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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 53, 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 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/53\hfil Existence of positive solutions]
{Existence of positive solutions and  eigenvalues intervals
for nonlinear Sturm Liouville  problems  with a singular interface}

\author[D. K. K. Vamsi, P. K. Baruah \hfil EJDE-2012/53\hfilneg]
{D. K. K. Vamsi, Pallav Kumar Baruah}  % in alphabetical order

\address{Dasu Krishna Kiran Vamsi \newline
 Department of Mathematics and computer Science,
 Sri Sathya Sai Institute of Higher Learning,
 Prasanthi Nilayam, Puttaparthi,
 Ananthapur, Andhra-Pradesh, 515134, India}
\email{dkkvamsi@gmail.com}

\address{Pallav Kumar Baruah \newline
 Department of Mathematics and computer Science,
 Sri Sathya Sai Institute of Higher Learning,
 Prasanthi Nilayam, Puttaparthi,
 Ananthapur, Andhra-Pradesh, 515134, India}
\email{pkbaruah@sssihl.edu.in}


\thanks{Submitted February 1, 2012. Published March 30, 2012.}
\thanks{Supported by grant ERIP/ER/0803728/M/01/1158 from
 DRDO, Ministry of Defence, \hfill\break\indent Govt. of India}
\subjclass[2000]{34B09, 34B27, 34L15, 47J25}
\keywords{Regular problems; singular problems; singular interface problems;
\hfill\break\indent time scale; dynamic equation; Green's matrix}

\begin{abstract}
 In this article, we define the  Green's matrix for a nonlinear Sturm Liouville problem
 associated with a pair of dynamic equations on time scales with a singularity at
 the point of interface. Then using iterative techniques, we obtain eigenvalue
 intervals for which there exist positive solutions. Then we present iterative schemes
 for approximating the solutions, and discus an  example that illustrates the
 the results obtained.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}

Solving boundary-value problems with different types of
singularities has remained a challenge for mathematicians over the
ages. While ``regular''  problems, those over finite intervals
with well-behaved coefficients pose no difficulties.
There are applications  where  either the domain of the problem is not
well defined, or the continuity and/or smoothness of the
functions, coefficients involved are not guaranteed in some parts
of the domain, sometimes in the boundary or  parts of the
boundary. In all such cases the problem is considered to be a
``singular'' problem.
The definition of the problem and therefore
the description of the solution becomes a highly difficult task.
Here are quite a number of different approaches that we come across in
the literature to tackle these singular problems \cite{36,37,38,39,40,41,42}.

In the literature we find a  class of interface problems, termed
as mixed pair of equations, discussed in the papers
\cite{13,18,14,15,12,20,19,16,26,21,22,25,24,27,23} where two different
differential equations are defined on two adjacent intervals  and
the solutions satisfy a matching condition at the point  of
interface.  These problems are called as matching interface
problems. If the boundary is well defined then  we call the
problem to be a regular interface problem.  These interface
problems with singularities in the domain are always of great
interest.

We see that these interface problems for regular case has been
discussed in \cite{13,14,21,22,25,24,27,23}  and the problem of having
singularity at the boundary is discussed in \cite{18,15,12,20,19,16,26}.

From the above we see that the regular interface problems and interface
problems with singularity at the boundary are dealt in detail.
But the  problem of having a singularity at the point of interface seems
to be less explored.
 Study of these problems using classical analytical
tools is tedious. We term these problems as singular interface
problems.

The singularity at the point of interface in the domain of
definition of the mixed pair of equations could be of the
following three types satisfying certain matching conditions at
the  singular interface.

\setlength{\unitlength}{1mm}
\begin{picture}(90,27)(0,-1)
\put(0,21){Interface 1: $[a,c]\cup[\sigma(c),b]$}
\put(50,24){\line(1,0){20}}
\put(50,23){\line(0,1){2}}
\put(70,23){\line(0,1){2}}
\put(49,20){$a$}
\put(68.5,20){$c$}
\put(73,24){\line(1,0){17}}
\put(73,23){\line(0,1){2}}
\put(90,23){\line(0,1){2}}
\put(72.5,20){$\sigma(c)$}
\put(89,20){$b$}
%
\put(0,11){Interface 2: $[a,\rho(c)]\cup[c,b]$}
\put(50,14){\line(1,0){17}}
\put(50,13){\line(0,1){2}}
\put(67,13){\line(0,1){2}}
\put(49,10){$a$}
\put(62.5,10){$\rho(c)$}
\put(70,14){\line(1,0){20}}
\put(70,13){\line(0,1){2}}
\put(90,13){\line(0,1){2}}
\put(70,10){$c$}
\put(89,10){$b$}
%
\put(0,1){Interface 3: $[a,\rho(c)]\cup[\sigma(c),b]$}
\put(50,4){\line(1,0){17}}
\put(50,3){\line(0,1){2}}
\put(67,3){\line(0,1){2}}
\put(49,0){$a$}
\put(62.5,0){$\rho(c)$}
\put(73,4){\line(1,0){17}}
\put(73,3){\line(0,1){2}}
\put(90,3){\line(0,1){2}}
\put(72,0){$\sigma(c)$}
\put(89,0){$b$}
\end{picture}

To describe the singularities in the domain of definition we take
help of the terminology used on Time Scale \cite{2}.  The new
framework of the dynamic equations on time scale with facilities
of the two jump operators with  various definitions of continuity
and derivatives make one's job simple to study the interface
problems with mixed operators along with a singular interface.
Recently we have worked on the linear singular interface problems
as seen in \cite{17,63,64}, \cite{65,35}. Here we
discuss the corresponding nonlinear problem.


