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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 57, 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 2010 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2010/57\hfil Eigenvalue intervals]
{Solvability of a nonlinear third-order
 three-point general eigenvalue problem \\ on time scales}

\author[K. R. Prasad, N. V. V. S. S. Narayana\hfil EJDE-2010/57\hfilneg]
{Kapula R. Prasad, Nadakuduti V. V. S. S. Narayana}  % in alphabetical order

\address{Kapula R. Prasad \newline
Department of Applied Mathematics\\
Andhra University\\
Visakhapatnam, 530 003, India}
\email{rajendra92@rediffmail.com}

\address{Nadakuduti V. V. S. Suryanarayana \newline
Department of Mathematics\\
VITAM College of Engineering\\
 Visakhapatnam, 531 173, India}
\email{suryanarayana\_nvvs@yahoo.com}

\thanks{Submitted November 30, 2009. Published April 23, 2010.}
\subjclass[2000]{34B99, 39A99}
\keywords{Time scales; boundary value problem;
  eigenvalue interval; \hfill\break\indent positive solution; cone}

\begin{abstract}
 We study the existence of eigenvalue intervals for the third-order
 nonlinear three-point boundary value problem on time scales
 satisfying general boundary conditions. Values of a parameter
 are determined for which the boundary value problem has a
 positive solution by utilizing a fixed point theorem on a cone in a
 Banach space.
\end{abstract}

\maketitle
\numberwithin{equation}{section}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

The study of obtaining optimal eigenvalue
intervals for the existence of  positive solutions to boundary value
problems(BVPs) on time scales has gained prominence and is a rapidly
growing field, since it arises in many applications. By a time scale
we mean a nonempty closed subset of $\mathbb{R}$. For an excellent
introduction to the overall area of dynamic equations on time
scales, we refer to the text book by Bohner and Peterson \cite{mbacp}.

 In this paper, we focus on
determining the eigenvalue intervals for which there exists a
positive solution to the  third order boundary value problem on time
scales
\begin{equation}\label{e11}
y^{\Delta^{3}}(t)+\lambda f(t, y(t),y^{\Delta}(t),
y^{\Delta^{2}}(t))=0,\quad t\in[t_1, \sigma^{3}(t_3)]
\end{equation}
satisfying  the general three point boundary conditions
\begin{equation}\label{e12}
\begin{gathered}
\alpha_{11}y(t_1)+\alpha_{12}y^{\Delta}(t_1)
 +\alpha_{13}y^{\Delta^{2}}(t_1)=0\\
\alpha_{21}y(t_2)+\alpha_{22}y^{\Delta}(t_2)
 +\alpha_{23}y^{\Delta^{2}}(t_2)=0\\
\alpha_{31}y(\sigma^{3}(t_3))
 +\alpha_{32}y^{\Delta}(\sigma^{2}(t_3))
 +\alpha_{33}y^{\Delta^{2}}(\sigma(t_3))=0
\end{gathered}
\end{equation}
where  $t_1<t_2<\sigma^{3}(t_3)$ and $ \alpha_{ij}$, for
$i,j=1,2,3$ are real constants. The BVPs of this form arise in the
modelling of nonlinear diffusion via nonlinear sources, thermal
ignition of gases, and in chemical concentrations in biological
problems.  In these applied settings, only positive solutions are
meaningful.

 Optimal eigenvalue intervals  were obtained for the existence of
 positive solutions of  boundary value problems for ordinary
differential equations, as well as for finite difference equations
using the Krasnosel'skii  fixed point theorem \cite{mak} on a cone.
A few papers along these lines are Agarwal, Bohner and Wang
\cite{abwo}, Anderson and Davis \cite{ad}, Davis, Eloe  and
Henderson \cite{deh}, Davis, Henderson, Prasad and Yin \cite{dhpyk,
dhpy}, Eloe and Henderson \cite{pwej}, Erbe and Tang \cite{et},
Henderson and Wang \cite{jhhw}, Jiang and Liu \cite{jl}, Prasad and
Murali \cite{pm}. Recently, Prasad and Rao \cite{kr} extended these
results to third order general three point boundary value problem.
\par In order to unify the results on differential equations and 
difference equations, the
theory of dynamical equations on time scales is being developed. It
has a great potential in nonlinear analysis and its applications in
the modeling of physical and biological systems. Some papers on
boundary value problems on time scales are Chyan and Henderson
\cite{ch}, Chyan, Davis, Henderson and Yin \cite{cdhy}, DaCunha,
Davis and Singh \cite{dds} and Erbe and Peterson \cite{ep, epe,
erpe}. This paper generalizes many papers in the literature. By
choosing different values to the constants in the boundary
conditions we get various three point BVPs.

