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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 31, pp. 1--13.\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-2012/31\hfil Multiple positive solutions]
{Multiple positive solutions for integro-differential equations 
 with integral boundary conditions and sign changing nonlinearities}

\author[M. Jia, P. Wang \hfil EJDE-2012/31\hfilneg]
{Mei Jia, Pingyou Wang}  % in alphabetical order

\address{Mei Jia \newline
College of Science, University of Shanghai for Science and Technology,
 Shanghai 200093, China}
\email{jiamei-usst@163.com}

\address{Pingyou Wang \newline
College of Science, University of Shanghai for Science and Technology,
 Shanghai 200093, China}
\email{wpingy2008@163.com}

\thanks{Submitted November 16, 2011. Published February 23, 2012.}
\thanks{Supported by grants 10ZZ93 from the Innovation Program of Shanghai Municipal
 Education \hfill\break\indent 
 Commission, and 11171220 from the National Natural Sciences Foundation of China}
\subjclass[2000]{34B15, 34B18}
\keywords{Boundary value problems; $p$-Laplacian operator; \hfill\break\indent
 integral boundary conditions; fixed point theorem;
  multiple positive solutions}

\begin{abstract}
 In this article, we show the existence of multiple positive solutions
 for integro-differential equations with one-dimensional $p$-Laplacian operator,
 sign changing nonlinearities, and integral boundary conditions.
 By using the Schauder fixed point theorem and the Krasnosel'skii
 fixed point theorem, we obtain sufficient conditions for the existence
 of at least two positive solutions.
\end{abstract}

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

\section{Introduction}

In this article, we study the existence of positive solutions for the
following integro-differential equation with integral boundary
conditions, and sign changing nonlinearities:
\begin{equation} \label{e1.1}
\begin{gathered}
      (\varphi_p(u'(t)))'+w(t)f(t,u(t),Au(t),Bu(t))=0, \quad 0<t<1, \\
      u(0)=Au(\xi),\quad  u(1)=-Bu(\eta),
   \end{gathered}
\end{equation}
where $Au(t)=\int_0^tg(t,s)u'(s)\,\mathrm{d}s$,
$Bu(t)=\int_t^1h(t,s)u'(s)\,\mathrm{d}s$, $0<\xi\leq \eta <1$,
$\varphi_p(u)=|u|^{p-2}u$ is the one-dimensional $p$-Laplacian operator wiht
$p>1$, $\varphi_q=(\varphi_p)^{-1}$, and
$\frac{1}{q}+\frac{1}{p}=1$. 
By using the Schauder fixed point
theorem and the Krasnosel'skii fixed point theorem, we obtain
sufficient conditions for the
 existence of at least two positive solutions under
suitable conditions assumed on the nonlinear terms $f$ and $w$.


The theory of boundary-value problems for integro-differential
equations arises in different areas of applied mathematics, fluid
dynamics, plasma physics, biological sciences and chemical kinetics
(for details, see \cite{alm,as,waz} and the references therein).
Since boundary-value problems with integral boundary conditions
include two, three, multi-point and nonlocal boundary-value problems
as special cases, the existence and multiplicity of positive
solutions for such problems have  been put  emphasis on continuously
 (see \cite{g1,ma,wi1,wi2,x1,kt,lj,jw,l1} and
references therein). Because of the wide mathematical and physical
background, the existence of positive solutions for nonlinear
boundary-value problems with $p$-Laplacian has also received wide
attention. For details, we can refer to see
\cite{b1,lg,f2,c1,m1,w1,lh,lj,jw,jtg,Ge}. The main tools for such
problems are various kinds of fixed-point theorem in cones (see
\cite{b1,lg,f2,w1,lh,lj,jw,jtg}), the monotone iterative technique
(see \cite{m1}) and the fixed point index theory (see
\cite{c1,jtg}). If the nonlinear term is nonnegative, we can apply
the concavity of solutions in the proofs. Under the assumption that
the nonlinear term is nonnegative, authors obtained the existence of
at least one positive solutions or multiple positive solutions, see
\cite{b1,lg,f2,m1,w1,lh,jw}.


By using the upper and lower solution approach and the growth
restriction approach, in \cite{ar} the author presented some general
existence theorems   second-order boundary-value problems with
sign changing nonlinearities:
\begin{equation}\label{1.2}
    \begin{gathered}
       y''+q(t)f(t,y)=0, \quad 0<t<1, \\
       y(0) = 0 = y(1),
   \end{gathered}
\end{equation}
and
    \begin{gather*}
      y''+q(t)f(t,y)=0, \quad 0<t<1, \\
      y(0) = 0,\; \theta(y(1)) + y(1) = 0,
   \end{gather*}
where the nonlinear term $f$ is allowed to change sign and $\theta$ may
be nonlinear. Moreover, in  \cite{arls}, the authors
discussed the singular Dirichlet boundary-value problem \eqref{1.2}
and established existence results, where nonlinearity $f$ is allowed
to change sign and may be singular at $y = 0$.

Guo \cite{g2}  established a new fixed point
theorem in double cones and discussed the existence of positive
solutions for the second-order three-point boundary-value problem
    \begin{gather*}
      x''+f(t,x)=0, \quad 0\leq t\leq 1,\\
      x(0)-\beta x'(0)=0,\quad x(1)=\alpha x(\eta),
   \end{gather*}
where $f$ is allowed to change sign. Sufficient conditions of the
existence of at least two positive solutions for the boundary-value 
problems above are obtained by imposing growth conditions on $f$. By
applications of fixed point index theory, Cheung and Ren \cite{c1}
proved the existence of two positive solutions for the problem
$$
      (\Phi_p(u'))'+h(t)f(t,u)=0, \quad 0<t<1,
$$
with each of the following two sets of boundary conditions
\begin{gather*}
      u'(0)=0, \quad u(1)=\sum_{i=1}^{m-2}\alpha_iu(\xi_i);\\
u(0)=\sum_{i=1}^{m-2}\alpha_iu(\xi_i), \quad u'(1)=0,
\end{gather*}
where $h:[0,1]\to \mathbb{R}^+$ and $f:[0,1]\times [0,\infty)\to
\mathbb{R}$ are continuous functions.
Ji \cite{jtg} studied the existence of positive solutions for the
one-dimensional $p$-Laplacian equation
 \begin{gather*}
      (\Phi_p(u'))'+f(t,u,u')=0, \quad 0<t<1,\\
      u'(0)=\sum_{i=1}^{m-2}\alpha_iu'(\xi_i), \quad
      u(1)=\sum_{i=1}^{m-2}\beta_iu(\xi_i),
   \end{gather*}
where $f$ may change sign. They show that it has at least one or two
positive solutions under some assumptions by applying the fixed
point theorem and fixed point index theory. Liu, Jia and Tian \cite{lj}
studied the existence of positive solutions for the boundary-value
problem, with integral boundary conditions and sign changing
nonlinearities of one-dimensional $p$-Laplacian
    \begin{gather*}
      (\Phi_p(u'))'+f(t,u)=0, \quad 0<t<1,\\
      au(0)-bu'(0)=\sum_{i=1}^{m-2}a_iu(\xi_i), \quad
      u(1)=\int_0^1g(s)u(s)\,\mathrm{d}s,
   \end{gather*}
where $a, b\in[0,+\infty)$, $a_i\in(0,+\infty)$, 
$i = 1, 2, \dots ,m$, $0 < \xi_1 < \xi_2\dots < \xi_{m-2} < 1$, $m\geq 3$. The
sufficient conditions for the existence of at least two positive
solutions were obtained by using a fixed point theorem in double
cones given in \cite{g2}.


