\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 102, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/102\hfil Multiple symmetric positive solutions]
{Multiple symmetric positive solutions for systems of higher order
 boundary-value problems on time scales}

\author[P. V. S. Anand,  P. Murali, K. R. Prasad
 \hfil EJDE-2011/102\hfilneg]
{Putcha. V. S. Anand, Penugurthi Murali, Kapula R. Prasad}  % in alphabetical order

\address{Putcha V. S. Anand \newline
CR Rao Advanced Institute of Mathematics,
Statistics and Computer Science\\
University of Hyderabad Campus\\
Hyderabad, 500 046, India}
\email{anand\_putcha@yahoo.com}

\address{Penugurthi Murali \newline
Department of Applied Mathematics \\
Andhra University \\
Visakhapatnam, 530003, India}
\email{murali\_uoh@yahoo.co.in}

\address{Kapula Rajendra Prasad \newline
Department of Applied Mathematics \\
Andhra University \\
Visakhapatnam, 530003, India}
\email{rajendra92@rediffmail.com}

\thanks{Submitted January 7, 2011. Published August 10, 2011.}
\subjclass[2000]{39A10, 34B15, 34A40}
\keywords{Boundary value problem; cone;
symmetric positive solution; \hfill\break\indent
symmetric time scale}

\begin{abstract}
 In this article, we find multiple symmetric positive solutions
 for a system of higher order two-point boundary-value problems
 on time scales by determining growth conditions and applying
 a fixed point theorem in cones under suitable conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks


\section{Introduction}

Symmetry creates beauty in nature and in nature every thing is
almost symmetric. One can observe that symmetry in the structure of
fruits, the structure of human body, the revolution of planets and
the structure of atoms. Due to the importance of symmetric
properties in both theory and applications, the study of existence
of symmetric solutions of boundary value problems gained momentum.

 In this paper, we address the question of the existence of at
least three symmetric positive solutions for the system of dynamical
equations on symmetric time scales,
\begin{equation}\label{e1}
\begin{gathered}
(-1)^n y_1^{(\Delta\nabla)^n}=f_1(t,y_1,y_2),\quad t\in [a,b]_{\mathbb{T}} \\
(-1)^m y_2^{(\Delta\nabla)^m}=f_2(t,y_1,y_2),\quad t\in [a,b]_{\mathbb{T}}
\end{gathered}
\end{equation}
subject to the two-point boundary conditions
\begin{equation}\label{e2}
\begin{gathered}
y_1^{(\Delta\nabla)^i}(a)=0=y_1^{(\Delta\nabla)^i}(b),\quad
 i=0,1,2,\dots,n-1,\\
 y_2^{(\Delta\nabla)^j}(a)=0=y_2^{(\Delta\nabla)^j}(b),\quad
 j=0,1,2,\dots,m-1,
\end{gathered}
\end{equation}
where $f_i:[a,b]_{\mathbb{T}}\times\mathbb{R}^{2}\to [0,\infty)$  are
continuous and $f_i(t,y_1,y_2)=f_i(a+b-t,y_1,y_2)$ for $i=1,2$,
$a\in \mathbb{T}_{k}$, $b\in\mathbb{T}^{k}$ for a time scale $\mathbb{T}$, and also
$\sigma(a)<\rho(b)$.

 By an interval time scale,
we mean the intersection of a real interval with a given time
scale; i.e.,
$$
[a,b]_{\mathbb{T}}=[a,b]\cap\mathbb{T}.
$$
For time scale calculus, we refer the reader to Bohner and Peterson
\cite{bop,bopp}.

An interval time scale $\mathbb{T}=[a,b]_{\mathbb{T}}$  is said to be
a symmetric time scale if
$t\in \mathbb{T}\Leftrightarrow a+b-t\in\mathbb{T}$.

 If $\mathbb{T}=\mathbb{R} $ or $\mathbb{T}=h\mathbb{Z},(h>0)$ then the symmetry definition is always
satisfied. In addition to, the interval time scale
$\mathbb{T}=[1,2]\cup\{3,4,5\}\cup[6,7]\cup\{8\}\cup[9,10]\cup\{11,12,13\}
\cup[14,15]$
has the symmetrical property.  But the time scale
$\mathbb{T}=\{0\}\cup\{\frac{1}{n}:n\in\mathbb{N}\}$ is not a symmetric time scale.

 By a  \emph{symmetric solution} $(y_1,y_2)$
of the system of boundary value problem
\eqref{e1}-\eqref{e2}, we mean $(y_1,y_2)$ is a solution of
 \eqref{e1}-\eqref{e2} and satisfies
$$
y_1(t)=y_1(b+a-t)~{\rm and}~y_2(t)=y_2(b+a-t),
\quad t\in [a,  b ]_{\mathbb{T}}.
$$
The development of the theory has gained attention by many
researchers; To mention a few, we list some papers,  Erbe and Wang
\cite{lhh},  Eloe and Henderson \cite{pwj,pwe},  Eloe,  Henderson
and  Sheng \cite{pqj},  Henderson and Thompson \cite{jt}, Avery and
Henderson \cite{ah,aah, rj}, Avery, Davis and Henderson \cite{adh},
Davis and Henderson \cite{jh}, Davis, Henderson and Wong \cite{dhw},
Anderson \cite{dra}, Henderson and Wong \cite {jhw}, and Henderson,
Murali and Prasad \cite{jmp}.

 This article is organized as follows.
In Section 2, we establish certain
 lemmas and inequalities on Green's function which are needed later.
In Section 3, by using the cone theory techniques, we establish
the existence of at least three symmetric positive solutions to
 \eqref{e1}-\eqref{e2}.
 The main tool in this paper is an application of the Avery's
generalization of the Leggett-Williams fixed point theorem for
operator leaving a Banach space cone invariant.

\section{Green's function and bounds}

 In this section, we construct the Green's function for the
homogeneous  SBVP corresponding to  \eqref{e1}-\eqref{e2}.
We estimate bounds of the Green's function, and  establish some
lemmas, in which we prove some inequalities  on the Green's
function, which are needed in our main result.

