{0}^{L}\left(\int\limits_{0}^{t}m^{2}(t-s)ds\right)\left(\int\limits_{0}^{t}\phi^{2}(x,s)ds\right)dx$
$\displaystyle\leq$
$\displaystyle\frac{\epsilon_{3}}{2}\int\limits_{0}^{L}\phi^{2}(x,t)dx+\frac{T}{2\epsilon_{3}}\sup\limits_{0\leq
t\leq T}m^{2}(t)\int\limits_{0}^{L}\int\limits_{0}^{t}\phi^{2}dxd\tau,$
$\displaystyle\int\limits_{0}^{t}\int\limits_{0}^{L}\left(\int\limits_{0}^{\tau}m^{\prime}(\tau-s).\phi(x,s)ds\right)\phi
dxd\tau$ (4.19) $\displaystyle\leq$
$\displaystyle\frac{\epsilon_{4}}{2}\int\limits_{0}^{t}\int\limits_{0}^{L}\phi^{2}dxd\tau+\frac{1}{2\epsilon_{4}}\int\limits_{0}^{t}\int\limits_{0}^{L}\left(\int\limits_{0}^{\tau}m^{\prime
2}(\tau-s)ds\right)\left(\int\limits_{0}^{\tau}\phi^{2}(x,s)ds\right)dxd\tau$
$\displaystyle\leq$
$\displaystyle\frac{\epsilon_{4}}{2}\int\limits_{0}^{t}\int\limits_{0}^{L}\phi^{2}dxd\tau+\frac{T}{2\epsilon_{4}}\sup\limits_{0\leq
t\leq T}m^{\prime
2}(t)\int\limits_{0}^{t}\int\limits_{0}^{L}\left(\int\limits_{0}^{\tau}\phi^{2}(x,s)ds\right)dxd\tau$
$\displaystyle=$
$\displaystyle\frac{\epsilon_{4}}{2}\int\limits_{0}^{t}\int\limits_{0}^{L}\phi^{2}dxd\tau+\frac{T}{2\epsilon_{4}}\sup\limits_{0\leq
t\leq T}m^{\prime
2}(t)\int\limits_{0}^{L}\left[\left(\tau\int\limits_{0}^{\tau}\phi^{2}(x,s)ds\right)_{0}^{t}-\int\limits_{0}^{t}\tau\phi^{2}d\tau\right]dx$
$\displaystyle=$
$\displaystyle\frac{\epsilon_{4}}{2}\int\limits_{0}^{t}\int\limits_{0}^{L}\phi^{2}dxd\tau+\frac{T}{2\epsilon_{4}}\sup\limits_{0\leq
t\leq T}m^{\prime
2}(t)\int\limits_{0}^{L}\left[\int\limits_{0}^{t}(t-\tau)\phi^{2}(x,\tau)d\tau\right]dx$
$\displaystyle\leq$
$\displaystyle\frac{\epsilon_{4}}{2}\int\limits_{0}^{t}\int\limits_{0}^{L}\phi^{2}dxd\tau+\frac{T^{2}}{2\epsilon_{4}}\sup\limits_{0\leq
t\leq T}m^{\prime 2}(t)\int\limits_{0}^{L}\int\limits_{0}^{t}\phi^{2}d\tau dx$
$\displaystyle=$
$\displaystyle\left(\frac{\epsilon_{4}}{2}+\frac{T^{2}}{2\epsilon_{4}}\sup\limits_{0\leq
t\leq T}m^{\prime
2}(t)\right)\int\limits_{0}^{t}\int\limits_{0}^{L}\phi^{2}dxd\tau,\ \ $
$-\left(F,\mathcal{I}_{x}^{2}\mathcal{\theta}_{\tau}\right)_{L^{2}(0,t;L^{2}(0,L))}\leq\frac{\epsilon_{5}}{2}\|F\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\frac{L^{2}}{2\epsilon_{5}}\|\mathcal{I}_{x}\mathcal{\theta}_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2},$
(4.20)
$-\left(G,\mathcal{I}_{x}^{2}\phi_{\tau}\right)_{L^{2}(0,t;L^{2}(0,L))}\leq\frac{\epsilon_{6}}{2}\|G\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\frac{L^{2}}{4\epsilon_{6}}\|\mathcal{I}_{x}\phi_{\tau}\|_{L^{2}(0,t;L^{2}(0,L)),}^{2}$
(4.21)

Combination of (4.16)-(4.21) and (4.14), leads to

$\displaystyle\frac{\rho_{1}}{2}D^{\alpha-1}\left\|\mathcal{I}_{x}\mathcal{\theta}_{t}\right\|_{L^{2}(0,L)}+\frac{\rho_{2}}{2}D^{\alpha-1}\left\|\mathcal{I}_{x}\phi_{t}\right\|_{L^{2}(0,L)}+\frac{\kappa_{1}}{2}\|\mathcal{\theta}(.,t)\|_{L^{2}(0,L)}^{2}$
(4.22)
$\displaystyle+\frac{\kappa_{2}}{2}\|\phi(.,t)\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{x}\mathcal{\theta}_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\frac{1}{2}\|\mathcal{I}_{x}\phi(.,t)\|_{L^{2}(0,L)}^{2}$
$\displaystyle\leq$
$\displaystyle\frac{\rho_{1}T^{1-\alpha}L^{2}}{4(\Gamma(1-\alpha)(1-\alpha))}\left\|\psi\right\|_{L^{2}(0,L)}^{2}+\frac{\kappa_{1}}{2}\|\varphi\|_{L^{2}(0,L)}^{2}$
$\displaystyle+\frac{\rho_{2}T^{1-\alpha}L^{2}}{4(\Gamma(1-\alpha)(1-\alpha))}\left\|g\right\|_{L^{2}(0,L)}^{2}+\left(\frac{\kappa_{2}}{2}+\frac{L^{2}}{4}\right)\|f\|_{L^{2}(0,L)}^{2}$
$\displaystyle+\left(\frac{\kappa_{1}\epsilon_{1}}{2}+\frac{T}{2\epsilon_{3}}\sup\limits_{0\leq
t\leq
T}m^{2}(t)+\frac{\epsilon_{4}}{2}+\frac{T^{2}}{2\epsilon_{4}}\sup\limits_{0\leq
t\leq T}m^{\prime 2}(t)+m(0)\right)\|\phi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}$
$\displaystyle+\left(\frac{\kappa_{1}}{2\epsilon_{1}}+\frac{L^{2}}{2\epsilon_{5}}\right)\|\mathcal{I}_{x}\mathcal{\theta}_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\left(\frac{\kappa_{1}}{2\epsilon_{2}}+\frac{L^{2}}{4\epsilon_{6}}\right)\|\mathcal{I}_{x}\phi_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}$
$\displaystyle+\frac{\kappa_{1}\epsilon_{2}}{2}\|\mathcal{\theta}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\frac{\epsilon_{3}}{2}\|\phi(.,t)\|_{L^{2}(0,L)}^{2}+\frac{\epsilon_{5}}{2}\|F\|_{L^{2}(0,t;L^{2}(0,L))}^{2}$
$\displaystyle+\frac{\epsilon_{6}}{2}\|G\|_{L^{2}(0,t;L^{2}(0,L))}^{2}.$