The method of lower and upper solutions is one of the commonly
used  methods for dealing with the second order initial and
boundary value problems. It has its origin  as early as 1893
\cite{66}. Also this method of lower and upper solutions clubbed
with the monotone iterative technique is used in the existence
theory  for nonlinear problems.  A good introduction  covering
different aspects for  the  monotone iterative methods is given by
Lakshmikantham and others in \cite{67}.


Lower and upper solutions give bounds for solutions which are
improved iteratively using monotone iterative process. This method
of lower and upper solutions for separated BVPs on time scales was
developed recently by Akin in \cite{68}.

Off late iterative methods have  been used to prove the existence of positive
solutions of nonlinear boundary value problems for ordinary differential
equations \cite{70,71,72,73}. By applying iterative methods, we not only obtain
the existence of positive solutions, but also establish iterative schemes for
approximating  the solutions.

In this paper we define the Green's matrix for a nonlinear non-homogenous Sturm Liouville
boundary value problem associated with singular interface problems(NN-SL-BVP-SIP)
on time scales.  Using the Green's matrix we obtain   eigenvalue intervals for which
 positive solutions exist for the NN-SL-BVP-SIP on time scales using  iterative methods.
We also  establish iterative schemes for approximating  the solutions.
 We present an example that illustrates the results obtained.


\section{Preliminaries}

An introduction on Time scale and Dynamic equations can be found in \cite{2}.
In the following section we introduce few definitions for our usage.
  \begin{definition} \label{def2.1} \rm
Let $ \mathbb{T}$ be a time scale(an arbitrary closed subset of real numbers).
For $t \in \mathbb{T} $ we define the  forward jump operator
 $\sigma : \mathbb{T} \to \mathbb{T}$ by
 \[
\sigma(t) := \inf  \{s \in \mathbb{T} : s > t \},
\]
 while the  backward jump operator
 $ \rho : \mathbb{T} \to \mathbb{T}$ is defined by
\[
\rho(t) := \sup  \{s \in \mathbb{T} : s < t \}.
\]
If $\sigma(t)> t,$ we say that $t$  is   right-scattered,  while $\rho(t) < t$
we say that $t$ is  left-scattered. Points that are right-scattered and left-scattered
at the same time are called  isolated.

Also, if $t <  \sup   \mathbb{T} $ and $\sigma(t) = t,$ then $t$ is called  right-dense,
 and if $t > \  \inf \ \mathbb{T}$ and $\rho(t) = t,$ then $t$ is called  left-dense.
 Points that are right-dense and left-dense at the same time are called  dense.

Finally, the graininess function    $\mu : \mathbb{T} \to [0,\infty)$ is  defined by
$ \mu(t) := \sigma(t) - t$.
 \end{definition}


\begin{definition} \label{def1}\rm
$$ 
\mathbb{T}^\kappa =   \begin{cases}
 \mathbb{T} - \{m\} &    \text{if }\sup  \mathbb{T} <\infty \\
  \mathbb{T} &   \text{if }\sup  \mathbb{T}  = \infty
\end{cases}  
$$
where $m$ is the left scattered maximum of $\mathbb{T}$.
\end{definition}

\begin{definition} \label{def2.4}\rm
  Let f be a function defined on $ \mathbb{T} $.    We say that $f$ is
 delta differentiable  at $ t \in \mathbb{T} ^ \kappa $ provided there exists
an $\alpha$ such that for all  $\epsilon > $ 0 there is a neighborhood
$\mathcal{N}$ around $t$ with
\[
|f(\sigma(t)-f(s)-\alpha(\sigma(t)-s)  | \leq \epsilon
|\sigma(t)-s | \quad \text {for all }  s \in   \mathcal{N}.
\]
\end{definition}

\begin{definition} \label{higher} \rm
For a function $f : \mathbb{T} \to \mathbb R$ we shall talk about the second
derivative ${{f}^{\Delta\Delta}}
$ provided $f^\Delta$ is differentiable on
${\mathbb{T}^\kappa}^2 = {(\mathbb{T} ^ \kappa)}^\kappa$ with derivative
 ${{f}^{\Delta\Delta}} = {(f^{\Delta})}^\Delta : {\mathbb{T}^\kappa}^2 \to \mathbb R$.
 Similarly we define the higher order derivatives
 ${{f}^{\Delta}}^n : {\mathbb{T}^{\kappa}}^n \to \mathbb R$.
\end{definition}

\begin{theorem}[Arzela-Ascoli Theorem]
A subset $M$ of $\mathcal{C}([a,b],{\mathbb R}^n)$ is relatively compact if and only
if it is bounded and equicontinuous.
\end{theorem}

\begin{theorem}[\cite{69}] \label{Existence}
Let $K$ be  a normal cone of a Banach space $E$ and $v_0 \leq w_0$.
Let us suppose that
\begin{itemize}
\item[(A1)] $T : [v_0,w_0] \to E$  is completely continuous;

\item[(A2)] $T$ is monotone increasing on $[v_0,w_0]$;

\item[(A3)] $v_0$ is a lower solution of $T,$ that is, $v_0 \leq Tv_0$;

\item[(A4)] $w_0$ is an upper solutions of $T,$ that is, $Tw_0 \leq w_0$.
\end{itemize}
Then the iterative sequences
$v_n = Tv_{n-1}$ and $w_n = Tw_{n-1}$  ($n = 1,2,3 \dots$)
satisfy
\[
v_0 \leq v_1 \leq \dots \leq v_n \leq \dots \leq w_n \dots \leq w_1 \leq w_0
\]
and converge to $v$ and $w \in [v_0,w_0]$, respectively, which are fixed points of $T$.
\end{theorem}

