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\begin{document}
		
		\title{Radial solutions \\for a nonlocal boundary value problem}
		\author{Ricardo Enguiça$^*$, Luís Sanchez$^\sharp$} 
		\date{July, 2005}
		\maketitle
		
		
		\vspace*{.5cm}
\begin{center}
$^*$Área Científica de Matemática, Instituto Superior de Engenharia de Lisboa, \\
Rua Conselheiro Emídio Navarro, 1 - 1950-062 Lisboa, Portugal.\\
E-mail : rroque@dec.isel.ipl.pt\\
  $^\sharp$ Faculdade de Ciências da Universidade de Lisboa, \\
Avenida Professor Gama Pinto 2, 1649-003 Lisboa, Portugal.\\
E-mail : sanchez@ptmat.fc.ul.pt
\end{center}
		\begin{abstract} 
			We consider the boundary value problem for the nonlinear 
			Poisson equation with a nonlocal term
				\begin{equation*} 
					-\Delta u = f\left(u,\int_U g\left(u\right)\right),\quad u|_{\partial U} = 0.
				\end{equation*}
			We prove the existence of a positive radial solution when $f$ grows linearly in $u$,
			using Krasnoselskii's fixed point Theorem together with eigenvalue theory. 
			In presence of upper and lower solutions, we consider monotone approximation to solutions.
		\end{abstract}	
					\textbf{Keywords and phrases}: Second order boundary value problem; 
				Nonlocal; Exis\-tence of positive solution; Krasnoselskii fixed point theorem; 
				Lower and Upper solutions; Monotone method.
				\\[10pt]			
				This work was  supported by Fundação para a Ciência e Tecnologia, projects 
				POCTI/Mat/57258/2004 and POCTI-ISFL-1-209  (Centro de Matem\'atica e
Aplica\c{c}\~{o}es Fundamentais). 
				
		\section{Introduction}
		
		Let us consider the following nonlocal BVP in a ball $U = B(0,R)$ of $\mathbb{R}^n$:
			\begin{equation} \label{pinic}
				-\Delta u = f\left(u,\int_U g \left(u\right)\right)
		\end{equation}
		\begin{equation}\label{cinic}
				u|_{\partial U} = 0,
		\end{equation}
		where $f$ and $g$ are continuous functions.
	%the annulus $B(0,R)\backslash \overline{B}(0,\rho)$ with $0<~\rho<~R$, or, in the case $\rho = 0$,
		 For simplicity we shall take $R=1$. We want to study the existence of positive radial solutions
			\begin{equation}
				u(x) = v(\left\|x\right\|).
			\end{equation}
		of (\ref{pinic})-(\ref{cinic}). This may be seen as  the stationary problem corresponding to a class of nonlocal evolution (parabolic) boundary value problems related to relevant phenomena in Engineering and Physics. The literature dealing with such problems has been growing in the last decade. The reader may find some hints on the motivation for the study of this mathematical model e.g. in the paper by Bebernes and Lacey \cite{jBaL}. For more recent developments, see \cite{nCmC04} and the references therein. 

Here we are considering a nonlocal term inserted in the right-hand side of the equation. Note, however, that it is also of interest to study boundary value problems where the nonlocal expression appears in a boundary condition. We refer the reader to the recent paper by Yang \cite{Y} and its references.

When dealing with a nonlinear term with rather general dependence on the nonlocal functional as in (\ref{pinic}) new difficulties arise with respect to the treatment of standard boundary value problems. Differences of behaviour which are met in general elliptic and parabolic problems are already present in simple models as those we shall analyse in this paper. For instance, the use of the powerful lower and upper solutions method (good accounts of which can be consulted in the monographs of  Pao \cite{P} and De Coster and Habets \cite{DCH}) is limited by the absence of general maximum principles. Even for linear problems with nonlocal terms the issue of positivity is far from trivial and may require a detailed study via the analysis of the Green's operator, as in Freitas and Sweers \cite{FS98}.



The purpose of this paper is twofold. First, we want to improve a quite recent result of P. Fijalkowski 
		and B.~Przeradski \cite{pFbP03}: these authors have obtained existence of positive radial solutions of (\ref{pinic})-(\ref{cinic}) by using a Krasnoselski's fixed point theorem in cones; the main assumption is that $f$ may grow at most like $Au+B$, the bound on $A$ being computed by means of a Green's function. By using a similar theoretical background, together with the consideration of the eigenvalues of the underlying linear problem, we show that an improvement of that bound is possible. This is done in Theorem \ref{normax}. Second, while remaining in the same simple general setting, we shall handle (\ref{pinic})-(\ref{cinic}) from the point of view of the upper and lower solution method. We establish a nonlocal maximum principle (Lemma \ref{princmaxseml}) and we use it as a device to obtain a monotone approximation scheme for the radial solutions of (\ref{pinic})-(\ref{cinic}) in presence of lower and upper solutions (Theorem \ref{subsobre2}). We follow an idea used by 
				D. Jiang, W. Gao, A. Wan \cite {dJwGaW02} in studying a fourth order periodic problem.


Note that we could use similar methods to  consider the case where $U~=~B(0,1)\backslash \overline{B}(0,\rho)$,
		  with $0<~\rho<~1$. Similar results could then be reached. We remark also that for special classes of functions $f$ and $g$ different approaches are needed. For instance, in \cite{jGlS} variational methods have been used to study existence and multiplicity when $f(u,v)=g(u)/v^p$ ($p>0$) and $g$ behaves as an exponencial function.

The authors wish to thank the referee for carefully reading the manuscript and hints to improve its final form.
		%A physichal motivation is behind the study of this problem.

		
			
	\section{Some auxiliary results}
	
			It is well known that the existence of a solution for some 
			boundary value problems is equivalent 
			to the existence of a fixed point of a certain operator. 
			For our purpose we need to 
			consider a second order ordinary differential equation of the form
				\begin{equation} \label{comp}
					-\left(p\left(t\right)u'\left(t\right)\right)'=p(t)f\left(t,u(t)\right),
				\end{equation}	
			with boundary conditions
				%\begin{align} \label{cfcomp1}
					%u(0)\,&=\,u(1)\,=\,0,\\ \notag
				  %&\;\;\text{ or}\\ \label{cfcomp2}
				%	u'(0)\,&=\,u(1)\,=\,0, 
			%	\end{align}
				\begin{equation}\label{cfcomp2}
					u'(0)=u(1)=0,
				\end{equation}
			where $f$ is a continuous function in $[0,1]\times \mathbb{R}$ and $p\in C[0,1]$
			is positive and increasing in $]0,1]$. 
			\\[5pt]
			If $p>0$ in $[0,1]$, it is well known that the problem is fully regular, 
			having a standard reduction to a fixed point problem:
				\[
					u=Tf(\cdot, u(\cdot)) \quad \text{in } C[0,1],
				\]
			where $T$ is the linear operator that takes $v\in C[0,1]$ into the unique solution
			$u$	of 
				\begin{equation}\label{linprob}
					-\left(p\left(t\right)u'\left(t\right)\right)'=p(t)v(t), \;\;\;u'(0)=u(1)=0,
				\end{equation}
						
			In addition we can write explicitly
				\[
					Tv(t)=\int_0^1 G(t,s)\,v(s)\,ds,
				\]
			where $G(t,s)$ is the Green's function associated to the problem. 
			The Green's function is continuous in $[0,1]\times[0,1]$ , 
			so $T$ is a completely continuous linear operator in $C[0,1]$.
			
