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\def\rightheadline{EJDE--2001/13\hfil Multiple positive solutions \hfil\folio}
\def\leftheadline{\folio\hfil G. L. Karakostas \& P. Ch. Tsamatos
\hfil EJDE--2001/13}
\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
Electronic Journal of Differential Equations,
Vol. {\eightbf 2001}(2001), No. 13, pp. 1--10.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break
ftp ejde.math.swt.edu (login: ftp)\bigskip} }
\topmatter
\title
Multiple positive solutions for a nonlocal boundary-value problem with
response function quiet at zero
\endtitle
\thanks
{\it Mathematics Subject Classifications:} 34B18. \hfil\break\indent
{\it Key words:} Multiple positive solutions, nonlocal boundary
value problems, \hfil\break\indent
functions quiet at zero, Krasnoselskii's fixed point theorem.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted January 23, 2001. Published February 19, 2001.
\endthanks
\author G. L. Karakostas \& P. Ch. Tsamatos \endauthor
\address G. L. Karakostas \hfill\break\indent
Department of Mathematics, University of Ioannina,
451 10 Ioannina, Greece
\endaddress
\email gkarako\@cc.uoi.gr
\endemail
\address P. Ch. Tsamatos \hfill\break\indent
Department of Mathematics, University of Ioannina,
451 10 Ioannina, Greece
\endaddress
\email ptsamato\@cc.uoi.gr \endemail
\abstract
The existence of positive solutions of a nonlocal boundary value problem for a
second order differential equation is investigated. By assuming that the
response function is quiet at zero, in a sense introduced here, and it satisfies
some easy conditions, existence results for a countable set of positive
solutions are given.
\endabstract \endtopmatter
\document
\head 1. Introduction \endhead
In a recent paper the authors gave sufficient conditions for the existence
of a positive solution of
the nonlocal boundary value problem
$$ \gather
(p(t)x')'+q(t)f(x)=0,\enskip \hbox{a.a.}\enskip t\in [0,1]\tag 1.1 \\
x(0)=0,\tag 1.2 \\
x'(1)=\int_{\eta}^{1}x'(s)dg(s)\tag 1.3
\endgather
$$
where $\eta \in (0,1),$ see [25]. Among these conditions, the monotonicity
of the response
function $f$ seemed to be crucial in the proof. In this paper we
weaken the
monotonicity condition on $f$ by assuming that this function is
quiet at zero in the
following sense: Given any pair of sequences $(x_n )$, $(y_n )$ with $0\le
x_n\le y_n$
converging to zero it holds $f(x_n )=O (f(y_n ))$, (where $O$ stands for
the big-$O$
symbol). This definition, which is introduced here, refers to functions $f$
which do not
vanish at least on $(0, +\infty )$. It is not difficult to see that if
$f(0)>0$, or, if $f$ is
increasing in a right neighborhood of zero, then $f$ is quiet at zero.
\par
Moreover we extend the results of [25] and show that our boundary value
problem can
admit a countable family of positive solutions.
Here we have to mention that boundary value problems of the form
$(1.1)$, $(1.2)$,
$(1.3)$ are mainly motivated by the works of Bitsadze [8], Bitsadze and
Samarskii [9]
and Il'in and Moiseev [23] and includes as special cases multipoint
boundary value problems
considered in [19] and [20]. Moreover, the authors in [25-28] proved
recently existence
results for some relative nonlocal boundary value problems. On the other
hand the problem
of the existence of multiple solutions (at least two) for various types of
boundary value
problems is recently the subject of many papers. Among others we refer to
[1-4, 6, 7, 10-12, 14-17, 21, 22, 24, 29, 32]. The technique in these
papers is based on
fixed point results in cones. Most of them are based on the following well
known
fixed point theorem due to Krasnoselskii [30].
\proclaim{Theorem 1.1}
Let ${\Cal B}$ a Banach space and let $\Bbb K$ be a cone in ${\Cal B}$.