 For simplicity we make the following notation:
$\beta_{i}=\alpha_{i1}t_{i}+\alpha_{i2}$,
$\gamma_{i}=\alpha_{i1}t_{i}^{2}+\alpha_{i2}(t_{i}+\sigma(t_{i}))+2\alpha_{i3}$,
for $i=1,2$,
 $\beta_3=\alpha_{31}\sigma^{3}(t_3)+\alpha_{32}$ and
 $\gamma_3=\alpha_{31}(\sigma^{3}(t_3))^{2}
+\alpha_{32}(\sigma^{2}(t_3)+\sigma^{3}(t_3))+2\alpha_{33}$.
 We define
$$
m_{ij}=\frac{\alpha_{i1}\gamma_{j}-\alpha_{j1}\gamma_{i}}{2(\alpha_{i1}
 \beta_{j}-\alpha_{j1}\beta_{i})};\quad
M_{ij}=\frac{\beta_{i}\gamma_{j}-\beta_{j}\gamma_{i}}{\alpha_{i1}\beta_{j}
 -\alpha_{j1}\beta_{i}}.
$$
Also let
\begin{gather*}
m_1=\max\{m_{12}, m_{13},m_{23}\},\\
m_2=\min\{m_{23}+\sqrt{m_{23}^{2}-M_{23}};m_{13}
 +\sqrt{m_{13}^{2}-M_{13}}\},\\
d=\alpha_{11}(\beta_2\gamma_3-\beta_3\gamma_2)
 -\beta_1(\alpha_{21}\gamma_3-\alpha_{31}\gamma_2)+\gamma_1
 (\alpha_{21}\beta_3-\alpha_{31}\beta_2),\\
l_{i}=\alpha_{i1}\sigma(s)\sigma^{2}(s)-(\sigma(s)
+\sigma^{2}(s))\beta_{i}+\gamma_{i}\quad
\text{for } i=1, 2, 3.
\end{gather*}
Let us assume that
\begin{itemize}
\item[(A1)] $f:[t_1, \sigma^{3}(t_3)]\times\mathbb{R}^{+^{3}}
\to \mathbb{R}^{+}$
  is continuous;

\item[(A2)] $\alpha_{11}>0$, $\alpha_{21}>0$, $\alpha_{31}>0$
  and  $\frac{\alpha_{12}}{\alpha_{11}}
 >\frac{\alpha_{22}}{\alpha_{21}}>\frac{\alpha_{32}}{\alpha_{31}}$;

\item[(A3)] $m_1\leq t_1<t_2<t_3\leq m_2$;
 $2\alpha_{13}\alpha_{11}>\alpha_{12}^2$,
 $2\alpha_{23}\alpha_{21}>\alpha_{22}^2$,
 $2\alpha_{33}\alpha_{31}>\alpha_{32}^2$;

\item[(A4)] $m_{23}^{2}>M_{23}$,  $m_{12}^{2}< M_{12}$,
$m_{13}^{2}>M_{13}$, $d>0$ and

\item[(A5)] The point $t\in[t_1, \sigma^{3}(t_3)]$ is not left
dense and right scattered at the same time.

\end{itemize}
Define the nonnegative extended real numbers $f_{0}, f^{0},
f_{\infty}, f^{\infty}$ by
\begin{gather*}
f_{0}=\lim_{y\to0^{+},y^{\Delta}\to0^{+},y^{\Delta^{2}}\to 0^{+}}
 \min_{t\in[t_1, \sigma^{3}(t_3)]}\frac{f(t,y,y^{\Delta},
 y^{\Delta^{2}})}{y},\\
f^{0}=\lim_{y\to0^{+},y^{\Delta}\to0^{+},y^{\Delta^{2}}\to0^{+}}
 \max_{t\in[t_1, \sigma^{3}(t_3)]}\frac{f(t,y,y^{\Delta},
 y^{\Delta^{2}})}{y},\\
f_{\infty}=\lim_{y\to\infty,y^{\Delta}\to\infty,y^{\Delta^{2}}\to
 \infty}\min_{t\in[t_1, \sigma^{3}(t_3)]}\frac{f(t,y,
 y^{\Delta},y^{\Delta^{2}})}{y},\\
f^{\infty}=\lim_{y\to\infty,y^{\Delta}\to\infty,y^{\Delta^{2}}\to
 \infty}\max_{t\in[t_1,
 \sigma^{3}(t_3)]}\frac{f(t,y,y^{\Delta},y^{\Delta^{2}})}{y}
\end{gather*}
and assume that they will exist.
By an interval we mean the
intersection of the real interval with a given time scale.

 This paper is organized as follows.
In Section 2, we construct Green's function  for the corresponding
homogeneous problem of  \eqref{e11}-\eqref{e12} and estimate bounds
of the Green's function. In Section 3, we present a lemma which is
needed in further discussion and determine eigenvalue intervals for
which   \eqref{e11}-\eqref{e12}  has at least one positive solution,
by using Krasnosel'skii fixed point theorem. Finally as an
application, we give an example to demonstrate our result.

\section{Green's function and Bounds}

 In this section, we construct the Green's function for the
corresponding homogeneous problem of  \eqref{e11}-\eqref{e12} in six
different intervals and we estimate the bounds for the Green's
function.