			We are interested in the case where $p(t)>0$ in $]0,1]$ only. Under certain assumptions we still have a continuous Green's function for the linear problem (\ref{linprob}). The reader can find a more general approach in \cite{hgz}, but for completeness we include here a simple version which is sufficient for our purpose:
			
				\begin{lema}\label{l1}
				Let $p$ be continuous, increasing in $[0,1]$, $p(0)=0$ and $p>0$ in $]0,1]$.
					If the function
						\[
							p(s)\int_s^1 \frac{1}{p(\tau)}\,d\tau
						\]
					is continuously extendible to $[0,1]$,
					%and 
					%	\[
					%		\lim_{t\to 0}\frac{p(t)}{p'(t)}=0,
					%	\]
					then the operator $T\colon C[0,1]\to C[0,1]$ previously considered 
					is well defined, linear and completely continuous.
				\end{lema}
				\begin{proof}
				
					Consider the equation
					 	\begin{equation} \label{compv}
							-\left(p\left(t\right)u'\left(t\right)\right)'=p(t)v(t), 
						\end{equation}
					with boundary conditions \eqref{cfcomp2}. Integrating both sides we get 
						\[
							-p(t)u'(t)=\int_0^t p(s)v(s)\,ds.
						\]
					Integrating again, we obtain
						\begin{align*}
							u(t)&=\int_t^1 \frac{d\tau}{p(\tau)}\int_0^\tau p(s)v(s)\,ds \\
							&=\int_0^t p(s)v(s)\,ds\int_t^1 \frac{1}{p(\tau)}\,d\tau\, 
									+ \int_t^1 p(s)v(s)\,ds\int_s^1 \frac{1}{p(\tau)}\,d\tau\,  \\
							&=\int_0^1 G(t,s)\,v(s)\,ds,
						\end{align*}
				where 
					\[
						G(t,s)=
								\begin{cases}
									p(s)\int_t^1 \frac{1}{p(\tau)}\,d\tau, &t\geq s \\
									p(s)\int_s^1 \frac{1}{p(\tau)}\,d\tau, &t\leq s 
								\end{cases}
					\]
				is clearly continuous in $[0,1]\times[0,1]$, so that	
				the operator
					\[
						Tv(t)=\int_0^1 G(t,s)\,v(s)\,ds=\int_0^t p(s)\int_t^1 \frac{1}{p(\tau)}\,d\tau\, v(s)\,ds
									+ \int_t^1 p(s)\int_s^1 \frac{1}{p(\tau)}\,d\tau\, v(s)\,ds,
					\]
				is completely continuous in $C[0,1]$.
								
				It is trivial to see that $Tv(1)=0$  and if we differentiate 
				the expression for $Tv(t)$ we obtain
					\begin{align*}
						(Tv)'(t)&=p(t)\int_t^1 \frac{1}{p(\tau)}\,d\tau \,v(t) +\int_0^t -
						\frac{p(s)v(s)}{p(t)}\,ds-p(t)\int_t^1 \frac{1}{p(\tau)}\,d\tau\, v(t) \\
						&=-\int_0^t \frac{p(s)v(s)}{p(t)}\,ds,
					\end{align*}
				and thus 
					\[
						(Tv)'(0)=\lim_{t\to 0}-\int_0^t \frac{p(s)v(s)}{p(t)}\,ds=
						-\lim_{t\to 0} v(0)\frac{\int_0^t p(s)}{p(t)}=0.
					\]
				%We can now prove the complete continuity of $T$ as before since  $G(t,s)$ is continuous in
				% $[0,1]\!\!\!~\times\!\!\!~[0,1]$.
			\end{proof}

			\begin{nota}
				The continuous functions $p(t)=t^n$ with $n>0$, 
				satisfy the assumptions of the lemma.
			\end{nota}
		
			\

The following fixed point theorem of M. Krasnoselskii will be used in the next section
(see Deimling \cite{kD85}).

\
		
			\begin{teo}\label{tkrasno}
			Let $P$ be a cone in a Banach space and $S\colon P\to P$ a completely continuous operator. 
			If there exist positive constants $r<R$ such that (compression case)
				\[
					\left\|Sx\right\| \geq \left\|x\right\|, \quad \text{for all } 
					x\in P \text{ such that } \left\|x\right\|=r,
				\]
				\[
					\left\|Sx\right\| \leq \left\|x\right\|, \quad \text{for all } 
					x\in P \text{ such that } \left\|x\right\|=R,
				\]
			then $S$ has a fixed point $x$ in $P$ such that $r<\left\|x\right\|<R$.
		\end{teo}
			
	\section{Nonlinearities with linear growth in $u$: a positive solution}	
	
		Let $f\colon \mathbb{R}^+ \times \mathbb{R}\rightarrow \mathbb{R}^+$ and
		$g\colon \mathbb{R}^+ \rightarrow \mathbb{R}$ be continuous functions. The radial solutions $v$ 
		of the problem \eqref{pinic}--\eqref{cinic} solve the ordinary differential equation
			\begin{equation} \label{bvp}
			 	-v''(r)	- \frac{n-1}{r}	v'(r) = f\left(v(r),
				 \omega_n \int_0^1 s^{n-1} g(v(s))\,ds\right)	
			\end{equation}
		which is equivalent to
			\[
				-\left(r^{n-1}v'(r)\right)' = 
				r^{n-1}f\left(v(r), \omega_n \int_0^1 s^{n-1} g(v(s))\,ds\right),
			\]
		with boundary conditions
			%		\begin{equation} \label{cf1}
			%	v(\rho) = v(R) = 0 \quad \text{when}\; \rho>0
			%\end{equation}
			%			or
			\begin{equation} \label{cf2}
				\lim_{r \to 0^+} v'(r) = v(1) = 0,
			\end{equation}
		where $\omega_n$ is the measure of the unit sphere in $\mathbb{R}^n$.
	