The choice $\varepsilon_{1}=\kappa_{1},$ $\varepsilon_{5}=L^{2}/2,$
$\varepsilon_{3}=\kappa_{2}/2,$
$\varepsilon_{2}=\varepsilon_{4}=\varepsilon_{6}=1,$ and cancellation of the
last term on the left-hand side of (4.22) reduce it to the following estimate

$\displaystyle
D^{\alpha-1}\left\|\mathcal{I}_{x}\mathcal{\theta}_{t}\right\|_{L^{2}(0,L)}+D^{\alpha-1}\left\|\mathcal{I}_{x}\phi_{t}\right\|_{L^{2}(0,L)}+\|\mathcal{\theta}(.,t)\|_{L^{2}(0,L)}^{2}+\|\phi(.,t)\|_{L^{2}(0,L)}^{2}$
$\displaystyle\leq$
$\displaystyle\mathcal{W}^{\ast}\left(\|\mathcal{I}_{x}\mathcal{\theta}_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{x}\phi_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{\theta}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\phi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right.$
$\displaystyle+\|\varphi\|_{L^{2}(0,L)}^{2}+\left\|\psi\right\|_{L^{2}(0,L)}+\|f\|_{L^{2}(0,L)}^{2}+\|g\|_{L^{2}(0,L)}^{2}+\|F\|_{L^{2}(0,t;L^{2}(0,L))}^{2}$
$\displaystyle\left.+\|G\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right),$

where

$\displaystyle\mathcal{W}^{\ast}$ $\displaystyle=$
$\displaystyle\max\left(\frac{\kappa_{1}^{2}}{2}+\frac{T}{\kappa_{2}}\sup\limits_{0\leq
t\leq T}m^{2}(t)+\frac{1}{2}+\frac{T^{2}}{2}\sup\limits_{0\leq t\leq
T}m^{\prime 2}(t)+m(0),\text{ }\frac{3}{2},\text{
}\frac{\kappa_{1}}{2}+\frac{L^{2}}{4},\frac{\kappa_{2}}{4},\right.$ (4.24)
$\displaystyle\left.\frac{\rho_{1}T^{1-\alpha}L^{2}}{4(\Gamma(1-\alpha)(1-\alpha))},\frac{\rho_{2}T^{1-\alpha}L^{2}}{4(\Gamma(1-\alpha)(1-\alpha))}\right)/\min\left(\frac{\rho_{1}}{2},\frac{\rho_{2}}{2},\frac{\kappa_{1}}{2},\frac{\kappa_{2}}{2},\frac{1}{2}\right).$

By omitting the first and second term on the left-hand side of (4), and
applying the Gronwall-Bellman lemma ( [46]), where

$\left\\{\begin{array}[]{c}E(t)=\|\mathcal{\theta}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\phi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\\\
\frac{dE}{dt}=\|\mathcal{\theta}(.,t)\|_{L^{2}(0,L)}^{2}+\|\phi(.,t)\|_{L^{2}(0,L)}^{2},\\\
\mathcal{Q}(0)=0.\end{array}\right.$ (4.25)

we obtain

$\displaystyle y(t)$ $\displaystyle\leq$
$\displaystyle\mathcal{W}^{\ast}e^{\mathcal{W}^{\ast}t}\left(\|\mathcal{I}_{x}\mathcal{\theta}_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{x}\phi_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right.$
$\displaystyle+\|\varphi\|_{L^{2}(0,L)}^{2}+\left\|\psi\right\|_{L^{2}(0,L)}+\|f\|_{L^{2}(0,L)}^{2}+\|g\|_{L^{2}(0,L)}^{2}$
$\displaystyle+\left.\|F\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|G\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right).$

Then by omitting the last two terms on the left-hand side of (4), and using
(4), we have

$\displaystyle
D^{\alpha-1}\left(\left\|\mathcal{I}_{x}\mathcal{\theta}_{t}\right\|_{L^{2}(0,L)}+\left\|\mathcal{I}_{x}\phi_{t}\right\|_{L^{2}(0,L)}\right)$
$\displaystyle\leq$
$\displaystyle\mathcal{W}^{\ast}(\mathcal{W}^{\ast}e^{\mathcal{W}^{\ast}T}+1)\left(\|\mathcal{I}_{x}\mathcal{\theta}_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{x}\phi_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right.$
$\displaystyle+\|\varphi\|_{L^{2}(0,L)}^{2}+\left\|\psi\right\|_{L^{2}(0,L)}+\|f\|_{L^{2}(0,L)}^{2}+\|g\|_{L^{2}(0,L)}^{2}$
$\displaystyle+\left.\|F\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|G\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right).$

Now, Lemma 3.2, can be applied to remove the first two terms on the right-hand
side of (4), by taking

$\left\\{\begin{array}[]{c}\mathcal{Q}(t)=\|\mathcal{I}_{x}\mathcal{\theta}_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{x}\phi_{\tau}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\\\
{}^{C}\partial_{t}^{\beta}\mathcal{Q}(t)=D^{\alpha-1}\left(\left\|\mathcal{I}_{x}\mathcal{\theta}_{t}\right\|_{L^{2}(0,L)}+\left\|\mathcal{I}_{x}\phi_{t}\right\|_{L^{2}(0,L)}\right)\\\
\mathcal{Q}(0)=0.\end{array}\right.$ (4.28)