Assume $\Omega _1 $,
$\Omega _2 $ are open subsets of $E$, with $0\in\Omega _1 \subset \overline
{\Omega _1
}\subset \Omega _2$, and let
$$A\colon \Bbb K\cap (\Omega _2\setminus \overline {\Omega _1 } )\to \Bbb K$$
be a completely continuous operator such that either
$$\|Au\|\le \|u\|,\enskip u\in \Bbb K \cap \partial \Omega _1 ,\enskip
\|Au\|\ge \|u\|,\enskip
u\in \Bbb K \cap \partial \Omega _2 $$
or
$$\|Au\|\ge \|u\|,\enskip u\in \Bbb K \cap \partial \Omega _1 ,\enskip
\|Au\|\le \|u\|,\enskip
u\in \Bbb K \cap \partial \Omega _2 .$$
Then $A$ has a fixed point in $\Bbb K\cap (\Omega _2\setminus \overline
{\Omega _1 } ).$
\endproclaim
In this norm form and in its degree form (see [18]), Theorem 1.1 is applied in
[1, 2, 3, 11, 14-17, 22, 24, 29, 32]. Some interesting versions of the
theorem (see, e.g.,
[5, 13, 31]) are also applied in [4, 6, 7, 12, 21]. Finally, we mention
that by using a different
fixed point theorem due to Ricceri [33] a multiplicity existence result is
obtained in [10].
\par Here we apply Theorem 1.1 to obtain existence results for a countable
set of positive
solutions of the boundary value problem $(1.1)$, $(1.2)$, $(1.3)$, where
the main hypothesis is
that the function $f$ is a quiet at zero function. This meaning is given in
the following
section.
\head 2. Quietness at zero\endhead
We introduce the following definition:
\par {\bf Definition.} A continuous function
$f:[0,+\infty )\to\Bbb R$, with $f(x)>0$ when $x>0$, is said to be quiet at
zero, if for any pair of
sequences $(x_n )$, $(y_n )$ with $0\le x_n\le y_n$, $n=1,2,...$, which
converge to zero, it
holds $$f(x_n )=O (f(y_n )).$$
\par
This means that there is a $K>0$ such that $f(x_n )\le Kf(y_n )$ for all $n$.
An equivalent form of this definition, which will be used in our proofs, is
given by the
following lemma:
\vskip 0.2cm
\proclaim{ Lemma 2.1} A continuous function $f:[0,+\infty )\to \Bbb R$,
with $f(x)>0$
when $x>0$ is quiet at zero, if and only if for each $T>0$ there is a
$\mu\ge1$ such that
for all $\tau\in(0,T)$ it holds
$$\sup\{f(x):x\in [0,\tau ]\}\le\mu\inf\{f(x):x\in [\tau ,T]\}.\tag 2.1$$
\endproclaim
\demo{ Proof}
Assume that $f$ is quiet at zero and there is a $T>0$ such that for each
positive integer $\mu$
there is a point $\tau_{\mu}\in (0,T)$, with
$$\sup\{f(x):x\in[0,\tau]\}>\mu\inf\{f(x):x\in [\tau ,T]\}.$$
This implies that there are sequences $(x_{\mu})$, $(y_{\mu})$, with
$0\mu f(y_{\mu})$ for
all $\mu$.
Since $f$ is bounded on $[0, T]$, taking limits as $\mu\to +\infty$, we get
$f(y_{\mu})\to
0$, thus $y_{\mu}\to 0$ and, so, also $x_{\mu}\to 0$. These facts
contradict to our
assumption.
\par
For the "if" part of the proof, we assume that there are sequences $(x_n
)$, $(y_n )$, with
$0\le x_n < y_n$ and $x_n\to 0$, $y_n\to 0$ and, moreover,
$$\displaystyle{\lim_{n\rightarrow\infty}}\frac{f(x_n )}{f(y_n )}=+\infty
.\tag 2.2$$
Set $T:= max\{y_n\}$. Then, by assumption, there is a $\mu\ge 1$ such that
for all $\tau\in
(0,T)$ it holds $(2.1)$. From $(2.2)$ there is an index $n_0$ such that
$f(x_{n_ {_0}})\ge
(\mu +1)f(y_{n_{_0}}).$ Set $\tau:=x_{n_{_0}}$ and observe that
$$ \aligned \sup\{f(x): x\in [0,\tau ]\}&\ge f(x_{n_{_0}})\ge (\mu
+1)f(y_{n_{_0}})\\
&\ge (\mu+1)\inf\{f(x):x\in [\tau ,T]\},
\endaligned
$$
contradicting to 2.1. The proof is complete.