\section{Definition of Problem}

Let $\mathbb{T}_1 = {[a,\rho(c)]}_\mathbb{T}$,
$\mathbb{T}_2 = {[\sigma(c),b]}_\mathbb{T} $  where
$-\infty < a, \rho(c), \sigma(c), b < +\infty$. Also let $(f_1,f_2)$ be
nonlinear function tuple in
$ \mathcal{C}(\mathbb{T}_1 \times  \mathbb{T}_1, \mathbb R)
\times \mathcal{C}(\mathbb{T}_2 \times \mathbb{T}_2, \mathbb R)$.
Let $\lambda \in \mathbb R$.
The nonlinear nonhomogenous Sturm Liouville boundary-value problem
associated with singular interface problems (NN-SL-BVP-SIP) is defined by
\begin{gather}
y_1^{\Delta\Delta}(t)  =  \lambda f_1(t,y_1^{\sigma}), \quad t \in \mathbb{T}_1^{\kappa^2}
\label{eqnbvp1-c} \\
y_2^{\Delta\Delta}(t)  =  \lambda f_2(t,y_2^{\sigma}), \quad t \in {\mathbb{T}_2}^{\kappa^2}
  \label{eqnbvp3-c}
\end{gather}
with the boundary conditions
\begin{eqnarray}
y_1(a)   =  0 = y_2(b)   \label{eqn5bvp-c}
\end{eqnarray}
followed by the matching interface conditions
\begin{gather}
y_1(\rho(c))  =  y_2(\sigma(c))  \label{bvpeqn7-c} \\
y_1^{\Delta}(\rho(c))  =  y_2^{\Delta}(\sigma(c)).  \label{bvpeqn9-c}
\end{gather}

\section{Green's Matrix associated with NN-SL-BVP-SIP}

A proof for the following theorem can be found in \cite{62}.

\begin{theorem} \label{theorem1}
Let $Y = (y_1,y_2)$,  $F = (f_1,f_2)$. Then the NN-SL-BVP-SIP has a unique solution
 $Y(t)$ for which the formula
\[
Y(t) = \lambda\int^b_{a}G(t,s)F(s,y^{\sigma})  \Delta s
\]
 holds, where $G(t,s)$ is the Green's matrix associated with NN-SL-BVP-SIP
 given by
 $\begin{pmatrix}
 G_{11}(t,s) & G_{12}(t,s) \\
  G_{21}(t,s) & G_{22}(t,s)    \end{pmatrix}$
where
\begin{gather*}
G_{11}(t,s)  =    \begin{cases}
                              u_1 = a-t, &  a \leq t \leq s \leq \rho(c) \\
                              v_1 = a-s, &  a \leq s \leq t \leq \rho(c)
                            \end{cases}   \\
  G_{22}(t,s)  =  \begin{cases}
                              u_2 = s-b, &  \sigma(c) \leq t \leq s \leq b \\
                              v_2 = t-b, & \sigma(c) \leq s \leq t \leq b
                            \end{cases}   \\
 G_{12}(t,s)  =  \begin{cases}
                (a-t)(b-s), & a \leq t \leq \rho(c),\;  \sigma(c) \leq s \leq b
                            \end{cases}    \\
 G_{21}(t,s)  =  \begin{cases}
                 (a-s)(b-t), \quad a \leq s \leq \rho(c),  \;   \sigma(c) \leq t \leq b
                            \end{cases}
    \end{gather*}
   provided $f_1$ and $f_2$ satisfy  the following conditions:
\begin{gather}
  \int_a^{\rho(c)}((a+1)-s) f_1(s,y_1^{\sigma}) \Delta s
 = \int_{\sigma(c)}^{b}(s-(b+1))f_2(s,y_2^{\sigma}) \Delta s  \label{C1} \\
  [(\sigma(c)+1)-b]\int_a^{\rho(c)} (a-s)f_1(s,y_1^{\sigma}) \Delta s
  =  [(a+1)-\rho(c)]\int_{\sigma(c)}^{b}(s-b)f_2(s,y_2^{\sigma}) \Delta s. \label{C2}
   \end{gather}
 \end{theorem}

 By $Y(t) = \lambda\int^b_{a}G(t,s)F(s,y^{\sigma})  \Delta s $,
 we mean
\begin{align*}
y_1(t) &=  \lambda\Big[\int_a^b G_{11}(t,s)f_1(s,y_1^{\sigma}) \Delta s
 + \int_a^b G_{12}(t,s)f_2(s,y_2^{\sigma}) \Delta s\Big] \\
       &=  \lambda \int_a^{\rho(c)} G_{11}(t,s)f_1(s,y_1^{\sigma}) \Delta s
+ \lambda \int_{\sigma(c)}^{b}G_{12}(t,s)f_2(s,y_2^{\sigma}) \Delta s,
\end{align*}
for  $t \in \mathbb{T}_1$;
and
\begin{align*}
y_2(t) &=  \lambda\Big[\int_a^b G_{21}(t,s)f_1(s,y_1^{\sigma}) \Delta s
  + \int_a^b G_{22}(t,s)f_2(s,y_2^{\sigma}) \Delta s\Big] \\
       &=  \lambda \int_a^{\rho(c)} G_{21}(t,s)f_1(s,y_1^{\sigma}) \Delta s
 + \lambda \int_{\sigma(c)}^{b}G_{22}(t,s)f_2(s,y_2^{\sigma}) \Delta s,
\end{align*}
for $t \in \mathbb{T}_2$.