Recently, by using the expansion and compression fixed point theorem
of norm in cone under suitable conditions imposed on the nonlinear
term $f$ and $w$, Jia and Wang \cite{jw} established  sufficient
conditions for the existence of at least one positive solutions for
 \eqref{e1.1}, where the nonlinear term
$f$ and $w$ are nonnegative.
However, there are a few works devoted to the integro-differential boundary-value problems 
with integral boundary conditions, one-dimensional
$p$-Laplacian operator and sign changing nonlinearities.

Motivated by the above, we obtain some meaningful conclusions by
considering the existence of multiply positive solutions for 
\eqref{e1.1}, with
integral boundary conditions and sign changing nonlinearities of
one-dimensional $p$-Laplacian.

The following hypotheses will be assumed throughout this paper:
\begin{itemize}

\item[(H1)] $f:[0,1] \times [0,+\infty)\times \mathbb{R}^2\to \mathbb{R}$ is
continuous;

\item[(H2)] $f(t,0,\cdot,\cdot)\geq 0$, $w\in L^1[0,1]$, $f(t,0,\cdot,\cdot)\neq 0$, 
$w(t)\geq 0$ and $w\neq 0$ a.e. on [0,1];

\item[(H3)] $g$, $h\in C([0,1]\times[0,1],[0,+\infty))$, $g(\xi,s)$
 is monotone decreasing with respect to $s\in[0,1]$
 and $h(\eta, s)$ is monotone increasing with respect to $s\in[0,1]$.
\end{itemize}

\section{Preliminaries}


For any $y\in L^1[0,1]$, $y(t)\geq 0$ and $y(t)\not\equiv 0$ for
$t\in [0,1]$, we denote
\begin{align*}
   H(C)&=
   \int_0^{\xi}g(\xi,s)\varphi_q\Big(C-\int_0^sy(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
 &\quad +\int_\eta^{1}h(\eta,s)\varphi_q\Big(C-\int_0^sy(\tau)\,\mathrm{d}\tau\Big)\,
 \mathrm{d}s
  +\int_0^1\varphi_q\Big(C-\int_0^sy(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s.
\end{align*}

\begin{lemma} \label{lem2.3}
Suppose that {\rm (H3)} holds. Then for each $y\in L^1[0,1]$,
$y(t)\geq 0$ and $y(t)\not\equiv 0$ for $t\in [0,1]$, the equation
$H(C)=0$ has a unique solution in $(-\infty,+\infty)$ and the
solution $C_y\in (0,\int_0^1y(\tau)\,\mathrm{d}\tau)$. Moreover, there exists
$\sigma\in(0,1)$ such that $C_y=\int_0^{\sigma}y(\tau)\,\mathrm{d}\tau$.
\end{lemma}

The proof of Lemma \ref{lem2.3} is similar to that of
\cite[Lemma 2.2]{jw}. In this article, we take
\begin{equation}\label{2.1}
  \sigma_y=\inf\big\{\sigma\in(0,1)\;:\;C_y=\int_0^{\sigma}y(\tau)\,\mathrm{d}\tau\big\}.
\end{equation}
For the convenience, we recall the following results (see \cite{jw}).



\begin{lemma}[{\cite[Lemma 2.3]{jw}}]  \label{lem2.2}
 Assume {\rm (H3)} holds.
Then for each $y\in L^1[0,1]$, $y(t)\geq 0$ and $y(t)\not\equiv 0$
for $t\in [0,1]$, the boundary-value problem
\begin{equation} \label{e2.1}
    \begin{gathered}
      (\varphi_p(u'(t)))'+y(t)=0, \quad 0<t<1, \\
      u(0)=Au(\xi),\quad  u(1)=-Bu(\eta),
   \end{gathered}
\end{equation}
has a unique solution of the form
\begin{equation} \label{e2.2}
  u(t)=\int_0^{\xi}g(\xi,s)\varphi_q\Big(C_y-\int_0^sy(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s
       +\int_0^t\varphi_q\Big(C_y -\int_0^sy(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s,
\end{equation}
or
\begin{equation} \label{e2.3}
  u(t)=-\int_\eta^{1}h(\eta,s)\varphi_q\Big(C_y-\int_0^sy(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s
       -\int_t^1\varphi_q\Big(C_y -\int_0^sy(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s,
\end{equation}
where $C_y$ satisfies $H(C_y)=0$.
\end{lemma}

\begin{remark} \rm
By Lemma \ref{lem2.3} and Lemma \ref{lem2.2}, \eqref{e2.2} and
\eqref{e2.3} can be changed into \eqref{2.4} and
\eqref{2.5}, respectively.
\begin{equation} \label{2.4}
  u(t)=\int_0^{\xi}g(\xi,s)\varphi_q\Big(\int_s^{\sigma_y}y(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s
       +\int_0^t\varphi_q\Big(\int_s^{\sigma_y}y(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s,
\end{equation}
and
\begin{equation} \label{2.5}
  u(t)=\int_\eta^{1}h(\eta,s)\varphi_q\Big(\int_{\sigma_y}^sy(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s
       +\int_t^1\varphi_q\Big(\int_{\sigma_y}^sy(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s.
\end{equation}
\end{remark}

\begin{lemma}[{\cite[Lemma 2.4]{jw}}] \label{lem2.4}
 Suppose {\rm (H3)} holds. If $y\in L^1[0,1]$, $y(t)\geq 0$ and $y(t)\not\equiv 0$ for
 $t\in [0,1]$. Then the solution of
boundary-value problem \eqref{e2.1} has the following properties:
\begin{itemize}
\item[(1)]  $u(t)$ is a concave function;
\item[(2)] $u(t)\geq 0$, $t\in [0,1]$;
\item[(3)] $u(\sigma_y)=\max_{0\leq t\leq 1}u(t)$ and $u'(\sigma_y)=0$, where
$\sigma_y$ is defined in \eqref{2.1}.
\end{itemize}
\end{lemma}


\begin{lemma}[\cite{Ge}] \label{lem2.5}
 If $u\in C[0,1]$ and $u(t)\geq 0$ is
a concave function. Then for each $\gamma\in(0,\frac{1}{2})$, we
have
$$
  \min_{\gamma\leq t\leq {1-\gamma}}u(t)\geq \gamma \| u\|.
$$
\end{lemma}



Let $X=C[0,1]$ and $\|u\|=\max_{0\leq t\leq 1}|u(t)|$, take
$0<\delta<\min\{\frac{1}{2},\xi,\eta\}$ and denote
$$
  P=\{u\in X:u(t)\geq 0, t\in [0,1]\}
$$
and
$$
  K=\{u\in P: u(t)\text{ is a concave function on $[0,1]$ and }
  \min_{\delta\leq t\leq {1-\delta}}u(t)\geq \delta \| u\|\}.
$$
Obviously, $P,\;K\subset X$ are two cones of $X$ with $K\subset P$.

