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\begin{document}

\title[existence of periodic solutions for singular systems]
{ON THE EXISTENCE OF PERIODIC SOLUTIONS FOR SINGULAR
NON-AUTONOMOUS THIRD ORDER SYSTEMS}%
\author{B. Mehri}%
\address{Department of Mathematics, Sharif University of Technology, Tehran, Iran}%
\email{Mehri@sharif.edu}%
\author{M. A. Niksirat}%
\address{Department of Mathematics, Sharif University of Technology, Tehran, Iran}%
\email{Niksirat@yahoo.com}%
%\thanks{}%
\subjclass{34C25, 34A05}%
\keywords{singular non-autonomous systems, periodic solutions, parametric nonlinear systems, topological degree}%

%\date{}%
%\dedicatory{}%
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\begin{abstract}
Here we are concerned with the existence of periodic solution for
nonlinear non-autonomous third order  system of ordinary
differential equations with singular terms. Our method here is
based on the topological method in the sense that we infer some
results for the focal system from its nonsingular homotopy. The
aim is to obtain sufficient conditions for which the system has
periodic solution whenever the value of deformation with respect
to the first variation of the nonsingular subsystem is
sufficiently small. The method presented here is constructive in
the sense that the existence of periodic orbits can be verified
numerically as well as computed if any. For this, we will show how
classical methods like the Newton for solving algebraic equations,
can be applied to the equations obtained analytically in this
paper. Finally we treat a definite singular system numerically in
order to verify the obtained results.
\end{abstract}
\maketitle
% ----------------------------------------------------------------
\section{Introduction}

There have been extensive researches on the existence of periodic
solution for the nonlinear non-autonomous system of ODEs. One of
the standard methods employed for this, is to transforming the
system into a Fredholm's type integral operator by making use of
a Green or generalized Green function with a suitable periodic or
semi-periodic boundary conditions, and then to apply the
topological fixed point theorems like the Banach or Leray-Schauder
to prove the existence of a solution for the integral operator in
a suitable Banach space, see for example~\cite{laz68, cron64,
mehnik97}. When the system has a trivial solution, applying fixed
point theorems to establish the existence of a aperiodic solution
for the system is more complicated than the case in which the
system doesn't have any trivial solution, and for this reason
many authors have employed the topological degree theory  to treat
these cases; see for example~\cite{mehnik01, mehnik00,
mehemam78}. However, the analysis of differential equations with
a singular term is more complicated than the analysis of regular
differential equations where defining the topological degree is
simply possible. Singular terms are usually encountered in the
studies of celestial mechanics specially the studies of periodic
behaviour of heavens and
galaxies.\\\\
Here we are concerned with the following non-autonomous system:
\begin{equation}\label{syst}
x'''+\Lambda x'=f(x,x',x'',t)+\gamma{x\over \norm{x}^3}
\end{equation}
where $x\in\Real^n$, $f$ is periodic with respect to $t$ with
period $\omega$, $\Lambda$ is a constant positive diagonal
$n\times n$ matrix as $\diag[\lambda_1^2\cdots \lambda_n^2]$ and
$\gamma$ is also a diagonal matrix. As usual it is assumed that
$f:\Real^{3n+1}\to\Real^n$ is smooth enough to guarantee the
existence and uniqueness of the solution. We first prove the
existence of a periodic solution for the nonsingular subsystem of
the Eq.(\ref{syst}) where $\gamma=0$ just for the sufficiently
small norm of $f$. For this let us consider the following
parametric system:
\begin{equation}\label{NSsyst}
  x'''+\Lambda x'=\epsilon f(x,x',x'',t)
\end{equation}
where $\epsilon$ is considered as a parameter. Note that when
$\omega$ is equal to one of the ratios ${2\pi\over\lambda_i}$ for
$\lambda_i\in\set{\lambda_1,\cdots,\lambda_n}$, then the solution
of the Eq.(\ref{NSsyst}) for linear systems is of the type of
resonance which would be unbounded when $t\to \infty$. What is
interesting here is that we can obtain some conditions under which
the nonlinear system has a periodic solution if
$\set{\lambda_1,\cdots,\lambda_n}$ has a greatest common divisor
$\lambda$, i.e $\lambda=\gcd\set{\lambda_1,\cdots,\lambda_n}$ and
$\omega={2\pi\over\lambda}$. Considering the nonlinearity as a
term dependent on a parameter has this advantage that makes it
possible to study the deformation, which is caused by the change
of the value of the parameter in the system, of a periodic orbit
which is related to the first variation of the system.