\section{Preliminary Results}

 We define the integral operator
$T : \mathcal{C}({\mathbb{T}_1} \cup{\mathbb{T}_2},
\mathbb R) \to \mathcal{C}({\mathbb{T}_1} \cup{\mathbb{T}_2}, \mathbb R)$ by
\begin{align*}
&(Ty)(t) \\
&=
\begin{cases}
 (Ty_1)(t) =  \lambda\big[\int_a^b G_{11}(t,s)f_1(s,y_1^{\sigma}) \Delta s
+ \int_a^b G_{12}(t,s)f_2(s,y_2^{\sigma}) \Delta s\big], & t \in \mathbb{T}_1 \\
(Ty_2)(t) =  \lambda\big[\int_a^b G_{21}(t,s)f_1(s,y_1^{\sigma}) \Delta s
+ \int_a^b G_{22}(t,s)f_2(s,y_2^{\sigma}) \Delta s\big], & t \in \mathbb{T}_2.
\end{cases}
\end{align*}
We define the Banach space $E = \mathcal{C}({\mathbb{T}_1} \cup{\mathbb{T}_2}, \mathbb R)$
with the supremum norm
\[
\|y\| = {\sup}_{t \in \mathbb{T}_1}|y_1(t)| + {\sup}_{t \in \mathbb{T}_2}|y_2(t)|
\]
and the cone $K \subset E$ as
\[
K = \{y \geq 0  : y \in E\}.
\]

%\label{positive} assumption


\begin{lemma} \label{cc}
Let $f_1$ be positive on $\mathbb{T}_1$
and $f_2$ be positive on $\mathbb{T}_2$.
Also let $\lambda \in {\mathbb R}^{-}$.
 Then the operator $T : K \to K$ is completely continuous.
\end{lemma}

\begin{proof}
 We first show that $T$ is continuous. We prove it by showing that $T$ preserves
convergence.  Indeed let $y_n (=(y_{n1},y_{n2}))$ be a sequence of functions in
$\mathcal{C}({\mathbb{T}_1} \cup{\mathbb{T}_2}, \mathbb R) $ such that they
converge to $y (=(y_1,y_2))$. In other words
\[
\lim_{n \to \infty}\|y_n - y\| \to 0
\]
i.e.,
$ \lim_{n \to \infty}\|(y_{n1},y_{n2}) - (y_{1},y_{2})\| \to 0$.
 The above equation implies
\[
    \lim_{n \to \infty}\|(y_{n1}-y_1,y_{n2}-y_2)\| \to   0;
\]
i.e., $\lim_{n \to \infty} {\sup}_{t_1 \in \mathbb{T}_1}|(y_{n1} - y_1)(t_1)| \to 0$
and $\lim_{n \to \infty} {\sup}_{t_2 \in \mathbb{T}_2}|(y_{n2} - y_2)(t_2)| \to  0$.
Now with
$  \|T(y_{n})-T(y)\| = {\sup}_{t \in \mathbb{T}_1}|T(y_{n1}-y_1)(t)|
+ {\sup}_{t \in \mathbb{T}_2}|T(y_{n2}-y_2)(t)|$,
we see that
\begin{align*}
&{\sup}_{t \in \mathbb{T}_1}|T(y_{n1}-y_1)(t)|\\
&\leq  {\sup}_{t \in \mathbb{T}_1}\lambda|\int_a^b G_{11}(t,s)f_1(s,y_{n1}) \Delta s
 - \int_a^b G_{11}(t,s)f_1(s,y_{1}) \Delta s|  \\
&+\quad  {\sup}_{t \in \mathbb{T}_1}\lambda|\int_a^b G_{12}(t,s)f_2(s,y_{n2}) \Delta s
 - \int_a^b G_{12}(t,s)f_2(s,y_{2})| \\
&\leq  {\sup}_{t \in \mathbb{T}_1}\lambda \int_a^b G_{11}(t,s)|f_1(s,y_{n1})
 -f_1(s,y_1^{\sigma})| \Delta s  \\
&+\quad {\sup}_{t \in \mathbb{T}_1}\lambda \int_a^b G_{12}(t,s)|f_2(s,y_{n2})
-f_2(s,y_2^{\sigma})| \Delta s.
\end{align*}
 Similarly it can be sown that
\begin{align*}
 {\sup}_{t \in \mathbb{T}_2}|T(y_{n2}-y_2)(t)|
 &\leq  {\sup}_{t \in \mathbb{T}_2}\lambda \int_a^b G_{21}(t,s)|f_1(s,y_{n1})
 -f_1(s,y_1^{\sigma})| \Delta s \\
&\quad + {\sup}_{t \in \mathbb{T}_2}\lambda \int_a^b G_{22}(t,s)|f_2(s,y_{n2})
 -f_2(s,y_2^{\sigma})| \Delta s.
\end{align*}
 Since $(f_1,f_2)$ is continuous on
 $ \mathcal{C}(\mathbb{T}_1 \times  \mathbb{T}_1, \mathbb R)
 \times \mathcal{C}(\mathbb{T}_2 \times \mathbb{T}_2, \mathbb R)$ we have
 \begin{gather*}
 \lim_{n \to \infty}|f_1(s,y_{n1})-f_1(s,y_1^{\sigma})|  \to   0, \\
 \lim_{n \to \infty}|f_2(s,y_{n2})-f_1(s,y_2)|  \to   0.
\end{gather*}
 Hence, $ \lim_{n \to \infty} \|T(y_{n})-T(y)\| \to 0$ proving that $T$ is continuous.
  Let
   \begin{align*}
   f_1(s,y_1^{\sigma}) &\leq  M_1, \ \text{for some} \ M_1 > 0, \forall s \in  \mathbb{T}_1, \\
   f_2(s,y_2^{\sigma}) &\leq  M_2, \ \text{for some} \ M_2 > 0, \forall s \in  \mathbb{T}_2.
    \end{align*}
We now  show that $T(\mathcal{C}({\mathbb{T}_1} \cup{\mathbb{T}_2}, \mathbb R)) $
is bounded and equicontinuous subset of
 $\mathcal{C}({\mathbb{T}_1} \cup{\mathbb{T}_2}, \mathbb R)$.
Let us assume that
$y(=(y_1,y_2)) \in \mathcal{C}({\mathbb{T}_1} \cup{\mathbb{T}_2}, \mathbb R)$  and
$\|y(=(y_1,y_2))\| \leq M'$. Then
\begin{align*}
\|Ty\|
 &\leq  \sup_{t_1 \in \mathbb{T}_1}\lambda\Big[\int_a^b |G_{11}(t,s)||f_1(s,y_1^{\sigma})|
 \Delta s + \int_a^b |G_{12}(t,s)||f_2(s,y_2^{\sigma})| \Delta s\Big]  \\
    &\quad + \sup_{t_2 \in \mathbb{T}_2}\lambda
 \Big[\int_a^b |G_{21}(t,s)||f_1(s,y_1^{\sigma})| \Delta s
 + \int_a^b |G_{22}(t,s)||f_2(s,y_2^{\sigma})| \Delta s\Big]
\end{align*}
  Since $(f_1,f_2)$ is bounded we can conclude that there exists  a $K' > 0$ independent
of choice of $ y (=(y_1,y_2)) $ such that $\|Ty(=(y_1,y_2))\| \leq K'$.
Hence, $T(\mathcal{C}({\mathbb{T}_1} \cup{\mathbb{T}_2}, \mathbb R)) $ is bounded.
 We next show that
    $T(\mathcal{C}({\mathbb{T}_1} \cup{\mathbb{T}_2}, \mathbb R)) $
is  equicontinuous subset of $\mathcal{C}({\mathbb{T}_1} \cup{\mathbb{T}_2}, \mathbb R)$.
 We need to show that   for all $\epsilon > 0$ there  exists   $\delta > 0$
such that whenever    $\| t-t'\|   < \delta $
we have $ \|Ty(t)-Ty(t')\| < \epsilon$.