Denote $B^+=\max\{B,0\}$.
For $u\in K$, we define $T^*:K\to K$ by
\begin{equation*}
  T^*u(t)=
  \begin{cases}
  \int_0^{\xi} g(\xi,s)\varphi_q\Big(\int_s^{\sigma_u}w(\tau)f^+
 (\tau,u(\tau),Au(\tau),Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  +\int_0^t\varphi_q\Big(\int_s^{\sigma_u}w(\tau)f^+(\tau,u(\tau),Au(\tau),Bu(\tau))
 \,\mathrm{d}\tau\Big)\,\mathrm{d}s,
  & 0\leq t\leq \sigma_u, \\[4pt]
  \int_\eta^{1}h(\eta,s)\varphi_q\Big(\int_{\sigma_u}^sw(\tau)f^+
 (\tau,u(\tau),Au(\tau),Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
 +\int_t^1\varphi_q\Big(\int_{\sigma_u}^sw(\tau)f^+(\tau,u(\tau),Au(\tau),Bu(\tau))
 \,\mathrm{d}\tau\Big)\,\mathrm{d}s, 
 &\sigma_u\leq t\leq 1,
 \end{cases}
\end{equation*}
where $\sigma_u$ is defined in \eqref{2.1}. It follows $T^*$
 is well definition from (H1)--(H3), Lemma \ref{lem2.4} and Lemma \ref{lem2.5}.
Define $T:P\to P$ by
\begin{equation*}
  Tu(t)=
  \begin{cases}
    \Big[\int_0^{\xi}g(\xi,s)\varphi_q\Big(\int_s^{\sigma_u}w(\tau)
    f(\tau,u(\tau),Au(\tau),Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
   +\int_0^t\varphi_q\Big(\int_s^{\sigma_u}w(\tau)f(\tau,u(\tau),Au(\tau),
   Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\Big]^+,
  & 0\leq t\leq \sigma_u, \\[4pt]
    \Big[\int_\eta^{1}h(\eta,s)\varphi_q\Big(\int_{\sigma_u}^sw(\tau)
    f(\tau,u(\tau),Au(\tau),Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
   +\int_t^1\varphi_q\Big(\int_{\sigma_u}^sw(\tau)f(\tau,u(\tau),Au(\tau),
   Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\Big]^+,
  &\sigma_u\leq t\leq 1\,.
 \end{cases}
\end{equation*}
Define $S:P\to X$ by
\begin{equation*}
  Su(t)=
  \begin{cases}
   \int_0^{\xi}g(\xi,s)\varphi_q\Big(\int_s^{\sigma_u}w(\tau)
 f(\tau,u(\tau),Au(\tau),Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
   +\int_0^t\varphi_q\Big(\int_s^{\sigma_u}w(\tau)f(\tau,u(\tau),Au(\tau),Bu(\tau))\,
  \mathrm{d}\tau\Big)\,\mathrm{d}s,
  & 0\leq t\leq \sigma_u, \\[4pt]
   \int_\eta^{1}h(\eta,s)\varphi_q\Big(\int_{\sigma_u}^sw(\tau)f(\tau,u(\tau),Au(\tau),
  Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
 +\int_t^1\varphi_q\Big(\int_{\sigma_u}^sw(\tau)f(\tau,u(\tau),Au(\tau),Bu(\tau))\,
 \mathrm{d}\tau\Big)\,\mathrm{d}s,
  &\sigma_u\leq t\leq 1\,.
 \end{cases}
\end{equation*}
From Lemma \ref{lem2.2}, we have the following result.

\begin{lemma} \label{lem2.7}
Suppose that {\rm (H1)}--{\rm (H3)} hold. Then a function $u(t)$ is
a solution of boundary-value problem \eqref{e1.1} if and only if
$u(t)$ is a fixed point of the operator $S$.
\end{lemma}

 We can easily prove that the following lemma  holds.


\begin{lemma} \label{lem2.6}
Suppose that {\rm (H1)}--{\rm (H3)} hold. Then $T^*:K\to K$ is
completely continuous.
\end{lemma}

For $u\in X$, denote $\theta : X\to P$ by $(\theta u)(t)=\max
\{u(t),0\}$, then $T=\theta \circ S$.


\begin{lemma}[{\cite[Lemma 2.2]{c1}}] \label{lem2.8}
  If $S:K\to X$ is completely
continuous, then $T=\theta \circ S :K\to K$ is also completely
continuous.
\end{lemma}


\begin{lemma} \label{lem2.9}
Suppose {\rm(H1)}--{\rm(H3)} hold. If $u$ is a fixed point of
operator $T$, then $u$ is also a fixed point of operator $S$.
\end{lemma}

\begin{proof} 
Let $u$ be a fixed point of the operator $T$, if we
prove $Su(t)\geq 0$ for $t\in[0,1]$, then $u(t)$ is a fixed point of
operator $S$.

Suppose $Su(t)\geq 0$ for $t\in[0,1]$ is not true, then there exists
a $t_0\in (0,1)$ such that $u(t_0)=0>Su(t_0)$. Let $(t_1,t_2)$ be
the maximal interval which contains $t_0$ and such that $Su(t)<0$,
$t\in (t_1,t_2)$. It follows $[t_1,t_2]\neq [0,1]$ from (H2).

Case 1: If $t_2<1$, we have $u(t)=0$ for $t\in [t_1,t_2]$, $Su(t)<0$
for $t\in (t_1,t_2)$ and $Su(t_2)=0$. Thus $(Su)'(t_2)\geq 0$. From
(H2), we know $[\varphi_p((Su)'(t))]'=-w(t)f(t,0,Au(t),Bu(t))\leq 0$
for $t\in[t_1,t_2]$ and we can get $(Su)'(t)$ is monotone decreasing
on $[t_1,t_2]$. So $t_1=0$, $Su(t_1)<0$ and
$$
  (Su)'(t)\geq (Su)'(t_2)\geq 0,\; t\in [0,t_2].
$$
On the other hand, if $\xi\leq t_2$, we have
$$
  0>Su(0)=\int_0^\xi g(\xi,s)(Su)'(s)\,\mathrm{d}s\geq 0,
$$
which is a contradiction.