From (4), it follows that

$\displaystyle\mathcal{Q}(t)$ $\displaystyle\leq$
$\displaystyle\mathcal{M}\left\\{D^{-1-\alpha}\left(\|F\|_{L^{2}(0,L)}^{2}+\|G\|_{L^{2}(0,L)}^{2}\right)\right.$
(4.29)
$\displaystyle+\left.\|\varphi\|_{L^{2}(0,L)}^{2}+\left\|\psi\right\|_{L^{2}(0,L)}+\|f\|_{L^{2}(0,L)}^{2}+\|g\|_{L^{2}(0,L)}^{2}\right\\},$

where

$\mathcal{M}=\Gamma(\alpha)E_{\alpha,\alpha}(\omega
t^{\alpha})\left(\max\left\\{1,\frac{T^{\alpha}}{\alpha\Gamma(\alpha)}\right\\}\right),$

with

$\omega=\mathcal{W}^{\ast}(\mathcal{W}^{\ast}e^{\mathcal{W}^{\ast}T}+1).$

Now since

$D^{-1-\alpha}\left(\|F\|_{L^{2}(0,L)}^{2}+\|G\|_{L^{2}(0,L)}^{2}\right)\leq\frac{T^{\alpha}}{\Gamma(\alpha+1)}\int\limits_{0}^{t}\left(\|F\|_{L^{2}(0,L)}^{2}+\|G\|_{L^{2}(0,L)}^{2}\right)d\tau,$
(4.30)

then, we infer from (4.29), (4.30), and (4) the following inequality

$\displaystyle
D^{\alpha-1}\left\|\mathcal{I}_{x}\mathcal{\theta}_{t}\right\|_{L^{2}(0,L)}+D^{\alpha-1}\left\|\mathcal{I}_{x}\phi_{t}\right\|_{L^{2}(0,L)}+\|\mathcal{\theta}(.,t)\|_{L^{2}(0,L)}^{2}+\|\phi(.,t)\|_{L^{2}(0,L)}^{2}$
(4.31) $\displaystyle\leq$
$\displaystyle\mathcal{F}^{\ast}\left(\|\psi\|_{L^{2}(0,L)}^{2}+\|\varphi\|_{L^{2}(0,L)}^{2}+\|f\|_{L^{2}(0,L)}^{2}+\|g\|_{L^{2}(0,L)}^{2}\right.$
$\displaystyle\left.+\|F\|_{L^{2}(0,T;L^{2}(0,L))}^{2}+\|G\|_{L^{2}(0,T;L^{2}(0,L))}^{2}\right).$

The first estimate (4.1) follows, if we disregard the first and second term on
the left-hand side of (4.31), and pass to the supremum with respect to $t$
over $(0,T).$ The second estimate (4.2) follows from (4.31) if we drop the
last two terms on the left-hand side of the inequality (4.31).

Since the range of the operator $\mathcal{G}$ is subset of $\mathcal{E}$, that
is $R\left(\mathcal{G}\right)\subset\mathcal{E}$, so we extend $\mathcal{G}$
so that inequality (4.31) holds for the extension, and
$R\left(\overline{\mathcal{G}}\right)=\mathcal{E}$. We can easily show that
the following

Proposition 3.2. The unbounded operator
$\mathcal{G}:\mathcal{B}\rightarrow\mathcal{E}$ admits a closure
$\overline{\mathcal{G}\text{ }}$ with domain of definition
$D(\overline{\mathcal{G}\text{ }}).$

Definition 3.3. The solution of the equation $\overline{\mathcal{G}\text{
}}\mathcal{Z=}H=(\\{F,\varphi,\psi\\},\\{G,f,g\\})$ is called a strong
solution of problem (2.1), (2.3), (2.4).

The a priori estimate (4.1), can be extended to

$\|\mathcal{Z}\|_{\mathcal{B}}^{2}\leq\mathcal{F}^{\ast}\|\overline{\mathcal{G}\text{
}}\mathcal{Z}\|_{\mathcal{E}}^{2},\text{ \ \
}\forall\mathcal{Z\in}D(\overline{\mathcal{G}\text{ }}).$ (4.32)

The estimate (4.32) shows that the operator $\overline{\mathcal{G}\text{ }}$
is one to one and that $\overline{\mathcal{G}\text{ }}^{-1}$ is continuous
from $R(\overline{\mathcal{G}\text{ }})$ onto $\mathcal{B}.$ Consequently if a
strong solution of problem (2.1), (2.3), (2.4) exists, it is unique and
depends continuously on the input data $\ \varphi,\psi,f,g$ and the external
forces $F$ and $G.$ Also as a consequence of (4.32), the set
$R(\overline{\mathcal{G}\text{ }})\subset\mathcal{E}$ is closed and
$R(\overline{\mathcal{G}\text{ }})=\overline{R(\mathcal{G})}.$

##  5\. Existence of solution

Theorem 4.1. Problem (2.1), (2.3), (2.4) admits a unique strong solution
$\mathcal{Z}=\left(\overline{\mathcal{G}\text{
}}\right)^{-1}(\\{F,\varphi,\psi\\},\\{G,f,g\\})=\overline{\mathcal{G}^{-1}}(\\{F,\varphi,\psi\\},\\{G,f,g\\}),$which
depend continuously on the given data, for all $F,G\in
L^{2}(0,T;L^{2}(0,L)),$and $\varphi,\psi,f,g\in L^{2}(0,L).$

Proof. It follows from above that in order to prove the existence of the
strongly generalized solution of problem (2.1), (2.3), (2.4), it suffices to
prove that $\overline{R(\mathcal{G})}=\mathcal{E}$. To this end, we first
prove the density in the following special case.

Theorem 4.2. If for some function
$W(x,t)=(\Lambda_{1}(x,t),\Lambda_{2}(x,t))\in(L^{2}(0,T;L^{2}(0,L)))^{2}$ and
for elements $\ \mathcal{Z}\in D_{0}(\mathcal{G})=\\{\mathcal{Z}\ :\
\mathcal{Z}\in D(\mathcal{G})\ $and$\
\Gamma_{i}\mathcal{\theta}=\Gamma_{i}\phi=0\ ,\ i=1,2\ \\}$ we have

$(\mathcal{S}_{1}(\mathcal{\theta},\phi),\Lambda_{1})_{L^{2}(0,T;L^{2}(0,L))}+(\mathcal{S}_{2}(\mathcal{\theta},\phi),\Lambda_{2})_{L^{2}(0,T;L^{2}(0,L))}=0,$
(5.1)

then $W(x,t)=(\Lambda_{1}(x,t),\Lambda_{2}(x,t))=(0,0)$ a.e in $Q^{T}$.

