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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 68, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/68\hfil Existence of positive solutions]
{Existence of positive solutions for  boundary-value problems
for singular higher-order functional differential equations}
\author[C. Bai,  Q. Yang, J. Ge\hfil EJDE-2006/68\hfilneg]
{Chuanzhi Bai,  Qing Yang, Jing Ge}

\address{Chuanzhi Bai \newline
Department of  Mathematics \\
 Huaiyin Teachers College\\
  Huaian, Jiangsi 223001,  China; \newline
  and  Department of Mathematics\\
  Nanjing University \\
  Nanjing 210093,  China}
\email{czbai8@sohu.com}

\address{Qing Yang  \hfill\break
Department of  Mathematics \\
 Huaiyin Teachers College\\
  Huaian, Jiangsi 223001,  China}
  \email{yangqing3511115@163.com}

\address{Jing Ge  \hfill\break
Department of  Mathematics \\
 Huaiyin Teachers College\\
  Huaian, Jiangsi 223001,  China}
  \email{gejing0512@163.com}

\date{}
\thanks{Submitted April 21, 2006. Published July 6, 2006.}
\thanks{Supported by the Natural Science Foundation of
 Jiangsu Education Office  and by \hfill\break\indent
  Jiangsu Planned Projects  for Postdoctoral Research Funds}
\subjclass[2000]{34K10, 34B16}
\keywords{Boundary value problem;  higher-order;
positive solution; \hfill\break\indent
functional differential equation; fixed point}

\begin{abstract}
 We study  the existence of positive  solutions  for the
 boundary-value problem of the singular higher-order functional
 differential equation
 \begin{gather*}
 (L y^{(n-2)})(t) +  h(t) f(t, y_t) = 0, \quad  \text{for }
  t \in [0, 1], \\
 y^{(i)}(0) = 0, \quad  0 \leq i \leq n - 3, \\
 \alpha y^{(n-2)}(t) - \beta  y^{(n-1)} (t) = \eta (t),
 \quad  \text{for } t \in [- \tau, 0],\\
 \gamma y^{(n-2)}(t) + \delta  y^{(n-1)}(t) = \xi (t),
 \quad \text{for } t \in [1, 1 + a],
 \end{gather*}
 where $ Ly := -(p y')' + q y$, $p \in C([0, 1],(0, + \infty))$,
 and  $q \in C([0, 1], [0, + \infty))$. Our main tool is the fixed
 point theorem on a cone.
\end{abstract}

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

\section{Introduction}

As pointed out in \cite{e1}, boundary-value problems associated with
functional differential equations  arise from problems in physics,
from variational problems in control theory, and from applied
mathematics; see for example \cite{g1,g3}. Many authors have
investigated the existence of solutions for boundary-value problems
of functional differential equations; see \cite{d1, h1, n1, t2}.
Recently an increasing interest in studying the existence of
positive solutions for such problems has been observed. Among others
publication, we refer to \cite{b1, b2, h2, h3, k1, w1}.


In this paper, we investigate the existence of positive solutions
for singular  boundary-value problems (BVP) of an $n$-th order
($n \geq 3$) functional differential equation (FDE) of the form
\begin{gather}
 (L y^{(n-2)})(t) +  h(t) f(t, y_t) = 0, \quad  \text{for }
 t \in [0, 1],   \label{e1.1}\\
 y^{(i)}(0) = 0, \quad  0 \leq i \leq n - 3, \label{e1.2} \\
 \alpha y^{(n-2)}(t) - \beta  y^{(n-1)} (t) = \eta (t),
\quad  \text{for } t \in [- \tau, 0], \label{e1.3}\\
 \gamma y^{(n-2)}(t) + \delta  y^{(n-1)}(t) = \xi (t),
\quad \text{for } t \in [1, 1 + a],  \label{e1.4}
\end{gather}
 where $ Ly := -(p y')' + q y$, $p \in C([0, 1],(0, + \infty))$,
 and  $q \in C([0, 1], [0, + \infty))$;
 $\alpha,\beta, \gamma, \delta \geq 0$, and
$\alpha \delta + \alpha \gamma + \beta \gamma > 0$;
$ \eta \in C([- \tau, 0], \mathbb{R})$,
$\xi \in C([1, b], \mathbb{R})$ ($b = 1 + a$), and $\eta (0) = \xi (1) = 0$;
$h \in C((0, 1), \mathbb{R})$ ($h(t)$ is allowed to have singularity
at $t = 0$ or $1$); $f \in C([0, 1] \times D,  \mathbb{R})$,
$D = C([- \tau, a], \mathbb{R})$, for every $t \in [0, 1]$, $y_t \in D$  is
defined by $y_t(\theta) = y(t + \theta), \theta \in [- \tau, a]$.


 The study of higher-order functional differential
equation has received also some  attention; see for
example \cite{d1, h2, t1}.  Recently, Hong et al. \cite{h4}
imposed conditions on $f(t, y^t)$ to yield at least one
positive solution to  \eqref{e1.1}-\eqref{e1.4} for the special
case $ h(t) \equiv 1$, $p(t) \equiv 1$, and $ q(t) \equiv 0$.
They applied the Krasnosel'skii fixed-point theorem.

The purpose of this paper is to establish the existence  of positive
solutions of the singular higher-order functional differential
equation \eqref{e1.1} with boundary conditions
\eqref{e1.2}-\eqref{e1.4} under suitable conditions on $f$.

\section{Preliminaries}

To abbreviate our discussion, we assume the following hypotheses:
\begin{itemize}
\item[{\rm (H1)}] $G(t, s)$ is the Green's function of the
differential equation
$$ (Ly^{(n - 2)})(t) = 0, \quad  0 < t < 1
$$
subject to the boundary condition \eqref{e1.2}-\eqref{e1.4} with
$\tau = a = 0$.
\item[{\rm (H2)}] $g(t, s)$ is the Green's function of the
differential equation
 $$
 Ly(t) = 0 \quad  t \in (0, 1)
 $$
 subject to the boundary conditions
 \[
 \alpha y(0) - \beta y' (0) = 0,\quad
 \gamma y(1) + \delta  y'(1) = 0,
\]
where $\alpha, \beta, \gamma$ and $ \delta$ are as in \eqref{e1.3} and
\eqref{e1.4}.
\item[{\rm (H3)}] $h \in C((0, 1), [0, + \infty))$ and
satisfies
 $$ 0 < \int_0^1 g(s, s)
h(s) ds < + \infty.$$
\item[{\rm (H4)}] $f \in C([0, 1] \times D^+, \, [0,
\infty))$,
  where   $D^+ = C([- \tau, a], [0, + \infty))$.
\item[{\rm (H5)}] $ \eta \in C([- \tau, 0], [0, +\infty))$,
$ \xi \in C([1, 1 + a], [0, + \infty))$, and
$\eta (0) = \xi (1) =0$.
\end{itemize}