		The homogeneous equation $-v''-(n-1)v'/r = 0$, with the boundary conditions~\eqref{cf2}, 
		has only the trivial solution, and therefore there exists a Green's function 
		associated to the linear problem. In fact, the Green's function may be written according to
		lemma \ref{l1} (see also \cite{pFbP03})
		
		\begin{itemize}
		  %\item[(i)]for $\rho>0$ and $n>2$,
		  %	\[
		  %		G(r,t)=\frac{t(R^{n-2}-\max(r,t)^{n-2})(\min(r,t)^{n-2}-\rho^{n-2})}
		  %		{(n-2)(R^{n-2}-\rho^{n-2})r^{n-2}};
		  %	\]
		  %\item[(ii)]for $\rho>0$ and $n=2$,
			%	\[
		  %		G(r,t)=\frac{t\left(\ln R- \ln\left(\max\left(r,t\right)\right)\right)
		  %		\left(\ln\left(\min\left(r,t\right)\right)-\ln \rho\right)}
		  %		{\ln R -\ln \rho};
		  %	\]
		  \item[(i)]for $n>2$,
				\[
		  		G(r,t)=\frac{t^{n-1}}{n-2}\left(\frac{1}{\max(r,t)^{n-2}}-1\right);
		  	\]	
		  \item[(ii)]and for $n=2$,
				\[
		  		G(r,t)=-t\ln\left(\max\left(r,t\right)\right).
		  	\]
		  \end{itemize}
				
		Hence the boundary value problem \eqref{bvp}--\eqref{cf2} is equivalent
		to the integral equation
		 	\begin{equation} \label{eq.int}
				v(r) = \int_0^1 G(r,t) f\left(v(t),
				 \omega_n \int_0^1 s^{n-1} g(v(s))\,ds\right)dt.
			\end{equation}
			
		In $C\left[0,1\right]$, the Banach space of continuous functions in
		$\left[0,1\right]$ with the usual norm, let $P$  be the cone of the non negative functions.
		The radial solutions of	\eqref{pinic}--\eqref{cinic} are exactly the fixed points of the 
		com\-ple\-te\-ly conti\-nuous operator
		$S\colon P~\rightarrow~P$, defined by
			\begin{equation} \label{ptfixo}	
		 		S(v)(r) = \int_0^1 G(r,t) f\left(v(t),
				 \omega_n \int_0^1 s^{n-1} g(v(s))\,ds\right)dt.
			\end{equation}
	
	In \cite{pFbP03}, the following theorem is proved:
		
		
		
		\begin{teo}\label{veror}
			Let $f\colon \mathbb{R}^+ \times \mathbb{R}\rightarrow \mathbb{R}^+$ and
			$g\colon \mathbb{R}^+ \rightarrow \mathbb{R}$ be continuous functions, and 
				\[
					\gamma=\sup_{r\in[0,1]}\int_0^1G(r,s)\,ds.
				\]
		 	Suppose there exist constants $A,B\in \mathbb{R}$ such that $0 \leq A < \gamma^{-1}$ and
		 		\[
		 			f(v,y) \leq Av + B
		 		\]
		 	for all $v \geq 0$ and $y \in \mathbb{R}$.
		 	
		 	Then the problem \eqref{pinic}--\eqref{cinic} has a positive radial solution.
		 \end{teo}	
		 	
			We will show that the 
			estimate on the constant $A$ in the previous result can be improved.
			\\[10pt]
			Consider the problem \eqref{bvp}--\eqref{cf2} and the associated eigenvalue problem:
		
				\begin{equation} \label{v.pp}
					-v''(r) - \frac{n-1}{r}v'(r) = \lambda v(r), \text{ with }
					\lim_{r\to 0^+}v'(r) = 0 \text{ and } v(1) = 0.
				\end{equation}
					
			We have
				\[
					\begin{array}{crcl}
						&\quad	-v''(r) - \frac{n-1}{r}v'(r)&= &\lambda v(r) \\ 
						%\Leftrightarrow & -r^{n-1}v''(r) -(n-1)r^{n-2}v'(r)&=& \lambda r^{n-1}v(r) \\ 
						%\Leftrightarrow &-\left(r^{n-1}v'\left(r\right)\right)'&=&\lambda r^{n-1}v(r) \\ 
						\Leftrightarrow		&\left(r^{n-1}v'\left(r\right)\right)'+ \lambda r^{n-1}v(r)&=&0.	 
					\end{array}
				\] 
					\\
			To find the eigenvalues, it is useful to consider the auxiliar initial value problem:
						
				\begin{equation} \label{paux}
					\left(r^{n-1}v'\left(r\right)\right)'+ r^{n-1}v(r) = 0,\quad 
					v(0)=1 \text{ and } v'(0)=0.
				\end{equation}
			The solution $v$ to this problem is well defined in $[0,+\infty[$, 
			oscillates, and has zeros $\left\{\xi_n \mid n\in \mathbb{N}\right\}$ such that 
			 $0<\xi_1<\xi_2<\ldots\rightarrow+\infty$, 
			 with $\xi_{n+1}-\xi_n\rightarrow \pi$ (see \cite{wW98}).
			 
			 Define $u(r) = v(\beta r)$. Then
			 	\[
			 		u'(r) = \beta v'(\beta r) \qquad \text{and} \qquad
			 		u''(r) = \beta^2 v''(\beta r).
			 \]
			 Using \eqref{paux} we have
			 \[
			 	\begin{array}{crcl}
			 		& (n-1)(\beta r)^{n-2}v'(\beta r) + (\beta r)^{n-1}v''(\beta r) 
			 			+(\beta r)^{n-1}v(\beta r) &=&0 \\[7pt]
			 		%\Leftrightarrow & (n-1)r^{n-2} \frac{v'(\beta r)}{\beta} + 
			 		%	r^{n-1}v''(\beta r) + r^{n-1}v(\beta r) &=&0 \\[7pt]
			 		%\Leftrightarrow & (n-1)r^{n-2}\frac{u'(r)}{\beta^2}
			 		%	+ r^{n-1}\frac{u''(r)}{\beta^2} + r^{n-1}u(r) &=&0 \\[7pt]
			 		 &\Leftrightarrow \left(r^{n-1}u'\left(r\right)\right)' +
			 		\beta^2 r^{n-1} u(r) &=&0.
			 	\end{array}\\[10pt]
			 \]
			 
		It is obvious that $u'(0)=0$, so it remains to find $\beta$ such that $u(1)=0$.
		As $u(1)=v(\beta)$, we get $\beta = \xi_n$ 
		for some $n \in \mathbb{N}$, hence $\beta = \xi_n$ 
		and, therefore, the eigenvalues of \eqref{v.pp} are 
			\[
				\lambda_n = \beta^2 ={\xi_n}^2.
			\]
				\\
		Let us identify the zeros of the unique solution of \eqref{paux}. We have
			\[
				\begin{array}{crcl}
					& \left(r^{n-1}v'\left(r\right)\right)'+ r^{n-1}v(r) &=& 0 \\
				%	\Leftrightarrow & (n-1)r^{n-2}v' + r^{n-1}v'' + r^{n-1}v &=&0 \\
					\Leftrightarrow & r^{n-3}\left(r^2 v'' + (n-1)rv' + r^2v\right) &=&0, 
				\end{array}
			\]
		and the last equation has the form
			\[
				t^2u'' +atu'+ \left(b+ct^m\right)u = 0,
			\]
		which is easily reduced to a Bessel equation (cf.\cite{vS70}). Using the new independent variable
			\[
				y = r^{\frac{n-2}{2}}v
			\]
		we obtain the transformed equation
			\[
				r^2y'' + ry' + \left(r^2 - \left(\frac{n-2}{2}\right)^2\right)y = 0,
			\]	
		whose solutions are well known, and thus we get:
			\[
				\begin{array}{ll}
					(i) \quad v(r)= c_1 r^{-\frac{n-2}{2}}J_{\frac{n-2}{2}}(r)+
						c_2 r^{-\frac{n-2}{2}} K_{\frac{n-2}{2}}(r)
						& \text{if}\,\,n\,\, \text{is even, \,or}\\[15pt]
					(ii) \quad v(r)= c_1 r^{-\frac{n-2}{2}}J_{\frac{n-2}{2}}(r)+
						c_2 r^{-\frac{n-2}{2}} J_{\frac{2-n}{2}}(r)
						& \text{if}\,\,n\,\, \text{is odd,}
				\end{array}
			\]
		where $c_1$, $c_2$ are constants and $J_{i}$, $K_{i}$
		are Bessel functions of order $i$, of the first and the second kind respectively.
		