Proof. The identity (5.1) is equivalent to

$\displaystyle\int\limits_{0}^{T}(\rho_{1}\partial_{t}^{\alpha+1}\mathcal{\theta},\Lambda_{1})_{L^{2}(0,L)}dt-\kappa_{1}\int\limits_{0}^{T}(\mathcal{\theta}_{xx},\Lambda_{1})_{L^{2}(0,L)}dt-\kappa_{1}\int\limits_{0}^{T}(\phi_{x},\Lambda_{1})_{L^{2}(0,L)}dt+\int\limits_{0}^{T}(\mathcal{\theta}_{t},\Lambda_{1})_{L^{2}(0,L)}dt$
$\displaystyle+\int\limits_{0}^{T}(\rho_{2}\partial_{t}^{\alpha+1}\phi,\Lambda_{2})_{L^{2}(0,L)}dt-\kappa_{2}\int\limits_{0}^{T}(\phi_{xx},\Lambda_{2})_{L^{2}(0,L)}dt+\kappa_{1}\int\limits_{0}^{T}((\mathcal{\theta}_{x},\Lambda_{2})_{L^{2}(0,L)}dt$
$\displaystyle+\kappa_{1}\int\limits_{0}^{T}(\phi,\Lambda_{2})_{L^{2}(0,L)}dt+\int\limits_{0}^{T}(\int\limits_{0}^{t}m(t-s)\phi_{xx}(x,s)ds,\Lambda_{2})_{L^{2}(0,L)}dsdt$
$\displaystyle=$ $\displaystyle 0.\ $

Assume that the functions $\xi(x,t),$ $\eta(x,t)$and satisfy the boundary and
the initial conditions (2.3), and (2.4) and such that $\xi,\eta,$
$\xi_{x},\eta_{x},$ $\mathcal{I}_{t}\xi,$ $\mathcal{I}_{t}\eta,$
$\mathcal{I}_{t}\mathcal{I}_{x}^{2}\xi,$
$\mathcal{I}_{t}\mathcal{I}_{x}^{2}\eta,$ $\mathcal{I}_{t}^{2}\xi,$
$\mathcal{I}_{t}^{2}\eta$ and $\partial_{t}^{\beta+1}\xi,$
$\partial_{t}^{\beta+1}\eta\in L^{2}(0,T;L^{2}(0,L))$, we then set

$\mathcal{\theta}(x,t)=\mathcal{I}_{t}^{2}\xi=\int\limits_{0}^{t}\int\limits_{0}^{s}\xi(x,z)dzds,\text{
}\phi(x,t)=\mathcal{I}_{t}^{2}\eta=\int\limits_{0}^{t}\int\limits_{0}^{s}\eta(x,z)dzds,$
(5.3)

and introduce the functions

$\Lambda_{1}(x,t)=\mathcal{I}_{t}\xi+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\xi,\text{
}\Lambda_{2}(x,t)=\mathcal{I}_{t}\eta+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\eta.$
(5.4)

Equation (5) then reduces to

$\displaystyle\int\limits_{0}^{T}(\rho_{1}\partial_{t}^{\alpha+1}\mathcal{I}_{t}^{2}\xi,\mathcal{I}_{t}\xi+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\xi)_{L^{2}(0,L)}dt-\kappa_{1}\int\limits_{0}^{T}(\mathcal{I}_{t}^{2}\xi_{xx},\mathcal{I}_{t}\xi+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\xi)_{L^{2}(0,L)}dt$
(5.5)
$\displaystyle-\kappa_{1}\int\limits_{0}^{T}(\mathcal{I}_{t}^{2}\eta_{x},\mathcal{I}_{t}\xi+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\xi)_{L^{2}(0,L)}dt+\int\limits_{0}^{T}(\mathcal{I}_{t}\xi,\mathcal{I}_{t}\xi+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\xi)_{L^{2}(0,L)}dt$
$\displaystyle+\int\limits_{0}^{T}(\rho_{2}\partial_{t}^{\alpha+1}\mathcal{I}_{t}^{2}\eta,\mathcal{I}_{t}\eta+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\eta)_{L^{2}(0,L)}dt-\kappa_{2}\int\limits_{0}^{T}(\mathcal{I}_{t}^{2}\eta_{xx},\mathcal{I}_{t}\eta+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\eta)_{L^{2}(0,L)}dt$
$\displaystyle+\kappa_{1}\int\limits_{0}^{T}((\mathcal{I}_{t}^{2}\xi_{x},\mathcal{I}_{t}\eta+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\eta)_{L^{2}(0,L)}dt+\kappa_{1}\int\limits_{0}^{T}(\mathcal{I}_{t}^{2}\eta,\mathcal{I}_{t}\eta+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\eta)_{L^{2}(0,L)}dt$
$\displaystyle+\int\limits_{0}^{T}(\int\limits_{0}^{t}m(t-s)\mathcal{I}_{s}^{2}\eta_{xx}(x,s)ds,\mathcal{I}_{t}\eta+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\eta)_{L^{2}(0,L)}dt$
$\displaystyle=$ $\displaystyle 0.$

Invoking boundary integral conditions and carrying out appropriate
integrations by parts of each inner product term, we have