It is easy to see that
$$
\frac{\partial^{n-2}}{\partial
 t^{n-2}}G(t, s)  = g(t, s), \quad  t, s \in [0, 1].
 $$
It is also well known that the Green's function $g(t, s)$ is
\[
g(t, s) = \frac{1}{c}
\begin{cases}
\phi (s) \psi(t), &  \text{if} \  0 \leq s \leq t \leq 1, \\
\phi (t) \psi(s), &  \text{if}  \ 0 \leq t \leq s \leq 1,
\end{cases}
\]
 where $\phi$ and $\psi$ are solutions, respectively, of
\begin{gather}
 L \phi = 0, \quad  \phi(0) = \beta, \quad
 \phi' (0) = \alpha, \label{e2.1} \\
 L \psi = 0, \quad  \psi(1) = \delta,
\quad  \psi' (1) = - \gamma. \label{e2.2}
\end{gather}
 One can show that $c = - p(t) (\phi (t) \psi' (t)
 - \phi' (t) \psi (t)) > 0$ and $\phi' (t) > 0$ on
$(0, 1]$ and $\psi' (t) < 0$ on $[0, 1)$.
 Clearly
\begin{equation}
  g(t, s) \leq g(s, s), \quad
0 \leq t, s \leq 1.  \label{e2.3}
\end{equation}
 By {\rm (H3)}, there exists $t_0 \in (0, 1)$ such that
  $h(t_0) > 0$.   We may choose $\varepsilon \in (0, 1/2)$ such that
   $t_0 \in (\varepsilon, 1 - \varepsilon)$.
Then for $\varepsilon \leq t \leq 1 - \varepsilon$ we have $\phi
(\varepsilon) \leq \phi (t) \leq \phi (1 - \varepsilon)$
 and $\psi (1 - \varepsilon) \leq \psi (t) \leq \psi(\varepsilon)$.
 Also for $(t, s) \in [\varepsilon, 1 - \varepsilon] \times (0, 1)$
\begin{equation}
 \frac{g(t, s)}{g(s, s)} \geq \min \big\{ \frac{\psi (1 -
\varepsilon)}{ \psi (s)},  \frac{\phi (\varepsilon)}{\phi (s)}
\big\} \geq \min \big\{ \frac{\psi (1 - \varepsilon)}{ \psi
(0)},   \frac{\phi (\varepsilon)}{\phi (1)} \big\} := \sigma.
\label{e2.4}
\end{equation}

Let $E = C^{(n-2)}([- \tau, b]; \mathbb{R})$ with a norm
$\|u\|_{[-\tau, b]} = \sup_{-\tau \leq t \leq b}|u^{(n-2)}(t)|$ for
$u \in E$. Obviously, $E$ is a Banach space. And let $C =
C^{(n-2)}([- \tau, a], \mathbb{R})$ be a space with norm $\|\psi \|_{[-
\tau, a]} = \sup_{- \tau \leq t \leq a} |\psi^{(n-2)}(x)|$ for
$\psi \in C$. Let
$$
C^+ = \{\psi \in C: \psi(x) \geq 0, x \in [- \tau, a]\}.
$$
 It is easy to see that $C^+$  is a subspace of $C$.

 Define a cone $K \subset E$ as follows:
\begin{equation}
 K = \{ y \in E : y(t) \geq 0, \; \min _{t \in [\varepsilon,
1 - \varepsilon]}
 y^{(n-2)}(t) \geq \overline{\sigma}  \|y\|_{[- \tau, b]} \},\label{e2.5}
\end{equation}
 where $\overline{\sigma} = \frac{1}{b} \min \{\varepsilon, \sigma\}$,
 $\sigma$ is as in \eqref{e2.4}.

For each $\rho > 0$, we define
$K_{\rho} = \{y \in K : \|y\|_{[-\tau, b]} < \rho \}$.
 Furthermore, we define  a set $\Omega_{\rho} $ as follows:
$$
\Omega_{\rho} = \big\{y \in K : \min_{\varepsilon \leq t \leq 1
- \varepsilon} y^{(n - 2)}(t) < \overline{\sigma} \rho \big\}.
$$

Similar to the \cite[Lemma 2.5]{l1}, we have

\begin{lemma}\label{lem2.1}   $\Omega_{\rho}$ defined above has the
following properties:

\begin{itemize}
\item[{\rm (a)}]  $\Omega_{\rho}$ is open relative to $K$.


\item[{\rm (b)}]  $K_{\overline{\sigma} \rho } \subset \Omega_{\rho} \subset
K_{\rho}$.

\item[{\rm (c)}] $y \in \partial \Omega_{\rho}$ if and only if
$\min _{\varepsilon \leq t \leq 1 - \varepsilon} y^{(n-2)}(t)
= \overline{\sigma} \rho. $

\item[{\rm (d)}] If $y \in \partial \Omega_{\rho}$, then
$\overline{\sigma} \rho \leq y^{(n-2)}(t) \leq \rho$ for $t \in
[\varepsilon, 1 - \varepsilon]$.
\end{itemize}
\end{lemma}

To obtain the positive solutions of \eqref{e1.1}-\eqref{e1.4}, the
following fixed point theorem in cones will be fundamental.

\begin{lemma}\label{lem2.2}
 Let $K$ be a cone in a Banach space $E$. Let $D$ be an open
 bounded subset of $E$ with $D_K = D \cap K \neq  \emptyset$
 and $\overline{D}_K \neq  K$.
Assume that $A : \overline{D}_K \to K$ is a compact map such that $x
\neq  Ax$ for $x \in \partial D_K$. Then the following results hold.

\begin{itemize}
\item[{\rm (1)}]  $ \ \|Ax\| \leq \|x\|, \quad  x \in
\partial D_K,$ then $i_K (A, D_K) = 1$.

\item[{\rm (2)}]  If there exists $ e \in K \backslash \{0\}$ such
that $ x \neq  Ax + \lambda e$ for all $ x \in \partial D_K$ and all
$ \lambda > 0$, then $i_K (A, D_K) = 0$.