If $\xi> t_2$, by using mean value theorem of integral, we have
\begin{align*}
     Su(0)
  & =\int_0^{t_2} g(\xi,s)(Su)'(s)\,\mathrm{d}s+\int_{t_2}^\xi
     g(\xi,s)(Su)'(s)\,\mathrm{d}s\\
  & =g(\xi,\xi_1)\int_0^{t_2}
  (Su)'(s)\,\mathrm{d}s+g(\xi,t_2)\int_{t_2}^{\xi_2}
     (Su)'(s)\,\mathrm{d}s\\
  & =
     g(\xi,\xi_1)(Su(t_2)-Su(0))+g(\xi,t_2)(Su(\xi_2)-Su(t_2))\\
  & =
     g(\xi,\xi_1)(-Su(0))+g(\xi,t_2)Su(\xi_2),
\end{align*}
and
\begin{equation}\label{2.6}
  0>(1+g(\xi,\xi_1))Su(0)=g(\xi,t_2)Su(\xi_2),
\end{equation}
where $\xi_1\in [0,t_2]$ and $\xi_2\in [t_2,\xi]$.

It follows that \eqref{2.6} is a contradiction if $Su(\xi_2)\geq 0$.

If $Su(\xi_2)< 0$, let $(t_3,t_4)$ be the maximal interval which
contains $\xi_2$ and such that $Su(t)<0$, $t\in (t_3,t_4)$. It is
obvious that $[t_3,t_4]\subset [t_2,1]$. If $t_4<1$, we have
$u(t)=0$ for $t\in [t_3,t_4]$, $Su(t)<0$ for $t\in (t_3,t_4)$ and
$Su(t_3)=0$. Thus $(Su)'(t_3)\leq 0$. From (H2), we know
$[\varphi_p((Su)'(t))]'=-w(t)f(t,0,Au(t),Bu(t))\leq 0$ and
$\varphi_p((Su)'(t))$ is monotone decreasing on $[t_3, t_4]$, we can
obtain $(Su)'(t)$ is monotone decreasing on $[t_3, t_4]$. It is easy
to show that
$$
  (Su)'(t)\leq (Su)'(t_3)\leq 0,\; t\in [t_3, t_4].
$$
Hence, $t_4=1$ and $Su(1)<0$. Since $\xi\leq\eta$, we have
$(Su)'(t)\leq 0,\; t\in [\eta,1]$ and
$$
  0>Su(1)=-\int_\eta^1 h(\eta,s)(Su)'(s)\,\mathrm{d}s\geq 0,
$$
which is a contradiction.

Therefore, $t_2<1$ is not true.
We have $t_2=1$.

Case 2: If $t_1>0$, we have $Su(t)=0$ for $t\in [t_1,1]$, $Su(t)<0$
for $t\in (t_1,1)$ and $Su(t_1)=0$. Thus $(Su)'(t_1)\leq 0$. We have
$[\varphi_p((Su)'(t))]'=-f(t,0,Au(t),Bu(t))\leq 0$ by (H2). This
implies $(Su)'(t)\leq0$ and $Su(t)<0$ for $t\in (t_1,1]$ and
$Su(1)=\min_{t\in [t_1,1]}Su(t)$.


We can prove that
\begin{equation}\label{2.08}
  Su(t)\geq 0\;\mathrm{for}\;t\in [0,t_1].
\end{equation}
  If there exists a $t_5\in [0,t_1]$ such
that $Su(t_5)<0$ and there is a maximal interval $[t_6,t_7]$ which
contains $t_5$ such that $Su(t)<0$ for $t\in (t_6,t_7)$. Obviously
$[t_6,t_7)\cap [t_1,1]=\emptyset$, so $1\not\in(t_6,t_7)$; i.e.,
$t_7<1$, this is a contradiction with the above discussion. Thus we
can show $Su(t)\geq 0$ for $t\in [0,t_1]$.

For $Su(1)<0$, we have
$$
Su(1)=-\int_\eta^1 h(\eta,s)(Su)'(s)\,\mathrm{d}s.
$$
Then, if $\eta\geq t_1$, we have
$$
  0>Su(1)=-\int_\eta^1 h(\eta,s)(Su)'(s)\,\mathrm{d}s\geq 0,
$$
which is a contradiction.

If $\eta<t_1$, by using mean value theorem of integral, there exist
$\eta_1\in [\eta,t_1]\subset[0,t_1]$ and $\eta_2\in [t_1,1]$ such
that
\begin{align*}
     Su(1)
  & =-\int_\eta^{t_1} h(\eta,s)(Su)'(s)\,\mathrm{d}s
     -\int_{t_1}^1 h(\eta,s)(Su)'(s)\,\mathrm{d}s\\
  & =-h(\eta,t_1)\int_{\eta_1}^{t_1}(Su)'(s)\,\mathrm{d}s
     -h(\eta,\eta_2)\int_{t_1}^1 (Su)'(s)\,\mathrm{d}s\\
  & =
     -h(\eta,t_1)(Su(t_1)-Su(\eta_1))-h(\eta,\eta_2)(Su(1)-Su(t_1))\\
  & =
     h(\eta,t_1)Su(\eta_1))-h(\eta,\eta_2)Su(1),
\end{align*}
and
\begin{equation}\label{2.7}
  0>(1+h(\eta,\eta_2))Su(1)=h(\eta,t_1)Su(\eta_1).
\end{equation}

By \eqref{2.08}, we have $Su(\eta_1)\geq 0$. Hence, \eqref{2.7} is a
contradiction.
Therefore $t_1=0$.

The above  also  contradicts $[t_1,t_2]\neq [0,1]$.
Thus $Su(t)\geq 0$ for $t\in[0,1]$. That is $u(t)$ is a fixed point
of operator $S$.
\end{proof}

Next we state the Krasnosel'skii Fixed Point Theorem \cite{gl}.

\begin{lemma} \label{lemma2.10}
 Let  $E$ be a Banach space and $K \subset E$ be a cone in $E$.
Assume $\Omega_1$ and $\Omega_2$ are open subsets of $E$ with $0 \in
\Omega_1$ and $\overline{\Omega}_1 \subset \Omega_2$ and $A: K \cap
(\overline{\Omega}_2 \backslash \Omega_1) \rightarrow K$ be a
completely continuous operator. In addition, suppose either
$$
 \|Au\|\leq\|u\|,\indent u \in K \cap \partial \Omega_1,\;and \;
 \|Au\|\geq \|u\|,\indent  u \in K \cap \partial \Omega_2;
$$
or
$$
 \|Au\|\geq\|u\|, \indent u \in K \cap \partial \Omega_1,\;
 and\;\|Au\|\leq \|u\|,\indent u \in K \cap \partial \Omega_2.
$$
hold. Then $A$ has a fixed point in $K \cap (\overline{\Omega}_2
\backslash \Omega_1)$.
\end{lemma}