$\displaystyle(\rho_{1}\partial_{t}^{\alpha+1}\mathcal{I}_{t}^{2}\xi,\mathcal{I}_{t}\xi+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\xi)_{L^{2}(0,L)}$
(5.6) $\displaystyle=$
$\displaystyle(\rho_{1}\partial_{t}^{\alpha}\mathcal{I}_{t}\xi,\mathcal{I}_{t}\xi)+(\rho_{1}\partial_{t}^{\alpha}\mathcal{I}_{x}\mathcal{I}_{t}\xi,\mathcal{I}_{x}\mathcal{I}_{t}\xi)_{L^{2}(0,L)},$
$\displaystyle-\kappa_{1}(\mathcal{I}_{t}^{2}\xi_{xx},\mathcal{I}_{t}\xi+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\xi)_{L^{2}(0,L)}$
(5.7) $\displaystyle=$ $\displaystyle\frac{\kappa_{1}\partial}{2\partial
t}\|\mathcal{I}_{t}^{2}\xi_{x}\|_{L^{2}(0,L)}^{2}+\frac{\kappa_{1}\partial}{2\partial
t}\|\mathcal{I}_{t}^{2}\xi\|_{L^{2}(0,L)}^{2},$
$\displaystyle-\kappa_{1}(\mathcal{I}_{t}^{2}\eta_{x},\mathcal{I}_{t}\xi+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\xi)_{L^{2}(0,L)}$
(5.8) $\displaystyle=$
$\displaystyle-\kappa_{1}(\mathcal{I}_{t}^{2}\eta_{x},\mathcal{I}_{t}\xi)_{L^{2}(0,L)}+\kappa_{1}(\mathcal{I}_{t}^{2}\eta,\mathcal{I}_{x}\mathcal{I}_{t}\xi)_{L^{2}(0,L)},$
$\displaystyle(\mathcal{I}_{t}\xi,\mathcal{I}_{t}\xi+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\xi)_{L^{2}(0,L)}$
(5.9) $\displaystyle=$
$\displaystyle\|\Im_{t}\xi\|_{L^{2}(0,L)}^{2}-\|\mathcal{I}_{x}\Im_{t}\xi\|_{L^{2}(0,L)}^{2},$
$\displaystyle(\rho_{2}\partial_{t}^{\alpha+1}\mathcal{I}_{t}^{2}\eta,\mathcal{I}_{t}\eta+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\eta)_{L^{2}(0,L)}$
(5.10) $\displaystyle=$
$\displaystyle(\rho_{2}\partial_{t}^{\alpha}\mathcal{I}_{t}\eta,\mathcal{I}_{t}\eta)+(\rho_{2}\partial_{t}^{\alpha}\mathcal{I}_{x}\mathcal{I}_{t}\eta,\mathcal{I}_{x}\mathcal{I}_{t}\eta)_{L^{2}(0,L)},$
$\displaystyle-\kappa_{2}(\mathcal{I}_{t}^{2}\eta_{xx},\mathcal{I}_{t}\eta+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\eta)_{L^{2}(0,L)}$
(5.11) $\displaystyle=$ $\displaystyle\frac{\kappa_{2}\partial}{2\partial
t}\|\mathcal{I}_{t}^{2}\eta_{x}\|_{L^{2}(0,L)}^{2}+\frac{\kappa_{2}\partial}{2\partial
t}\|\mathcal{I}_{t}^{2}\eta\|_{L^{2}(0,L)}^{2}$
$\displaystyle\kappa_{1}(\mathcal{I}_{t}^{2}\xi_{x},\mathcal{I}_{t}\eta+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\eta)_{L^{2}(0,L)}$
(5.12) $\displaystyle=$
$\displaystyle\kappa_{1}(\mathcal{I}_{t}^{2}\xi_{x},\mathcal{I}_{t}\eta)_{L^{2}(0,L)}-\kappa_{1}(\mathcal{I}_{t}^{2}\xi,\mathcal{I}_{x}\mathcal{I}_{t}\eta)_{L^{2}(0,L)},$
$\displaystyle(\int\limits_{0}^{t}m(t-s)\mathcal{I}_{s}^{2}\eta_{xx}(x,s)ds,\mathcal{I}_{t}\eta+\mathcal{I}_{x}^{2}\mathcal{I}_{t}\eta)_{L^{2}(0,L)}$
(5.13) $\displaystyle=$
$\displaystyle-(\int\limits_{0}^{t}m(t-s)\mathcal{I}_{s}^{2}\eta_{x}(x,s)ds,\mathcal{I}_{t}\eta_{x})_{L^{2}(0,L)}+(\int\limits_{0}^{t}m(t-s)\mathcal{I}_{s}^{2}\eta(x,s)ds,\mathcal{I}_{t}\eta)_{L^{2}(0,L)}$
$\displaystyle=$
$\displaystyle-\frac{d}{dt}\int\limits_{0}^{L}\mathcal{I}_{t}^{2}\eta_{x}\left(\int\limits_{0}^{t}m\left(t-s\right)(\mathcal{I}_{s}^{2}\eta_{x})(x,s)ds\right)dx$
$\displaystyle+\int\limits_{0}^{L}\mathcal{I}_{t}^{2}\eta_{x}\left(\int\limits_{0}^{t}m^{\prime}\left(t-s\right)(\mathcal{I}_{s}^{2}\eta_{x})(x,s)ds\right)dx+\int\limits_{0}^{L}m\left(0\right)\left(\mathcal{I}_{t}^{2}\eta_{x}\right)^{2}dx$
$\displaystyle+(\int\limits_{0}^{t}m(t-s)\mathcal{I}_{s}^{2}\eta(x,s)ds,\mathcal{I}_{t}\eta)_{L^{2}(0,L)}.$