\item[{\rm (3)}]  Let $U$ be an open set in $E$ such that $\overline{U}
\subset D_K$.
 If $i_K(A, D_K) = 1$  and  $i_K(A, U_K) = 0$, then $A$ has a fixed point
  in $D_K \backslash \overline{U}_K$.
   The same results holds if $i_K(A, D_K) = 0$
   and $i_K(A, U_K) = 1$.
\end{itemize}
\end{lemma}


Suppose that $y(t)$ is a solution of \eqref{e1.1}-\eqref{e1.4}, then
it can be written as
\[
 y(t) = \begin{cases}
         y(- \tau; t), &   - \tau \leq t \leq 0,\\
         \int_0^1 G(t, s) h(s) f(s, y_s)ds, &  0 \leq t \leq 1,\\
         y(b; t), &  1 \leq t \leq b,
        \end{cases}
\]
where $y(- \tau; t)$ and $y(b; t)$ satisfy
\[
 y^{(n-2)}(- \tau; t) =  \begin{cases}
          e^{\frac{\alpha}{\beta}t} \left(\frac{1}{\beta}
          \int_t^0 e^{- \frac{\alpha}{\beta}s}\eta(s)ds + y^{(n-2)}(0)\right), &
       t \in [- \tau, 0], \; \beta \neq  0, \\
          \frac{1}{\alpha}\eta(t), & t \in [- \tau, 0], \;\beta = 0,
\end{cases}
\]
and
\[
 y^{(n-2)}(b; t) = \begin{cases}
          e^{- \frac{\gamma}{\delta}t} \left(\frac{1}{\delta}
          \int_1^t e^{\frac{\gamma}{\delta}s}\xi(s)ds + e^{\frac{\gamma}{\delta}}y^{(n-2)}(1)\right),
           &  t \in [1, b], \; \delta \neq  0, \\
          \frac{1}{\gamma}\xi(t), & t \in [1, b], \;\delta = 0.
\end{cases}
\]
  Throughout this paper, we assume that  $u_0(t)$ is the solution of
   \eqref{e1.1}-\eqref{e1.4}    with $f \equiv 0$, and
 $\|u_0\|_{[- \tau, b]} =: M_0$.
 Clearly, $u_0^{(n-2)}(t)$ can be expressed as follows:
\[
u_0^{(n-2)}(t) = \begin{cases}
  u_0^{(n-2)}(- \tau; t), &  - \tau \leq t \leq 0, \\
 0, &   0 \leq t \leq 1, \\
 u_0^{(n-2)}(b; t), &  1 \leq t \leq b.
\end{cases}
\]
 where
\[
u_0^{(n-2)}(- \tau; t) = \begin{cases}
  \frac{1}{\beta} e^{\frac{\alpha}{\beta}t}
  \int_t^0 e^{- \frac{\alpha}{\beta}s}\eta(s)ds, & t \in [- \tau, 0],
  \; \beta \neq  0, \\
 \frac{1}{\alpha} \eta(t), & t \in [- \tau, 0], \; \beta = 0,
 \end{cases}
\]
 and
\[
u_0^{(n-2)}(b; t) = \begin{cases}
 \frac{1}{\delta} e^{- \frac{\gamma}{\delta}t} \int_1^t
e^{\frac{\gamma}{\delta}s} \xi(s)ds, & t \in [1, b], \;
 \delta \neq  0,\\
\frac{1}{\gamma} \xi(t), & t \in [1, b], \; \delta = 0.
\end{cases}
\]

Let $y(t)$ be a solution of BVP \eqref{e1.1}-\eqref{e1.4} and $u(t)
= y(t) - u_0(t)$. Noting that $u(t) \equiv y(t)$ for $0 \leq t \leq
1$, we have
\[
u^{(n-2)}(t) = \begin{cases}
u^{(n-2)}(- \tau; t),&  - \tau \leq t \leq 0,\\
   \int_0^1 g(t, s) h(s) f(s, u_s + (u_0)_s)ds, & 0 \leq t \leq 1,\\
u^{(n-2)}(b; t),& 1 \leq t \leq b,
 \end{cases}
\]
where
\[
u^{(n-2)}(- \tau; t) = \begin{cases}
e^{\frac{\alpha}{\beta}t} \int_0^1 g(0, s) h(s) f(s, u_s +
(u_0)_s)ds,& t \in [- \tau, 0],\; \beta \neq  0,\\
0, & t \in [- \tau, 0], \; \beta = 0,
\end{cases}
\]
 and
\[
u^{(n-2)}(b; t) = \begin{cases}
e^{- \frac{\gamma}{\delta}(t - 1)} \int_0^1 g(1, s) h(s)
    f(s, u_s + (u_0)_s)ds,& t \in [1, b],\;
  \delta \neq  0,\\
0, & t \in [1, b], \; \delta = 0.
\end{cases}
\]
It is easy to see that $y(t)$ is a solution of BVP
\eqref{e1.1}-\eqref{e1.4}
 if and only if $u(t) = y(t) - u_0(t)$ is a solution of the
 operator equation
\begin{equation}
 u(t) = Au(t) \quad   \text{for }  t \in [- \tau, b]. \label{e2.6}
\end{equation}
Here, operator $A : E \to E$ is  defined by
\[
 Au(t) :=    \begin{cases}
   B_1 u(t), &  - \tau \leq t \leq 0, \\
   \int_0^1 G(t, s) h(s) f(s, u_s + (u_0)_s)ds, &  0 \leq t \leq 1,\\
 B_2 u(t), & 1 \leq t \leq b,
     \end{cases}
\]
where
\[
 B_1 u(t) := \begin{cases}
   \big(\frac{\beta}{\alpha}\big)^{n-2} e^{\frac{\alpha}{\beta}t}
   \int_0^1 g(0, s) h(s) f(s, u_s + (u_0)_s)ds,
     &  \beta \neq  0,\; \alpha \neq  0,\\
  \frac{t^{n-2}}{(n-2)!} \int_0^1 g(0, s) h(s) f(s, u_s + (u_0)_s)ds,
   &  \beta \neq  0,\; \alpha = 0,\\
 0,  & \beta = 0
 \end{cases}
\]
for each $t \in [- \tau, 0]$,  and
\[
 B_2 u(t) :=    \begin{cases}
\big(-\frac{\delta}{\gamma}\big)^{n-2}e^{-\frac{\gamma}{\delta}(t - 1)}
\int_0^1 g(1, s) h(s)f(s, u_s + (u_0)_s)ds,
      & \delta \neq  0,\; \gamma \neq  0, \\
 \frac{t^{n-2}}{(n-2)!} \int_0^1 g(1, s) h(s)f(s, u_s + (u_0)_s)ds,
   & \delta \neq  0,\; \gamma = 0, \\
 0,  & \delta = 0
 \end{cases}
\]
 for any $t \in [1, b]$.  Obviously,
\[
(Au)^{(n-2)}(t) := \begin{cases}
   (B_1 u)^{(n-2)}(t), & - \tau \leq t \leq 0,\\
   \int_0^1 g(t, s) h(s) f(s, u_s + (u_0)_s)ds, &  0 \leq t \leq 1,\\
    (B_2 u)^{(n-2)}(t), & 1 \leq t \leq b,
     \end{cases}
\]
 where
\[
(B_1 u)^{(n-2)}(t) :=  \begin{cases}
 e^{\frac{\alpha}{\beta}t}
   \int_0^1 g(0, s) h(s) f(s, u_s + (u_0)_s)ds,
    &  t \in [- \tau, 0], \  \beta \neq  0,\\
 0, & t \in [- \tau, 0],\; \beta = 0,
\end{cases}
\]
 and
\[
(B_2 u)^{(n-2)}(t) :=  \begin{cases}
  e^{-\frac{\gamma}{\delta}(t - 1)}
    \int_0^1 g(1, s) h(s)f(s, u_s + (u_0)_s)ds,
     &  t \in [1, b], \;  \delta \neq  0, \\
     0, & t \in [1, b], \; \delta = 0.
     \end{cases}
\]