 Let us denote the
Green's function of the problem
\begin{gather*}
-y^{\Delta\nabla}(t)=0,\quad t\in[a,b]_{\mathbb{T}},\\
y(a)=0=y(b),
\end{gather*}
as $G_1(t,s)$, and it is given by
$$
G_1(t,s)=\begin{cases}
\frac{(b-s)(t-a)}{(b-a)}, \quad t\leq s \\
\frac{(b-t)(s-a)}{(b-a)},\quad  s\leq t
\end{cases}
$$
for all $t,s\in [a,b]_{\mathbb{T}}$. Then,  we can recursively define
\begin{equation}\label{e3}
G_j(t,s)=\int_{a}^{b}G_{j-1}(t,r)G_1(r,s)\nabla r, \quad\text{for all }
t,s\in [a,b]_{\mathbb{T}},
\end{equation}
for $j=2,3,\dots,p$, and $p=\max\{m,~n\}$, where $G_j(t,s)$ is the
Green's function for the problem
\begin{gather*}
 (-1)^jy^{(\Delta\nabla)^{j}}(t)=0,\quad t\in [a,b]_{\mathbb{T}},\\
  y^{(\Delta\nabla)^{i}}(a)=y^{(\Delta\nabla)^{i}}(b)=0,\quad
i=0,1,2,\dots, j-1,
\end{gather*}
 and $G_j(t,s)\geq 0$ for all $t,s\in [a,b]_{\mathbb{T}}$.
For details we refer to \cite{dra,jmp}.

 The following lemmas are needed to establish our main result.

\begin{lemma}\label{lem1}
Let $l\in[\frac{b-a}{8},\frac{b-a}{2}]_{\mathbb{T}}$ and $(t,s)\in
[a+l,b-l]_{\mathbb{T}}\times[a,b]_{\mathbb{T}}$,
\begin{equation}\label{e110}
|G_j(t,s)|\geq
L_l^j\phi_l^{j-1}\frac{(b-s)(s-a)}{b-a},\quad\text{for }
j=1,2,\dots,p,
\end{equation}
where $p$ is maximum of $\{m,n\}$, $L_l=\frac{l}{b-a}$ and
$\phi_l=\int_{a+l}^{b-l}\frac{(b-s)(s-a)} {b-a}\nabla s$.
\end{lemma}

\begin{proof}
For $j=1$ the inequality \eqref{e110} holds provided that
$L_l=\frac{l}{b-a}$.
 Next
for fixed $j$, assuming that \eqref{e110} is true, from \eqref{e3}
we have for $(t,s)\in[a+l,b-l]_{\mathbb{T}}\times[a,b]_{\mathbb{T}}$,
\begin{align*}
|G_{j+1}(t,s)|
&=|\int_a^bG_j(t,r)G_1(r,s)\nabla r|\\
&\geq|\int_{a+l}^{b-l}G_j(t,r)G_1(r,s)\nabla r|\\
&\geq\int_{a+l}^{b-l}L_l^j\phi_l^{j-1}\frac{(b-r)(r-a)}
{b-a}\times L_l \frac{(b-s)(s-a)}{b-a}\nabla r\\
&=L_l^{j+1}\phi_l^{j}\frac{(b-s)(s-a)}{b-a}.
\end{align*}
Hence, by induction the result is true for all $j\leq p-1$.
\end{proof}

\begin{lemma}\label{lem2}
For $(t,s)\in [a,b]_{\mathbb{T}}\times[a,b]_{\mathbb{T}}$,
\begin{equation}\label{e111}
|G_j(t,s)|\leq \phi_0^{j-1}\frac{(b-s)(s-a)}{b-a},\quad
\text{for }j=1,2,\dots,p,
\end{equation}
where $\phi_0=\int_{a}^{b}\frac{(b-s)(s-a)}{b-a}\nabla s$.
\end{lemma}

\begin{proof}
For $j=1$ the inequality \eqref{e111} is obvious. Next for fixed
$j$, assume that \eqref {e111} is true, then from \eqref{e3} we have
\begin{align*}
|G_{j+1}(t,s)|&=|\int_a^{b}G_j(t,r)G_1(r,s)\nabla r|\\
&\leq\int_{a}^{b}\phi_0^{j-1}\frac{(b-r)(r-a)}
{b-a}\times  \frac{(b-s)(s-a)}{b-a}\nabla r\\
&=\phi_0^j\frac{(b-s)(s-a)}{b-a}.
\end{align*}
Hence, by induction the result is true for all $j\leq p-1$.
\end{proof}

\begin{lemma}\label{lem3}
Let $t_{k}=\frac{b+a}{2}$ and $t_i\in[a, \frac{b+a}{2}]_{\mathbb{T}}$,
$1\leq i\leq 3$ with $t_1\leq t_2$. For $s\in [a,b]_{\mathbb{T}}$,
$$
\frac{G_1(t_1,s)}{G_1(t_2,s)}\geq
\frac{t_1-a}{t_2-a}\quad\text{and}\quad
\frac{G_1(t_{k},s)}{G_1(t_3,s)}\leq \frac{t_{k}-a}{t_3-a}.
$$
\end{lemma}

\begin{proof}
For $t\leq s$, we have
$\frac{G_1(t_1,s)}{G_1(t_2,s)}=\frac{t_1-a}{t_2-a}$. And for $s\leq
t$, we have $\frac{G_1(t_1,s)}{G_1(t_2,s)}=\frac{b-t_1}{b-t_2}$.
Since $t_1\leq t_2$, we get $\frac{b-t_1}{b-t_2}\geq
\frac{t_1-a}{t_2-a}$.

 Similarly, for $t\leq s$, we have
$\frac{G_1(t_{k},s)}{G_1(t_3,s)}=\frac{t_{k}-a}{t_3-a}$. And for
$s\leq t$, we have
$\frac{G_1(t_{k},s)}{G_1(t_3,s)}=\frac{b-t_{k}}{b-t_3}$. Since
$t_1\leq t_2$, we get $\frac{b-t_{k}}{b-t_3}\leq
\frac{t_{k}-a}{t_3-a}$.
\end{proof}

\begin{lemma}\label{lem4}
For $t, s \in [a, b]_{\mathbb{T}}$, the Green's function  $G_j(t, s)$
satisfies the symmetric property,
\begin{equation}\label{e122}
G_j(t, s)=G_j(b+a-t,~ b+a-s),\quad\text{for }j=1,2,\dots,p.
\end{equation}
\end{lemma}

\begin{proof}
By the definition of $G_j(t,s)$, $(j=2,3,\dots,p-1)$,
$$
G_j(t,s)=\int_a^bG_{j-1}(t,r)G_1(r,s)\nabla r, \quad
\text{for~all } t,s\in [a,b]_{\mathbb{T}}.
$$
Clearly, $G_1(t,s)=G_1(a+b-t,a+b-s)$. Now, the proof is by induction.
 For $j=2$  the inequality \eqref{e122} is obvious. Next, assume that
\eqref {e122} is true,  for fixed $j$ $(j=1,2,\dots,p-1)$, then from
\eqref{e3} we have
\begin{align*}
G_{j+1}(t,s)
&=\int_a^{b}G_j(t,r)G_1(r,s)\nabla r\\
&=\int_a^{b}G_j(a+b-t,a+b-r)G_1(a+b-r,a+b-s)\nabla r\\
&=\int_a^{b}G_j(a+b-t,r_1)G_1(r_1,a+b-s)\nabla r_1\\
&=G_{j+1}(a+b-t,a+b-s),
\end{align*}
by using a transformation $r_1=a+b-r$.
\end{proof}