Now
\begin{align*}
\|Ty(t)-Ty(t')\|
&=  \sup_{t \in \mathbb{T}_1}|Ty_1(t)-Ty_1(t')|
 + \sup_{t_2 \in \mathbb{T}_2}|Ty_2(t)-Ty_2(t')| \\
&\leq   \sup_{t_1 \in \mathbb{T}_1}\Big[|\lambda\int_a^b (G_{11}(t,s)
 -G_{11}(t',s))f_1(s,y_1^{\sigma})\Delta s|\Big] \\
&\quad+ \sup_{t \in \mathbb{T}_1}\Big[|\lambda\int_a^b (G_{12}(t,s)
 -G_{12}(t',s)f_2(s,y_2^{\sigma}))\Delta s|\Big] \\
&\quad+ \sup_{t \in \mathbb{T}_2}\Big[|\lambda\int_a^b (G_{21}(t,s)
 -G_{21}(t',s)f_1(s,y_1^{\sigma}))\Delta s|\Big] \\
&\quad+ \sup_{t \in \mathbb{T}_2}\Big[|\lambda\int_a^b (G_{22}(t,s)
 -G_{22}(t',s)f_2(s,y_2^{\sigma}))\Delta s|\Big]
\end{align*}
Let $M = \max\{M_1,M_2\}$. Then we have
\begin{align*}
\|Ty(t)-Ty(t')\|
&\leq  M \sup_{t \in \mathbb{T}_1}\Big[|\lambda\int_a^b (G_{11}(t,s)
 -G_{11}(t',s))\Delta s|\Big] \\
&\quad+ M \sup_{t \in \mathbb{T}_1}\Big[|\lambda\int_a^b (G_{12}(t,s)
 -G_{12}(t',s))\Delta s|\Big] \\
&\quad+ M \sup_{t \in \mathbb{T}_2}\Big[|\lambda\int_a^b (G_{21}(t,s)
 -G_{21}(t',s))\Delta s|\Big] \\
&\quad+ M \sup_{t \in \mathbb{T}_2}\Big[|\lambda\int_a^b (G_{22}(t,s)
 -G_{22}(t',s))\Delta s|\Big]
\end{align*}
We see that
\begin{align*}
&M \sup_{t \in \mathbb{T}_1}\Big[|\lambda\int_a^b (G_{11}(t,s)-G_{11}(t',s))
 \Delta s|\Big]\\
&=  M\sup_{t \in \mathbb{T}_1}\Big[|\lambda \int_a^t (a-s)\Delta s 
   -\lambda \int_a^{t'}(a-s)\Delta s + \int_t^{\rho(c)} (a-t)\Delta s \\
& \quad -\lambda\int_{t'}^{\rho(c)}(a-t')\Delta s| \Big] \\
&=  M\sup_{t \in \mathbb{T}_1}\Big(|\lambda(t-t')\Big[a-\frac{1}{2}(t+t')\Big]
 +\lambda(t-t')[(t+t')-(a+\rho(c))]|\Big) \\
&\leq  M \sup_{t \in \mathbb{T}_1}\Big(|t-t'||\lambda \Big[\frac{1}{2}(t+t')
 -\rho(c)\Big]|\Big)
\end{align*}
Also 
\begin{align*}
&M \sup_{t \in \mathbb{T}_1}\Big[|\lambda\int_a^b (G_{12}(t,s)-G_{12}(t',s))
\Delta s|\Big]\\
&=  M \sup_{t \in \mathbb{T}_1}\Big[|\lambda\int_{\sigma(c)}^b (a-t)(b-s) \Delta s 
 - \lambda\int_{\sigma(c)}^b (a-t')(b-s)\Delta s|\Big] \\
&\leq   M |\lambda| \sup_{t \in \mathbb{T}_1} \Big[\int_{\sigma(c)}^b|t-t'|(b-s)
 \Delta s\Big] \\
&=  M |\lambda|\sup_{t \in \mathbb{T}_1}|t-t'|\int_{\sigma(c)}^b\Delta s
\end{align*}
We observe that 
\begin{align*}
&M \sup_{t \in \mathbb{T}_2}\Big[|\lambda\int_a^b (G_{21}(t,s)-G_{21}(t',s))
 \Delta s|\Big]\\
&=  M \sup_{t \in \mathbb{T}_2} \Big[|\lambda \int_a^{\rho(c)} (a-s)(b-t) \Delta s 
 - \lambda \int_a^{\rho(c)} (a-s)(b-t') \Delta s \\
&\leq   M |\lambda|\sup_{t \in \mathbb{T}_2} |t-t'|\int_a^{\rho(c)} (a-s) \Delta s
\end{align*}
Finally we have 
\begin{align*}
&M \sup_{t \in \mathbb{T}_2}\Big[|\lambda\int_a^b (G_{22}(t,s)-G_{22}(t',s))\Delta s|\Big]\\
&\leq  M|\lambda|\sup_{t \in \mathbb{T}_2}\Big[|\lambda\int_a^b (G_{22}(t,s)
 -G_{22}(t',s))\Delta s|\Big] \\
&=  M|\lambda|\sup_{t \in \mathbb{T}_2} \Big[|\int_{\sigma(c)}^t (t-b) \Delta s 
 -\int_{\sigma(c)}^{t'} ({t}'-b) \Delta s \\
&\quad + \int_t^b (s-b) \Delta s  -\int_{t'}^b (s-b) \Delta s| \Big] \\
&\leq  M|\lambda|\sup_{t \in \mathbb{T}_2} \Big[|t-t'|[t+t'-(b+\sigma(c))] 
 +\int_t^{t'}|s-b|\Delta s\Big] \\
&\leq  M|\lambda|\sup_{t \in \mathbb{T}_2} \Big[|t-t'|[t+t'-(b+\sigma(c))] 
 + |t'-t|b\Big] \\
&=  M|\lambda|\sup_{t \in \mathbb{T}_2} \Big[|t-t'|((t+t')-\sigma(c))\Big].
\end{align*}
From the above it is clear that 
 $ \|Ty(t)-Ty(t')\| < \epsilon$ whenever $\| t-t'\| < \delta$. Hence 
 $T(\mathcal{C}({\mathbb{T}_1} \cup{\mathbb{T}_2}, \mathbb R)) $  
is  equi-continuous subset of $\mathcal{C}({\mathbb{T}_1} \cup{\mathbb{T}_2}, \mathbb R)$.
 Hence from the Arzela-Ascoli theorem we see that $T$ is completely continuous. 
\end{proof}