\begin{lemma}\label{lem2.3}  With the above notation, $A(K) \subset K$.
\end{lemma}

\begin{proof}  By the assumptions of {\rm (H1)-(H5)}, it
is easy to know that $Au \in E$ and $Au \geq 0$ for any $u \in K$.
Moreover, it follows from
\begin{gather*}
0 \leq (Au)^{(n-2)}(t) \leq (Au)^{(n-2)}(0)
\quad  \text{for } - \tau \leq t \leq 0
\\
0 \leq (Au)^{(n-2)}(t) \leq (Au)^{(n-2)}(1) \quad  \text{for }
1 \leq t \leq b
\end{gather*}
 that $\|Au\|_{[- \tau, b]} = \|Au\|_{[0, 1]}$. By
\eqref{e2.3} we have, for any $u \in K$ and $t \in [0, 1]$ that
\begin{equation}
 \|Au\|_{[- \tau, b]} = \|Au\|_{[0,
1]} \leq \int_0^1 g(s, s) h(s)f(s, u_s + (u_0)_s) ds. \label{e2.7}
\end{equation}
 From \eqref{e2.4}, we get
\begin{equation}
\begin{aligned}
\min _{\varepsilon \leq t \leq 1 - \varepsilon}
(Au)^{(n-2)}(t) &= \min _{\varepsilon \leq t \leq 1 -
\varepsilon }
 \int_0^1 g(t, s) h(s) f(s, u_s + (u_0)_s) ds\\
 &\geq \sigma \int_0^1 g(s,
s) h(s) f(s, u_s + (u_0)_s) ds\\
  &\geq \overline{\sigma} \int_0^1 g(s,
s) h(s) f(s, u_s + (u_0)_s) ds.
\end{aligned} \label{e2.8}
\end{equation}
In view of \eqref{e2.7} and \eqref{e2.8}, we obtain
$$
\min _{\varepsilon \leq t \leq 1 - \varepsilon}
(Au)^{(n-2)}(t) \geq  \overline{\sigma} \|Au\|_{[- \tau, b]},
\quad  u \in K,
$$
 which implies  $A(K) \subset K$.
\end{proof}

Let
\begin{gather*}
 C_{[k, r]}^+ = \{\varphi \in C^+ :  \  k \leq \|\varphi\|_{ [-
\tau, a]} \leq r \}, \\
 C_{[k, \infty)}^+ = \{\varphi \in C^+ :  \  k \leq \|\varphi\|_{[-
\tau, a]} < \infty \},
\end{gather*}
  where $ 0 \leq  k < r$.

\begin{lemma}\label{lem2.4}
  $A : K \to K$ is completely continuous.
\end{lemma}

\begin{proof}    We apply a truncation technique
(cf. \cite{s1}). We define the function $h_m$ for $m \geq 2$, by
\[
h_m(t) = \begin{cases}
\min \big\{h(t), h(\frac{1}{m})\big\}, &   0 < t \leq \frac{1}{m},\\
h(t), & \frac{1}{m} < t < 1 - \frac{1}{m}, \\
\min \big\{h(t), h(\frac{m - 1}{m})\big\}, &  \frac{m - 1}{m}
\leq t < 1.
\end{cases}
\]
  It is clear that $h_m(t)$ is nonnegative and
continuous on $[0, 1]$. We define the operator $A_m$ by
\[
 A_m u(t) := \begin{cases}
   B_{1 m} u(t), &  - \tau \leq t \leq 0, \\
   \int_0^1 G(t, s) h_m(s) f(s, u_s + (u_0)_s)ds, &  0 \leq t \leq 1,\\
 B_{2 m} u(t), & 1 \leq t \leq b,
     \end{cases}
\]
where
\[
 B_{1 m} u(t) :=   \begin{cases}
   \left(\frac{\beta}{\alpha}\right)^{n-2} e^{\frac{\alpha}{\beta}t}
   \int_0^1 g(0, s) h_m (s) f(s, u_s + (u_0)_s)ds,
     &  \beta \neq  0,\; \alpha \neq  0,\\
  \frac{t^{n-2}}{(n-2)!} \int_0^1 g(0, s) h_m(s) f(s, u_s + (u_0)_s)ds,
   &  \beta \neq  0,\; \alpha = 0,\\
 0,  & \beta = 0
 \end{cases}
\]
for each $t \in [- \tau, 0]$,  and
\[
 B_{2 m} u(t) :=  \begin{cases}
\big(-\frac{\delta}{\gamma}\big)^{n-2}e^{-\frac{\gamma}{\delta}(t - 1)}
\int_0^1 g(1, s) h_m(s)f(s, u_s + (u_0)_s)ds,
      & \delta \neq  0,\; \gamma \neq  0, \\
 \frac{t^{n-2}}{(n-2)!} \int_0^1 g(1, s) h_m(s)f(s, u_s + (u_0)_s)ds,
   & \delta \neq  0,\; \gamma = 0, \\
 0,  & \delta = 0
 \end{cases}
 \]
 for any $t \in [1, b]$.  By Lemma \ref{lem2.3},  it is easy to check
 that $A_m : K \to K$. And, $A_m$ is continuous, the proof is similar
 to that of \cite[Theorem 2.1]{h3}.