 Let $D=\{v\mid v:[a,b]_{\mathbb{T}}\to \mathbb{R} \text{is
continuous function}\}$. We  define the operator
$F_j:D\to D$ by
$$
(F_jv)(t)=\int_a^{b}G_j(t,s)v(s)\nabla s,\quad
t\in [a,b]_{\mathbb{T}},\textup{ for } j=1,2,\dots, p-1.
$$
 By the construction of $F_j$ and
properties of  $G_j(t, s)$, it is clear that
\begin{gather*}
(-1)^j(F_jv)^{(\Delta\nabla)^{j}}(t)=v(t),\quad t\in [a,b]_{\mathbb{T}},\\
(F_jv)^{(\Delta\nabla)^{i}}(a)=(F_jv)^{(\Delta\nabla)^{i}}(b)=0,\quad
i=0,1,\dots,j-1.
\end{gather*}

\begin{lemma}\label{lem5}
For $t \in [a,b]_{\mathbb{T}}$, the operator $F_j$ satisfies the symmetric
property
$$
F_jy(t)=F_jy(b+a-t)\quad\text{for }j=1,2,\dots,p-1.
$$
\end{lemma}

\begin{proof}
By definition of $F_j$, and using the transformation $s_1=b+a-s$,
\begin{align*}
F_jy(t)
&=\int_{a}^{b}G_j(t,s)v(s)\nabla s \\
&=\int_{a}^{b}G_j(a+b-t,a+b-s)v(s)\nabla s\\
&=\int_{a}^{b}G_j(a+b-t,s_1)v(s_1)\nabla s_1 \\
&=F_jy(b+a-t),
\end{align*}
from Lemma \ref{lem4}).
\end{proof}

By using the above transformations and lemmas, we can reduce
the SBVP \eqref{e1},\eqref{e2} into  SBVP \eqref{e6}-\eqref{e7} and
vice-versa.

 Hence, we see that SBVP \eqref{e1}-\eqref{e2} has a solution
if and only if the following problem has a solution:
\begin{equation}\label{e6}
\begin{gathered}
v_1^{\Delta\nabla}+f_1(t,F_{n-1}v_1,F_{m-1}v_2)=0,\quad
 t\in [a,b]_{\mathbb{T}} \\
v_2^{\Delta\nabla}+f_2(t,F_{n-1}v_1,F_{m-1}v_2)=0,\quad
 t\in [a,b]_{\mathbb{T}},
\end{gathered}
\end{equation}
with boundary conditions
\begin{equation}
\label{e7}
 v_1(a)=0=v_1(b),\quad
_2(a)=0=v_2(b).
\end{equation}
Indeed, if $(y_1,y_2)$  is a solution of
\eqref{e1}-\eqref{e2}, then
$(v_1=y_1^{(\Delta\nabla)^{(n-1)}},v_2=y_2^{(\Delta\nabla)^{(m-1)}})$
is a solution of  \eqref{e6}-\eqref{e7}.  Conversely, if
$(v_1,v_2)$ is a solution of  \eqref{e6}-\eqref{e7}, then
$(y_1=F_{n-1}v_1,y_2=F_{m-1}v_2)$ is a solution of
\eqref{e1}-\eqref{e2}. In fact, we have the representation
$$
y_1(t)=\int_a^bG_{n-1}(t,s)v_1(s)\nabla s,\quad
y_2(t)=\int_a^bG_{m-1}(t,s)v_2(s)\nabla s,
$$
where
\begin{gather*}
v_1(s)=\int_a^bG_1(s,\tau)f_1(\tau,F_{n-1}v_1 ,F_{m-1}v_2)\nabla
\tau,\\
v_2(s)=\int_a^bG_1(s,\tau)f_2(\tau,F_{n-1}v_1
,F_{m-1}v_2)\nabla \tau.
\end{gather*}
It is also noted that a solution $(v_1,v_2)$ of
 \eqref{e6}-\eqref{e7} is symmetric; i. e.,
$$
v_1(t)=v_1(b+a-t)\quad\text{and}\quad
v_2(t)=v_2(b+a-t), \quad t\in [a,  b ]_{\mathbb{T}},
$$
and it gives rise to a symmetric solution
$(y_1,y_2)$ of  \eqref{e1}-\eqref{e2}.

\section{Existence of multiple symmetric  positive solutions}

In this section, we establish the existence of at least
three symmetric  positive
solutions for \eqref{e1}-\eqref{e2}, by using Avery's
generalization of the Leggett-Williams fixed point theorem.
Let $B$ be a real Banach space with cone $P$. We consider the
nonnegative continuous convex functionals $\gamma,\beta,\theta$
and nonnegative continuous concave functionals $\alpha,\psi$  on
$P$, for nonnegative numbers $a',b',c',d'$ and $h'$, we define
the following sets
\begin{gather*}
P(\gamma,c')=\{y\in P: \gamma(y)<c'\},\\
P(\gamma,\alpha,a',c')=\{y\in P:a'\leq\alpha(y),\;\gamma(y)\leq c'\},\\
Q(\gamma,\beta,d',c')=\{y\in P:\beta(y)\leq d',~\gamma(y)\leq c'\},\\
P(\gamma,\theta,\alpha,a',b',c')=\{y\in P:a'\leq\alpha(y),\;
\theta(y)\leq b',\;\gamma(y)\leq c'\},\\
Q(\gamma,\beta,\psi,h',d',c')=\{y\in P|h'\leq \psi(y),\;\beta(y)\leq
d',~\gamma(y)\leq c'\}.
\end{gather*}

For obtaining multiple symmetric
positive solutions of \eqref{e1}-\eqref{e2}, we state the
following fundamental theorem the so called Five Functionals Fixed
Point Theorem \cite{ria}.