   Let $G(t, s)$ be the Green's function for the problem
$-y^{\Delta^{3}}(t)=0$ satisfying \eqref{e12}.
 After computation, the Green's function $G(t, s)$ can be obtained as
\begin{equation}\label{e21}
G(t,s)= \begin{cases}
G_{11}(t,s), & {  t_1\leq t<s<t_2<\sigma^{3}(t_3)},\\
G_{12}(t,s), & {  t_1<\sigma(s)<t\leq t_2<\sigma^{3}(t_3)},\\
G_{13}(t,s), & { t_1\leq t< t_2<s<\sigma^{3}(t_3)},\\
G_{21}(t,s), & {t_1<t_2\leq t<s<\sigma^{3}(t_3)},\\
G_{22}(t,s), & { t_1<t_2<\sigma(s)<t\leq \sigma^{3}(t_3)},\\
G_{23}(t,s), & {t_1\leq \sigma(s)< t_2<t<\sigma^{3}(t_3)},
\end{cases}
\end{equation}
where
\begin{align*}
G_{11}(t, s)=&\frac{1}{2d}[-(\beta_1\gamma_3-\beta_3\gamma_1)
+t(\alpha_{11}\gamma_3-\alpha_{31}\gamma_1)
-t^{2}(\alpha_{11}\beta_3-\alpha_{31}\beta_1)]l_2\\
&+\frac{1}{2d}[(\beta_1\gamma_2-\beta_2\gamma_1)
-t(\alpha_{11}\gamma_2-\alpha_{21}\gamma_1)+t^{2}(\alpha_{11}\beta_2
-\alpha_{21}\beta_1)]l_3\\
G_{12}(t,
s)=&\frac{1}{2d}[-(\beta_2\gamma_3-\beta_3\gamma_2)
+t(\alpha_{21}\gamma_3-\alpha_{31}\gamma_2)-t^{2}(\alpha_{21}\beta_3
-\alpha_{31}\beta_2)] l_1\\
G_{13}(t,s)=&\frac{1}{2d}[(\beta_1\gamma_2-\beta_2\gamma_1)
-t(\alpha_{11}\gamma_2-\alpha_{21}\gamma_1)+t^{2}(\alpha_{11}\beta_2
-\alpha_{21}\beta_1)]l_3\\
G_{21}(t,s)=&\frac{1}{2d}[(\beta_1\gamma_2-\beta_2\gamma_1)
-t(\alpha_{11}\gamma_2-\alpha_{21}\gamma_1)+t^{2}(\alpha_{11}\beta_2
-\alpha_{21}\beta_1)]l_3\\
G_{22}(t,s)=&\frac{1}{2d}[-(\beta_2\gamma_3-\beta_3\gamma_2)
+t(\alpha_{21}\gamma_3-\alpha_{31}\gamma_2)-t^{2}(\alpha_{21}\beta_3
-\alpha_{31}\beta_2)]l_1\\
&+\frac{1}{2d}[(\beta_1\gamma_3-\beta_3\gamma_1)
-t(\alpha_{11}\gamma_3-\alpha_{31}\gamma_1)+t^{2}(\alpha_{11}\beta_3
-\alpha_{31}\beta_1)]l_2\\ G_{23}(t, s)=&\frac{1}{2d}[-(\beta_2\gamma_3
-\beta_3\gamma_2)+t(\alpha_{21}\gamma_3-\alpha_{31}\gamma_2)
-t^{2}(\alpha_{21}\beta_3-\alpha_{31}\beta_2)]l_1\\
\end{align*}
 Figure \ref{fig1} indicates that the Green's function for
\eqref{e11}-\eqref{e12} should take the form of \eqref{e21}, where
$s\in[t_1,t_3]$.

\begin{figure}[ht] \label{fig1}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1} %{Eigen_3c.md}\\
\includegraphics[width=0.6\textwidth]{fig2} %{Eigen_3d.md}
\end{center}
\caption{Representation of Green's function in six intervals}

\end{figure}

\begin{theorem}\label{t1}
Assume that the conditions {\rm (A2)-(A4)} are  satisfied. Then
\begin{equation}\label{e22}
\gamma G(\sigma(s),s)\leq G(t,s)\leq G(\sigma(s),s),\quad
\text{for all }(t,s)\in[t_1,\sigma^{3}(t_3)]\times[t_1,t_3],
\end{equation}
where
$$
0<\gamma=\min\big\{\frac{G_{12}(\sigma^{3}(t_3),s)}{G_{12}(t_1,s)},
 \frac{G_{13}(t_1,s)}{G_{13}(\sigma^{3}(t_3),s)},
 \frac{G_{11}(t_1,s)}{G_{11}(\sigma^{3}(t_3),s)},
 \frac{G_{11}(\sigma^{3}(t_3),s)}{G_{11}(t_1,s)}\big\}<1.
$$
\end{theorem}

\begin{proof}
The Green's function $G(t,s)$ is given in \eqref{e21} in six
different cases. In each case we prove the inequality as in
\eqref{e22}. Clearly
\begin{equation}\label{e23}
G(t,s)>0\quad \text{on }[t_1,\sigma^{3}(t_3)]\times[t_1,t_3].
\end{equation}
\textbf{Case (i).} For $t_1<\sigma(s)<t\leq t_2<\sigma^{3}(t_3)$,
\begin{align*}
\frac{G(t,s)}{G(\sigma(s),s)}&=\frac{G_{12}(t,s)}{G_{12}(\sigma(s),s)}\\
&=\frac{[-(\beta_2\gamma_3-\beta_3\gamma_2)+t(\alpha_{21}
\gamma_3-\alpha_{31}\gamma_2)-t^2(\alpha_{21}\beta_3
-\alpha_{31}\beta_2)]}
{[-(\beta_2\gamma_3-\beta_3\gamma_2)
+\sigma(s)(\alpha_{21}\gamma_3-\alpha_{31}
\gamma_2)-(\sigma(s))^{2}(\alpha_{21}\beta_3-\alpha_{31}\beta_2)]},
\end{align*}
from (A3) and (A4), we have
$G_{12}(t,s)\leq G_{12}(\sigma(s),s)$. Therefore,
$$
G(t,s)\leq G(\sigma(s),s),\quad \text{for all }
(t,s)\in[t_1,\sigma^{3}(t_3)]\times[t_1,t_3].
$$
And also, from (A2), we have
$$
\frac{G(t,s)}{G(\sigma(s),s)}
=\frac{G_{12}(t,s)}{G_{12}(\sigma(s),s)}
\geq\frac{G_{12}(t,s)}{G_{12}(t_1,s)}
\geq\frac{G_{12}(\sigma^{3}(t_3),s)}{G_{12}(t_1,s)}.
$$
Therefore,
$$
G(t,s)\geq\frac{G_{12}(\sigma^{3}(t_3),s)}{G_{12}(t_1,s)}
G(\sigma(s),s),\quad\text{for all }(t,s)\in[t_1,
\sigma^{3}(t_3)]\times[t_1,t_3].
$$