Insertion of equations (5.6)-(5.13) into (5.5), and using Lemma 2.2, yields

$\displaystyle\frac{\rho_{1}}{2}^{C}\partial_{t}^{\alpha}\left\|\mathcal{I}_{t}\xi)\right\|_{L^{2}(0,L)}^{2}+\frac{\rho_{1}}{2}^{C}\partial_{t}^{\alpha}\left\|\mathcal{I}_{x}\mathcal{I}_{t}\xi)\right\|_{L^{2}(0,L)}^{2}+\frac{\kappa_{1}\partial}{2\partial
t}\|\mathcal{I}_{t}^{2}\xi_{x}\|_{L^{2}(0,L)}^{2}$
$\displaystyle+\frac{\kappa_{1}\partial}{2\partial
t}\|\mathcal{I}_{t}^{2}\xi\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{t}\xi\|_{L^{2}(0,L)}^{2}+\frac{\rho_{2}}{2}^{C}\partial_{t}^{\alpha}\left\|\mathcal{I}_{t}\eta)\right\|_{L^{2}(0,L)}^{2}$
$\displaystyle+\frac{\rho_{2}}{2}^{C}\partial_{t}^{\alpha}\left\|\mathcal{I}_{x}\mathcal{I}_{t}\eta)\right\|_{L^{2}(0,L)}^{2}+\frac{\kappa_{2}\partial}{2\partial
t}\|\mathcal{I}_{t}^{2}\eta_{x}\|_{L^{2}(0,L)}^{2}+\frac{\kappa_{2}\partial}{2\partial
t}\|\mathcal{I}_{t}^{2}\eta\|_{L^{2}(0,L)}^{2}$
$\displaystyle+\int\limits_{0}^{L}h\left(0\right)\left(\mathcal{I}_{t}^{2}\eta_{x}\right)^{2}dx$
$\displaystyle\leq$ $\displaystyle\
\kappa_{1}(\mathcal{I}_{t}^{2}\eta_{x},\mathcal{I}_{t}\xi)_{L^{2}(0,L)}-\kappa_{1}(\mathcal{I}_{t}^{2}\eta,\mathcal{I}_{x}\mathcal{I}_{t}\xi)_{L^{2}(0,L)}+\|\mathcal{I}_{x}\mathcal{I}_{t}\xi\|_{L^{2}(0,L)}^{2}$
(5.14)
$\displaystyle-\kappa_{1}(\mathcal{I}_{t}^{2}\xi_{x},\mathcal{I}_{t}\eta)_{L^{2}(0,L)}+\kappa_{1}(\mathcal{I}_{t}^{2}\xi,\mathcal{I}_{x}\mathcal{I}_{t}\eta)_{L^{2}(0,L)}$
$\displaystyle+\frac{d}{dt}\int\limits_{0}^{L}\mathcal{I}_{t}^{2}\eta_{x}\left(\int\limits_{0}^{t}m\left(t-s\right)(\mathcal{I}_{s}^{2}\eta_{x})(x,s)ds\right)dx$
$\displaystyle-\int\limits_{0}^{L}\mathcal{I}_{t}^{2}\eta_{x}\left(\int\limits_{0}^{t}m^{\prime}\left(t-s\right)(\mathcal{I}_{s}^{2}\eta_{x})(x,s)ds\right)dx$
$\displaystyle-(\int\limits_{0}^{t}m(t-s)\mathcal{I}_{s}^{2}\eta(x,s)ds,\mathcal{I}_{t}\eta)_{L^{2}(0,L)}.$

By using Cauchy $\epsilon$ -inequality, we estimate each term of the right-
hand side of previous relations to get

$\kappa_{1}(\mathcal{I}_{t}^{2}\eta_{x},\mathcal{I}_{t}\xi)_{L^{2}(0,L)}\leq\frac{\kappa_{1}}{2}\|\mathcal{I}_{t}^{2}\eta_{x}\|_{L^{2}(0,L)}^{2}+\frac{\kappa_{1}}{2}\|\mathcal{I}_{t}\xi\|_{L^{2}(0,L)}^{2},\
$ (5.15)
$-\kappa_{1}(\mathcal{I}_{t}^{2}\eta,\mathcal{I}_{x}\mathcal{I}_{t}\xi)_{L^{2}(0,L)}\leq\frac{\kappa_{1}}{2}\|\mathcal{I}_{t}^{2}\eta\|_{L^{2}(0,L)}^{2}+\frac{\kappa_{1}}{2}\|\mathcal{I}_{x}\mathcal{I}_{t}\xi\|_{L^{2}(0,L)}^{2},\
$ (5.16)
$-\kappa_{1}(\mathcal{I}_{t}^{2}\xi_{x},\mathcal{I}_{t}\eta)_{L^{2}(0,L)}\leq\frac{\kappa_{1}}{2}\|\mathcal{I}_{t}^{2}\xi_{x}\|_{L^{2}(0,L)}^{2}+\frac{\kappa_{1}}{2}\|\mathcal{I}_{t}\eta\|_{L^{2}(0,L)}^{2},$
(5.17)
$\kappa_{1}(\mathcal{I}_{t}^{2}\xi,\mathcal{I}_{x}\mathcal{I}_{t}\eta)_{L^{2}(0,L)}\leq\frac{\kappa_{1}}{2}\|\mathcal{I}_{t}^{2}\xi\|_{L^{2}(0,L)}^{2}+\frac{\kappa_{1}}{2}\|\mathcal{I}_{x}\mathcal{I}_{t}\eta\|_{L^{2}(0,L)}^{2},$
(5.18)
$\displaystyle-\int\limits_{0}^{L}\mathcal{I}_{t}^{2}\eta_{x}\left(\int\limits_{0}^{t}m^{\prime}\left(t-s\right)(\mathcal{I}_{s}^{2}\eta_{x})(x,s)ds\right)dx$
(5.19) $\displaystyle\leq$ $\displaystyle\frac{1}{2}\underset{0\leq t\leq
T}{\sup}\left|m^{\prime}\right|\left(1+\frac{T^{2}}{2}\right)\|\mathcal{I}_{t}^{2}\eta_{x}\|_{L^{2}(0,L)}^{2},$
$\displaystyle-(\int\limits_{0}^{t}m(t-s)\mathcal{I}_{s}^{2}\eta(x,s)ds,\mathcal{I}_{t}\eta)_{L^{2}(0,L)}$
$\displaystyle\frac{1}{2}\underset{0\leq t\leq
T}{\sup}\left|m\right|\|\mathcal{I}_{t}\eta\|_{L^{2}(0,L)}^{2}+\frac{T^{2}}{2}\underset{0\leq
t\leq T}{\sup}\left|m\right|\|\mathcal{I}_{t}^{2}\eta\|_{L^{2}(0,L)}^{2}.$
(5.20)

By combining equality (LABEL:e4.14) and inequalities (5.15)-(5.20), we obtain

${}^{C}\partial_{t}^{\alpha}\left\|\mathcal{I}_{t}\xi)\right\|_{L^{2}(0,L)}^{2}+^{C}\partial_{t}^{\alpha}\left\|\mathcal{I}_{t}\eta)\right\|_{L^{2}(0,L)}^{2}+^{C}\partial_{t}^{\alpha}\left\|\mathcal{I}_{x}\mathcal{I}_{t}\xi)\right\|_{L^{2}(0,L)}^{2}$
$\displaystyle+^{C}\partial_{t}^{\alpha}\left\|\mathcal{I}_{x}\mathcal{I}_{t}\eta)\right\|_{L^{2}(0,L)}^{2}+\frac{\partial}{\partial
t}\|\mathcal{I}_{t}^{2}\xi_{x}\|_{L^{2}(0,L)}^{2}+\frac{\partial}{\partial
t}\|\mathcal{I}_{t}^{2}\eta_{x}\|_{L^{2}(0,L)}^{2}$
$\displaystyle+\frac{\partial}{\partial
t}\|\mathcal{I}_{t}^{2}\xi\|_{L^{2}(0,L)}^{2}+\frac{\partial}{\partial
t}\|\mathcal{I}_{t}^{2}\eta\|_{L^{2}(0,L)}^{2}+\int\limits_{0}^{L}\left(\mathcal{I}_{t}^{2}\eta_{x}\right)^{2}dx+\|\mathcal{I}_{t}\xi\|_{L^{2}(0,L)}^{2}$
(5.21) $\displaystyle\leq$
$\displaystyle\mathcal{W}\left(\|\mathcal{I}_{t}\xi\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{t}\eta\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{x}\mathcal{I}_{t}\xi\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{x}\mathcal{I}_{t}\eta\|_{L^{2}(0,L)}^{2}\right.$
$\displaystyle\left.+\|\mathcal{I}_{t}^{2}\xi_{x}\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{t}^{2}\eta_{x}\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{t}^{2}\xi\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{t}^{2}\eta\|_{L^{2}(0,L)}^{2}\right),$