\begin{theorem}\label{thm1}
Let $P$ be a cone in a real Banach space $E$.  Suppose $\alpha$ and
$\psi $ are nonnegative continuous concave functionals on $P$ and
$\gamma,\beta$ and $\theta$ are nonnegative continuous convex
functionals on $P$ such that, for some positive numbers $c'$ and
$g'$,
$$
\alpha(y)\leq \beta(y)\quad\text{and}\quad \| y\|\leq
g'\gamma(y)\quad\text{for all }y\in\overline{P(\gamma,c')}.
$$
Suppose further that $T:\overline{P(\gamma,c')}\to
\overline{P(\gamma,c')} $ is completely continuous and there exist
constants $h',d',a',b'\geq 0$ with $ 0<d'<a'$ such that each of
the following is satisfied.
\begin{itemize}
\item[(B1)]
$\{y\in P(\gamma,\theta,\alpha,a',b',c')|\alpha(y)>a'\}\neq
\emptyset$ and $\alpha(Ty)>a'$ for \\
$y\in P(\gamma,\theta,\alpha,a',b',c')$,

\item[(B2)] $\{y\in Q(\gamma,\beta,\psi,h',d',c')|\beta(y)<d'\}\neq
\emptyset$ and $\beta(Ty)<d'$ for \\
 $y\in Q(\gamma,\beta,\psi,h',d',c')$,

\item[(B3)] $\alpha(Ty)>a'$ provided $y\in P(\gamma,\alpha,a',c') $
with $\theta(Ty)>b' $,

\item[(B4)] $\beta(Ty)<d'$ provided $y\in Q(\gamma,\beta,d',c')$ with
$\psi(Ty)<h'$.

\end{itemize}
Then $T$ has at least three fixed points $y_1,y_2,y_3\in
\overline{P(\gamma,c')}$ such that
$\beta(y_1)<d',\quad a<\alpha(y_2)$, and
$d'<\beta(y_3)$ with $\alpha(y_3)<a'$.
\end{theorem}

To apply the fixed point theorem for our problem we need the
space
$$
C_0=\{(v_1,v_2)|v_1,v_2:[a,b]_{\mathbb{T}}\to \mathbb{R}
\text{ are continuous functions }\}
$$
 equipped with the norm
$$
\| (v_1,v_2)\|= \| v_1\|_0+\| v_2\|_0
$$
where $\| v\|_0=\max_{t\in[a,b]_{\mathbb{T}}}|v(t)|$. For a
fixed $k_0\in[\frac{b-a}{8},\frac{b-a}{2}]_{\mathbb{T}}$, define the cone
$P\subset C_0$ by
\begin{align*}
P&=\{(v_1,v_2)\in C_0|v_1(t),v_2(t)\text{ are nonnegative convex
symmetric functions for}\\
&\quad t\in[a,b]_{\mathbb{T}}\text{ and }
\min_{t\in[a+k_0,b-k_0]_{\mathbb{T}} }(|v_1(t)|+|v_2(t)|)\geq
\frac{k_0}{t_{k}-a}\|( v_1,v_2)\|\},
\end{align*}
where $t_{k}=\frac{b+a}{2}$. Let $k_i\in [\frac{b-a}{8},
\frac{b-a}{2}]_{\mathbb{T}}$, $1\leq i\leq 3$, be fixed and
$k_1<k_2$.  Also let $t_i=a+k_i,~0\leq i\leq 3$. Clearly, $t_1<t_2$
and $t_i\leq t_{k},~i=0,1,2,3$. Define the nonnegative continuous
concave functionals $\alpha,\psi$ and the nonnegative continuous
convex functionals $\beta,\theta,\gamma$ on $P$ by
\begin{gather*}
\gamma(v_1,v_2)=\max_{t\in[a,a+k_0]_{\mathbb{T}}\cup[b-k_0,b]_{\mathbb{T}}}
(|v_1(t)|+|v_2(t)|)=|v_1(t_0)|+|v_2(t_0)|,\\
\psi(v_1,v_2)=\min_{t\in [a+k_3,b-k_3]_{\mathbb{T}}}(|v_1(t)|+|v_2(t)|)
=|v_1(t_3)|+|v_2(t_3)|,\\
\beta(v_1,v_2)=\max_{t\in [a+k_3,b-k_3]_{\mathbb{T}}}(|v_1(t)|+|v_2(t)|)
=|v_1(t_{k})|+|v_2(t_{k})|,\\
\alpha(v_1,v_2)=\min_{t\in[a+k_1,a+k_2]_{\mathbb{T}}\cup[b-k_2,
b-k_1]_{\mathbb{T}}}(|v_1(t)|+|v_2(t)|)=|v_1(t_1)|+|v_2(t_1)|,\\
\theta(v_1,v_2)=\max_{t\in[a+k_1,a+k_2]_{\mathbb{T}}\cup[b-k_2,
b-k_1]_{\mathbb{T}}}(|v_1(t)|+|v_2(t)|)=|v_1(t_2)|+|v_2(t_2)|.
\end{gather*}
We observe that for any $(v_1,v_2)\in P$,
\begin{equation}\label{e112}
\alpha(v_1,v_2)=|v_1(t_1)|+|v_2(t_1)|\leq
|v_1(t_{k})|+|v_2(t_{k})|=\beta(v_1,v_2),
\end{equation}
and
\begin{equation}\label{e113}
\| (v_1,v_2)\|=|v_1(t_{k})|+|v_2(t_{k})|\leq
\frac{t_{k}-a}{t_0-a}(|v_1(t_0)|+|v_2(t_0)|)=\frac{t_{k}-a}{t_0-a}
\gamma(v_1,v_2).
\end{equation}
Let us denote
$$
\overline{\phi_z}=\phi_0-\phi_z=\int_{s\in[a,b]_{\mathbb{T}}
\backslash[a+z,b-z]_{\mathbb{T}}}\frac{(b-s)(s-a)} {b-a}\nabla s.
$$
 We are now ready to present the main theorem  of the paper.