\textbf{Case (ii).} For $t_1\leq t< t_2<s<\sigma^{3}(t_3)$
\begin{align*}
\frac{G(t,s)}{G(\sigma(s),s)}
&=\frac{G_{13}(t,s)}{G_{13}(\sigma(s),s)}\\
& =\frac{[(\beta_1\gamma_2-\beta_2\gamma_1)
 -t(\alpha_{11}\gamma_2-\alpha_{21}\gamma_1)
 +t^2(\alpha_{11}\beta_2-\alpha_{21}\beta_1)]}{[(\beta_1\gamma_2
 -\beta_2\gamma_1)-\sigma(s)(\alpha_{11}\gamma_2
 -\alpha_{21}\gamma_1)+(\sigma(s))^{2}(\alpha_{11}\beta_2
 -\alpha_{21}\beta_1)]},
\end{align*}
from, (A3) and (A4), we have
$G_{13}(t,s)\leq G_{13}(\sigma(s),s)$. Therefore,
$$
G(t,s)\leq G(\sigma(s),s)\quad\text{for all }
 (t,s)\in[t_1,\sigma^{3}(t_3)]\times[t_1,t_3].
$$
Also, from (A2), we have
$$
\frac{G(t,s)}{G(\sigma(s),s)}=\frac{G_{13}(t,s)}{G_{13}(\sigma(s),s)}
\geq\frac{G_{13}(t,s)}{G_{13}(\sigma^{3}(t_3),s)}
\geq\frac{G_{13}(t_1,s)}{G_{13}(\sigma^{3}(t_3),s)}.
$$
Therefore,
$$
G(t,s)\geq\frac{G_{13}(t_1,s)}{G_{13}(\sigma^{3}(t_3),s)}
 G(\sigma(s),s),\quad\text{for all }
 (t,s)\in[t_1,\sigma^{3}(t_3)]\times[t_1,t_3].
$$

\textbf{Case (iii).} For  $t_1\leq t<s< t_2<\sigma^{3}(t_3)$.
 From (A3) and Case (ii), we have
 $G_{11}(t,s)\leq G_{11}(\sigma(s),s)$. Therefore,
$$
G(t,s)\leq G(\sigma(s),s)\quad\text{ for all }
(t,s)\in[t_1,\sigma^{3}(t_3)]\times[t_1,t_3].
$$
Also, from (A2), we have
$$
\frac{G(t,s)}{G(\sigma(s),s)}\geq \min
\big\{\frac{G_{11}(\sigma^{3}(t_3),s)}{G_{11}(t_1,s)},
\frac{G_{11}(t_1,s)}{G_{11}(\sigma^{3}(t_3),s)},
 \frac{G_{13}(t_1,s)}{G_{13}(\sigma^{3}(t_3),s)}\big\}.
$$
Therefore,
$$
G(t,s)\geq
\min\big\{\frac{G_{11}(\sigma^{3}(t_3),s)}{G_{11}(t_1,s)},
\frac{G_{11}(t_1,s)}{G_{11}(\sigma^{3}(t_3),s)},
\frac{G_{13}(t_1,s)}{G_{13}(\sigma^{3}(t_3),s)}\big\}
G(\sigma(s),s),
$$
for all $(t,s)\in[t_1,\sigma^{3}(t_3)]\times[t_1,t_3]$.

\textbf{Case (iv).} For $t_1<t_2< \sigma(s)<t\leq \sigma^{3}(t_3)$.
 From Case (i) and Case (ii), we have
$$
G(t,s)\leq G(\sigma(s),s)\quad\text{ for all }
(t,s)\in[t_1,\sigma^{3}(t_3)]\times[t_1,t_3],
$$
and
$$
G(t,s)\geq\frac{G_{12}(\sigma^{3}(t_3),s)}{G_{12}(t_1,s)}
 G(\sigma(s),s),\quad\text{for all }
  (t,s)\in[t_1,\sigma^{3}(t_3)]\times[t_1,t_3].
$$

\textbf{Case (v).} For $t_1<t_2\leq t<s<\sigma^{3}(t_3)$.
 From Case (ii), we have
$$
G(t,s)\leq G(\sigma(s),s)\quad\text{ for all }
(t,s)\in[t_1,\sigma^{3}(t_3)]\times[t_1,t_3],
$$
and
$$
G(t,s)\geq\frac{G_{13}(t_1,s)}{G_{13}(\sigma^{3}(t_3),s)}
G(\sigma(s),s),\quad\text{for all }(t,s)\in[t_1,\sigma^{3}(t_3)]
 \times[t_1,t_3].
$$