		Taking into consideration the boundary conditions,  
		the constant $c_2$ must be zero in both cases
		(otherwise we would have $\lim_{r \to{0^+}}v(r) = \infty$), so that
			\[
				v(r)= c_1 r^{-\frac{n-2}{2}}J_{\frac{n-2}{2}}(r).
			\] 
		For our boun\-dary value problem we know 
		that $\gamma^{-1}~=~2n$ (see \cite{pFbP03}). If we compare $\sqrt{2n}$ 
		with $\xi_1$ -- the zeros of these Bessel functions are well known~-- we can see that
			\[
		 		\sqrt{2n} < \xi_1
			 \] 
		 and hence,
		 	\[
		 	\gamma^{-1} < \lambda_1 \quad \text{(first eigenvalue of \eqref{v.pp}).}
		 \]
		 	
		 For instance, for $n=2$ or $n=4$ we have
		 \begin{align}
		 	 \sqrt 4=2,000 \quad & \notag < \quad \xi_1 (J_0)\approx2,404, \\ \notag
		 	 \sqrt 8\approx2,828 \quad & < \quad \xi_1 (J_1)\approx3,832. %\\ \notag
		 	 %n=6\rightarrow \sqrt 12=3,464 \quad & <\quad \xi_1 (J_2)=5,135, \\ \notag
		 	 %n=8\rightarrow \sqrt 16=4,000 \quad & < \quad \xi_1 (J_3)=6,379, \\ \notag
		 	 %n=10\rightarrow \sqrt 20=4,472 \quad & < \quad \xi_1 (J_4)=7,586, \\ \notag
		 	 %n=12\rightarrow \sqrt 24=4,899 \quad & < \quad \xi_1 (J_5)=8,780.
		 \end{align}
		 By adapting the approach of \cite{pFbP03}
we shall prove the following improved version of theorem \ref{veror}:
			
			\begin{teo}\label{normax}
			Let $f\colon \mathbb{R}^+ \times \mathbb{R}\rightarrow \mathbb{R}^+$ and
			$g\colon \mathbb{R}^+ \rightarrow \mathbb{R}$ be continuous functions, 
			and $\lambda_1$ defined as above.
			
			Suppose there exist constants $A,B\in \mathbb{R}$ such that $0 \leq A <  \lambda_1$, and
		 		\[
		 			f(v,y) \leq Av + B, \quad  \text{ for all $v \geq 0$ and $y \in \mathbb{R}$}.
		 		\]
		 			 	
		 	Then the problem \eqref{pinic}--\eqref{cinic} has a positive radial solution.
		\end{teo}	 

		
		Let $\phi$ be an eigenfunction associated with the first eigenvalue $\lambda_1$. We have
			\begin{equation}
				-\phi'' - \frac{n-1}{r}\phi' = \lambda_1 \phi, \text{ and } \phi'(0)=0=\phi(1).
			\end{equation}
				
		Since our computation above shows that we may assume that $\phi(t)=v(\xi_1r)$ 
		where $v(r)=r^{-\frac{n-2}{2}}J_{\frac{n-2}{2}}(r)$, it is clear that $\phi>0$ in $[0,1[$, 
		(and, by the way, $\phi'(1)<0$). We may therefore consider the norm
			\[
				\left\|v(r)\right\|_X = \sup_{[0,1[}\frac{|v(r)|}{\phi(r)},
			\]
			in the Banach  space 
			\[
				X=\left\{v \in C([0,1]) : \frac{|v(r)|}{\phi(r)} \:\text{bounded}\right\}.
			\]
			Then, as stated before, we can write problem \eqref{bvp}--\eqref{cf2} as 
		$v=Sv$, where
			\begin{equation} \label{kf}
				S(v)(r) = \int_0^1 G(r,t) f\left(v(t),
				 \omega_n \int_0^1 s^{n-1} g(v(s))\,ds\right)dt,\text{ for } v\in X
			\end{equation}
				
				Let $T$ denote the operator introduced in section 2, with $p(s)=s^{n-1}$. This operator acts in $C[0,1]$. Let $K$ be the restriction of $T$ to $X$ and
				$v \in X$. Since 
					\[
						\left|v(t)\right|\leq \left\|v\right\|_X\phi (t),
					\]
				and
					\[
						\int_0^1 G(r,t) \phi (t)\,dt=\frac{\phi(r)}{\lambda_1},
					\]
				we have
				\[
					\left|K\left(v\right)(r)\right| \leq \int_0^1 G(r,t) 		
					\left|v(t)\right|\,dt
					\leq \left\|v\right\|_X \int_0^1 G(r,t) \phi (t)\,dt
				\]
			so that
				\[
					\frac{|K(v)(r)|}{\phi (r)} \leq \frac{\left\|v\right\|_X}{\lambda_1}.
				\]
			Taking the least upper bound in the left hand side of last inequality, we obtain
				\begin{equation}\label{ineK}
					\left\|K(v)\right\|_X \leq \frac{\left\|v\right\|_X}{\lambda_1}.
				\end{equation}
This estimate, which is the main reason to work in the functional space $X$, will be used in the proof of theorem 3.2 in a crucial way.
							
			\begin{lema}\label{compac}
					The operator $S\colon X\to X$ is completely continuous.
			\end{lema}
			\begin{proof}
				Since the imbedding $i_1 \colon X \to C[0,1]$ is continuous, the
				Nemytskii operator
				$N\colon X\to C[0,1]$ given, for each $v\in X$, by
					\[
						N(v)=f\left(v, \omega_n\int_0^1s^{n-1}g(v(s))\,ds\right),
					\]
				is continuous. Moreover it takes bounded sets into bounded sets.
				
				Now let us consider the following decomposition of $T$:
					\begin{equation}
						C[0,1]\stackrel{T_*}{\longrightarrow} C^2_*[0,1]
						\stackrel{i_2}{\longrightarrow} C^1_*[0,1]\stackrel{i_3}{\longrightarrow} X,
					\end{equation}
				where $$C^2_*[0,1]=\left\{u\in C^2[0,1]\colon u'(0)=u(1)=0\right\},$$ 
				$$C^1_*[0,1]=\left\{u\in C^1[0,1]\colon u(1)=0\right\},$$ 
				$i_2,i_3$ are imbeddings, and $T_*$ is the
				operator $T$ acting between those two spaces.	
				