\begin{theorem}\label{thm2}
Suppose there exist $0<a'<b'<\frac{(t_2-a)}{t_1-a}b'\leq c'$ such
that $f_1$ and $f_2$ satisfy the following conditions:
\begin{itemize}
\item[(A1)] $|f_i(t,u_{n-1},w_{m-1})|<(\frac{a'-c'(k_3^2-k_3)
 [(t_0-a)(b-t_0)]^{-1}}{(t_{k}-a)
(b-t_{k})+k_3-k_3^2})$ for all $~(t,|u_{n-1}|,|w_{m-1}|)$ in
\begin{align*}
&[a,b]_{\mathbb{T}}\times[
\frac{a'(t_3-a)}{t_{k}-a}L_{k_3+1}^{n-1}\phi_{k_3+1}^{n-1},
a'\phi_0^{n-2}\phi_{k_3+1}+\frac{c'(t_{k}-a)}{t_0-a}\phi_0^{n-2}
\overline{\phi}_{k_3+1}]\\
&\times [ \frac{a'(t_3-a)}{t_{k}-a}L_{k_3+1}^{m-1}\phi_{k_3+1}^{m-1},
a'\phi_0^{m-2}\phi_{k_3+1}+\frac{c'(t_{k}-a)}{t_0-a}\phi_0^{m-2}
\overline{\phi}_{k_3+1}],\quad
 i=1, 2.
\end{align*}
\item[(A2)] $|f_i(t,u_{n-1},w_{m-1})|>\frac{b'}{k_1(k_2+1-k_1)}$
for all $(t,|u_{n-1}|,|w_{m-1}|)$ in
\begin{align*}
&[a+l,b-l]_{\mathbb{T}}\times[b'L_{k_1+1}^{n-1}\phi_{k_1+1}^{n-2}
(\phi_{k_1+1}-\phi_{k_2+2}),\frac{b'(t_2-a)}
{t_1-a}\phi_0^{n-2}(\phi_{k_1+1}-\phi_{k_2+2})\\
&+\frac{c'(t_{k}-a)}{t_0-a}\phi_0^{n-2}(\overline{\phi}_{k_1+1}
 +\phi_{k_2+2}) ]\times
[b'L_{k_1+1}^{m-1}\phi_{k_1+1}^{m-2}(\phi_{k_1+1}-\phi_{k_2+2}),\\
&\frac{b'(t_2-a)} {t_1-a}\phi_0^{m-2}(\phi_{k_1+1}-\phi_{k_2+2})
+ \frac{c'(t_{k}-a)}{t_0-a}\phi_0^{m-2}(\overline{\phi}_{k_1+1}
+\phi_{k_2+2})],
\end{align*}
 either $i=1$ or $i=2$.

\item[(A3)] $|f_i(t,u_{n-1},w_{m-1})|<\frac{c'}{(t_0-a)(b-t_0)}$
 for all $(t,|u_{n-1}|,|w_{m-1}|)$ in
\begin{align*}
&[a,b]_{\mathbb{T}}\times [0, \frac{c'(t_{k}-a)}{t_0-a}\phi_0^{n-1}]\times
[0,\frac{c'(t_{k}-a)}{t_0-a}\phi_0^{m-1}] ,\quad
 i=1, 2.
\end{align*}
\end{itemize}
Then  \eqref{e1}-\eqref{e2} has at least three symmetric
positive solutions.
\end{theorem}