\section{Eigenvalue Intervals}

\begin{theorem} \label{temp}
Let $f_1$ be positive on $\mathbb{T}_1$
and $f_2$ be positive on $\mathbb{T}_2$.
Also let $\lambda \in {\mathbb R}^{-}$.
 Let us assume that there exists $\Omega, K > 0$ such that for 
$u=(u_1,u_2), v=(v_1,v_2)$ we have
\begin{gather*}
f_1(t,u_1^{\sigma}) \leq   f_1(t,v_1^{\sigma}) \leq \Omega K, \quad t \in \mathbb{T}_1, \\
f_2(t,u_2^{\sigma}) \leq  f_2(t,u_2^{\sigma})  \leq \Omega K, \quad t \in \mathbb{T}_2,
\end{gather*}
whenever $0 \leq u^{\sigma} \leq v^{\sigma} \leq \Omega K$; i.e.,
\[
0 \leq u_1^{\sigma} \leq v_1^{\sigma} < \Omega K,\quad 
0 \leq u_2^{\sigma} \leq v_2^{\sigma}  < \Omega K.
\]
Then for all $\lambda$ satisfying
\begin{align*}
\lambda \leq \frac{1}{\Omega\big[\int_a^b G_{11}(t,s) \Delta s 
 + \int_a^b G_{12}(t,s)\Delta s \big]} \\
\lambda \leq \frac{1}{\Omega\big[\int_a^b G_{11}(t,s) \Delta s 
 + \int_a^b G_{12}(t,s)\Delta s \big]},
\end{align*}
there exists  positive solutions for NN-SL-BVP-SIP.
\end{theorem}

\begin{proof} 
Let $v_0(t) = 0$ and $w_0(t) = K$ for all $t \in \mathbb{T}_1 \cup \mathbb{T}_2$. 
Then from Lemma \ref{cc} it is clear that $T : [v_0,w_0] \to K$ is completely continuous.