				The operator ${\left(T_*\right)}^{-1}$ takes $u$ into $-u''-\frac{(n-1)}{r}u'$;
				it is obviously linear continuous and bijective 
				and, therefore, using the Open Map Theorem, we get that	$T_*$ is conti\-nuous. 
				The imbedding $i_2$ is a well known completely continuous operator
				and using L'Hospital's rule we can prove that $i_3$ is also conti\-nuous. 
				Since $S=i_3i_2T_*i_1$ the conclusion of the lemma is now straightforward.
			\end{proof}
					
		
		\begin{proof}[Proof of theorem \ref{normax}]
The proof is similar to that of theorem \ref{veror} and so we only outline it.
			If $f(0,\omega_n \frac{g(0)}{n})= 0$, then $v\equiv 0$ 
			is obviously a fixed point of the operator~$S$,
			so let us suppose that $f(0,\omega_n \frac{g(0)}{n})> 0$.
			Then there exist positive constants $M$ and $\delta$ such that
				\[
					f\left(v(t), \omega_n \int_0^1 s^{n-1} g(v(s))\,ds\right)\geq M,
					\quad \text{for all } \left\|v\right\|_X \leq \delta.
				\]
			A simple computation yields
							 
					\[
					\left\|Sv\right\|_X \geq M 
					\sup_{r \in ]0,1[}\int_0^1 \frac{G(r,t)}{\phi(r)}\,dt
					= M\epsilon,
				\]
			if $\left\|v\right\|_X \leq \delta$, where we have set
			 $\epsilon:=\sup_{r \in ]0,1[}\int_0^1 \frac{G(r,t)}{\phi(r)}\,dt$.
			 
			 \

			If we define $\Omega_1 = \left\{v \in X \mid \left\|v\right\|_X
			< \min(M\epsilon/2,\delta)\right\}$, in $\partial\Omega_1$ we have
			$$\left\|Sv\right\|_X \geq M\epsilon\!>\left\|v\right\|_X.$$
			
			Defining $\Omega_2 = \left\{v \in X \mid \left\|v\right\|_X 
			< \left\|TB\right\|_X/(1-A'/\lambda_1)\right\}$ with $A<A'<\lambda_1$, then for 
			$v\in P\cap \partial \Omega_2$ we have (using the positivity of  $T$ and the estimate (\ref{ineK}))
			 \begin{align*}
			 	\left\|Sv\right\|_X 			 	&\leq \left\|T(Av + B)\right\|_X
			 	\leq \left\|AKv\right\|_X + \left\|TB\right\|_X \\ 
			 	&<\frac{A'/\lambda_1\left\|TB\right\|_X}{1-A'/\lambda_1}
			 	+ \frac{\left\|TB\right\|_X-A'/\lambda_1\left\|TB\right\|_X}{1-A'/\lambda_1}
			 	=\left\|v\right\|_X.
			 \end{align*}
		%Here we have used the estimate that preceedes lemma ?????REF
		%	Note that despite the fact that$B$ does not belong to $K$, 
		%	the function $KB$ exists and belongs to $X$. 
				
			Applying Krasnoselskii's fixed point Theorem \ref{tkrasno} (compression version) 
			we find a fixed point of $S$,
			and therefore a positive radial solution of
			\eqref{pinic}--\eqref{cinic}.
			
		\end{proof}
		
		In both theorems above, as mentioned in \cite{pFbP03}, 
		the condition on $f$ does not depend on the second variable, 
		and, therefore, nothing is restraining
		the behaviour of~$g$. The arguments used there are also valid for the same problem with
		$f(v(r),\alpha(v))$, for any continuous functional $\alpha$ in $X$.
		
		A similar procedure allows us to us prove a result in the spirit of the one considered in \cite{pFbP03}
where $g$ is restrained, but the condition on $f$ is weakened:
		
		\begin{teo}
			Let $f\colon \mathbb{R}^+ \times \mathbb{R}\rightarrow \mathbb{R}^+$ and
			$g\colon \mathbb{R}^+ \rightarrow \mathbb{R}$ be continuous functions.
			
			Suppose there exist positive constants $A<\lambda_1$, $B$,
		 	$C$, $D$, $p$ and $q$ with $pq \leq 1$ such that
		 		\begin{equation*}
		 			f(v,y) \leq Av + B + C\left|y\right|^p 
		 			\quad \text{for all}\quad v \geq 0 \text{ and } y\in \mathbb{R} 
		 		\end{equation*}and
		 		\begin{equation*}
		 			\left|g(v)\right| \leq D \left|v\right|^q
		 			\quad \text{for all}\quad v\in\mathbb{R}
		 		\end{equation*}
		 	where $\phi$ is the eigenfunction associated with $\lambda_1$.
		 	
		 	Then the problem \eqref{pinic}--\eqref{cinic} has a positive radial solution.
		\end{teo}
		 
		%\begin{proof}
		 	%The proof is an analogue of that of Theorem~\ref{normax}, as we only have to change 
		 	%the definition of the set $\Omega_2$.
		 			 	
		 	%Before we define $\Omega_2$, note that, from \eqref{kf}, if we set 
		 	%$D_1=D\left(\sup\phi\right)^q$,
		 	
		 	
		 	%	\begin{align}
		 	%		\left\|Sv\right\|_X &= \notag
			%		\left\|T\left[f\left(v(t), \omega_n \int_0^1
			%		 s^{n-1} g(v(s))\,ds\right)\right]\right\|_X \\[8pt] \notag
			%		& \leq \left\|T\left[Av + B +
			%			C\left(\omega_n \int_0^1
			%		 	s^{n-1} |g(v(s))|\,ds\right)^p\,\right]\right\|_X \\[8pt] \notag
			%		&\leq \frac{A}{\lambda_1}\left\|v\right\|_X +
			%			\left\|TB\right\|_X +
			%			\left\|T\left[C {\omega_n}^p
			%			\left(\int_{0}^1 s^{n-1} D_1 
			%			\left(\frac{\left|v(s)\right|}{\phi(s)}\right)^q\,ds
			%			\right)^p\,\right]\right\|_X \\[8pt] \notag
			%		&\leq \frac{A}{\lambda_1}\left\|v\right\|_X +
			%			\left\|TB\right\|_X +
			%			\left\|T\left[C \omega_{n}^p
			%			D_1^p {\left\|v\right\|_X}^{pq}\left(\frac
			%			{1}{n}\right)^p\,\right]\right\|_X \\[8pt] \notag
			%		&\leq \frac{A}{\lambda_1}\left\|v\right\|_X +
			%			\left\|TB\right\|_X +
			%			\left\|T\left[C \omega_{n}^p
			%			D_1^p \left(\frac
			%			{1}{n}\right)^p\,\right]\right\|_X
			%			{\left\|v\right\|_X}^{pq} \\[8pt] \notag
			%		&\leq \left(\frac{A}{\lambda_1} +
			%			\left\|TB\right\|_X
			%			{(\left\|v\right\|_X)}^{-1} +
			%			\left\|T\left[C \omega_{n}^p
			%			D_1^p \left(\frac
			%			{1}{n}\right)^p\,\right]\right\|_X
			%			{\left\|v\right\|_X}^{pq-1}\right)
			%			\left\|v\right\|_X.
			%	\end{align}
			%Hence, for $M$ large enough (considering the fact that $A/\lambda_1<1$),
			%if $\left\|v\right\|_X =M$, then
			