 Next let $B \subset K$ be a bounded subset of $K$, and $M_1 > 0$ be a
constant such that $\|u\|_{[- \tau, b]} \leq M_1$ for $u \in B$.
Noting that if $x_t \in C = C^{n-2}([- \tau, a], \mathbb{R})$, then
$x_t^{(n-2)} \in C([- \tau, a], \mathbb{R})$, and $x_t^{(n-2)} (\theta)
= x^{(n-2)}(t + \theta)$, $\theta \in [- \theta, a]$. Thus
\begin{equation}
\begin{aligned}
\|u_s + (u_0)_s\|_{[- \tau, a]}
&= \sup_{- \tau \leq \theta \leq a}
|(u^s + u_0^s)^{(n-2)}(\theta)|\\
&\leq \sup_{- \tau \leq \theta \leq a} |(u^{(n-2)}(s + \theta)|  +
\sup_{- \tau \leq \theta \leq a}|u_0^{(n-2)}(s + \theta)| \\
&\leq \sup _{- \tau \leq t \leq b} |u^{(n-2)}(t)| + \sup
_{- \tau \leq t \leq b} |u_0^{(n-2)}(t)| = \|u\|_{[- \tau,
b]} + \|u_0\|_{[- \tau, b]}\\
&\leq M_1 + M_0 := M_2
\end{aligned} \label{e2.9}
\end{equation}
 for $ u \in B$ and $s \in [0, 1]$.  Hence,  there exists a
constant $M_3 > 0$ such that
 \begin{equation}
 |f(s, u_s + (u_0)_s)| \leq M_3, \quad  {\rm on} \ [0, 1] \times
C_{[0, M_3]}^+, \label{e2.10}
\end{equation}
 since $f$ is continuous on $[0, 1] \times C^+$. For
 $u \in B$ we have
\[
(A_m u)^{(n-2)}(t) := \begin{cases}
   (B_{1 m} u)^{(n-2)}(t), & - \tau \leq t \leq 0,\\
   \int_0^1 g(t, s) h_m(s) f(s, u_s + (u_0)_s)ds, &  0 \leq t \leq 1,\\
    (B_{2 m} u)^{(n-2)}(t), & 1 \leq t \leq b,
     \end{cases}
\]
where
\[
(B_{1 m} u)^{(n-2)}(t) := \begin{cases}
 e^{\frac{\alpha}{\beta}t}
   \int_0^1 g(0, s) h_m(s) f(s, u_s + (u_0)_s)ds,
    &  t \in [- \tau, 0], \;  \beta \neq  0,\\
 0, & t \in [- \tau, 0],\; \beta = 0,
\end{cases}
\]
 and
\[
(B_{2 m} u)^{(n-2)}(t) := \begin{cases}
  e^{-\frac{\gamma}{\delta}(t - 1)}
    \int_0^1 g(1, s) h_m(s)f(s, u_s + (u_0)_s)ds,
     &  t \in [1, b], \;  \delta \neq  0, \\
     0, & t \in [1, b], \; \delta = 0.
     \end{cases}
\]
 These and \eqref{e2.10} imply $(A_m u)^{(n-2)} (t)$ is
continuous and uniformly bounded for $u \in B$. So $(A_m u)'
(t)$ is continuous and uniformly bounded for $u \in B$ also. The
Ascoli-Arzela Theorem implies that $A_m$ is a completely continuous
operator on $K$ for any $m \geq 2$.

Moreover, $A_m$ converges uniformly to $A$ as $m \to \infty$ on any
bounded subset of $K$. To see this, note that if $u \in K$ with
$\|u\|_{[- \tau, b]} \leq M$, then from {\rm (H3)} and $0 \leq
h_n(s) \leq h(s)$,
\begin{align*}
 |(A_m u)^{(n-2)}(t) - (A u)^{(n-2)}(t)|
& = \Big|\int_0^{\frac{1}{m}} g(t, s) [h(s) - h_m(s)] f(s,
u_s + (u_0)_s)ds  \ \\
 & \quad  +    \int_{\frac{m - 1}{m}}^1 g(t, s)
    [h(s) - h_m(s)] f(s, u_s + (u_0)_s)ds \Big|\\
  & \leq  \int_0^{\frac{1}{m}}
      g(s, s)|h(s) - h_m(s)| f(s, u_s + (u_0)_s)ds \\
& \quad + \int_{\frac{m - 1}{m}}^1
 g(s, s) |h(s) - h_m(s)| f(s, u_s + (u_0)_s)ds \\
  & \leq 2 \overline{M}
 \Big[\int_0^{\frac{1}{m}} g(s, s) h(s) ds
 + \int_{\frac{m - 1}{m}}^1 g(s, s) h(s) ds\Big]\\
 & \to  0 \quad \text{as }  m \to \infty,
\end{align*}
 where   $\overline{M} :=  \max_{t \in [0, 1], \varphi \in C_{[0, M + M_0]}^+} f(t,
\varphi)$.  Thus, we have
$$
\|A_m u - A u\|_{[- \tau, b]} = \|A_m u - A
u\|_{[0, 1]} \to 0, \quad  n \to \infty,
$$
 for each $u \in K$ with $\|u\|_{[- \tau, b]} \leq M$. Hence, $A_m$ converges
uniformly to $A$ as $m \to \infty$ and therefore $A$ is completely
continuous also.  This completes the proof of Lemma \ref{lem2.4}.
\end{proof}