$\bullet$ We claim that $T$ is monotone increasing on $[v_0,w_0]$.
Let us suppose that $u=(u_1,u_2), v=(v_1,v_2) \in [v_0,w_0]$ such that 
$u^{\sigma} \leq v^{\sigma}$. Then  clearly
 $0 \leq u^{\sigma}(t) \leq v^{\sigma}(t) \leq \Omega K,$ for all 
$t \in \mathbb{T}_1 \cup \mathbb{T}_2$. We have
\begin{align*}
&(Tu)(t) \\
&= \begin{cases} 
(Tu_1)(t) =  \lambda\big[\int_a^b G_{11}(t,s)f_1(s,u_1^{\sigma}) \Delta s 
 + \int_a^b G_{12}(t,s)f_2(s,u_2^{\sigma}) \Delta s\big], & t \in \mathbb{T}_1 \\
 (Tu_2)(t) =  \lambda\big[\int_a^b G_{21}(t,s)f_1(s,u_1^{\sigma}) \Delta s 
 + \int_a^b G_{22}(t,s)f_2(s,u_2^{\sigma}) \Delta s\big], & t \in \mathbb{T}_2.
\end{cases}
\end{align*}
From the hypothesis it is clear that $(Tu)(t) \leq (Tv)(t)$ where
\begin{align*}
&(Tv)(t) \\
&=\begin{cases}
  (Tv_1)(t) =  \lambda\big[\int_a^b G_{11}(t,s)f_1(s,v_1^{\sigma}) \Delta s 
+ \int_a^b G_{12}(t,s)f_2(s,v_2^{\sigma}) \Delta s\big], & t \in \mathbb{T}_1 \\
  (Tv_2)(t) =  \lambda\big[\int_a^b G_{21}(t,s)f_1(s,v_1^{\sigma}) \Delta s 
+ \int_a^b G_{22}(t,s)f_2(s,v_2^{\sigma}) \Delta s\big], & t \in \mathbb{T}_2.
\end{cases}
\end{align*}
Hence  $T$ is monotone increasing on $[v_0,w_0]$.

$\bullet$ We claim that $v_0$ is an lower solution of $T$.
We see that
\begin{align*}
&(Tv_0)(t) \\
&=\begin{cases}
  (Tv_{01})(t) =  \lambda\big[\int_a^b G_{11}(t,s)f_1(s,0) \Delta s 
+ \int_a^b G_{12}(t,s)f_2(s,0) \Delta s\big] \geq 0, & t \in \mathbb{T}_1 \\
 (Tv_{02})(t) =  \lambda\big[\int_a^b G_{21}(t,s)f_1(s,0) \Delta s
 + \int_a^b G_{22}(t,s)f_2(s,0) \Delta s\big] \geq 0, & t \in \mathbb{T}_2.
\end{cases}
\end{align*}
which implies that $v_0 \leq Tv_0$.

$\bullet$ We claim that $w_0$ is an upper solution of $T$.
We see that
\begin{align*}
&(Tw_0)(t) \\
&=  \begin{cases}
 (Tw_{01})(t) =  \lambda\big[\int_a^b G_{11}(t,s)f_1(s,w_0) \Delta s 
+ \int_a^b G_{12}(t,s)f_2(s,w_0) \Delta s\big], & t \in \mathbb{T}_1 \\
  (Tw_{02})(t) =  \lambda\big[\int_a^b G_{21}(t,s)f_1(s,w_0) \Delta s 
+ \int_a^b G_{22}(t,s)f_2(s,w_0) \Delta s\big], & t \in \mathbb{T}_2.
\end{cases}
\end{align*}
Let $t \in \mathbb{T}_1$. Then
\begin{align*}
&(Tw_{01})(t) \\
&\leq  \Omega K \lambda\Big[\int_a^b G_{11}(t,s) \Delta s \Big] 
 +  \Omega K \lambda\Big[\int_a^b G_{12}(t,s) \Delta s \Big] \\
&=  K \Omega  \lambda\Big[\int_a^b G_{11}(t,s) \Delta s \Big] 
 + K \Omega  \lambda\Big[\int_a^b G_{12}(t,s) \Delta s \Big] \\
&\leq  K \Omega   \Big[\int_a^b G_{11}(t,s) \Delta s
  + \int_a^b G_{12}(t,s) \Delta s \Big]
  \frac{1}{\Omega\big[\int_a^b G_{11}(t,s) \Delta s + \int_a^b G_{12}(t,s)\Delta s \big]} 
 \\
&=  K = w_0.
\end{align*}
We now let  $t \in \mathbb{T}_2$. Then
\begin{align*}
&(Tw_{02})(t) \\
&\leq  \Omega K \lambda\Big[\int_a^b G_{21}(t,s) \Delta s \Big] 
 +  \Omega K \lambda\Big[\int_a^b G_{22}(t,s) \Delta s \Big] \\
&=  K \Omega  \lambda\Big[\int_a^b G_{21}(t,s) \Delta s \Big] 
 + K \Omega  \lambda\Big[\int_a^b G_{22}(t,s) \Delta s \Big] \\
&\leq  K \Omega   \Big[\int_a^b G_{21}(t,s) \Delta s 
 + \int_a^b G_{22}(t,s) \Delta s \Big] \frac{1}{\Omega\big[\int_a^b G_{21}(t,s) \Delta s
  + \int_a^b G_{22}(t,s)\Delta s \big]}  \\
&=  K = w_0.
\end{align*}
Hence $Tw_0 \leq w_o$ proving that $w_0$ is an upper solution of $T$.
 We now construct sequences ${\{v_n\}}_{n=1}^{\infty}$ and 
${\{u_n\}}_{n=1}^{\infty}$ as follows:
\[
v_n = Tv_{n-1},\quad  w_n = Tw_{n-1}, \quad\text{for } n = 1,2,3,\dots
\]
Then from theorem \ref{Existence} we have that
\[
v_0 \leq v_1 \leq \dots \leq v_n \leq \dots w_n \leq \dots \leq w_1 \leq w_0,
\]
and ${\{v_n\}}_{n=1}^{\infty}$ and ${\{u_n\}}_{n=1}^{\infty}$ converge to,
 $v$ and $w$ in $[v_0,w_0]$, which are the fixed points of the operator $T$.
In other words $v$ and $w$ are the  positive solutions of the NN-SL-BVP-SIP.
\end{proof}