\textbf{Case (vi).} For   $t_1\leq \sigma(s)< t_2<t<\sigma^{3}(t_3)$.
 From Case (i), we have
 $$
G(t,s)\leq G(\sigma(s),s)\quad\text{ for all }
(t,s)\in[t_1,\sigma^{3}(t_3)]\times[t_1,t_3],
$$
and
$$
G(t,s)\geq\frac{G_{12}(\sigma^{3}(t_3),s)}{G_{12}(t_1,s)}
 G(\sigma(s),s),\text{for all }(t,s)\in[t_1,
 \sigma^{3}(t_3)]\times[t_1,t_3].
$$
By consolidating  all the above cases, we have
$$
\gamma G(\sigma(s),s)\leq G(t,s)\leq G(\sigma(s),s),\quad
 \text{for all }(t,s)\in[t_1,\sigma^{3}(t_3)]\times[t_1,t_3],
$$
where
$$
0<\gamma=\min\big\{\frac{G_{12}(\sigma^{3}(t_3),s)}{G_{12}(t_1,s)},
\frac{G_{13}(t_1,s)}{G_{13}(\sigma^{3}(t_3),s)},
\frac{G_{11}(t_1,s)}{G_{11}(\sigma^{3}(t_3),s)},
\frac{G_{11}(\sigma^{3}(t_3),s)}{G_{11}(t_1,s)}\big\}<1.
$$
\end{proof}

\section{Existence of Positive Solutions}

  In this section, first we prove a lemma which is needed in
our main result and  establish a criteria to determine eigenvalue
intervals for which there exists at least one positive solution of
 \eqref{e11}-\eqref{e12}.

\begin{definition} \label{def3.1} \rm
Let $X$ be a Banach space. A nonempty closed convex set $\kappa $ is
called a \emph{cone} of $X$,  if it satisfies the following
conditions:
 \begin{itemize}
\item[(1)] $\alpha_1u+\alpha_2v\in\kappa$, for all
 $u, v\in \kappa$ and $\alpha_1,\alpha_2\geq0$,
\item[(2)] $u\in \kappa$ and $-u\in \kappa$, implies  $u=0$.
\end{itemize}
\end{definition}

 Let $y(t)$  be the solution of \eqref{e11}-\eqref{e12},
 given by
\begin{equation}\label{e31}
y(t)=\lambda\int_{t_1}^{\sigma(t_3)}G(t,s)f(s,y(s),y^{\Delta}(s),
y^{\Delta^{2}}(s))\Delta s,\quad\text{for~all}~t\in[t_1,
\sigma^{3}(t_3)].
\end{equation}
 Define
$$
X=\big\{u \in C^{3}[t_1: \sigma^{3}(t_3)]\big\},
$$
with norm
$ \| u\|=\max_{t\in[t_1, \sigma^{3}(t_3)]} | u(t) |$.
Then $(X,\|\cdot \|)$ is
a Banach space. Define a set
\begin{equation}\label{e32}
\kappa=\big\{u\in X:u(t)\geq0\text{ on }[t_1,\sigma^{3}(t_3)]
\text{ and } \min_{t\in [t_1,
\sigma^{3}(t_3)]}u(t)\geq \gamma \| u \| \big\}.
\end{equation}
Then it is easy to see that $\kappa$ is a positive cone in $X$.

\begin{definition} \label{def3.2} \rm
Let $X$ and $Y$ be Banach spaces and $T:X\to Y$. $T$ is said
to be completely continuous, if $T$ is continuous, and for each
bounded sequence $\{x_{n}\}\subset X$, $\{Tx_{n}\}$ has a convergent
subsequence.
\end{definition}

 Now we define the operator $T:\kappa\to X$ by
\begin{equation}\label{e33}
(Ty)(t)=\lambda\int_{t_1}^{\sigma(t_3)}G(t,s)
f(s,y(s),y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s, \quad
\text{for all }t \in [t_1, \sigma^{3}(t_3)].
\end{equation}
 If $y\in\kappa$ is a fixed point of $T$, then $y$
satisfies \eqref{e31} and hence $y$ is a positive solution of
\eqref{e11}-\eqref{e12}.  We seek a fixed point of the operator
$T$ in the cone $\kappa$.

\begin{lemma} \label{lem3.1}
The operator $T$  defined in \eqref{e33} is a self map on $\kappa$.
\end{lemma}

\begin{proof}
Let $y\in\kappa$.  From \eqref{e23}, we have $(Ty)(t)\geq0$, for all
$t\in[t_1, \sigma^{3}(t_3)]$, and
$$
\begin{aligned}
(Ty)(t)&
=\lambda \int_{t_1}^{\sigma(t_3)}G(t, s)
f(s, y(s),y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s \\
& \leq \lambda\int_{t_1}^{\sigma(t_3)}
G(\sigma(s), s)f(s,y(s),y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s \\
\end{aligned}
$$
so that
$$
\| Ty \| \leq \lambda \int_{t_1}^{\sigma(t_3)}
 G(\sigma(s), s)  f(s, y(s),y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s
$$
Next, if $y\in\kappa$, then by the above inequality we have
$$
\begin{aligned}
(Ty)(t)&
=\lambda \int_{t_1}^{\sigma(t_3)}G(t, s)f(s, y(s),
 y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s \\
&\geq \gamma \lambda \int_{t_1}^{\sigma(t_3)}G(\sigma(s), s)
 f(s, y(s),y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s\\
&\geq\gamma \| Ty \|.
\end{aligned}
$$
Hence $T:\kappa \to \kappa$. Standard arguments involving
the Arzela-Ascoli theorem shows that $T$ is completely continuous.
\end{proof}

To establish eigenvalue intervals we will employ the following fixed
point theorem due to Krasnosel'skii \cite{mak}.

\begin{theorem}\label{t31}
Let $X$ be a Banach space,
$K\subseteq X$ be a cone, and suppose that $\Omega_1, \Omega_2$
are open subsets of $X$ with $0\in\Omega_1$ and
$\overline{\Omega}_1\subset\Omega_2 $. Suppose further that
$T:K\cap(\overline{\Omega}_2 \backslash \Omega_1)\to K$
is completely continuous operator such that either
\begin{itemize}
\item[(i)] $\| Tu\|\leq\| u\|$, $u\in K\cap\partial\Omega_1$
and $\| Tu\|\geq\|u\|$, $u\in K\cap\partial\Omega_2$, or