\section{Main result}

Denote
\begin{gather*}
  M=\min\Big\{\int_{\delta}^{1/2}\varphi_q\Big(\int_s^{1/2}
  w(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s,\int_{1/2}^{1-\delta}
  \varphi_q\Big(\int_{1/2}^sw(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s\Big\},
\\
\begin{aligned}  N=\max\Big\{&(1+\int_0^{1}g(\xi,s)\,\mathrm{d}s)\varphi_q
\Big(\int_0^{1}w(\tau)\,\mathrm{d}\tau\Big), \\
&(1+\int_0^{1}h(\eta,s)\,\mathrm{d}s)\varphi_q\Big(\int_0^{1}w(\tau)\,\mathrm{d}\tau\Big)
    \Big\}.
\end{aligned}
\end{gather*}
For the next theorem, we assume that $f$ satisfies the following growth conditions:
\begin{itemize}
\item[(H4)]  $f(t,u,x,y)\geq 0$ for
   $(t,u,x,y)\in [0,1]\times [c_1,c_3]\times\mathbb{R}^2$;

\item[(H5)]  $f(t,u,x,y)<\varphi_p(\frac{c_2}{N})$ for
  $(t,u,x,y)\in [0,1]\times [0,c_2]\times\mathbb{R}^2$;

\item[(H6)] $f(t,u,x,y)\geq \varphi_p(\frac{c_3}{M})$ for
  $(t,u,x,y)\in [\delta,1-\delta]\times
  [\delta c_3,c_3]\times\mathbb{R}^2$.
\end{itemize}

\begin{theorem} \label{thm3.1}
Suppose {\rm (H1)--(H6)} hold. There exist constants $c_1, c_2, c_3$
such that
$$
0<c_1\leq\min\Big\{\frac{g(\xi,\xi)}{1+g(\xi,0)},
  \frac{h(\eta,\eta)}{1+h(\eta,1)}\Big\}\delta c_2,\quad 
  c_2<\delta c_3<c_3\,.
$$
Then  \eqref{e1.1} has at least two
positive solutions $u_1$ and $u_2$ such that
$$
0<\|u_1 \|<c_2\leq \|u_2 \|\leq c_3.
$$
\end{theorem}

\begin{proof} Let
$\Omega_1=\{u\in K:\|u\|<c_2\}$.
For any $u\in \overline{\Omega}_1$, we have $u\in K$ and $\|u\|\leq
c_2$. Denote
$$
\| Tu \|=\max_{0\leq t \leq 1}|Tu(t)|=Tu(\overline{t}).
$$
If $\overline{t}<\sigma _u$, it follows from $(H_5)$ that
\begin{align*}
Tu(\overline{t})
 &=
    \Big[\int_0^{\xi}g(\xi,s)\varphi_q\Big(\int_s^{\sigma_u}w(\tau)
    f(\tau,u(\tau),Au(\tau),Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  &\quad +
    \int_0^{\bar{t}}\varphi_q\Big(\int_s^{\sigma_u}w(\tau)f(\tau,u(\tau),Au(\tau),
     Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\Big]^+\\
  &<
    \int_0^{1}g(\xi,s)\varphi_q\Big(\int_0^{1}w(\tau)
    \varphi_p(\frac{c_2}{N})\,\mathrm{d}\tau\Big)\,\mathrm{d}s
   +\int_0^{1}\varphi_q\Big(\int_0^{1}w(\tau)\varphi_p(\frac{c_2}{N})\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  &=
    \frac{c_2}{N}\varphi_q\Big(\int_0^{1}w(\tau)\,\mathrm{d}\tau\Big))
    \Big[1+\int_0^{1}g(\xi,s)\,\mathrm{d}s\Big]\\
  &\leq 
  c_2,
\end{align*}
and if $\bar{t}>\sigma _u$, we have
\begin{align*}
Tu(\overline{t})
  &=
    \Big[\int_\eta^{1}h(\eta,s)\varphi_q\Big(\int_{\sigma_u}^sw(\tau)
    f(\tau,u(\tau),Au(\tau),Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  &\quad +
    \int_{\bar{t}}^1\varphi_q\Big(\int_{\sigma_u}^sw(\tau)f(\tau,u(\tau),Au(\tau),
   Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\Big]^+\\
  &<
    \int_0^{1}h(\eta,s)\varphi_q\Big(\int_0^{1}w(\tau)
    \varphi_p(\frac{c_2}{N})\,\mathrm{d}\tau\Big)\,\mathrm{d}s
   +\int_0^{1}\varphi_q\Big(\int_0^{1}w(\tau)\varphi_p(\frac{c_2}{N})\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  &=
    \frac{c_2}{N}\varphi_q\Big(\int_0^{1}w(\tau)\,\mathrm{d}\tau\Big)
    \Big[1+\int_0^{1}h(\eta,s)\,\mathrm{d}s\Big]\\
  &\leq 
  c_2.
\end{align*}
 We have
\begin{equation}\label{3.1}
  \|Tu\|<c_2=\|u\|\;\mathrm{for}\;u\in \overline{\Omega}_1.
\end{equation}
By using Schauder fixed point theorem,
 we can get the $T$ has at least one fixed point $u_1$ in $\Omega_1$. That is,
 $Tu_1=u_1$ and $\|u_1\|<c_2$. If $\|u_1\|=0$, we have
 $u_1\equiv 0$, $t\in[0,1]$, which is a contradiction with (H2). Hence, 
$0<\|u_1  \|<c_2$.

It follows that \eqref{e1.1} has at least one
positive solutions $u_1$ such that $0<\|u_1 \|<c_2$ from Lemma
\ref{lem2.9}.
Let
$$
  \Omega_2=\{u\in K:\|u\|<c_3\}.
$$
For any $u\in \partial\Omega_2$, we have $u\in K$ and
 $\|u\|=c_3$ and $\delta c_3\leq u(t)\leq c_3$ for
$\delta \leq t\leq {1-\delta}$. By Lemma \ref{lem2.4}, we have
$$
  \| T^*u\|= T^*u(\sigma_u)\geq T^*u\big(\frac{1}{2}\big).
$$
If $\sigma_u\geq 1/2$, it follows from (H6) that
\begin{align*}
  \| T^*u\|
  &\geq 
    \int_0^{\xi}g(\xi,s)\varphi_q\Big(\int_s^{\sigma_u}w(\tau)
    f^+(\tau,u(\tau),Au(\tau),Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  &\quad +
    \int_0^{1/2}\varphi_q\Big(\int_s^{\sigma_u}w(\tau)f^+(\tau,u(\tau),Au(\tau),
     Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  &\geq
    \int_{\delta}^{1/2}\varphi_q\Big(\int_s^{1/2}w(\tau)f^+(\tau,u(\tau),Au(\tau),
     Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  &\geq
    \int_{\delta}^{1/2}\varphi_q\Big(\int_s^{1/2}w(\tau)
    \varphi_p(\frac{c_3}{M})\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  &=
    \frac{c_3}{M}\int_{\delta}^{1/2}\varphi_q\Big(\int_s^{1/2}w(\tau)
    \,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  &\geq     c_3.
\end{align*}