\begin{proof}
Define a completely continuous operator $T:C_0\to C_0$ by
\begin{equation}\label{e114}
T(v_1,v_2) :=(T_1(v_1, v_2),T_2(v_1, v_2)),
\end{equation}
where
$$
T_{i}(v_1, v_2):=\int_a^bG_1(t,s)f_i(s,F_{n-1}v_1,
F_{m-1}v_2)\nabla s,\quad\text{for } i=1,2.
$$
It is obvious that a fixed point of $T$ is a solution of
\eqref{e6}-\eqref{e7}.  We seek three fixed points
$(x_1, x_2), (y_1, y_2), (z_1, z_2)\in P$ of $T$.
First, we show that $T$ is self map on $P$. Let $(v_1,v_2)\in P$,
then $T_1(v_1, v_2)(t)\geq 0$, $T_2(v_1, v_2)(t)\geq 0$
for $t\in[a,b]_{\mathbb{T}}$, and
$T^{\Delta\nabla}_1(v_1, v_2)(t)\leq 0$,
$T^{\Delta\nabla}_2(v_1, v_2)(t)\leq 0$ for
$t\in[a,b]_{\mathbb{T}}$. Further  $G_1(t,s)$ is symmetric, it follows
that $T_1(v_1, v_2)(t)=T_1(v_1,
v_2)(b+a-t),~T_2(v_1, v_2)(t)=T_2(v_1, v_2)(b+a-t)$,
for $t\in[a, b]_{\mathbb{T}}$. Also, noting that
$T_1(v_1,v_2)(a)=0=T_1(v_1, v_2)(b)$,
$T_2(v_1,v_2)(a)=0=T_2(v_1, v_2)(b)$ and
$\|T(v_1,v_2)\|=|T_1(v_1, v_2)(t_{k})|+|T_2(v_1,v_2)(t_{k})|$,
we have
\begin{align*}
&\min_{t\in[a+k_0,b-k_0]_{\mathbb{T}}}(|T_1(v_1,
v_2)(t)|+|T_2(v_1, v_2)(t)|)\\
&=\min_{t\in[a+k_0,t_{k}]_{\mathbb{T}}}
(|T_1(v_1, v_2)(t)|+|T_2(v_1, v_2)(t)|)\\
&\geq \min_{t\in[a+k_0,t_{k}]_{\mathbb{T}}}\frac{t-a}{t_{k}-a}\| T(v_1,v_2)\|\\
&=\frac{k_0}{t_{k}-a}\| T(v_1,v_2)\|.
\end{align*}
Thus $T:P\to P$.  Next, for all $(v_1,v_2)\in P$, and using
\eqref{e112},\eqref{e113}, $\alpha(v_1,v_2)\leq \beta(v_1,v_2)$ and
$\| (v_1,v_2)\|\leq\frac{t_{k}-a}{t_0-a}\gamma(v_1,v_2)$.
To show that
$T:\overline{P(\gamma,c')}\to \overline{P(\gamma,c')}$,
let $(v_1,v_2)\in \overline{P(\gamma,c')}$ and hence
$\|(v_1,v_2)\|\leq\frac{t_{k}-a}{t_0-a}c'$.
Using Lemma \ref{lem2} and for $t\in [a,b]_{\mathbb{T}}$,
\begin{align*}
|F_{n-1}v_1(t)|
&=|\int_a^bG_{n-1}(t,s)v_1(s)\nabla s| \\
&\leq\frac{c'(t_{k}-a)}{t_0-a}\int_a^b|G_{n-1}(t,s)|\nabla s\\
&\leq\frac{c'(t_{k}-a)}{t_0-a}\phi_0^{n-2}\int_a^b
\frac{(b-s)(s-a)}{(b-a)}\nabla s\\
&=\frac{c'(t_{k}-a)}{t_0-a}\phi_0^{n-1}.
\end{align*}
Similarly, for $t\in [a,b]_{\mathbb{T}}$, we have
$$
|F_{m-1}v_2(t)|\leq\frac{c'(t_{k}-a)}{t_0-a}\phi_0^{m-1}.
$$
By condition (A3),
$$
\gamma(T_1(v_1, v_2),T_2(v_1, v_2))=|T_1(v_1, v_2)
(t_0)|+|T_2(v_1, v_2)(t_0)|.
$$
and
\begin{align*}
|T_1(v_1,v_2)(t_0)|
&=|\int_a^bG_1(t_0,s)f_1(s,F_{n-1}v_1,F_{m-1}v_2)\nabla  s|\\
&< \frac{c'}{(t_0-a)(b-t_0)}\int_a^b|G_1(t_0,s)|\nabla  s
=\frac{c'}{2}.
\end{align*}
 Similarly, $|T_2(v_1, v_2)(t_0)|< c'/2$, and hence
$T:\overline{P(\gamma,c')}\to\overline{P(\gamma,c')}$.
It is obvious that
$$
\{(v_1,v_2)\in P(\gamma,\theta,\alpha,b',\frac{b'(t_2-a)}{t_1-a},c')|
\alpha(v_1,v_2)>b'\}\neq \emptyset .
$$
Next, let $(v_1,v_2)\in P(\gamma,\theta,\alpha,b',
\frac{b'(t_2-a)}{t_1-a},c')$, denote the
set $D_1=[a+k_1,a+k_2]_{\mathbb{T}}\cup[b-k_2,\sigma(b)-k_1]_{\mathbb{T}}$. It
follows that
\begin{gather}\label{e115}
|v_1(s)|,|v_2(s)|\in [b',\frac{b'(t_2-a)}{t_1-a}],\quad s\in D_1,\\
\label{e116}
|v_1(s)|,|v_2(s)|\in[0,\frac{c'(t_{k}-a)}{t_0-a}],\quad
s\in[a,b]_{\mathbb{T}}\setminus D_1.
\end{gather}
Using \eqref{e115}, \eqref{e116}, Lemma \ref{lem1}
Lemma \ref{lem2},
$$
|F_{n-1}v_1(s)|=|\int_{\tau\in D_1}G_{n-1}(s, \tau)v_1(\tau)\nabla
\tau+\int_{\tau\in[a,b]_{\mathbb{T}}\setminus
D_1}G_{n-1}(s,\tau)v_1(\tau)\nabla \tau|
$$
and
$$
|F_{m-1}v_2(s)|=|\int_{\tau\in D_1}G_{m-1}(s, \tau)v_2(\tau)\nabla
\tau+\int_{\tau\in[a,b]_{\mathbb{T}}\setminus
D_1}|G_{m-1}(s,\tau)v_2(\tau)\nabla \tau|,
$$
for $s\in D_1$,
 we have
\begin{align*}
&|F_{n-1}v_1(s)|\\
&\leq\frac{b'(t_2-a)}{t_1-a}\int_{\tau\in D_1}|G_{n-1}(s, \tau)|\nabla
\tau+\frac{c'(t_{k}-a)}{t_0-a}\int_{\tau\in[a,b]_{\mathbb{T}}\setminus
D_1}|G_{n-1}(s,\tau)|\nabla \tau,\\
&\leq\frac{b'(t_2-a)}{t_1-a}\phi_0^{n-2}(\phi_{k_1+1}-\phi_{k_2+2})+
\frac{c'(t_{k}-a)}{t_0-a}\phi_0^{n-2}(\overline{\phi}_{k_1+1}
+\phi_{k_2+2})
\end{align*}
and
\begin{align*}
|F_{n-1}v_1(s)|
&\geq\int_{\tau\in D_1}|G_{n-1}(s, \tau)v_1(\tau)|\nabla \tau\\
&\geq b'\int_{\tau\in D_1}|G_{n-1}(s, \tau)|\nabla \tau \\
&\geq b'L_{k_1+1}^{n-2}\phi_{k_1+1}^{n-2}(\phi_{k_1+1}-\phi_{k_2+2}).
\end{align*}
Similarly,
$$
|F_{m-1}v_2(s)|\leq\frac{b'(t_2-a)}{t_1-a}\phi_0^{m-2}
(\phi_{k_1+1}-\phi_{k_2+2})+ \frac{c'(t_{k}-a)}{t_0-a}
\phi_0^{m-2}(\overline{\phi}_{k_1+1}+\phi_{k_2+2})
$$
and
$$
|F_{m-1}v_2(s)|\geq b'L_{k_1+1}^{m-2}\phi_{k_1+1}^{m-2}
(\phi_{k_1+1}-\phi_{k_2+2}),~{for~s\in D_1}.
$$
Applying  (A2) we obtain
\begin{align*}
\alpha(T_1(v_1, v_2),&T_2(v_1, v_2))
=|T_1(v_1, v_2)(t_1)| +|T_2(v_1, v_2)(t_1)|\geq|T_1(v_1, v_2)(t_1)|\\
&=|\int_a^bG_1(t_1,s)f_1(s,F_{n-1}v_1, F_{m-1}v_2)\nabla s|\\
&\geq \int_{s\in D_1}|G_1(t_1,s)f_1(s,F_{n-1}v_1, F_{m-1}v_2)|\nabla s\\
&>\frac{b'}{k_1(k_2+1-k_1)}\int_{s\in D_1}|G_1(t_1,s)|\nabla s
=b'.
\end{align*}
Similarly,
$\alpha(T_1(v_1, v_2),T_2(v_1, v_2))\geq|T_2(v_1, v_2)(t_1)|$ and
from (A2) we have
 $$
\alpha(T_1(v_1, v_2),T_2(v_1, v_2))\geq b'.
$$
Clearly,
$$
\{(v_1,v_2)\in Q(\gamma,\beta,\psi,\frac{a'(t_3-a)}{t_{k}-a},a',c')
|\beta(v_1,v_2)<a'\}\neq \emptyset.
$$
Let $(v_1,v_2)\in Q(\gamma,\beta,\psi,\frac{a'(t_3-a)}{t_{k}-a},a',c')$,
and define the set $E_1=[a+k_3,b-k_3]_{\mathbb{T}}$, then
\begin{gather}\label{e117}
|v_1(s)|,|v_2(s)|\in [\frac{a'(t_3-a)}{t_{k}-a},a'],\quad s\in E_1,\\
\label{e118}
|v_1(s)|,|v_2(s)|\in[0,\frac{c'(t_{k}-a)}{t_0-a}],~~s\in[a,b]_{\mathbb{T}}\setminus
E_1.
\end{gather}
Then
\begin{gather*}
|F_{n-1}v_1(s)|=|\int_{\tau\in E_1}G_{n-1}(s, \tau)v_1(\tau)\nabla
\tau+\int_{\tau\in[a,b]_{\mathbb{T}}\setminus
E_1}G_{n-1}(s,\tau)v_1(\tau)\nabla \tau|,\\
|F_{m-1}v_2(s)|=|\int_{\tau\in E_1}G_{m-1}(s, \tau)v_2(\tau)\nabla
\tau+\int_{\tau\in[a,b]\setminus E_1}G_{m-1}(s,\tau)v_2(\tau)\nabla
\tau|.
\end{gather*}
Also using \eqref{e117},\eqref{e118}, Lemma
\ref{lem1}, and Lemma \ref{lem2}, we see that for $s\in E_1$,
\begin{align*}
|F_{n-1}v_1(s)|
&\leq a'\int_{\tau\in E_1}|G_{n-1}(s, \tau)|\nabla
\tau+\frac{c'(t_{k}-a)}{t_0-a}\int_{\tau\in[a,b]_{\mathbb{T}}\setminus
E_1}|G_{n-1}(s,\tau)|\nabla\tau,\\
&\leq a'\phi_0^{n-2}\phi_{k_3+1}+
\frac{c'(t_{k}-a)}{t_0-a}\phi_0^{n-2}\overline{\phi}_{k_3+1}
\end{align*}
and
\begin{align*}
|F_{n-1}v_1(s)|
&\geq\int_{\tau\in E_1}|G_{n-1}(s, \tau)v(\tau)|\nabla \tau\\
&\geq \frac{a'(t_3-a)}{t_{k}-a}\int_{\tau\in E_1}|G_{n-1}
 (s, \tau)|\nabla \tau \\
&\geq\frac{a'(t_3-a)}{t_{k}-a}L_{k_3+1}^{n-2}\phi_{k_3+1}^{n-1}.
\end{align*}
Similarly, for $s\in E_1$, we obtain
\begin{gather*}
|F_{n-1}v_2(s)|\leq a'\phi_0^{n-2}\phi_{k_3+1}+
\frac{c'(t_{k}-a)}{t_0-a}\phi_0^{n-2}\overline{\phi}_{k_3+1},\\
|F_{n-1}v_2(s)|\geq\frac{a'(t_3-a)}{t_{k}-a}L_{k_3+1}^{n-2}
\phi_{k_3+1}^{n-1}.
\end{gather*}
Thus, by (A1) and (A2), we obtain
\begin{align*}
&\beta((T_1(v_1, v_2),T_2(v_1, v_2))\\
&=|T_1(v_1, v_2)(t_{k})|+|T_2(v_1, v_2)(t_{k})|\\
&=\int_a^b|G_1(t_{k},s)f_1(s,F_{n-1}v_1,F_{m-1}v_2)|\nabla s \\
&\quad +\int_a^b|G_1(t_{k},s)f_2(s,F_{n-1}v_1,F_{m-1}v_2)\nabla s \\
&=\int_{s\in E_1}|G_1(t_{k},s)f_1(s,F_{n-1}v_1,F_{m-1}v_2)|\nabla
s\\
&\quad +\int_{s\in [a,b]_{\mathbb{T}}\setminus
E_1}|G_1(t_{k},s)f_1(s,F_{n-1}v_1,F_{m-1}v_2)|\nabla s\\
&\quad +\int_{s\in E_1}|G_1(t_{k},s)f_2(s,F_{n-1}v_1,F_{m-1}v_2)
|\nabla s\\
&\quad +\int_{s\in [a,b]_{\mathbb{T}}\setminus
E_1}|G_1(t_{k},s)f_2(s,F_{n-1}v_1,F_{m-1}v_2)|\nabla s\\
&<2[a'-\frac{c'(k_3^2-k_3)}{(t_0-a)(b-t_0)}][(t_{k}-a)(b-t_{k})
 +k_3-k_3^2]^{-1}\\
&\quad \times\int_{s\in E_1}|G_1(t_{k},s)|\nabla s
+\frac{2c'}{(t_0-a)(b-t_0)}\int_{s\in[a,b]_{\mathbb{T}}\setminus
E_1}|G_1(t_{k},s)|\nabla s=a'.
\end{align*}
Let $(v_1,v_2)\in P(\gamma,\alpha,b',c')$ with
$\theta(T_1(v_1,v_2),T_2(v_1, v_2))>\frac{b'(t_2-a)}{t_1-a}$.
Using Lemma \ref{lem3}, we obtain
\begin{align*}
&\alpha(T_1(v_1, v_2), T_2(v_1, v_2))\\
&=|T_1(v_1, v_2)(t_1)|+|T_2(v_1, v_2)(t_1)|\\
&=\int_a^b|\frac{G_1(t_1,s)}{G_1(t_2,s)}G_1(t_2,s)f_1(s,F_{n-1}v_1,
F_{m-1}v_2)|\nabla  s \\
&\quad +\int_a^b|\frac{G_1(t_1,s)}{G_1(t_2,s)}G_1(t_2,s)
 f_2(s,F_{n-1}v_1,F_{m-1}v_2)|\nabla  s \\
&\geq\frac{t_1-a}{t_2-a}\int_a^b|G_1(t_2,s)f_1(s,F_{n-1}v_1,
 F_{m-1}v_2)|\nabla s\\
&\quad +\frac{t_1-a}{t_2-a}\int_a^b|G_1(t_2,s)f_2(s,F_{n-1}v_1,
 F_{m-1}v_2)|\nabla s\\
&=\frac{t_1-a}{t_2-a}\theta(T_1(v_1, v_2),T_2(v_1, v_2))
>b'.
\end{align*}