% ----------------------------------------------------------------
\section{Main Theorem}
Let's consider Eq.(\ref{NSsyst}) with $\Lambda=I$, the identity
matrix just for notation simplicity. Applying the method for the
case when $\Lambda=\diag[\lambda_1,\cdots,\lambda_n]$ has a
greatest common divisor as $\lambda$ is similar. Now note that
$\omega=2\pi$ in this case and the necessary and sufficient
condition for the existence of a periodic solution for
Eq.(\ref{NSsyst}) is:
\begin{equation}%\label{}
  x_i^{(j)}(0)=x_i^{(j)}(2\pi),~~i=1,\cdots,n,~~j=0,1,2
\end{equation}
The solution of the Eq.(\ref{NSsyst}) can be written as:
\begin{equation}
x(t)=A+(\cos t) B+(\sin t) C+\epsilon\int_0^t{[1-\cos(t-s)]
f(x(s),x'(s),x''(s))\,ds}
\end{equation}
where $A, B$ and $C$ are vectors of dimension $n$ which determine
in turn the initial condition of Eq.(\ref{NSsyst}). Now define
$\Gamma_\epsilon$ as
\begin{equation}
\Gamma_\epsilon=(
x(0)-x(2\pi),~x'(0)-x'(2\pi),~x''(0)-x''(2\pi))^\star
\end{equation}
where notation $\star$ denotes the transpose of a matrix or
vector. It is obvious that
$\Gamma_\epsilon=\Gamma_\epsilon(A,B,C)$. Now the existence of a
periodic solution for Eq.(\ref{NSsyst}) is equivalent to the
existence of the triple $(A^\epsilon, B^\epsilon, C^\epsilon)$
such that:
\begin{equation}\label{GamEq}
\Gamma_\epsilon(A^\epsilon, B^\epsilon, C^\epsilon)=0
\end{equation}
For this we first obtain conditions under which $\Gamma_0$, the
first variation of $\Gamma_\epsilon$ has a periodic solution.
Defining the function $\Phi=(\phi_1, \phi_2, \phi_3)^\star$ as
the following:
\begin{eqnarray*}
\phi_1(A,B,C)=\int_0^{2\pi}{(1-\cos{s})\,\tilde{f}(s)\,ds}\\
\phi_2(A,B,C)=\int_0^{2\pi}{\sin{s}\,\tilde{f}(s)\,ds}\\
\phi_3(A,B,C)=\int_0^{2\pi}{\cos{s}\,\tilde{f}(s)\,ds}
\end{eqnarray*}
where
\begin{equation}\label{fsEq}
\tilde{f}(s)=f(A+\cos{s}~B+\sin{s}~C,-\sin{s}~B+\cos{s}~C,
-\cos{s}~B-\sin{s}~C,s)
\end{equation}
we have the following theorem.
\begin{thm}\label{NSthm}
Assume there are $(A^0,B^0,C^0)$ such that
\begin{equation}\label{FiEq}
\Phi(A^0,B^0,C^0)=0
\end{equation}
and the Jacobian of the mapping $\Phi$ with respect to $(A,B,C)$
is nonzero, i.e;
\begin{equation}\label{JacEq}
J_{\Phi}(A^0,B^0,C^0)\not =0
\end{equation}
then there exist $\epsilon_0>0$ such that for all
$|\epsilon|<\epsilon_0$ the Eq.(\ref{NSsyst}) has a nontrivial
periodic solution.