			%	\[
			%		\left(\frac{A}{\lambda_1} +
			%		\left\|TB\right\|_X
			%		(\left\|v\right\|_X)^{-1} +
			%		\left\|T\left[C \omega_{n}^p
			%		D_1^p \left(\frac
			%		{1}{n}\right)^p\,\right]\right\|_X
			%		{\left\|v\right\|_X}^{pq-1}	\right)
			%		<1.
			%	\]
			%Defining $\Omega_2=\left\{v \in X
			%\mid \left\|v\right\|_X<M\right\}$, the result follows as before.
		%\end{proof}
	
		\begin{nota}
			We could have considered in equation (\ref{bvp}) a right-hand side of the form\\
			$f\left(r,v(r),\omega_n\int_0^1s^{n-1}g(v(s))\,ds\right)$, continuous 
			in $[0,1]\times\mathbb{R}\times\mathbb{R}$. Indeed we might even work with a nonlinear
			non-negative function $f(r,v,w)$ continuous in $(v,w)$ for  a.e. $r\in[0,1]$, and
			measurable in $r$ for all $(v,w)\in \mathbb{R}\times\mathbb{R}$. However in this case, defining
				\[
					L_k^p(0,1)=\left\{u \colon u \text{ is measurable in } ]0,1[,\,
					\int_0^1s^k\left|u(s)\right|\,ds<+\infty\right\}
				\]
				we should
			confine ourselves to $L_{n-1}^p(0,1)$ Carathéodory functions $f$, i.e.
				\[
					\forall M>0 \;\sup_{\left|v\right|+\left|w\right|\le M}\left|f\left(\cdot,v,w\right)\right|\in
					L_{n-1}^p(0,1),\] where $p>n$ is fixed.
					
			Under this restriction, it can still be shown that the analogue of Lemma \ref{compac} holds,
			because we can obtain an analogue of $T$ acting compactly from $L_{n-1}^p(0,1)$
			to $C^1_*[0,1]$.					
		\end{nota} 
	
	\section{Lower and upper solutions and monotone approximation}
	
		We will now apply the lower and upper solution 
		method to find solutions of the boundary value problem
		\mbox{\eqref{bvp}--\eqref{cf2}}. 
		%(we are only going to use $R=1$, 
		%but the results also stand for any $R>0$) 
		
		We should point that in \cite{P} (p.695) a monotone method approach using lower and upper solutions
		is applied to an epidemic problem with diffusion. The problem considered in there is a second order
		system  of two PDE with a nonlocal term, under assumptions related to those we use below
		(in particular a Lipschitz condition) and where uniqueness is obtained as well.
		
		We will use two different types of conditions 
		concerning the given functions $f$ and $g$, and construct monotone convergent sequences 
		to solutions of the	problem.
	
		
		Let us define the linear operator 
			\[
				Lu(r)=-u''(r)-\frac{n-1}{r}u'(r)+\lambda u(r).
			\]
%		or its equivalent form (multiplying by $r^{n-1}$)
%			\[
%				L^*u(r)=-\left(r^{n-1}u'(r)\right)' +\lambda r^{n-1} u(r),
%				\text{ with } r\in\,]0,1].
%			\]
%		\begin{nota}
%			In the sequel, we shall call both operators $L$, as the statements 
%			we will make are allways true for both of them. 
%		\end{nota}
		
		\begin{lema}\label{princmaxL}
			Let $\lambda\geq0$, and $u\in C^1[0,1]\cap C^2]0,1[$ 
			be such that $Lu(r)\geq 0$ in $]0,1]$, $u'(0)\leq0$ and $u(1)\geq0$. 
			Then $u(r)\geq 0$ for all $r\in[0,1]$.
		\end{lema}
		
		\begin{proof}
			Towards a contradiction, assume that $u(r_0)<0$ for some $r_0\in\,]0,1[$. There are two cases to consider:
				\begin{itemize}
					\item[(i)] $u(r)<0$ in some interval $]c,d[\subset[0,1]$, with $u(c)=u(d)=0$.
					
						Let us consider first the case where $\lambda>0$.
						Then there must exist $p\in \,]c,d[$ such that 
						$u'(p)=0$, and $u''(p)\geq 0$, and since
						$u(p)<0$, we get $Lu(p)<0$, which is a contradiction.
						
						If $\lambda=0$, integrating in $[c,d]$, we get the contradiction
							\[
								0<d^{n-1}u'(d)-c^{n-1}u'(c)\leq 0.
							\]  
					\item[(ii)] $u(r)<0$ in some interval $[0,c[\subset[0,1]$, with $u(c)=0$.
				
						If $u'(0)<0$, the same argument applies.
						If $u'(0)=0$, integrating in $[0,c]$, we get
							\[
								0>-c^{n-1}u'(c)+\lambda\int_0^c r^{n-1}u(r)\,dr\geq 0.
							\]
					\end{itemize}
			\end{proof}

		From now on we assume that $f\colon \mathbb{R} \times \mathbb{R}\rightarrow \mathbb{R}$ and
		$g\colon \mathbb{R} \rightarrow \mathbb{R}$ are continuous functions.
\\
			Consider the boundary value problem 
				\begin{equation}\label{fcomr}
						-u''(r)-\frac{n-1}{r}u'(r)=f\left(u(r),\omega_n
						\int_0^1 s^{n-1}g\left(u(s)\right)\,ds\right)
						\text{ for }0<r\leq1, 
				\end{equation}
			and
				\begin{equation}\label{fcomr2} 
							u'(0)=0=u(1).
				\end{equation}
		We say that $\alpha(r)$ is a \emph{lower solution} of 
		\eqref{fcomr}--\eqref{fcomr2} if
			\[
				-\alpha''(r)-\frac{n-1}{r}\alpha'(r)\leq f\left(\alpha(r),\omega_n\int_0^1 s^{n-1}
				g\left(\alpha\left(s\right)\right)\,ds\right),\,\text{ for }0<r\leq1,
			\]
			\[
				\alpha'(0)\geq 0 \text{ and } \alpha(1)\leq 0.
			\]
		A function $\beta$ satisfying the reversed inequalities is called an \emph{upper solution}.
		