\section{Main results}

 For convenience, we introduce the following notation.
Let
\begin{gather*}
\omega = \Big(\int_0^1 g(s, s) h(s) ds\Big)^{-1}; \quad
N = \Big(\min _{\varepsilon \leq t \leq 1 - \varepsilon}
 \int_{\varepsilon}^{1 - \varepsilon} g(t, s) h(s) ds \Big)^{- 1};
\\
f_{\overline{\sigma} \rho }^{\rho} =
\inf \big\{ \min _{t \in [\varepsilon,  1 - \varepsilon]}
 \frac{f(t, \varphi)}{\rho} :  \varphi  \in
C_{[\overline{\sigma} \rho, \rho + M_0]}^{+} \big\}; \\
f_0^{\rho} = \sup \big\{ \max _{t \in [0, 1]} \frac{f(t,
\varphi)}{\rho} :  \varphi  \in C_{[0,  \rho + M_0]}^{+} \big\};
\\
f^{\mu} = \lim _{\| \varphi \|_{[- \tau, a]} \to \mu} \sup
\max _{t \in [0, 1]} \frac{f(t, \varphi)}{\| \varphi \|_{[-
\tau, a]}}; \\
f_{\mu} = \lim _{\| \varphi \|_{[- \tau, a]} \to \mu}
 \inf \min _{t \in [\varepsilon, 1 - \varepsilon]}
  \frac{f(t, \varphi)}{\| \varphi \|_{[- \tau, a]}}, \quad
 (\mu := \infty \ {\rm or}  \ 0^+).
\end{gather*}
Now, we impose conditions on $f$ which we assure that
$i_K(A,K_{\rho}) = 1$.

\begin{lemma}\label{lem3.1}
Assume that
\begin{equation}
f_0^{\rho} \leq  \omega \quad \text{and} \quad
  u \neq  Au \quad  \text{for } u \in \partial K_{\rho}. \label{e3.1}
\end{equation}
Then $i_K(A, K_{\rho}) = 1$.
\end{lemma}

\begin{proof}  For $u \in \partial K_{\rho}$, we have
$\|u_s + (u_0)_s \|_{[- \tau, a]} \leq \rho + M_0,$ for all $s \in
[0, 1]$,  i.e., $u_s + (u_0)_s \in C_{[0, \rho + M_0]}$ for any $s
\in [0, 1]$. It follows from \eqref{e3.1} that for $t \in [0, 1]$,
\begin{align*}
(Au)^{(n-2)}(t) & =  \int_0^1 g(t, s) h(s) f(s, u_s +
(u_0)_s)ds\\
& \leq \int_0^1 g(s, s)h(s) f(s, u_s + (u_0)_s)ds\\
& < \rho \omega \int_0^1 g(s, s) h(s)ds \\
&= \rho = \|u\|_{[- \tau,b]}.
\end{align*}
 This implies that $\|Au\|_{[- \tau, b]} < \|u\|_{[-
\tau, b]}$ for $u \in \partial K_{\rho}$.  By Lemma \ref{e2.2} (1),
we have $i_K(A, K_{\rho}) = 1$.
\end{proof}

Let $u \in \partial \Omega_{\rho}$, then for any $s \in
[\varepsilon, 1 - \varepsilon]$, we have by  Lemma \ref{lem2.1}(c)
that
\begin{align*}
\|u_s + (u_0)_s\|_{[- \tau, a]}& = \sup _{\theta \in [- \tau,
a]} (u^{(n-2)}(s + \theta) + (u_0)^{(n-2)}(s + \theta))\\
& \geq \sup _{\theta \in [- \tau, a]} u^{(n-2)}(s + \theta) \
\text{(since $(u_0)^{(n-2)}(t) \geq 0$ for $t\in [- \tau,
b]$)}\\
& \geq u^{(n-2)}(s) \\
&\geq \min _{t \in [\varepsilon, 1 -
\varepsilon]} u^{(n-2)}(t)
  =  \overline{\sigma} \rho.
\end{align*}
 By Lemma \ref{lem2.1}(b), we have $\overline{\Omega}_{\rho}
\subset \overline{K}_{\rho}$, that is
 $ \|u\|_{[- \tau, b]} \leq \rho.$  Thus, from \eqref{e2.9} we get
$$
\|u_s + (u_0)_s\|_{[- \tau, a]} \leq
\|u\|_{[- \tau, b]} + \|u_0\|_{[- \tau, b]}
 \leq \rho + M_0,
\quad  \forall s \in [0, 1].
$$
  Hence
\begin{equation}
 u_s + (u_0)_s  \in C_{[\overline{\sigma} \rho, \
\rho + M_0]}^+, \quad  \text{for } u  \in  \partial
\Omega_{\rho} ,\;  s \in [\varepsilon, 1 - \varepsilon].  \label{e3.2}
\end{equation}


 Next, we impose conditions on $f$ which assure that
$i_K(A,\Omega_{\rho}) = 0$.

\begin{lemma}\label{lem3.2}
 If $f$ satisfies  the condition
\begin{equation}
f_{\overline{\sigma} \rho}^{\rho} \geq N \overline{\sigma}
\quad  \text{ and} \quad  u \neq  Au \quad
 \text{for } u \in \partial \Omega_{\rho}. \label{e3.3}
\end{equation}
Then $i_K(A, \Omega_{\rho}) = 0$.
\end{lemma}

\begin{proof}   Let
\[
e(t) = \begin{cases}
 - \frac{t^{n-1}}{(1 + \tau) (n-1)!},& n \text{ is  odd},\;
 - \tau \leq t \leq 0,\\
 \frac{t^n}{(1 + \tau)^2 n!},& n \text{ is  even },\;
- \tau \leq t \leq 0,\\
 \frac{t^{n-1}}{b (n-1)!}, &  0 \leq t \leq b.
\end{cases}
\]
It is easy to verify that $e \in C^{(n-2)}([- \tau, b], \mathbb{R})$,
$e(t) \geq 0$ for $t \in [- \tau, b]$, and
\[
e^{(n-2)}(t) = \begin{cases}
- \frac{t}{1 + \tau},& n \text{ is  odd},\;   - \tau \leq t \leq 0,\\
 \frac{t^2}{2 (1 + \tau)^2},& n \text{ is  even}, \;  - \tau \leq
t \leq 0,\\
 \frac{t}{b},    & 0 \leq t \leq b,
\end{cases}
\]
 which implies that $e \in K$ and $\|e\|_{[- \tau, b]} =1$,
that is   $e \in \partial K_1$. We claim that
$$
u \neq  Au + \lambda e, \quad  u \in
\partial \Omega_{\rho}, \quad  \lambda > 0.
$$
 In fact, if not, there exist
$u \in \partial \Omega_{\rho}$ and $\lambda_0 > 0$ such that
$ u = Au + \lambda_0 e$. Then by \eqref{e3.2} for
$t \in [\varepsilon, 1 - \varepsilon]$,
we get
\begin{align*}
u^{(n-2)} (t) & = (Au)^{(n-2)} (t) + \lambda_0 e^{(n-2)}(t) \\
& = \int_0^1 g(t, s) h(s) f(s, u_s + (u_0)_s ) ds + \lambda_0
e^{(n-2)}(t) \\
 & \geq \int_{\varepsilon}^{1 - \varepsilon} g(t, s) h(s) f(s, u_s
+ (u_0)_s ) ds + \lambda_0 \frac{\varepsilon}{b} \\
 & \geq \min _{\varepsilon \leq t \leq 1 - \varepsilon}
\int_{\varepsilon}^{1 - \varepsilon} g(t, s) h(s) f(s, u_s +
(u_0)^s ) ds + \lambda_0 \overline{\sigma}\\
& \geq \rho N \overline{\sigma} \min _{\varepsilon \leq t
\leq 1 - \varepsilon}
 \int_{\varepsilon}^{1 - \varepsilon} g(t, s) h(s) ds
 + \lambda_0 \overline{\sigma}\\
& \geq \overline{\sigma} \rho + \lambda_0 \overline{\sigma}.
\end{align*}
 This implies that  $\overline{\sigma} \rho \geq \overline{\sigma}
\rho + \lambda_0 \overline{\sigma}$,  a contradiction. Moreover, it
is easy to check that $u \neq  Au$ for
$u \in \partial \Omega_{\rho}$ from \eqref{e3.3}. Hence, by
Lemma \ref{lem2.2} (2), it follows that
$i_K(A,  \Omega_{\rho}) =0$.
\end{proof}