 Finally, we show that (B4) holds.
Let $(v_1,v_2)\in Q(\gamma,\beta,a',c')$ with
$$
\psi(T_1(v_1, v_2),T_2(v_1, v_2))<\frac{a'(t_3-a)}{t_{k}-a}.
$$
In view of Lemma \ref{lem3}, we have
\begin{align*}
\beta(T_1(v_1, v_2),T_2(v_1, v_2))
&=|T_1(v_1, v_2)(t_{k})|+|T_2(v_1, v_2)(t_{k})|\\
&=\int_a^b|\frac{G_1(t_{k},s)}{G_1(t_3,s)}G_1(t_3,s)
f_1(s,F_{n-1}v_1,F_{m-1}v_2)|\nabla  s \\
&\quad +\int_a^b|\frac{G_1(t_{k},s)}{G_1(t_3,s)}G_1(t_3,s)
f_2(s,F_{n-1}v_1,F_{m-1}v_2)|\nabla  s \\
&\leq\frac{t_{k}-a}{t_3-a}\int_a^b|G_1(t_3,s)f_1(s,F_{n-1}v_1,
 F_{m-1}v_2)|\nabla s \\
&\quad +\frac{t_{k}-a}{t_3-a}\int_a^b|G_1(t_3,s)f_2(s,
 F_{n-1}v_1,F_{m-1}v_2)|\nabla s\\
&=\frac{t_{k}-a}{t_3-a}\psi(T_1(v_1, v_2),T_2(v_1, v_2))
<a'.
\end{align*}
 We have thus proved that all the conditions of Theorem \ref{thm1} are
satisfied and so there exist at least three  symmetric positive
solutions for \eqref{e1}-\eqref{e2}.
\end{proof}