\section{An example}

In this section, an example is given to illustrate the main result of this paper.
Let $\mathbb{T}_1 = [1,5]_\mathbb{T}$,  $\mathbb{T}_2 = [6,10]_\mathbb{T}$. 
Let us consider the NN-SL-BVP-SIP-A
\begin{gather*}
y_1^{\Delta\Delta}(t) = \lambda \frac{1}{4} y_1^{\Delta^2}(\sigma(t)), \quad
  t \in {\mathbb{T}_1}^{\kappa^2}, \\
y_2^{\Delta\Delta}(t) = \lambda \Big(\frac{1}{2} y_2^{\Delta^2}(\sigma(t)) 
 + \frac{1}{4}y_2^{2}(t)\Big), \quad t \in {\mathbb{T}_2}^{\kappa^2},
\end{gather*}
along with the boundary and matching interface conditions
\begin{gather*}
y_1(1) = 0 = y_2(10)  \\
y_1(5) = y_2(6)  \\
y_1^{\Delta}(5) = y_2^{\Delta}(6).
\end{gather*}
Let $\Omega = 100$, $K = 10$. Also let $(u_1,u_2) = (t,t)$, $(v_1,v_2) = (t^2,t^2)$.
 We have
\begin{gather*}
u_1^{\Delta}(\sigma(t)) = 1, u_2^{\Delta}(\sigma(t)) = 1,\\
v_1^{\Delta}(\sigma(t)) = 2\sigma(t), v_2^{\Delta}(\sigma(t)) = 2\sigma(t).
\end{gather*}
Clearly
\begin{gather*}
u_1(\sigma(t)) = \sigma(t) \leq {\sigma(t)}^2 = v_1(\sigma(t)) < \Omega K, \\
u_2(\sigma(t)) = \sigma(t) \leq {\sigma(t)}^2 = v_2(\sigma(t)) < \Omega K.
\end{gather*}
Also
\begin{gather*}
f_1(t,u_1^{\sigma}) = \frac{1}{4} \leq \sigma^2(t) = f_1(t,v_1^{\sigma}), \\
f_2(t,u_2^{\sigma}) = \frac{1}{2} + \frac{1}{4}\sigma^2(t) \leq 2\sigma^2(t) 
 + \frac{1}{4}\sigma^4(t) = f_2(t,v_2^{\sigma}).
\end{gather*}
Hence from theorem \eqref{temp}, for all $\lambda$ satisfying
\begin{align*}
\lambda \leq \frac{1}{\Omega\big[\int_a^b G_{11}(t,s) \Delta s 
+ \int_a^b G_{12}(t,s)\Delta s \big]} \\
\lambda \leq \frac{1}{\Omega\big[\int_a^b G_{21}(t,s) \Delta s 
+ \int_a^b G_{22}(t,s)\Delta s \big]},
\end{align*}
there exists  positive solutions for NN-SL-BVP-SIP-A.
That is, for all $\lambda$ satisfying
\begin{align*}
\lambda 
&\leq  \frac{1}{\Omega \Big[\int_1^t (1-t) \Delta s + \int_t^5 (1-s) \Delta s}  \\
 &\quad + \int_1^5 (1-t)(10-s) \Delta s + \int_6^{10}(1-t)(10-s) \Delta s \Big] 
\end{align*}
and
\begin{align*}
\lambda
 &\leq  \frac{1}{\Omega \big[\int_1^5 (1-s)(10-t) \Delta s
 + \int_6^{10} (1-s)(10-t) \Delta s}  \\
 &\quad +  \int_6^{t}(t-10) \Delta s + \int_t^{10}(s-10) \Delta s  \big].
\end{align*}
\begin{gather*}
\lambda \leq  \frac{1}{100}\Big(-\frac{t^2}{2} -35t +\frac{55}{2}\Big), \quad
  t \in \mathbb{T}_1 \\
\lambda \leq   \frac{1}{100}\Big(\frac{t^2}{2} +30t -350\Big) , \quad
  t \in \mathbb{T}_2.
\end{gather*}
So for all $\lambda \leq -1/800$ there exists positive solutions for the NN-SL-BVP-SIP-A.

\begin{remark} \rm
We also note that the type of results embodied in 
\cite{13,18,14,15,12,20,19,16,26,21,22,25,24,27,23} when worked for second order 
are special cases of this work whenever $\rho(c) = c = \sigma(c)$. 
Also  the interfaces I and II  explained in introduction  can be studied 
as special cases of the results presented in this work.
\end{remark}

\subsection*{Acknowledgments}
The authors dedicate this work to the Founder chancellor of Sri Sathya Sai
Institute of Higher Learning, Bhagwan Sri Sathya Sai Baba.


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\end{document}