\item [(ii)] $\| Tu\|\geq\| u\|$, $u\in K\cap\partial\Omega_1$
and $\| Tu\|\leq\|u\|$, $u\in K\cap\partial\Omega_2$
\end{itemize}
 holds.
Then $T$ has a fixed
point in $K\cap(\overline{\Omega}_2 \backslash \Omega_1)$.
\end{theorem}

\begin{theorem}\label{t32}
Assume that conditions (A1)-(A5) are satisfied. Then, for each
$\lambda$ satisfying
\begin{equation}\label{e34} \frac{1} {
[\gamma^{2}\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)\Delta
s]f_{\infty}} < \lambda
<\frac{1}{[\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)\Delta
s]f^{0}},
\end{equation}
there exists at least one positive solution of
\eqref{e11}-\eqref{e12} that lies in $\kappa$.
\end{theorem}

\begin{proof}
Let $\lambda$ be given as in \eqref{e34}. Now, let $\epsilon>0$ be
chosen such that
$$
\frac{1}{[\gamma^{2}\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)\Delta
s](f_{\infty}-\epsilon)} \leq \lambda
\leq\frac{1}{[\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)\Delta
s](f^{0}+\epsilon)}.
$$
Let $T$ be the cone preserving, completely
continuous operator defined  in \eqref{e33}. By the definition of
$f^{0}$, there exists
 $H_{1i}>0$, $i=0, 1, 2$ such that
$$
\max_{t\in[t_1,\sigma^{3}(t_3)]}\frac{f(t,y,y^{\Delta},
 y^{\Delta^{2}})}{y} \leq(f^{0}+\epsilon)
$$
for $0<y\leq H_{10},0<y^{\Delta}\leq H_{11},0<y^{\Delta^{2}}\leq H_{12}$.
Let $H_1=\min\{H_{1i}:i=0, 1, 2\}$. It follows that,
 $f(t,y,y^{\Delta},y^{\Delta^{2}}) \leq(f^{0}+\epsilon)y$,
 for $0<y,y^{\Delta},y^{\Delta^{2}}\leq H_1$. So
 choosing  $y\in \kappa$ with $\| y\|=H_1$, then from \eqref{e22} we
have
\begin{align*}
(Ty)(t)
& =\lambda\int_{t_1}^{\sigma(t_3)}G(t,s)
 f(s,y(s), y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s \\
& \leq \lambda \int_{t_1}^{\sigma(t_3)}G(\sigma(s), s)
 f(s,y(s),y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s \\
& \leq\lambda \int_{t_1}^{\sigma(t_3)}G(\sigma(s), s)
 (f^{0}+\epsilon) y(s) \Delta s \\
& \leq\lambda \int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)
 (f^{0}+\epsilon)\| y \| \Delta s \\
& \leq\| y \|, \quad t\in [t_1, \sigma^{3}(t_3)].
\end{align*}
Consequently, $\| Ty\|\leq\| y\|$. So,
if we define
$\Omega_1=\{y\in X: \| y\|<H_1\}$,
then
\begin{equation}\label{e35}
\| Ty\|\leq\| y\|,\quad\text{for } y\in \kappa\cap\partial\Omega_1.
\end{equation}
By the definition of  $f_{\infty}$, there exists
$\overline{H}_{2i}>0$, $i=0, 1, 2$ such that
$$
\min_{t\in[t_1,\sigma^{3}(t_3)]}\frac{f(t,y,y^{\Delta},
y^{\Delta^{2}})}{y}\geq(f_{\infty}-\epsilon),
$$
for $y\geq\overline{H}_{20}$, $y^{\Delta}\geq\overline{H}_{21}$,
$y^{\Delta^{2}}\geq\overline{H}_{22}$. Let
$\overline{H}_2=\min\{\overline{H}_{2i}:i=0, 1, 2\}$. It follows
that,
$$
f(t,y,y^{\Delta},y^{\Delta^{2}})\geq(f_{\infty}-\epsilon)y,\quad
\text{for } y,y^{\Delta},y^{\Delta^{2}} \geq\overline{H}_2.
$$
Let
$$
H_2=\max\big\{2H_1,\frac{1}{\gamma}\overline{H}_2\big\},\quad
\Omega_2=\{y\in X: \| y\|<H_2\}.
$$
Now choose $y\in \kappa\cap\partial\Omega_2$ with $\|
y\|=H_2$, so that
$$
\min_{t\in [t_1,\sigma^{3}(t_3)]}y(t)\geq \gamma\|
y\|\geq\overline{H}_2.
$$
Consider
\begin{align*}
(Ty)(t)
&=\lambda\int_{t_1}^{\sigma(t_3)}G(t,s)f(s,y(s),y^{\Delta}(s),
 y^{\Delta^{2}}(s))\Delta s \\
& \geq\lambda\int_{t_1}^{\sigma(t_3)}\gamma G(\sigma(s),s)
 f(s,y(s),y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s \\
& \geq \gamma\lambda\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)
 (f_{\infty}-\epsilon)y(s)\Delta s \\
& \geq \gamma^{2} \lambda\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)
 (f_{\infty}-\epsilon)\| y\| \Delta s \\
& \geq\| y \|.
\end{align*}
Thus,
\begin{equation}\label{e36}
\| Ty\|\geq\| y\|,\quad\text{for }y\in \kappa\cap\partial\Omega_2.
\end{equation}
An  application  of Theorem \ref{t31} to \eqref{e35} and
 \eqref{e36} yields that $T$ has a  fixed point
$y(t)\in \kappa\cap(\overline{\Omega}_2 \backslash
\Omega_1)$.  This fixed point is the positive solution of
 \eqref{e11}-\eqref{e12} for the given $\lambda$.
\end{proof}