\end{thm}
\begin{proof} Let $d[f,U,p]$ denotes as usual  the Brouwer's degree;
see for example~\cite{cron64, loyd78}:
\[
d[f,U,p]=\sum_a{\sign J_f(a)}
\]
where $a$ is the root of equation $f=p$ on $U$. What we have to
show is the existence of an open bounded subset of $\Real^{3n}$ as
$\Omega$ so that
\[
\Gamma_0:\Omega\to \Real^{3n}
\]
and
\begin{equation}
d[\Gamma_0,\Omega,0]\neq 0
\end{equation}
Then by the homotopy invariance property of topological degree it
results that for small deformation, that is corresponds to small
values of $\epsilon$ in Eq.(\ref{NSsyst}), there is $(A^\epsilon,
B^\epsilon, C^\epsilon )$ that satisfies the Eq.(\ref{GamEq}). So
if there is an open bounded subset of $\Real^{3n}$ as $\Omega$
such that $d[\Gamma_0,\Omega,0] \not =0$ then there is
$\epsilon_0>0$ such that
\begin{equation}\label{degreeEq}
d[\Gamma_\epsilon,\Omega,0]\not =0,\,\epsilon\in[0,\epsilon_0]
\end{equation}
Now it is straightforward to note that the first variation of
$\Gamma_\epsilon$ at $\epsilon=0$ is equivalent to $\Phi$. Let
$p_0=(A^0, B^0, C^0)$ be one of the solutions of equation $\Phi=0$
as stated in the Eq.(\ref{FiEq}) and (\ref{JacEq}), then there is
an open bounded subset of $\Real^{3n}$ around $p_0$ as $\Omega$
by the Sard's theorem such that
\begin{equation}
d[\Gamma_0,\Omega,0]=d[\Phi,\Omega,0]=\sign J_\Phi
\end{equation}
So by Eq.(\ref{degreeEq}) there is $\epsilon>0$ such that for
$|\epsilon|<\epsilon_0$ the equation $\Gamma_\epsilon(A,B,C)=0$
has a solution in $\Omega$ as $(A^\epsilon, B^\epsilon,
C^\epsilon)$.
\end{proof}
We now compute the Jacobian of $\Phi$ at solution points of the
equation $\Phi=0$. The Jacobian easily can be computed as the
follows:
\[
J_\Phi=\left | \begin{tabular}{ccc} ${\partial\phi_1\over\partial
A}$ & ${\partial\phi_1\over\partial B}$ &
${\partial\phi_1\over\partial C}$ \\
\\
\hline
\\
${\partial\phi_2\over\partial A}$ & ${\partial\phi_2\over\partial
B}$ &
${\partial\phi_2\over\partial C}$ \\
\\
\hline
\\
${\partial\phi_3\over\partial A}$ & ${\partial\phi_3\over\partial
B}$ & ${\partial\phi_3\over\partial C}$
\end{tabular}\right |
\]
which we obtain by simple calculations the following:
\[\begin{array}{ll}\vspace{2mm}
{\partial\phi_1\over\partial A}=[-\int_0^{2\pi}
{(1-\cos{s}){\partial \tilde{f}_i\over\partial a_j}(s)\,ds}] &
{\partial\phi_2\over\partial A}=[\int_0^{2\pi} {\sin{s}{\partial
\tilde{f}_i\over\partial a_j}(s)\,ds}]
\\\vspace{2mm}
{\partial\phi_1\over\partial B}=[-\int_0^{2\pi}
{(1-\cos{s}){\partial \tilde{f}_i\over\partial b_j}(s)\,ds}] &
{\partial\phi_2\over\partial B}=[\int_0^{2\pi} {\sin{s}{\partial
\tilde{f}_i\over\partial b_j}(s)\,ds}]