\begin{theorem}\label{thm3.3}
 If one of the following conditions holds:

\begin{itemize}
\item[{\rm (H7)}]  There exist $\rho_1, \rho_2, \rho_3 \in
(0, \infty)$ with $\rho_1 < \overline{\sigma} \rho_2$ and $\rho_2 <
\rho_3$ such that
$$
f_0^{\rho_1} \leq \omega,  \quad
 f_{\overline{\sigma} \rho_2}^{\rho_2} \geq N \overline{\sigma}, \quad
u \neq  Au \quad \text{for} \ u \in \partial \Omega_{\rho_2} \quad
\text{and} \quad f_0^{\rho_3} \leq  \omega.
$$
\item[{\rm (H8)}] There exist $\rho_1, \ \rho_2, \ \rho_3 \in
(0, \infty)$ with $\rho_1 < \rho_2 < \overline{\sigma} \rho_3$ such
that
$$f_{\overline{\sigma} \rho_1}^{\rho_1} \geq N \overline{\sigma},
\quad
 f_0^{\rho_2} \leq \omega, \quad  u \neq  Au \quad
\text{for } u \in \partial K_{\rho_2} \quad  \text{and} \quad
f_{\overline{\sigma} \rho_3}^{\rho_3} \geq N \overline{\sigma}.
$$
\end{itemize}
 Then BVP \eqref{e1.1}-\eqref{e1.4} has two positive solutions.
 Moreover, if in {\rm (H7)}, $f_0^{\rho_1} \leq  \omega$ is replaced by
   $f_0^{\rho_1} < \omega$, then \eqref{e1.1}-\eqref{e1.4}
 has a third positive
    solution     $u_3 \in K_{\rho_1}$.
\end{theorem}

\begin{proof}  Suppose that {\rm (H7)} holds. We show that either
$A$ has a fixed point $u_1$ in $\partial K_{\rho_1}$ or in
$\Omega_{\rho_2} \setminus \overline{K_{\rho_1}}$. If $u \neq  Au$
for $u \in \partial K_{\rho_1} \cup \partial K_{\rho_3}$, by Lemmas
\ref{lem3.1}-\ref{lem3.2}, we have $i_K(A, K_{\rho_1}) = 1$, $i_K(A,
\Omega_{\rho_2}) = 0$ and $i_K(A, K_{\rho_3}) = 1$. Since $\rho_1 <
 \overline{\sigma} \rho_2$, we have $\overline{K_{\rho_1}} \subset
K_{\overline{\sigma} \rho_2} \subset \Omega_{\rho_2}$ by Lemma
\ref{lem2.1} (b). It follows from  Lemma \ref{lem2.2} that $A$ has a
fixed point $u_1 \in \Omega_{\rho_2} \setminus
\overline{K_{\rho_1}}$. Similarly, $A$ has a fixed point $u_2 \in
K_{\rho_3} \setminus \overline{\Omega_{\rho_2}}$. The proof is
similar when {\rm (H8)} holds.
\end{proof}


As a special case of Theorem \ref{thm3.3} we obtain the following
result.

\begin{corollary} \label{coro3.4}
 Let $\xi(t) \equiv 0$, $\eta(t) \equiv 0$.  If there exists
$\rho > 0 $ such that one of  the following conditions holds:
\begin{itemize}

\item[{\rm (H9)}]
$0 \leq f^0 <  \omega$, $f_{\overline{\sigma}
\rho}^{\rho} \geq N \overline{\sigma}$, $ u \neq  Au$
for $u \in \partial \Omega_{\rho}$ and $0 \leq f^{\infty} <  \omega$.

\item[{\rm (H10)}]
$ N < f_0 \leq \infty$, $f_0^{\rho} \leq  \omega$,
$u \neq  Au$  for $u \in \partial K_{\rho}$
  and  $N < f_{\infty} \leq \infty$.
\end{itemize}
 Then BVP \eqref{e1.1}-\eqref{e1.4} has two positive solutions.
\end{corollary}


\begin{proof}
 From $\xi(t) \equiv 0, \eta(t) \equiv 0$,
it is clear that $u_0(t) \equiv 0$ for $t \in [- \tau, b]$, thus
$M_0 = 0$.  We now show that ${\rm (H_9)}$ implies ${\rm (H_7)}$.
It is easy to verify that $0 \leq f^0 <  \omega$ implies that
there exists $\rho_1 \in (0, \overline{\sigma} \rho)$ such that
$f_0^{\rho_1} < \omega$.
 Let $k \in (f^{\infty}, \omega)$. Then there exists $r > \rho$ such that
  $\max_{t \in [0, 1]} f(t, \varphi) \leq k \|\varphi\|_{[- \tau, a]}$ for
  $\varphi  \in C_{[r, \infty)}^+$
since $0 \leq f^{\infty} <  \omega$. Let
$$
l = \max \{\max_{t \in [0, 1]} f(t, \varphi) : \varphi \in C_{[0,
r]}^+ \}, \quad  \text{and} \quad  \rho_3 >  \max
\big\{\frac{l}{\omega - k}, \ \rho \big\}.
$$
 Then we have
$$
 \max _{t \in [0, 1]} f(t,
\varphi) \leq k \|\varphi\|_{[- \tau, a]}
 + l \leq k \rho_3 + l <  \omega \rho_3 \quad  \text{for }
 \varphi \in  C_{[0, \rho_3]}^+.
$$
This implies that $f_0^{\rho_3} <  \omega$ and {\rm (H7)} holds.
 Similarly, {\rm (H10)} implies {\rm (H8)}.
\end{proof}