\begin{theorem}\label{t33}
Assume that conditions {\rm(A1)-(A5)}  are satisfied. Then, for each
$\lambda$ satisfying
\begin{equation}\label{e37} \frac{1}{
[\gamma^{2}\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)\Delta s]f_{0}}
< \lambda
<\frac{1}{[\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)\Delta
s]f^{\infty}},
\end{equation}
there exists at least one positive solution of
\eqref{e11}-\eqref{e12} that lies in $\kappa$.
\end{theorem}

\begin{proof}
Let $\lambda$ be given  in \eqref{e37}, and choose $\epsilon>0$
such that
$$
\frac{1}{ [\gamma^{2}\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)\Delta s
](f_{0}-\epsilon)} \leq \lambda \leq
\frac{1}{[\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)\Delta
s](f^{\infty}+\epsilon)}.
$$
Let $T$ be the cone preserving,
completely continuous operator that was defined by \eqref{e33}. By
the definition of $f_{0}$, there exists
 $J_{1i}>0$, $i=0, 1, 2$ such that
$$
\min_{t\in[t_1,\sigma^{3}(t_3)]}\frac{f(t,y,y^{\Delta},
 y^{\Delta^{2}})}{y} \geq (f_{0}-\epsilon),
$$
for  $0<y\leq J_{10}$, $0<y^{\Delta}\leq J_{11}$,
$0<y^{\Delta^{2}}\leq J_{12}$. Let $J_1=\min\{J_{1i}:i=0,1, 2\}$.
 It follows that,
$$
f(t,y,y^{\Delta},y^{\Delta \Delta})\geq(f_{0}-\epsilon)y,\quad
\text{for } 0<y,y^{\Delta},y^{\Delta^{2}}\leq J_1.
$$
So, choose $y\in \kappa$ with $\| y \|=J_1$, then
\begin{align*}
(Ty)(t)
&=\lambda\int_{t_1}^{\sigma(t_3)}G(t,s)f(s,y(s),y^{\Delta}(s),
 y^{\Delta^{2}}(s))\Delta s \\
& \geq\lambda\int_{t_1}^{\sigma(t_3)}\gamma G(\sigma(s),s)
 f(s,y(s),y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s \\
& \geq \gamma\lambda\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)
 (f_{0}-\epsilon)y(s) \Delta s \\
& \geq \gamma^{2} \lambda\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)
 (f_{0}-\epsilon)\| y\| \Delta s \\
& \geq\| y \|.
\end{align*}
Consequently, $\| Ty\|\geq\| y\|$. So,
if we define
$\Omega_1=\{y\in X:\| y\|<J_1\}$,  then
\begin{equation}\label{e38}
\| Ty\|\geq\| y\|,\quad\text{for } y\in\kappa\cap\partial\Omega_1.
\end{equation}
It remains for us to consider $f^{\infty}$.  By the definition of
$f^{\infty}$, there exists  $\overline{J}_{2i}>0$, $i=0, 1, 2$ such
that
$$
\max_{t\in[t_1,\sigma^{3}(t_3)]}\frac{f(t,y,y^{\Delta},
y^{\Delta^{2}})}{y}\leq(f^{\infty}+\epsilon),
$$
for  $y\geq\overline{J}_{20}$, $y^{\Delta}\geq\overline{J}_{21}$,
$y^{\Delta^{2}}\geq\overline{J}_{22}$, it follows that
$$
f(t,y,y^{\Delta},y^{\Delta^{2}})\leq(f^{\infty}+\epsilon)y,\quad
\text{for }y,y^{\Delta},y^{\Delta^{2}}\geq\overline{J}_2.
$$
There are  two possible cases.

 \textbf{Case(i)}. $f$ is bounded.  Suppose $L>0$ and
$\max_{t\in[t_1,
\sigma^{3}(t_3)]}f(t,y,y^{\Delta},y^{\Delta^{2}})\leq L$, for all
$0<y,y^{\Delta},y^{\Delta^{2}}<\infty$. Let
$$
J_2=\max\big\{2J_1,L\lambda
\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)\Delta s \big\}.
$$
Then, for $y\in \kappa$ with $\| y\|=J_2$, we have
\begin{align*}
(Ty)(t)
& =\lambda\int_{t_1}^{\sigma(t_3)}G(t,s)
 f(s,y(s),y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s \\
& \leq \lambda \int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)
 f(s,y(s),y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s\\
& \leq\lambda L\int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)\Delta s \\
& \leq\| y \|,~t\in [t_1,\sigma^{3}(t_3)],
\end{align*}
 so that $\| Ty\|\leq\| y\|$. So, if we
define $\Omega_2=\{y\in X:\| y\|<J_2\}$, then
\begin{equation}\label{e39}
\|Ty\|\leq\| y\|,\quad\text{for }y\in \kappa\cap\partial\Omega_2.
\end{equation}