where

$\mathcal{W}=\frac{\max\left\\{\frac{\kappa_{1}}{2}+(1+\frac{T^{2}}{2})\frac{1}{2}\sup\left|m^{\prime}\right|,\frac{\kappa_{1}}{2}+\frac{T^{2}}{2}\frac{1}{2}\sup\left|m\right|\right\\}}{\min\left\\{1,\frac{\rho_{1}}{2},\frac{\rho_{2}}{2},\frac{\kappa_{1}}{2},\frac{\kappa_{2}}{2},h(0)\right\\}}.$
(5.22)

By discarding the last two terms from the left hand side of (LABEL:e4.21*),
replacing $t$ by $\tau$ in (5.22) and then integrating with respect to $\tau$
over the interval $(0,t),$ we obtain

$\displaystyle
D_{t}^{\alpha-1}\left\|\mathcal{I}_{t}\xi)\right\|_{L^{2}(0,L)}^{2}+D_{t}^{\alpha-1}\left\|\mathcal{I}_{t}\eta)\right\|_{L^{2}(0,L)}^{2}+D_{t}^{\alpha-1}\left\|\mathcal{I}_{x}\mathcal{I}_{t}\xi)\right\|_{L^{2}(0,L)}^{2}$
(5.23)
$\displaystyle+D_{t}^{\alpha-1}\left\|\mathcal{I}_{x}\mathcal{I}_{t}\eta)\right\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{t}^{2}\xi_{x}\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{t}^{2}\eta_{x}\|_{L^{2}(0,L)}^{2}$
$\displaystyle+\|\mathcal{I}_{t}^{2}\xi\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{t}^{2}\eta\|_{L^{2}(0,L)}^{2}$
$\displaystyle\leq$
$\displaystyle\mathcal{W}\left(\|\mathcal{I}_{t}\xi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{t}\eta\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{x}\mathcal{I}_{t}\xi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{x}\mathcal{I}_{t}\eta\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right.$
$\displaystyle\left.+\|\mathcal{I}_{t}^{2}\xi_{x}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{t}^{2}\eta_{x}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{t}^{2}\xi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{t}^{2}\eta\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right).$

If we omit the first four terms on the left hand side of (5.23), and use
Gronwall-Bellman lemma, by taking

$\left\\{\begin{array}[]{c}\begin{array}[]{c}\mathcal{R}(t)=\|\mathcal{I}_{t}^{2}\xi_{x}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{t}^{2}\eta_{x}\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\\\
+\|\mathcal{I}_{t}^{2}\xi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{t}^{2}\eta\|_{L^{2}(0,t;L^{2}(0,L))}^{2},\end{array}\\\
\frac{\partial\mathcal{R}(t)}{\partial
t}=\|\mathcal{I}_{t}^{2}\xi_{x}\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{t}^{2}\eta_{x}\|_{L^{2}(0,L)}^{2}\\\
+\|\mathcal{I}_{t}^{2}\xi\|_{L^{2}(0,L)}^{2}+\|\mathcal{I}_{t}^{2}\eta\|_{L^{2}(0,L)}^{2},\\\
\mathcal{R}(t)=0,\end{array}\right.$ (5.24)

we obtain

$\displaystyle\mathcal{R}(t)$ $\displaystyle\leq$ $\displaystyle
Te^{T\mathcal{W}}\left(\|\mathcal{I}_{t}\xi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{t}\eta\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{x}\mathcal{I}_{t}\xi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right.$
(5.25)
$\displaystyle\left.+\|\mathcal{I}_{x}\mathcal{I}_{t}\eta\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right).$

Next, if we disregard the last four terms on the left-hand side and take into
account the inequality (5.25), we end with

$\displaystyle
D_{t}^{\alpha-1}\left\|\mathcal{I}_{t}\xi)\right\|_{L^{2}(0,L)}^{2}+D_{t}^{\alpha-1}\left\|\mathcal{I}_{t}\eta)\right\|_{L^{2}(0,L)}^{2}+D_{t}^{\alpha-1}\left\|\mathcal{I}_{x}\mathcal{I}_{t}\xi)\right\|_{L^{2}(0,L)}^{2}$
(5.26)
$\displaystyle+D_{t}^{\alpha-1}\left\|\mathcal{I}_{x}\mathcal{I}_{t}\eta)\right\|_{L^{2}(0,L)}^{2}$
$\displaystyle\leq$
$\displaystyle\mathcal{W}\left(Te^{T\mathcal{W}}+1\right)\left(\|\mathcal{I}_{t}\xi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{t}\eta\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right.$
$\displaystyle+\left.\|\mathcal{I}_{t}^{2}\xi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{t}^{2}\eta\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\right).$

Now, we are able to apply lemma 2.2, by letting

$\left\\{\begin{array}[]{c}\begin{array}[]{c}Q(t)=\|\mathcal{I}_{t}\xi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{t}\eta\|_{L^{2}(0,t;L^{2}(0,L))}^{2}\\\
\|\mathcal{I}_{t}^{2}\xi\|_{L^{2}(0,t;L^{2}(0,L))}^{2}+\|\mathcal{I}_{t}^{2}\eta\|_{L^{2}(0,t;L^{2}(0,L))}^{2},\end{array}\\\
{}^{C}\partial_{t}^{\alpha}Q(t)=D_{t}^{\alpha-1}\left\|\mathcal{I}_{t}\xi)\right\|_{L^{2}(0,L)}^{2}+D_{t}^{\alpha-1}\left\|\mathcal{I}_{t}\eta)\right\|_{L^{2}(0,L)}^{2}\\\
+D_{t}^{\alpha-1}\left\|\mathcal{I}_{x}\mathcal{I}_{t}\xi)\right\|_{L^{2}(0,L)}^{2}+D_{t}^{\alpha-1}\left\|\mathcal{I}_{x}\mathcal{I}_{t}\eta)\right\|_{L^{2}(0,L)}^{2},\\\
Q(0)=0,\end{array}\right.$ (5.27)

we infer from (5.26) that

$Q(t)\leq\Gamma(\alpha)E_{\alpha,\alpha}(\mathcal{W}\left(Te^{T\mathcal{W}}+1\right)t^{\alpha})D_{t}^{-\alpha}(0)=0.$
(5.28)