If $\sigma_u<1/2$, it follows from (H6) that
\begin{align*}
  \| T^*u\|
  &\geq 
    \int_\eta^{1}h(\eta,s)\varphi_q\Big(\int_{\sigma_u}^sw(\tau)
    f^+(\tau,u(\tau),Au(\tau),Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  &\quad+
    \int_{1/2}^1\varphi_q\Big(\int_{\sigma_u}^sw(\tau)f^+
    (\tau,u(\tau),Au(\tau),Bu(\tau))\,\mathrm{d}\tau\Big)
    \,\mathrm{d}s \\
  &\geq
    \int_{1/2}^{1-\delta}\varphi_q\Big(\int_{1/2}^sw(\tau)f^+(\tau,u(\tau),Au(\tau),
     Bu(\tau))\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  &\geq 
    \int_{1/2}^{1-\delta}\varphi_q\Big(\int_{1/2}^sw(\tau)
    \varphi_p(\frac{c_3}{M})\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  &=
    \frac{ c_3}{M}\int_{1/2}^{1-\delta}\varphi_q
    \Big(\int_{1/2}^sw(\tau)\,\mathrm{d}\tau\Big)\,\mathrm{d}s\\
  & \geq
   c_3.
\end{align*}
Hence, we can show that
\begin{equation}\label{3.2}
  \| T^*u\|\geq c_3=\|u\|\quad\mathrm{for }u\in\partial\Omega_2.
\end{equation}
As in the proof of \eqref{3.1}, we obtain
\begin{equation*}
  \|T^*u\|<c_2=\|u\|\quad\text{for }u\in \partial{\Omega}_1.
\end{equation*}
Therefore, by using Krasnosel'skii fixed point theorem,
$T^*$ has at least one fixed point
$u_2\in\overline{\Omega}_2\backslash\Omega_1$ with 
$c_2\leq\| u_2\|\leq  c_3$.
Subsequently, we prove $u_2$ is also a fixed point of $S$.

If $u\in\overline{\Omega}_2\backslash\Omega_1$ and $T^*u=u$, then
$u\in K$, $c_2\leq\|u\|\leq c_3$ and
$\min_{t\in[\delta,1-\delta]}u(t)\geq \delta\|u\|\geq\delta
c_2$. By Lemma \ref{lem2.4}, Lemma \ref{lem2.6} and the definition
of $T^*$, we obtain
\begin{gather*}
  (T^*u)'(t)=\varphi_q\big(\int_t^{\sigma_u}w(\tau)
   f^+(\tau,u(\tau),Au(\tau),Bu(\tau))\,\mathrm{d}\tau)\,\mathrm{d}s\big),
   \quad t\in[0,1]
\\
   (T^*u)'(\sigma_u)=0,\;(T^*u)'(t)\geq 0 \; \mathrm{if}\;
    t \in[0,\sigma_u],\; (T^*u)'(t)\leq 0 \; \mathrm{if}\;
    t \in[\sigma_u,1]
\end{gather*}
and
$$
\min_{0\leq t \leq 1}u(t)=\min\{u(0),u(1)\}\geq 0.
$$
If $\min_{0\leq t \leq 1}u(t)=u(0)$, when $\xi\leq \sigma_u$,
by (H3) and mean value theorem of integral, there exists a $\xi_1\in
[0,\xi]$ such that
$$
  u(0)=Au(\xi)=\int_0^{\xi}g(\xi,s)u'(s)\,\mathrm{d}s
  =g(\xi,\xi_1)\int_0^{\xi}u'(s)\,\mathrm{d}s
  \geq g(\xi,\xi)(u(\xi)-u(0))
$$
and
\begin{equation}\label{eq3.1}
    u(0)\geq\frac{g(\xi,\xi)u(\xi)}{1+g(\xi,\xi)}
  \geq\frac{\delta g(\xi,\xi)}{1+g(\xi,\xi)}\|u\|
  \geq\frac{\delta g(\xi,\xi)c_2}{1+g(\xi,\xi)}
  \geq\frac{\delta g(\xi,\xi)c_2}{1+g(\xi,0)}.
\end{equation}
When $\xi>\sigma_u$, by (H3) and mean value theorem of integral,
there exist $\xi_2\in [0,\sigma_u]$ and $\xi_3\in [\sigma_u,\xi]$
such that
\begin{align*}
  u(0)=
  & Au(\xi)=\int_0^{\sigma_u}g(\xi,s)u'(s)\,\mathrm{d}s
    +\int_{\sigma_u}^{\xi}g(\xi,s)u'(s)\,\mathrm{d}s\\
  &=
    g(\xi,\xi_2)\int_0^{\sigma_u}u'(s)\,\mathrm{d}s
    +g(\xi,\xi_3)\int_{\sigma_u}^{\xi}u'(s)\,\mathrm{d}s\\
  &=
    g(\xi,\xi_2)(u(\sigma_u)-u(0))+g(\xi,\xi_3)(u(\xi)-u(\sigma_u))\\
  \geq &
     g(\xi,\xi)u(\xi)-g(\xi,0)u(0)
\end{align*}
and
\begin{equation}\label{eq3.2}
    u(0)\geq\frac{g(\xi,\xi)u(\xi)}{1+g(\xi,0)}
  \geq\frac{\delta g(\xi,\xi)}{1+g(\xi,0)}\|u\|
  \geq\frac{\delta g(\xi,\xi)c_2}{1+g(\xi,0)}.
\end{equation}

If $\min_{0\leq t \leq 1}u(t)=u(1)$, when $\eta\leq\sigma_u$,
by (H3) and mean value theorem of integral, there exist $\eta_1\in
[\eta,\sigma_u]$ and $\eta_2\in [\sigma_u,1]$ such that
\begin{align*}
  u(1)&=   -Bu(\eta)=-\int_{\eta}^{\sigma_u}h(\eta,s)u'(s)\,\mathrm{d}s
    -\int_{\sigma_u}^{1}h(\eta,s)u'(s)\,\mathrm{d}s\\
  &=
    -h(\eta,\eta_1)\int_{\eta}^{\sigma_u}u'(s)\,\mathrm{d}s
    -h(\eta,\eta_2)\int_{\sigma_u}^{1}u'(s)\,\mathrm{d}s\\
  &=
    -h(\eta,\eta_1)(u(\sigma_u)-u(\eta))-h(\eta,\eta_2)(u(1)-u(\sigma_u))\\
  &=
    h(\eta,\eta_2)u(\eta)-h(\eta,\eta_1)u(1)
    +(h(\eta,\eta_2)-h(\eta,\eta_1))u(\sigma_u)\\
  &\geq 
     h(\eta,\eta)u(\eta)-h(\eta,1)u(1)
\end{align*}
and
\begin{equation}\label{eq3.3}
    u(1)\geq\frac{h(\eta,\eta)u(\eta)}{1+h(\eta,1)}
  \geq\frac{\delta h(\eta,\eta)}{1+h(\eta,1)}\|u\|
  \geq\frac{\delta h(\eta,\eta)c_2}{1+h(\eta,1)}.
\end{equation}
When $\eta>\sigma_u$, by (H3) and mean value theorem for integrals,
there exists $\eta_3\in [\eta,1]$ such that
\begin{align*}
  u(1)&=
  -Bu(\eta)=-\int_{\eta}^{1}h(\eta,s)u'(s)\,\mathrm{d}s\\
 &=
    h(\eta,\eta_3)(u(\eta)-u(1))
  \geq
     h(\eta,\eta)(u(\eta)-u(1))
\end{align*}
and
\begin{equation}\label{eq3.4}
    u(1)\geq\frac{h(\eta,\eta)u(\eta)}{1+h(\eta,\eta)}
  \geq\frac{\delta h(\eta,\eta)}{1+h(\eta,\eta)}\|u\|
  \geq\frac{\delta h(\eta,\eta)c_2}{1+h(\eta,\eta))}
  \geq\frac{\delta h(\eta,\eta)c_2}{1+h(\eta,1)}.
\end{equation}
Hence, it follows
$$
  \min_{0\leq t \leq 1}u(t)\geq\min\Big\{\frac{g(\xi,\xi)}{1+g(\xi,0)},
  \frac{h(\eta,\eta)}{1+h(\eta,1)}\Big\}\delta c_2\geq c_1
$$
from \eqref{eq3.1}--\eqref{eq3.4}.