By a similar argument to that of Theorem \ref{thm3.3}, we obtain the
following results on existence of at least one positive solution of
\eqref{e1.1}-\eqref{e1.4}.

\begin{theorem}\label{thm3.5}
If one of the following conditions holds:
\begin{itemize}
\item[{\rm (H11)}]   There exist $\rho_1,  \rho_2 \in (0,
\infty)$ with $\rho_1 < \overline{\sigma} \rho_2$ such that
$$
f_0^{\rho_1} \leq  \omega \quad  \text{and}\quad
f_{\overline{\sigma} \rho_2}^{\rho_2} \geq N
\overline{\sigma}.
$$
\item[{\rm (H12)}] There exist $\rho_1, \rho_2 \in (0,\infty)$ with
$\rho_1 < \rho_2$ such that
$$
f_{\overline{\sigma} \rho_1}^{\rho_1} \geq N
\overline{\sigma}  \quad  \text{and} \quad
f_0^{\rho_2} \leq  \omega.
$$
\end{itemize}
 Then BVP \eqref{e1.1}-\eqref{e1.4} has a positive solution.
\end{theorem}

As a special case of Theorem \ref{thm3.5}, we obtain the following
result.

\begin{corollary} \label{coro3.6}
Let $\xi(t) \equiv 0, \eta(t) \equiv 0$. If one of the following
conditions holds:
\begin{itemize}
\item[{\rm (H13)}]
$0 \leq f^0 <  \omega$  and $ N < f_{\infty} \leq \infty$.
\item[{\rm (H14)}]
 $0 \leq f^{\infty} <  \omega$  and  $ N < f_0 \leq \infty$.
\end{itemize}
Then BVP \eqref{e1.1}-\eqref{e1.4} has a positive solution.
\end{corollary}

\begin{thebibliography}{00}

\bibitem{b1}   C. Bai, J. Ma;  \emph{Eigenvalue criteria for existence
of multiple positive solutions to boundary-value problems of
second-order delay differential equations},
 J. Math. Anal. Appl. 301 (2005) 457-476.

\bibitem{b2}  C. Bai, J. Fang; \emph{Existence of
 multiple positive solutions for functional differential
 equations}, Comput. Math. Appl. 45(2003) 1797-1806.

\bibitem{d1}  J.M. Davis, K.R. Prasad, W.K.C. Yin; \emph{Nonlinear
eigenvalue problems involving two classa of functional differential
equations},  Houston J. Math.  26(2000) 597-608.

\bibitem{d2} K. Deimling;  \emph{Nonlinear Functional Analysis},
Springer-Verlag, New York, 1985.

\bibitem{e1}  L.H. Erbe, Q.K. Kong; \emph{Boundary value problems for
singular second order functional differential equations}, J. Comput.
Appl. Math. 53 (1994) 640-648.

\bibitem{g1}  L.J. Grimm, K. Schmitt;  \emph{Boundary value problem
problems for differential equations with deviating arguments},
Aequationes Math. 4 (1970) 176-190.

\bibitem{g2} D. Guo, V. Lakshmikantham;  \emph{Nonlinear Problems in
Abstract Cones}, Academic Press, New York, 1988.

\bibitem{g3} G.B. Gustafson, K. Schmitt; \emph{Nonzero solutions of
boundary-value problems for second order ordinary and
delay-differential equations}, J. Differential Equations 12 (1972)
129-147.

\bibitem{h1}   J. Henderson,  \emph{Boundary Value Problems for
Functional Differential Equations},  World Scientific, 1995.

\bibitem{h2}  J. Henderson, W.K.C. Yin;  \emph{Positive solutions and
nonlinear eigenvalue problems for functional differential
equations}, Appl. Math. Lett. 12 (1999) 63-68.

\bibitem{h3}  C.H. Hong,  C.C. Yeh, C.F. Lee, F. Hsiang,  F.H.
Wong;  \emph{Existence of positive solutions for  functional
 differential equations},  Comput. Math. Appl. 40 (2000) 783-792.

\bibitem{h4}   C.H. Hong, F.H. Wong, C.C. Yeh;
 \emph{Existence of positive solutions for higher-order  functional
 differential equations},  J. Math Anal. Appl.  297(2004) 14-23.

\bibitem{k1}  G. L. Karakostas, K. G. Mavridis, P. Ch. Tsamotos;
\emph{Multiple  positive solutions for a functional second-order
boundary-value problem},  J. Math. Anal. Appl. 282 (2003) 567-577.

\bibitem{l1}  K.Q. Lan; \emph{Multiple positive solutions of semilinear
differential equations with singularities}, J. London. Math. Soc.
63(2001) 690-704.

\bibitem{n1}  S.K. Ntouyas, Y. Sficas, P.Ch. Tsamatos; \emph{An
existence principle for boundary-value problems for second order
functional differential equations},  Nonlinear Anal. 20 (1993)
215-222.

\bibitem{s1} Y. Shi, S. Chen;  \emph{Multiple positive solutions of singular boundary
value problems}, Indian J. Pure Appl. Math. 30 (1999) 847-855.

\bibitem{t1}  D. Taunton, W.K.C. Yin;  \emph{Existence of some
functional differential equations}, Comm. Appl. Nonlinear Anal. 4
(1997) 31-43.

\bibitem{t2} \ P.Ch. Tsamatos;  \emph{On a boundary-value problem for a
system for functional differential equations with nonlinear boundary
conditions}, Funkcial. Ekvac. 42 (1999) 105-114.

\bibitem{w1}  P. Weng, D. Jiang;  \emph{Existence of some positive
solutions for boundary-value problem of second-order FDE},   J.
Comput. Math. Appl.  37(1999) 1-9.

\end{thebibliography}
\end{document}