\qed\enddemo
{\bf Remark.} We observe that if $f(0)>0$, then $f$ is quiet at zero.
Indeed, for each
$T>0$ the real number
$$\mu:= \frac{\sup\{f(x):x\in [0,T]\}}{\inf\{f(x):x\in [0,T]\}}$$
works in $(2.1)$ for all $\tau\in (0,T)$.
\par
Also, if $f$ is a nondecreasing in a right neihborhood of zero, then it is
quiet at zero. To see this
we assume that $f$ is nondecreasing on an interval $[0,\delta ]$ and
consider two sequences
$(x_{n}), (y_{n}),$ with $0\le x_n < y_n$ for all $n$ and $x_n\to 0$,
$y_n\to 0.$ These sequences
belong eventually in the interval $(0,\delta ],$ hence, by the monotonicity
of $f$,
$f(x_{n})\le f(y_{n})$ for all large $n$. This proves that the function $f$
is quiet at zero.
\par
\vskip 0.1cm
{\bf Example.} We give a simple example of a function, which is not quiet
at zero. Consider
sequences $(x_n )$, $(y_n )$ such that $x_n\to 0$, $y_n\to 0$, with $0
0$, when $x>0$ and
quiet at zero.
\item"(H2)" The functions $p, q$ belong to $C(I)$ and they are such that
$p>0$, $q\ge 0$ and
$\sup\{q(s): \eta \le {s}\le1\}>0$. It is clear that without loss of
generality we can assume that
$p(1)=1$.
\item"(H3)" The function $g\colon I\to \Bbb R$ is increasing and such that
$$g(\eta )=00$ such that $f_s (v)\le \theta v.$
\endroster
\par
Now we set
$$\Bbb X:=\left\{x\in C^1_0 (I):x\ge 0, x'\ge 0,
\enskip x \enskip \hbox{is concave and}\enskip \|x\| \le v.\right\}$$
and we give the following auxiliary results:
\vskip 0.2cm
\proclaim{ Lemma 3.1}
It holds $A\Bbb X\subset \Bbb X .$
\endproclaim
\demo{Proof}
Let $x\in \Bbb X$. Then $Ax(t)\ge 0$, $(Ax)'(t)\ge 0$, and
$(Ax)''(t)=-q(t)f(x(t))\le 0$
for all $t\in I$. Also, $x\in \Bbb X$ implies that $0\le x(t)\le v$ for all
$t\in I$. Then
$$
\aligned
\|Ax\|=(Ax)'(0) &=\frac{\alpha}{p(0)}\int_{\eta}^{1}\Phi
(f(x))(s)dg(s)+\frac{1}{p(0)}\int_{0}^{1}q(s)f(x(s))ds\\
&\le f_s (v)\left [\frac{\alpha H}{p(0)}+
\frac{1}{p(0)}\int_{0}^{1}q(s)ds\right ]\\
&\le \theta v\left [\frac{\alpha H +\|q\|_1}{p(0)}\right ]= v.
\endaligned
$$
\qed\enddemo
\vskip 0.2cm
\proclaim{ Lemma 3.2}
There exists a $\lambda_v >0$ such that for all $x\in \Bbb X$ it holds
$$\int_{\eta}^{1}\Phi (f(x))(s)dg(s)\ge
\frac{b_{0}}{\lambda_v}\int_{0}^{1}q(s)f(x(s))ds.$$
\endproclaim
\demo{Proof}
From the assumption $(H3)$ we have
$$g(s)\ge b_0 , \enskip s\in (\eta,1].\tag 3.1$$
Let $x\in \Bbb X$. Then $x$ is nondecreasing and $\|x\|\le v$. Since $f$ is
quiet at zero, for the
number $T_v :=v$, there is a $\mu_v\ge 1$ such that (2.1) holds for all
$\tau \in (0,T_v )$. Hence
(2.1) also holds for the real number $\tau:=x(\eta )<\|x\| \le v$.
