%\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \input amstex \documentstyle{amsppt} \loadmsbm \hcorrection{18mm} \vcorrection{5mm} \nologo \TagsOnRight \NoBlackBoxes \headline={\ifnum\pageno=1 \hfill\else% {\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi} \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