\subsection*{Acknowledgements}
P. Anand is supported by projects Lr.
No. SR/S4/MS:516/07 and Dt.21-04-2008 from the DST-CMS.
P. Murali is thankful to C.S.I.R. of India for awarding him an RA.
The authors  thank the anonymous referees for
their valuable suggestions.

\begin{thebibliography}{00}

\bibitem{rpd} R. P. Agarwal, D. O'Regan, and P. J. Y. Wong;
\emph{Positive Solutions of Differential, Difference and
Integral Equations},
Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.

\bibitem{dra} D. R. Anderson;
 Eigenvalue intervals for even order Sturm-Liouville dynamic equation,
\emph{Comm. Appl. Nonli. Anali.}, {\bf 12} (2005), no. 4, 1-13.

\bibitem {ria} R. I. Avery;
 A generalization of the Leggett-Williams fixed point theorem,
\emph{Math.Sci. Res. Hot-Line}, {\bf 3} (1999), 9-14.

\bibitem{ah} R. I. Avery and J. Henderson;
Existence of three positive
pseudo symmetric solutions for a one dimensional  p-Laplacian,
\emph{J. Math. Anal. Appl.,} {\bf 10} (2004), no. 6, 529-539.

\bibitem{aah} R. I. Avery and J. Henderson;
Existence of three positive
pseudo symmetric solutions for a one dimensional discrete
p-Laplacian, (preprint).

\bibitem{rj} R. I. Avery and J. Henderson;
 Three symmetric positive solutions for a second order boundary
vaue problem, \emph{Appl. Math. Letters.,}  {\bf 13} (2003), 1-7.

\bibitem{adh} R. I. Avery, J. M. Davis and  J. Henderson;
 Three symmetric positive solutions for Lidstone problems by a
generalization of the  Leggett-Williams theorem,
\emph{Electron. J. Diff. Equ.,} {\bf 2000} (2000), no 40, 1-15.

\bibitem{bop} M. Bohner and A. C. Peterson;
\emph{Dynamic Equations on Time scales, An Introduction with
Applications}, Birkhauser, Boston, MA, (2001).

\bibitem {bopp} M. Bohner and A. C. Peterson;
\emph{Advances in Dynamic Equations on Time scales}, Birkhauser,
Boston,  (2003).

\bibitem{jh} J. M. Davis and  J. Henderson;
Triple positive symmetric solutions for a Lidstone boundary
value problem,
\emph{Diff. Equn. Dyn. Systm.,} {\bf 7} (1999), 321-330.

\bibitem{dhw} J. M. Davis, J. Henderson and P. J. Y. Wong;
 General Lidstone problem: multiplicity and symmetric of solutions,
 \emph{J. Math. anal. Appl.,} {\bf 251} (2000), no. 2, 527-548.

\bibitem{pwj}P. W. Eloe and J. Henderson;
 Positive solutions for (n-1,1) conjugate boundary value problems,
\emph{Nonlinear Anal.} {\bf28} (1997), 1669-1680.

\bibitem {pwe} P. W. Eloe and J. Henderson;
Positive solutions and nonlinear (k,n-k) conjugate eigenvalue problems,  \emph{Diff. Equ.
Dyna. Syst.}, {\bf 6} (1998), 309-317.

\bibitem{pqj} P. W. Eloe, J. Henderson and Q. Sheng;
 Notes on Crossed  symmetry solutions of the two-point boundary
value problems on time scales, \emph{J. Difference Equ. Appl.},
{\bf 9} (2003), no. 1, 29-48.

\bibitem{lhh} L. H. Erbe and H. Wang;
On the existence of positive solutions of ordinary differential
equations, \emph{Proc. Amer. Math.Soc.} {\bf 120} (1994), 743-748.

\bibitem{gl} D. Guo and V. Lakshmikantam;
\emph{Nonlinear problems in Abstract Cones},
 Academic press, San Diego, 1988.

\bibitem{j} J. Henderson;
 Multiple symmetric  solution for discrete
Lidstone boundary value problems,
\emph{Dynam. Contin. Discrete Impuls. Systems}, {\bf 7} (2000), 577-585.

\bibitem{jmp} J. Henderson, P. Murali and K. R. Prasad;
 Multiple symmetric positive solutions for two-point even order
boundary value problems on time scales,
\emph{Math. Engi. Scie. Aero.,} {\bf1} (2010), no. 1, 105-117.

\bibitem{jhw} J. Henderson and P. J. Y. Wong;
 Double symmetric solutions for discrete Lidstone boundary value
problems, \emph{J. Difference Equ. and Appl.,} {\bf 7}(2001), 811-828.

\bibitem{jt} J. Henderson and H. B. Thompson;
 Multiple symmetric positive solutiions for a second order boundary
value problem, \emph{Proc. Amer. Math. Soc.,} {\bf 128} (2000),
 2373-2379.

\bibitem{lsk} V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan;
\emph{Dynanic Systems on Measure Chains}, Mathematics and its
Applications,  {\bf 370}, Kluwer, Dordrecht, 1996.

\bibitem{rwlrw} R. W. Leggett and L. R. Williams;
Multiple positive fixed points of nonlinear operations on ordered
Banach spaces, \emph{Indiana Univ. Math. J.,} {\bf 28} (1979),
673-688.

\bibitem{h} P. J. Y. Wong;
Multiple symmetric  solution for discrete Lidstone boundary value
problems, \emph{J. Difference Equ. and Appl.,}
{\bf 8} (2002), no. 9, 765-797.

\end{thebibliography}

\end{document}