Therefore, if $u\in\overline{\Omega}_2\setminus\Omega_1$ and
$T^*u=u$, we have
$$
c_1\leq u(t)\leq \|u\| \leq c_3.
$$
It follows $f(t,u(t),Au(t),Bu(t))\geq 0$, $t\in [0,1]$ from (H4).
Thus, $T^*u=Su$. That is, the fixed point of $T^*$ on
$\overline{\Omega}_2\setminus\Omega_1$ is also a fixed point of $S$.
We can get the boundary-value problem \eqref{e1.1} has at least one
positive solutions $u_2$ such that $c_2\leq\|u_2 \|\leq c_3$.
The proof is complete.
\end{proof}

For the next theorem, we have a new assumption:
\begin{itemize}
  \item[(H7)] For each $t\in [0,1]$, $g(t,s)$ and $h(t, s)$ are monotone
  with respect to $s$.
\end{itemize}
We denote
$$
  M_g=\max\{\max_{t\in[0,1]}g(t,0),\max_{t\in[0,1]}g(t,t)\},\quad
  M_h=\max\{\max_{t\in[0,1]}h(t,1),\max_{t\in[0,1]}h(t,t)\}.
$$
If (H7) holds, for each $t\in [0,1]$, when $g(t,s)$ is 
decreasing with respect to $s$, there exists a $\bar{\xi}_1\in[0,t]$
such that
$$
  |(Au)(t)|=|\int_0^{t}g(t,s)u'(s)\,\mathrm{d}s|
  =g(t,0)|\int_0^{\bar{\xi}_1}u'(s)\,\mathrm{d}s|
  \leq 2g(t,0)\|u\|\leq 2M_g\|u\|,
$$
and when $g(t,s)$ is monotone creasing with respect to $s$, there
exists a $\bar{\xi}_2\in[0,t]$ such that
$$
  |(Au)(t)|=|\int_0^{t}g(t,s)u'(s)\,\mathrm{d}s|
  =g(t,t)|\int_{\bar{\xi}_2}^{t}u'(s)\,\mathrm{d}s|
  \leq 2g(t,t)\|u\|\leq 2M_g\|u\|.
$$
As above, if (H7) holds, we can show that
$$
  |(Bu)(t)|\leq 2M_h\|u\|,\quad \text{for each } t\in[0,1].
$$
As in the proof of Theorem \ref{thm3.1}, we obtain the
following theorem under the assumption taht $f$ satisfies the following growth conditions:
\begin{itemize}
\item[\rm{(H8)}]  $f(t,u,x,y)\geq 0$ for
   $(t,u,x,y)\in [0,1]\times [c_1,c_3]\times[-2M_gc_3,2M_gc_3]\\
   \times [-2M_hc_3,2M_hc_3]$;

\item[\rm{(H9)}]  $f(t,u,x,y)<\varphi_p(\frac{c_2}{N})$ for
  $(t,u,x,y)\in [0,1]\times [0,c_2]\times[-2M_gc_2,2M_gc_2]
   \times[-2M_hc_2,2M_hc_2]$;

\item[\rm{(H10)}] $f(t,u,x,y)\geq \varphi_p(\frac{c_3}{M})$ for
  $(t,u,x,y)\in [\delta,1-\delta]\times
  [\delta c_3,c_3]\times[-2M_gc_3,2M_gc_3]
   \times[-2M_hc_3,2M_hc_3]$.
\end{itemize}

\begin{theorem} \label{thm3.2}
Suppose {\rm (H1)--(H3)} and {\rm (H7)--(H10)} hold. Then there exist
 constants $c_1, c_2, c_3$ such that such that
$$
0<c_1\leq\min\Big\{\frac{g(\xi,\xi)}{1+g(\xi,0)},
  \frac{h(\eta,\eta)}{1+h(\eta,1)}\Big\}\delta c_2,\; \mathrm{and}\;
  c_2<\delta c_3<c_3
$$
Then  \eqref{e1.1} has at least two positive solutions $u_1$ and
$u_2$ such that
$$
0<\|u_1 \|<c_2\leq \|u_2 \|\leq c_3.
$$
\end{theorem}

We have a new assumption:
\begin{itemize}
  \item[(H11)] For each $t,s\in [0,1]$, $g_s(t,s)$ and $h_s(t, s)$ are
   bounded.
\end{itemize}
We denote
\begin{gather*}
  N_g=\max\{\max_{t\in[0,1]}g(t,0),\max_{t\in[0,1]}g(t,t),
      \sup_{t,s\in[0,1]}g_s(t,s)\},
\\
  N_h=\max\{\max_{t\in[0,1]}h(t,1),\max_{t\in[0,1]}h(t,t),
      \sup_{t,s\in[0,1]}h_s(t,s)\}
\end{gather*}
If (H11) holds, for each $t\in [0,1]$, we have
\begin{align*}
  |(Au)(t)|&=|\int_0^{t}g(t,s)u'(s)\,\mathrm{d}s|\\
  &=|(g(t,t)u(t)-g(t,0)u(0))-\int_0^{t}g_s(t,s)u(s)\,\mathrm{d}s|
  \leq 3N_g\|u\|.
\end{align*}
As above, if (H11) holds, we can show that
$$
  |(Bu)(t)|\leq 3N_h\|u\|,\quad \text{for each } t\in[0,1].
$$
As in the proof of Theorem \ref{thm3.1}, we can get the
following theorem, and $f$ satisfies the following growth conditions:

\begin{itemize}
\item[\rm{(H12)}]  $f(t,u,x,y)\geq 0$ for
   $(t,u,x,y)\in [0,1]\times [c_1,c_3]\times[-3N_gc_3,3N_gc_3]\\
   \times[-3N_hc_3,3N_hc_3]$;