Therefore we have
$$
\aligned
\int_{0}^{1}q(s)f(x(s))ds
&=\int_{0}^{\eta}q(s)f(x(s))ds+\int_{\eta}^{1}q(s)f(x(s))ds\\
&\le\sup_{w\in [0,\tau
]}f(w)\int_{0}^{\eta}q(s)ds+\int_{\eta}^{1}q(s)f(x(s))ds\\
&\le\frac{\int_{0}^{\eta}q(s)ds}{\int_{\eta}^{1}q(s)ds}\enskip
\frac{\sup_{w\in [0,\tau]}f(w)}{\inf_{w\in [\tau ,T_v ]}f(w)}
\int_{\eta}^{1}q(s)f(x(s))ds\\
&+\int_{\eta}^{1}q(s)f(x(s))ds\\
&\le\left (\frac{\int_{0}^{\eta}q(s)ds}{\int_{\eta}^{1}q(s)ds}\mu_v +1\right
)\int_{\eta}^{1}q(s)f(x(s))ds\\
&=\xi \int_{\eta}^{1}q(s)f(x(s))ds,
\endaligned
$$
where $$\xi:=\frac{\int_{0}^{\eta}q(s)ds}{\int_{\eta}^{1}q(s)ds}\mu_v +1.$$
Next we use (3.1) and get
$$
\aligned
\int_{0}^{1}q(s)f(x(s))ds &\le\xi\int_{\eta}^1 q(s)f(x(s))ds\\
&\le\frac{\xi}{b_0}\int_{\eta}^1 q(s)f(x(s))g(s)ds\\
&=-\frac{\xi}{b_0}\int_{\eta}^1 d\left (\int_s^1q(r)f(x(r))dr\right ) g(s)\\
&=\frac{\xi}{b_0}\int_{\eta}^1 \int_s^1q(r)f(x(r))drdg(s)\\
&\le \frac{\lambda_v }{b_0}\int_{\eta}^1 \frac{1}{p(s)}\int_{s}^1
q(s)f(x(s))drdg(s),
\endaligned
$$
where
$$\lambda_v : =\left
(\frac{\int_{0}^{\eta}q(s)ds}{\int_{\eta}^{1}q(s)ds}\mu_v +1\right
)\sup_{s\in I}p(s).$$
The proof of the lemma is complete.
\qed\enddemo
\vskip 0.2cm
Now we set
$$D:=\int_{\eta}^{1}\Phi (P)(s)dg(s),$$
$$b:=min\left \{b_0 ,\frac{ \lambda_v H}{\alpha |D\eta p(0)-H|} \right \}$$
and
$$\sigma_v := \frac {\alpha bDp(0)}{\alpha b+\lambda_v}.$$
\proclaim{ Lemma 3.3}
It holds $$\sigma_v \eta \le H.$$
\endproclaim
\demo{ Proof}
If $D\eta p(0)-H>0$, then we have $b\le\frac{ \lambda_v H}{\alpha (D\eta
p(0)-H)}.$ Solving
with respect to $H$ we obtain the result. Also, if $D\eta p(0)-H<0$, then
$$\sigma_v \eta =\frac {\alpha bp(0)\eta} {\alpha
b +\lambda_v}D<\frac {\alpha bH} {\alpha b +\lambda_v}\le H.$$
\enddemo
\head 4. Main results\endhead
In this section we present our main results. Let us first define the function
$$f_i (w):=\inf \left\{f(z):\frac{\eta\sigma_v}{H}w\le z\le w\right\} ,$$
the cone
$$\Bbb K:=\left\{x\in C^1_0 (I):x\ge 0, x'\ge 0, \enskip x \enskip
\hbox{is concave
and}\enskip \int_{\eta}^{1}\Phi (x) (s)dg(s)\ge \sigma_v \|x\|\right\}$$
and let
$$\rho :=\frac{1}{\alpha H}.$$
\proclaim {Theorem 4.1}
Consider the functions $f,p,q$ and $g$ satisfying the assumptions
$(H1)-(H5)$ and the following
one:
\par
\quad
$(H6)$ \quad There exists $u>0$ such that $u