		Let $\alpha_0$ be a lower solution and $\beta_0$ an upper solution of
		 \eqref{fcomr}--\eqref{fcomr2}.
		Consider the restriction $L_0$ of the operator $L$ to the subspace 
		$\left\{u\in C^1[0,1]\cap C^2]0,1[ \colon u'(0)=u(1)=0\right\}$.
		%Then we obviously have 
		%	\begin{align*}
		%		&L\alpha_0\leq f\left(r,\alpha_0,\omega_n\int_0^1 s^{n-1}
		%		g\left(\alpha_0\left(s\right)\right)\,ds\right)~+~\lambda \alpha_0, \text{ and}\\
		%		&L\beta_0\geq f\left(r,\beta_0,\omega_n\int_0^1 s^{n-1}
		%		g\left(\beta_0\left(s\right)\right)\,ds\right)~+~\lambda \beta_0.
		%	\end{align*}
			With the notations above, to get a solution of the problem \eqref{fcomr}--\eqref{fcomr2} 
			is equivalent to find	a
			fixed point of the completely continuous operator in $C[0,1]$
				\[
					\Phi u\equiv L_0^{-1}\left(f\left(u,\omega_n\int_0^1 s^{n-1}
					g\left(u\left(s\right)\right)\,ds\right)+\lambda u\right).
				\]
		Let us define 
			\[
				R_f(u,v_1,v_2)=\frac{f(u,v_2)-f(u,v_1)}{v_2-v_1}\quad\text{ and}\quad 
				R_g(u_1,u_2)=\frac{g(u_2)-g(u_1)}{u_2-u_1}.
			\]	
				
		\begin{lema}\label{subsobre}
Let $\alpha_0$ be a lower solution and $\beta_0$ an upper solution of
		 \eqref{fcomr}--\eqref{fcomr2} such that $\alpha_0\le\beta_0$ in $[0,1]$.			
Suppose $f(u,v)$ is such that
				\[
					f(u_2,v)-f(u_1,v)\geq -\lambda (u_2-u_1),
				\]  
			for some  $\lambda\geq 0$,
			$v\in \mathbb{R}$, $u_1,u_2 \text{ such that for some} \,\,r\in[0,1], \,\,
			\alpha_0(r)\leq u_1\leq u_2\leq \beta_0(r)$, and 
			%\frac{\partial f}{\partial u}\geq -\lambda, \text{ for some } \lambda\geq 0,\, and 
			$R_f,R_g$ have the same sign
			for all $u_1,u_2$ such that
			\mbox{$\alpha_0(r)\leq u_1, u_2\leq\beta_0(r)$ for some $r\in[0,1]$}.
			%$\forall u,u_1,u_2,v_1,v_2,\, \text{ with } u_1\leq u_2,\,v_1\leq v_2.$
			%\frac{\partial f}{\partial v}\cdot\frac{\partial g}{\partial u}\geq 0.
				
			Then, for any two functions $u_1(r),\,u_2(r)\in C[0,1]$ such that 
				\[
					\alpha_0(r)\leq u_1(r)\leq u_2(r)\leq\beta_0(r), 
				\]
			we have
				\[
					\Phi u_1(r)\leq \Phi u_2(r).
				\]
		\end{lema}
		\begin{proof}
			The Green's function $G_\lambda$ associated with $L_0$ is non-negative 
			according to lemma \ref{princmaxL}. We have
				\begin{align*}
					&\Phi u_2(r)-\Phi u_1(r)=\\
					%&=\int_0^1G_\lambda(r,t)
					%\left[f\left(u_2,\omega_n\int_0^1 s^{n-1}
				%g\left(u_2\left(s\right)\right)\,ds\right)+\lambda u_2\right] dt\,-\\
			%&\qquad\qquad\qquad\qquad\qquad\quad\:\,
			%-\int_0^1 G_\lambda(r,t)\left[f\left(u_1,\omega_n\int_0^1 s^{n-1}
			%	g\left( u_1\left(s\right)\right)\,ds\right)+\lambda u_1\right]\,dt=\\
			&=\int_0^1\!G_\lambda(r,t)\!
				\left[f\!\left(u_2,\omega_n\!\int_0^1 s^{n-1}
				g\left( u_2\left(s\right)\right)\,ds\right)\!\!-\!\!
					f\!\left(u_1,\omega_n\!\int_0^1 s^{n-1}
				g\left( u_2\left(s\right)\right)\,ds\right)\right]dt+\\
			& +\int_0^1G_\lambda(r,t)\!
				\left[f\!\left(u_1,\omega_n\!\int_0^1 s^{n-1}
				g\left( u_2\left(s\right)\right)\,ds\right)\!\!-\!\!
				f\!\left(u_1,\omega_n\!\int_0^1 s^{n-1}
				g\left( u_1\left(s\right)\right)\,ds\right)\right]dt+\\
			&\qquad\qquad\qquad\qquad\qquad\qquad+\int_0^1G_\lambda(r,t)
				\lambda\left( u_2- u_1\right)\,dt\geq\\
			&\geq \int_0^1G_\lambda(r,t)\left[-\lambda\left( u_2- u_1\right)+
				\lambda\left( u_2- u_1\right)\right]\,dt\geq0.
				\end{align*}
				
		\end{proof}		
		\begin{nota}
			Clearly if $f$ and $g$ are $C^1$ functions, the hypotheses
			of the last theorem are
			satisfied provided that 
				\[
					\frac{\partial f}{\partial u}\geq -\lambda, \text{ and}\quad
					\frac{\partial f}{\partial v},\frac{\partial g}{\partial u}
					\text{ have the same sign}.
				\]
		\end{nota}	
		\begin{teo}\label{teosubsobre}
			Suppose that $f$ and $g$ satisfy the assumptions of lemma \ref{subsobre}.
			Let $\alpha_0,\beta_0$ be lower and upper solutions, respectively, of 
			 \eqref{fcomr}--\eqref{fcomr2}. If we put
				\[
					\alpha_{n+1}=\Phi\alpha_n\quad and \quad
					\beta_{n+1}=\Phi\beta_n,\quad\text{for all } n\in\mathbb{N}_0,
				\]
			we obtain
				\[
					\alpha_0\leq\alpha_1\leq\cdots\leq\alpha_n\leq\cdots\leq\beta_n\leq\cdots\leq
					\beta_1\leq\beta_0.
				\]
				
			The monotone bounded sequences $\left(\alpha_n\right)_{n\in\mathbb{N}_0}$ and
			$\left(\beta_n\right)_{n\in\mathbb{N}_0}$ defined above are convergent in $C[0,1]$
			respectively to the minimal and maximal solutions of \eqref{fcomr}--\eqref{fcomr2} in the interval $[\alpha_0,\beta_0]$.
		\end{teo}		
		\begin{proof}
			Since
				\[
					L\alpha_1=f\!\!\left(\alpha_0,\omega_n\!\!\int_0^1\!\! s^{n-1}
					g\left(\alpha_0\left(s\right)\right)\,ds\!\!\right)\!+\!\lambda\alpha_0,
			\text{ and }\;
					L\alpha_0\leq f\!\!\left(\alpha_0,\omega_n\!\!\int_0^1\!\! s^{n-1}
					g\left(\alpha_0\left(s\right)\right)\,ds\!\!\right)\!+\!\lambda\alpha_0, 
				\]
			we have $$L\left(\alpha_1-\alpha_0\right)\geq 0,\quad 
			\left(\alpha_1-\alpha_0\right)'(0)\leq 0,\;\; \left(\alpha_1-\alpha_0\right)(1)\geq 0,$$
			and therefore, by lemma \ref{princmaxL}, we have $\alpha_0\leq \alpha_1$.
			