\item[\rm{(H13)}]  $f(t,u,x,y)<\varphi_p(\frac{c_2}{N})$ for
  $(t,u,x,y)\in [0,1]\times [0,c_2]\times[-3N_gc_2,3N_gc_2]
   \times[-3N_hc_2,3N_hc_2]$;

\item[\rm{(H14)}] $f(t,u,x,y)\geq \varphi_p(\frac{c_3}{M})$ for
  $(t,u,x,y)\in [\delta,1-\delta]\times
  [\delta c_3,c_3]\times[-3N_gc_3,3N_gc_3]
   \times[-3N_hc_3,3N_hc_3]$.
\end{itemize}

\begin{theorem} \label{thm3.3}
Suppose {\rm (H1)--(H3)} and {\rm (H11--(H14))} hold. Then there  exist
nonnegative constants $c_1, c_2, c_3$ such that
$$
0<c_1\leq\min\Big\{\frac{g(\xi,\xi)}{1+g(\xi,0)},
  \frac{h(\eta,\eta)}{1+h(\eta,1)}\Big\}\delta c_2,\; \mathrm{and}\;
  c_2<\delta c_3<c_3
$$
Then  \eqref{e1.1} has at least two positive solutions $u_1$ and
$u_2$ such that
$$
0<\|u_1 \|<c_2\leq \|u_2 \|\leq c_3.
$$
\end{theorem}

\begin{thebibliography}{00}


\bibitem{ar}R.P. Agarwal, D. O'Regan;
\emph{Some new results for singular problems with sign changing
nonlinearities}, J. Comput. Appl. Math. 113 (2000), 1-15.

\bibitem{arls} R .P. Agarwal, D. O'Regan, V. Lakshmikantham,
 S. Leela,
\emph{Nonresonant singular boundary-value problems with sign
changing nonlinearities}, Appl. Math. Comput. 167 (2005), 1236-1248.

\bibitem{as} B. Ahmad, S. Sivasundaram;
\emph{Existence and approximation of solutions of forced duffing
type integro-differential equations with three-point nonlinear
boundary conditions}, Nonlinear Anal.: Real World Applications 11
(2010), 2905-2912.

\bibitem{alm} Qasem M. Al-Mdallal;
\emph{Monotone iterative sequences for nonlinear
integro-differential equations of second order}, Nonlinear Anal.:
Real World Applications 12 (2011), 3665-3673

\bibitem{b1} Z. Bai, Z. Gui, W. Ge;
\emph{Multiple positive solutions for some $p$-Laplacian boundary-value problems}, J. Math. Anal. Appl. 300 (2004), 477-490.

\bibitem{c1} W. Cheung, J. Ren;
\emph{Twin positive solutions for quasi-linear multi-point boundary-value problems}, Nonlinear Anal. 62 (2005), 167-177.

\bibitem{f2} H. Feng, W. Ge;
\emph{Existence of three positive solutions for m-point
boundary-value problems with one-dimensional $p$-Laplacian},
Nonlinear anal. 68 (2008), 2017-2026.


\bibitem{g1} J. M. Gallardo;
\emph{Second order differential operators with integral boundary
conditions and generation of semigroups}, Rocky Mountain J. Math. 30
(2000), 1265-1292.

\bibitem{Ge} W. Ge.
\emph{The boundary-value problems for nonlinear differential
equations}. Science Press, Beijing. 2007, 147- 228. (In Chinese)

\bibitem{gl}D. Guo, V. Lakshmikantham;
\emph{Nonlinear problems in abstract cones}, Academic Press, San
Diego, 1988.

\bibitem{g2} Y. Guo, W. Ge, S. Dong;
\emph{Two positive solutions for second order three point value
problems with sign change nonlinearities}, Acta. Math. Appl. Sinica.
27 (2004), 522-529. (In Chinese)


\bibitem{jtg}D. Ji, Y. Tian, W. Ge;
\emph{Positive solutions for one-dimensional p-Laplacian boundary-value problems with sign changing nonlinearity}, Nonlinear Anal., 71
(2009) 5406-5416.

\bibitem{jw} M. Jia, P. Wang;
\emph{Existence of positive solutions for boundary-value problems
with one-dimensional p-Laplacian}, Proceedings of the 5th
International Congress on Mathematical Biology (ICMB2011, Nanjing)
Vol. 1, 125-130.

\bibitem{kt} G. L. Karakostas, P. Ch. Tsamatos;
\emph{Multiple positive solutions of some Fredholm integral
equations arisen from nonlocal boundary-value problems}, Electron.
J. Differential Equations 30 (2002), 1-17.


\bibitem{lh} A. Lakmeche, A. Hammoudi;
\emph{Multiple positive solutions of the one-dimensional
p-Laplacian}, J. Math. Anal. Appl. 317 (2006), 43-49.

\bibitem{lg}H. Lian, W. Ge;
\emph{Positive solutions for a four-point boundary-value problem
with the $p$-Laplacian}, Nonlinear Anal., 68 (2008), 3493-3503

\bibitem{lj}S. Liu, M. Jia, Y. Tian;
\emph{Existence of positive solutions for boundary-value problems
with integral boundary conditions and sign changing nonlinearities},
Electron. J. Differential Equations, Vol. 2010(2010), No. 163, pp.
1-12.

\bibitem{l1}  A. Lomtatidze, L. Malaguti;
\emph{On a nonlocal boundary-value problems for second order
nonlinear singular differential equations}, Georgian Math. J. 7
(2000), 133-154.

\bibitem{m1} D. Ma, Z. Du, W. Ge;
\emph{Existence and iteration of monotone positive solutions for
multipoint boundary-value problem with $p$-Laplacian operator},
 Comput. Math. Appl. 50 (2005), 729-739.

\bibitem{ma} R. Ma;
\emph{Positive solutions for second-order functional differential
equations}, Dynam. Systems Appl. 10 (2001), 215--224.

\bibitem{w1} Y. Wang, C. Hou;
\emph{Existence of multiple positive solutions for one-dimensional
$p$-Laplacian}, J. Math. Anal. Appl. 315 (2006), 144-153.

\bibitem{waz} A. Wazwaz;
\emph{A comparison study between the modified decomposition method
and the traditional methods for solving nonlinear integral
equations}, Appl. Math. Comput. 181 (2006), 1703-1712.

\bibitem{wi1} J .R. L. Webb, G. Infante;
\emph{Positive solutions of nonlocal boundary-value problems
involving integral conditions}, NoDEA Nonlinear Differential
Equations Appl., 15(2008) 45--67.

\bibitem{wi2} J. R .L. Webb, G. Infante;
\emph{Positive solutions of nonlocal boundary-value problems: a
unified approach}, J. London Math. Soc. 74(2006) 673--693.

\bibitem{x1} S. Xi, M. Jia, H. Ji;
\emph{Positive solutions of boundary-value problems for systems of
second-order differential equations with integral boundary condition
on the half-line}, Electron. J. Qual. Theory Differ. Equ. 1 (2009),
1-12.

\end{thebibliography}

\end{document}