We conclude from (5.28), and (5.27) that $\xi=0,$ $\eta=0.$ Consequently,
$W(x,t)=(\Lambda_{1}(x,t),\Lambda_{2}(x,t))=(0,0)$ a.e in $Q^{T}.$

We now consider the general case for density

Since $\mathcal{E}$ is a Hilbert space, then
$\overline{R(\mathcal{G})}=\mathcal{E}$ $\Leftrightarrow
R(\mathcal{G})^{\bot}=\left\\{0\right\\}\Leftrightarrow(\mathcal{GZ},\mathcal{K})_{\mathcal{E}}=0,$
for all $\mathcal{Z\in B}$ , and $\mathcal{K\in E},$then
$\mathcal{K=(K}_{1},\mathcal{K}_{2})=\left\\{(J_{1},J_{3},J_{4}),(J_{2},J_{5},J_{6})\right\\}=(0,0),$
that is $J_{1}=J_{2}=J_{3}=J_{4}=J_{5}=J_{6}=0.$ So suppose that for some
element
$\mathcal{K=(K}_{1},\mathcal{K}_{2})=\left\\{(J_{1},J_{3},J_{4}),(J_{2},J_{5},J_{6})\right\\}\in
R(\mathcal{G})^{\bot}$

$\displaystyle(\mathcal{GZ},\mathcal{K})_{\mathcal{E}}$ (5.29)
$\displaystyle=$
$\displaystyle(\\{\mathcal{S}_{1}(\mathcal{\theta},\phi),\mathcal{S}_{2}(\mathcal{\theta},\phi\\},\\{\mathcal{K}_{1},\mathcal{K}_{2}\\})_{\mathcal{E}}$
$\displaystyle=$
$\displaystyle(\\{\mathcal{S}_{1}(\mathcal{\theta},\phi),\Gamma_{1}\mathcal{\theta},\Gamma_{2}\mathcal{\theta}\\},\\{\mathcal{S}_{2}(\mathcal{\theta},\phi),\Gamma_{1}\phi,\Gamma_{2}\phi\\}\\},\\{\\{J_{1},J_{2},J_{3}\\},\\{J_{4},J_{5},J_{6}\\}\\})_{\mathcal{E}}$
$\displaystyle=$
$\displaystyle(\mathcal{S}_{1}(\mathcal{\theta},\phi),J_{1})_{L^{2}(Q^{T})}+(\Gamma_{1}\mathcal{\theta},J_{2})_{L^{2}(0,L)}+(\Gamma_{2}\mathcal{\theta},J_{3})_{L^{2}(0,L)}$
$\displaystyle+(\mathcal{S}_{2}(\mathcal{\theta},\phi),J_{4})_{L^{2}(Q^{T})}+(\Gamma_{1}\phi,J_{5})_{L^{2}(0,L)}+(\Gamma_{2}\phi,J_{6})_{L^{2}(0,L)}$
$\displaystyle=$ $\displaystyle 0,$

where $\mathcal{Z}$ runs over the space $\mathcal{B}$, we have to prove that
$\mathcal{K}=0.$

Let $\mathcal{Z}\in D_{0}(\mathcal{G}),$ then equation (5.29) becomes

$(\mathcal{S}_{1}(\mathcal{\theta},\phi),J_{1})_{L^{2}(Q^{T})}+(\mathcal{S}_{2}(\mathcal{\theta},\phi),J_{4})_{L^{2}(Q^{T})}=0.$
(5.30)

Hence, by virtue of Theorem 4.2, it follows from (5.30) that $J_{1}=J_{4}=0$.
Consequently, equation (5.29) takes the form

$(\Gamma_{1}\mathcal{\theta},J_{2})_{L^{2}(0,L)}+(\Gamma_{2}\mathcal{\theta},J_{3})_{L^{2}(0,L)}+(\Gamma_{1}\phi,J_{5})_{L^{2}(0,L)}+(\Gamma_{2}\phi,J_{6})_{L^{2}(0,L)}=0.$
(5.31)

Since the four terms in (5.31) vanish independently and since the ranges
$R(\Gamma_{1}),R(\Gamma_{2})$ of the trace operators $\Gamma_{1},\Gamma_{2}$
are everywhere dense in the space $L^{2}(0,L)$, then it follows from (5.31)
that $J_{2}=J_{3}=J_{5}=J_{6}=0$. Consequently $\mathcal{K}=0$, that is
$R(\mathcal{G})^{\perp}=\\{0\\}$. Thus
$\overline{R(\mathcal{G})}=\mathcal{E}$.

Conclusion: In this article, we proved the well posedness of a nonhomogeneous
Timoshenko system with a viscoelastic damping term.The coupled two hyperbolic
equations were associated with initial conditions and a nonlocal boundary
conditions. The proofs of the results are mainly based on some energy and a
priori estimates and on some density arguments. The method uses functional
analysis tools such as operator theory and density arguments. It is found that
the method is efficient and powerful for solving initial boundary value
problems with non-local constraints.The a priori estimate for the solution can
be provided by constructing a suitable multiplicator and from which it is also
possible to establish the solvability of the stated problem. We notice that no
previous works were treated for Timoshenko systems with nonlocal conditions of
integral type.

Acknowledgment: The authors would like to extend their sincere appreciation to
the Deanship of Scientific Research at King Saud University for its funding
this Research group No (RG 117).

The second author would like to thank Shagrah University for giving her the
chance to further her PhD studies at King Saud University.

Availability of data and materials: Data sharing is not applicable to this
article as no data sets were generated or analysed during the current study.

Competing interests: The authors declare that they have no competing
interests.

Authors’ contributions: All authors contributed equally to this work. All
authors read and approved the final manuscript.

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