\\\vspace{2mm}
{\partial\phi_1\over\partial C}=[-\int_0^{2\pi}
{(1-\cos{s}){\partial \tilde{f}_i\over\partial c_j}(s)\,ds}] &
{\partial\phi_2\over\partial C}=[\int_0^{2\pi} {\sin{s}{\partial
\tilde{f}_i\over\partial c_j}(s)\,ds}]
\\\vspace{2mm}
{\partial\phi_3\over\partial A}=[-\int_0^{2\pi} {\cos{s}{\partial
\tilde{f}_i\over\partial a_j}(s)\,ds}] &
{\partial\phi_3\over\partial B}=[-\int_0^{2\pi} {\cos{s}{\partial
\tilde{f}_i\over\partial b_j}(s)\,ds}]
\\\vspace{2mm}
{\partial\phi_3\over\partial C}=[-\int_0^{2\pi}
{\cos{s}{\partial \tilde{f}_i\over\partial c_j}(s)\,ds}] \\
\end{array}
\]
where:
\[\begin{array}{l}\vspace{1mm}
{\partial \tilde{f}_i\over\partial a_j}= {\partial
f_i\over\partial x_j}\\\vspace{1mm} {\partial
\tilde{f}_i\over\partial b_j}= {\partial f_i\over\partial
x_j}\cos{s}- {\partial f_i\over\partial x'_j}\sin{s}- {\partial
f_i\over\partial x''_j}\cos{s}
\\\vspace{1mm}
{\partial \tilde{f}\over\partial c_j}= {\partial f_i\over\partial
x_j}\sin{s}+ {\partial f_i\over\partial x'_j}\cos{s}- {\partial
f_i\over\partial x''_j}\sin{s}
\end{array}\]
\begin{cor}\label{corCont}  For every solutiuon of (\ref{FiEq})
and (\ref{JacEq}), there is $\delta>0$ such that for $0\leq
|\epsilon|<\delta$ the path made by $(A^\epsilon, B^\epsilon,
C^\epsilon)$ is continuous with no branch.
\end{cor}
\begin{proof}
Let's consider the mapping
$\varphi=I-J_{\Phi}^{-1}~\Gamma_\epsilon$ on a sufficiently small
neighborhood $D_\delta$ of $(A^0, B^0, C^0)$. Since
$\Gamma_\epsilon$ is $C^1$ respect to $\epsilon$,
$||\Gamma_\epsilon-\Phi||$ is sufficiently small in $C^1$
topology, so that we have $||\varphi'||\leq \lambda<1$. Then
$\varphi$ is a contraction mapping and then has a unique fixed
point in $D_\delta$ as $(A^\epsilon, B^\epsilon, C^\epsilon)$.
Let's denote $(A^\epsilon, B^\epsilon, C^\epsilon)$ as
$u_s(\epsilon)$. Now consider $\varphi$ as a parameter family
functions of $\epsilon$ as $\varphi=\varphi(u,\epsilon)$ where
$u\in D_\delta$. We have
\[
||u_s(\epsilon)-u_s(0)||\leq{1\over
1-\lambda}||\varphi(u_s(0),\epsilon)-\varphi(u_s(0),0)||
\]
but $\varphi(u,\epsilon)$ is continuous respect to $\epsilon$,
then $u_s(\epsilon)$ is continuous and continuously converge to
$u_s(0)$.
\end{proof}
So it is obvious by the above lemma that when $\epsilon$ in
Eq.(\ref{NSsyst}) starts to increase or decrease,
$(A^\epsilon,B^\epsilon,C^\epsilon)$ makes form a curve in the
space spanned by $(A,B,C)$.
\begin{thm}
For sufficiently small norm of function $f$ in Eq.(\ref{syst}) and
under conditions of theorem~\ref{NSthm}, there is a $\delta>0$
such that for $\norm \gamma\leq\delta$, the Eq.(\ref{syst}) has a
periodic solution.