 \textbf{Case(ii)}. $f$ is unbounded. Let
$J_{2i}>\max\{2J_{1i},\overline{J}_{2i}\}$, $i=0, 1, 2$ be such that
$f(t,y,y^{\Delta},y^{\Delta^{2}})\leq f(t,J_{20},J_{21},J_{22})$,
for  $0<y\leq J_{20}$, $0<y^{\Delta}\leq J_{21}$,
$0<y^{\Delta^{2}}\leq J_{22}$. Let $J_2=\max\{J_{2i}:i=0, 1, 2\}$.
Let $y\in \kappa$ with $\| y\|=J_2$. Then
\begin{align*}
(Ty)(t)
& =\lambda\int_{t_1}^{\sigma(t_3)}G(t,s)
 f(s,y(s),y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s \\
& \leq \lambda \int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)
 f(s,y(s),y^{\Delta}(s),y^{\Delta^{2}}(s))\Delta s \\
&\leq\lambda \int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)
 f(s,J_{20},J_{21},J_{22})\Delta s\\
& \leq\lambda \int_{t_1}^{\sigma(t_3)}G(\sigma(s),s)
 (f^{\infty}+\epsilon) J_2  \Delta s \\
& \leq J_2\\
&=\| y \|, \quad t\in [t_1, \sigma^{3}(t_3)].
\end{align*}
Thus, $\| Ty\|\leq\| y\|$. For this case, if we
define
$\Omega_2=\{y\in X:\| y\|<J_2\}$,  then
\begin{equation}\label{e310}
\| Ty\|\leq\| y\|,\quad\text{for } y\in\kappa\cap\partial\Omega_2.
\end{equation}
Thus,  an application of  Theorem
\ref{t31} to \eqref{e38}, \eqref{e39} and
 \eqref{e310} yields that $T$  has fixed point
$y(t)\in\kappa\cap(\overline{\Omega}_2 \backslash \Omega_1)$.
This fixed point is the positive solution of
\eqref{e11}-\eqref{e12} for the given $\lambda$.
\end{proof}
\section{Example}
Now, we give an example to illustrate the above result. Consider the
eigenvalue problem
\begin{equation}\label{e41}
y^{\Delta^{3}}+\lambda y(20-19.5
e^{-7y})(30-29.5e^{-5y^{\Delta}})(61-60e^{-3y^{\Delta^{2}}})=0,~~t\in
[0,\sigma^{3}(1)]\cap \mathbb{T}
\end{equation}
where $\mathbb{T}=\{0\}\cup\{\frac{1}{2^{n+1}}:n\in \mathbb{N}\}
\cup [\frac{1}{2},
 \frac{3}{2}]$,
subject to the boundary conditions
\begin{equation}\label{e42}
\begin{gathered}
y(0)+\frac{4}{3}y^{\Delta}(0)+\frac{5}{4}y^{\Delta^{2}}(0)=0 \\
y(\frac{1}{2})+\frac{1}{2}y^{\Delta}(\frac{1}{2})
 +y^{\Delta^{2}}(\frac{1}{2})=0 \\
y(\sigma^{3}(1))+\frac{1}{4}y^{\Delta}(\sigma^{2}(1))
 +\frac{1}{2}y^{\Delta^{2}}(\sigma(1))=0
\end{gathered}
\end{equation}
The Green's function is
$$
G(t,s)=\begin{cases}
G_{11}(t,s), &  0\leq t<s<\frac{1}{2}<\sigma^{3}(1)\\
G_{12}(t,s), &   0<\sigma(s)<t\leq\frac{1}{2}<\sigma^{3}(1) \\
G_{13}(t,s), &   0\leq t< \frac{1}{2}<s<\sigma^{3}(1) \\
 G_{21}(t,s), &  0<\frac{1}{2}\leq t<s<\sigma^{3}(1) \\
 G_{22}(t,s),&   0<\frac{1}{2}<\sigma(s)<t\leq \sigma^{3}(1) \\
G_{23}(t,s), & 0\leq \sigma(s)< \frac{1}{2}<t<\sigma^{3}(1),
\end{cases}
$$
where
\begin{gather*}
\begin{aligned}
G_{11}(t,s)&=[\frac{-5}{6}+\frac{t^{2}}{3}]
 [6\sigma(s)\sigma^{2}(s)-6(\sigma(s)+\sigma^{2}(s))+\frac{33}{2}]\\
&\quad +[14-3t-4t^{2}][2\sigma(s)\sigma^{2}(s)-\frac{5}{2}(\sigma(s)
 +\sigma^{2}(s))+5]
\end{aligned}\\
G_{12}(t,s)=G_{23}(t,s)
 =[\frac{15}{4}-t-t^2][6\sigma(s)\sigma^{2}(s)
 -8(\sigma(s)+\sigma^{2}(s))+15]\\
G_{13}(t,s)=G_{21}(t,s)
 =[14-3t-4t^{2}][2\sigma(s)\sigma^{2}(s)-\frac{5}{2}(\sigma(s)
 +\sigma^{2}(s))+5]\\
\begin{aligned}
G_{22}(t,s)&=[\frac{15}{4}-t-t^{2}][6\sigma(s)\sigma^{2}(s)
 -8(\sigma(s)+\sigma^{2}(s))+15]\\
&\quad+[\frac{5}{6}-\frac{t^{2}}{3}][6\sigma(s)\sigma^{2}(s)
 -6(\sigma(s)+\sigma^{2}(s))+\frac{33}{2}].
\end{aligned}
\end{gather*}
 We found that $\gamma=0.4666$, $f_{\infty} =36600$, and $f^{0}=0.25$.
Employing Theorem \ref{t32}, we obtain the optimal eigenvalue interval
  $0.0000089125<\lambda<0.566972$, for which \eqref{e41}-\eqref{e42}
 has a positive solution.

\subsection*{Acknowledgements}
The authors thank the anonymous referee for his/her valuable suggestions.

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\end{document}