\end{thm}
\begin{proof}
 Let $P^\epsilon$ is the solution of
Eqs.(\ref{FiEq}),~(\ref{JacEq}) and $h_t$ be a $C^1$ homotopy on
$t\in[0,\delta]$ such that
\begin{equation}%\label{}
h_0=\epsilon f,~~~h_\delta=\epsilon f+\gamma {x\over\norm{x}^3}
\end{equation}
Let us denote by $\Phi^{h}$ the applying the mapping $\Phi$ on the
function $h_t$. To prove the theorem, we have to show that the
equation $\Phi^h=0$ has a solution. By the theorem~\ref{NSthm} we
have
\begin{equation}%\label{}
  d[\Phi,\Omega,0]=\sign J_\Phi(P^\epsilon)
\end{equation}
 So by homotopy invariance property of topological
degree it follows that
\begin{equation}%\label{}
   d[\Phi^h,\Omega,0]=d[\Phi,\Omega,0]=\sign J_\Phi(P^\epsilon)
\end{equation}
which implies that there is a $P^\gamma\in\Omega$ such that
$\Phi^h(P^\gamma)=0$.
\end{proof}
% ----------------------------------------------------------------
\section{Numerical Treatment}
Now let's consider an equation for the numerical analysis.
Consider the following system:
\begin{equation}\label{singsyst}
x'''+x'=\epsilon[L(x,x',x'')+f(t)]+[{\gamma_i
x_i\over(x_1^2+...+x_n^2)^{3/2}}]
\end{equation}
where $x\in\Real^n$ and $L(x,x',x'')$ is a linear term with $S_x,
S_{x'}, S_{x''}$ as its $n\times n$ coefficient matrices of
$x,x',x''$ respectively, i.e:
\[
L(x,x',x'')=S_x~x+S_{x'}~x'+S_{x''}~x''
\]
and $f$ is a $2\pi$ periodic vector function respect to $t$. The
difficulty arising in the computation of Eq.(\ref{FiEq}) and
~(\ref{JacEq}) for Eq.(\ref{singsyst}) is due to the singular
term presented in the equation. Let us denote by $\mathcal L$ the
linear subsystem of Eq.(\ref{singsyst}), i.e
\begin{equation}\label{linsyst}
{\mathcal L}(x,t)=L(x,x',x'')+f(t)
\end{equation}
Our aim is to infer the existence of a solution for
Eq.(\ref{singsyst}) from the existence of solution for equation
\begin{equation}%\label{}
  x'''+x'=\epsilon{\mathcal L}(x,t)
\end{equation}
by the theorem of small homotopy. Applying Eq.(\ref{FiEq}) to
${\mathcal L}(x,t)$ results in
\begin{equation}\label{FicalLEq}
 \Phi^{\mathcal L}=\Phi^L+\Phi^F
\end{equation}
where $\Phi^L=(\phi_1^L,\phi_2^L,\phi_3^L)$ is as:
\begin{eqnarray*}
  \phi_1^L=\int_0^{2\pi}{(1-\cos{s})~\tilde{L}(s)\,ds}\\
  \phi_2^L=\int_0^{2\pi}{\sin{s}~\tilde{L}(s)\,ds}\\
  \phi_3^L=\int_0^{2\pi}{\cos{s}~\tilde{L}(s)\,ds}
\end{eqnarray*}
and $\tilde{L}(s)$ is, similar to Eq.(\ref{fsEq}), as:
\[
\tilde{L}(s)=L(A+\cos{s}~B+\sin{s}~C,-\sin{s}~B+\cos{s}~C,
-\cos{s}~B-\sin{s}~C)
\]
and $\Phi^f=(\phi_1^f,\phi_2^f,\phi_3^f)$ is defined similarly as
\begin{eqnarray*}
  \phi_1^f=\int_0^{2\pi}{(1-\cos{s})~f(s)\,ds}\\
  \phi_2^f=\int_0^{2\pi}{\sin{s}~f(s)\,ds}\\
  \phi_3^f=\int_0^{2\pi}{\cos{s}~f(s)\,ds}
\end{eqnarray*}
Calculating of $\Phi^{\mathcal L}$ with respect to $A, B$ and $C$
results in:
\begin{equation}%\label{}
  \begin{pmatrix}
    2S_x & -S_x+S_{x''} & -S_{x'} \\
    0 & -S_{x'} & S_x-S_{x''} \\
    0 & S_x-S_{x''} & S_{x'} \
  \end{pmatrix}\begin{pmatrix}
    A \\
    B \\
    C \
  \end{pmatrix}=-{1\over\pi}\Phi^f
\end{equation}
It is known that if $\Phi^f\neq 0$ then the equation
$\Phi^{\mathcal L}=0$ has a unique nonzero solution in $A, B$ and
$C$, provided:
\begin{equation}\label{detCond}
  \det\{S_x\}[\det^2\{S_x-S_{x''}\}+\det^2\{S_{x'}\}]\neq 0
\end{equation}
It is interesting to note that the Jacobian of the
$\Phi^{\mathcal L}$ respect to $A, B, C$ is the same as
Eq.(\ref{detCond}), i.e:
\begin{equation}%\label{}
   J_{\Phi^{\mathcal L}}=\det\{S_x\}[\det^2\{S_x-S_{x''}\}+\det^2\{S_{x'}\}]
\end{equation}
\begin{thm}
If $S_x, S_{x'}, S_{x''}$ are the coefficient matrix of the
linear term $L(x,x',x'')$ in the Eq.(\ref{singsyst}) and $f(t)$
is a $2\pi$-periodic vector function, then Eq.(\ref{singsyst})
has a $2\pi$ periodic solution for sufficiently small norms of
matrix $\gamma=\diag[\gamma_i]$ if $\Phi^f\neq 0$ and furthermore
we have:
\begin{equation}
  \det\{S_x\}[\det^2\{S_x-S_{x''}\}+\det^2\{S_{x'}\}]\neq 0
\end{equation}
\end{thm}
As we mentioned earlier, one of the advantages of the method
presented in this paper is its suitability for numerical
treatment. This suitability is due to the applicability of the
classical methods as Newton's method, for solving the system of
algebraic-integral equations presented in this paper. Using the
primitive solutions corresponding to $\epsilon =0$, as an initial
guess, we are able to solve the perturbed system. For this we
obtain the initial conditions $(A^\epsilon,B^\epsilon,C^\epsilon)$
for nonzero $\epsilon$ through solving the equation
$\Gamma_\epsilon(A,B,C)=0$ for $(A, B, C)$ so that the $x(t),
x'(t)$ and $x''(t)$ are periodic with period $2\pi$. We do that by
using the following iterative formula:
\[
\begin{array}{l}
A^{(n+1)}=A^{(n)}+\delta A\\
B^{(n+1)}=B^{(n)}+\delta B\\
C^{(n+1)}=C^{(n)}+\delta C\
\end{array}
\]
for the following system:
\[
\Gamma_\epsilon(A^{(n+1)},B^{(n+1)},C^{(n+1)})\simeq 0
\]
and the following Jacobi matrix:
\[
J_n=\begin{pmatrix}
  I_1- x_A & I_1-x_B & -x_C \\
  -x'_A & -x'_B & I_1-x'_C \\
  -x''_A & -I_1-x''_B & -x''_C
\end{pmatrix}
\]
where $I_1$ is a unit vector and $x_V$ is the approximated
differential of $x$ respect to $V$ at the point $2\pi$. This
approximation can be done with the classical formula; for example
for $x_A$ we can use the following formula
\[x_A=
{x(A^{(n)}+h,B^{(n)},C^{(n)};2\pi)-
x(A^{(n)}-h,B^{(n)},C^{(n)};2\pi)\over 2h}
\]
The similar formula is defined for $x'_V$ and $x''_V$, for example
we have:
\[
x'_B={x(A^{(n)},B^{(n)}+h,C^{(n)};2\pi)-
x'(A^{(n)},B^{(n)}-h,C^{(n)};2\pi)\over 2h}
\]
and
\[
x''_C={x''(A^{(n)},B^{(n)},C^{(n)}+h;2\pi)-
x''(A^{(n)},B^{(n)},C^{(n)}-h;2\pi)\over 2h}
\]
We define also $Q_n$ as the following:
\[
Q_n=\left ( \begin{array}{c}
x(2\pi)-x(0)\\
x'(2\pi)-x'(0)\\
x''(2\pi)-x''(0)
\end{array} \right )
\]
The Newton method for the above problem can be formulated as
follows:
\begin{enumerate}
\item Calculate $Q_0$ at $P_0=(A^0,B^0,C^0)$, i.e., solve the system by the initial
conditions: $x(0)=A^0+B^0, x'(0)=C^0, x''(0)=-B^0$.
\item Calculate the matrix $J_0$ at $P_0$.
\item Solve the linear equation $J_0 \Delta P=Q_0$ and denote it by $\Delta P_0$.
\item Now put $P_1=P_0+\Delta P_0$ and repeat the process for $P_1$.
\end{enumerate}
For sufficiently small $\epsilon$ it is known by the classical
results of Newton method that $\Delta P_i \rightarrow 0$ and $P_i
\rightarrow P^*$ that satisfies the equation $Q=0$. For the
process to be run, we just need  a solver of nonlinear ODE and a
linear solver of algebraic equations.\\\\
Now consider the following system:
\begin{equation}
\begin{array}{l}
x'''+x'=2x-y+\frac{\gamma x}{(x^2+y^2)^{3/2}}+\cos(t)\\
y'''+y'=3y-2x+\frac{\gamma y}{(x^2+y^2)^{3/2}}+\sin(t)
\end{array}
\end{equation}
But for the following linear subsystem:
\begin{equation}
\begin{array}{l}
x'''+x'=2x-y+\cos(t)\\
y'''+y'=3y-2x+\sin(t)
\end{array}
\end{equation}
we know $\Phi^{\mathcal L}=0$ has a unique solution as
\[A=(0,0), B=(-0.75,-0,5), C=(-0.25,-0.5)\]
and for $\gamma=0.01$ we get the following solution by applying
the obtained results:
\[A=(0,0), B=(-0.726,-0.486), C=(-0.255,-0.489)\]
% ----------------------------------------------------------------------------------------
%\bibliographystyle{amsplain}
%\newpage
\begin{thebibliography}{99}
\bibitem{laz68} Lazer A. C., on Schauder's fixed point theorem and forced second order
nonlinear oscillation, J. Math. Anal. Appl., 21, (1968), 421-425,
\bibitem{cron64} Cronin, J. fixed point and topological degree in nonlinear
analysis, Math. Survey II, American Math. Society, Providence,
R.I., (1964)
\bibitem{mehnik97} Mehri, B. and Niksirat, M. A. on the existence of periodic
oscillation for vector nonlinear second-order system, Proceedings
of the 28th annual Iranian Mathematics Conference, (1997)
\bibitem{mehnik01} Mehri B. and Niksirat M. On the existence of periodic
solutions for the quasi-linear third order ODE., J. Math. Anl.
App. Vol 261 No. 1, (2001), 159-167,
\bibitem{mehnik00} Mehri, B. and Niksirat, M. The existence of periodic
solution for the nonlinear autonomous ODEs., Nonlinear Analysis
Forum 5, (2000), 163-171
\bibitem{mehemam78} Mehri, B. and Emamirad, H. A. on the existence of periodic
solutions for autonomous second-order systems, Nonlinear Analysis,
Theory, Method and Applications, Vol. 3, No. 5, (1978), 577-582
\bibitem{loyd78} Lloyd, N. G. degree theory, Cambridge University Press,
(1978)
\end{thebibliography}
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