			
			Using similar argumets, we can prove that $\alpha_1\leq \beta_0$. 
			
			%In fact,
			%	\begin{align*}
				%	L(\beta_0-\alpha_1)&=f\left(\beta_0,\omega_n\int_0^1 s^{n-1}
				%	g\left(\beta_0\left(s\right)\right)\,ds\right)+\lambda\beta_0\,-\\
				%	&\;\;-f\left(\alpha_1,\omega_n\int_0^1 s^{n-1}
				%	g\left(\alpha_1\left(s\right)\right)\,ds\right)-\lambda\alpha_1\geq 0, \text{ and}
				%\end{align*}
				%\[
				%	\left(\beta_0-\alpha_1\right)'(0)\leq 0,\;\; \left(\beta_0-\alpha_1\right)(1)\geq 0.
				%\]
			We are now able to apply Lemma \ref{subsobre} to $\alpha_0$ and $\alpha_1$
			which gives $\alpha_1\leq \alpha_2$. 
			By iteration of this argument, we prove that
			$\left(\alpha_n\right)_{n\in\mathbb{N}_0}$ is an increasing sequence 
			and stays below $\beta_0$.
			Analogously, we prove that 
			$\left(\beta_n\right)_{n\in\mathbb{N}_0}$ is a decreasing sequence so that
			\[
				\alpha_0\leq\alpha_1\leq\cdots\leq\alpha_n\leq\cdots\leq\beta_n\leq\cdots\leq
				\beta_1\leq\beta_0.
			\]
	
			Concerning the convergence of the sequences, 
			as the cone of positive functions in $C[0,1]$ is normal 
			(since $0\leq u\leq v$ implies $\left\|u\right\|\leq \left\|v\right\|$),
			we can use the standard argument (\cite{eZ86}, p.283), which
			gives the convergence of this iteration method to fixed points of $\Phi$, and
			these are exactly the smallest and largest fixed points 
			in $[\alpha_0,\beta_0]\subset C[0,1]$.
		\end{proof}				
						
		\begin{expl}
			Let us consider the non-local differential equation
				\begin{equation}\label{eqex}
					-u''(r)-\frac{2}{r}u'(r)=\frac{4}{3}\pi e^u\int_0^1 s^2\left(u(s)+1\right)\,ds
				\end{equation}
			with boundary conditions $u'(0)=u(1)=0$.
			
			In this case we have $n=3$, $f(u,v)=e^uv$, and $g(u)=u+1$.
			
			Consider $\alpha_0\equiv0$ and $\beta_0=1-r$. Then
				\[
					-\alpha_0''(r)-\frac{2}{r}\alpha_0'(r)=0\leq
					\frac{4}{9}\pi=\frac{4}{3}\pi e^0\int_0^1s^2\,ds
				\]
			and $\alpha_0'(0)=\alpha_0(1)=0$, so $\alpha_0$ is a lower solutions of
			\eqref{eqex}--\eqref{fcomr2}. 
			
			For $r\in[0,1]$ we have
				\[
					-\left(r^2\beta_0'\right)'=2r\geq 
					\frac{5}{9}\pi r^2 e^{1-r}=\frac{4}{3}\pi r^2e^{1-r}\int_0^1s^2\left(1-s+1\right)\,ds,
				\]
			$\beta_0'(0)=-1$ and $\beta_0(1)=0$. Therefore $\beta_0$ is an upper solution
			of \eqref{eqex}--\eqref{fcomr2}.
		
			The conditions in the Theorem \ref{subsobre} are satisfied for $\alpha_0$ and $\beta_0$,
			so there exists a solution $u$ of \eqref{eqex}--\eqref{fcomr2}, such that
				\[
					0\leq u(r)\leq 1-r, \text{ for all }r\in[0,1].
				\]

			This solution is the limit of a monotone sequence constructed as in the statement of the theorem.
		\end{expl}

\

		
		Let us now try another approach using the lower and upper solutions method,
		where we drop a part of the monotonicity assumptions.
		
		\begin{lema}\label{princmaxseml}
			Suppose that $u\in C^1[0,1]\cap C^2]0,1[$ satisfies
				\[
					-u''(r)-\frac{n-1}{r}u'(r)+\lambda u(r)+M\int_0^1s^{n-1}\left|u(s)\right|\,ds\geq 0
				\]
			for some $\lambda$, $M>0$ such that $\lambda+M<1$ 
			and $u'(0)\leq0$, $u(1)\geq0$. Then we have 
			$u(r)\geq0$ for all $r\in[0,1]$. 
		\end{lema}
		\begin{proof}
			Suppose by contradiction that there exists a function $u_0$ that satisfies the 
			assumptions above and is negative at some point.
			
			
			Normalising $u_0$, we can assume that 
			$\int_0^1s^{n-1}\left|u_0(s)\right|\,ds=1$ without loss of generality,
			which implies that $\left\|r^{n-1}u_0(r)\right\|_{\infty}\geq 1$.
			
			Let us consider the auxiliary problem
				\begin{equation}\label{mp1}
					-w''(r)-\frac{n-1}{r}w'(r)+M=0, \quad w'(0)=w(1)=0.	
				\end{equation}
			which is equivalent to
				\begin{equation}\label{mp2}
					\left(r^{n-1} w'(r)\right)'= r^{n-1}M, \quad w'(0)=w(1)=0. 
				\end{equation}
			Integrating \eqref{mp2}, we get
				\[
					w(r)=\frac{M}{2n}(r^2-1)\leq 0.
				\]
			As $u_0$ satisfies
				\[
					-u_0''(r)-\frac{n-1}{r}u_0'(r)+\lambda u_0(r)+
					M\geq 0,%\int_0^1s^{n-1}\left|u_0(s)\right|\,ds
				\]
			with $u_0'(0)\leq0$, $u_0(1)\geq0$,
			we have
				\[
					-(u_0-w)''-\frac{n-1}{r}(u_0-w)'+\lambda(u_0-w)\geq -\lambda w,\quad 
					(u_0-w)'(0)\leq0,\;(u_0-w)(1)\geq0,
				\]
			and, therefore, applying Lemma \ref{princmaxL}, we get $u_0\geq w$.
				
			We can easily see that $r^{n-1}u_0(r)\geq r^{n-1}w(r)\geq-\frac{M}{2n}>-1$,
			so the fact that \mbox{$\left\|r^{n-1}u_0(r)\right\|_{\infty}\!\geq1$}
			insures that there exists $a>0$ 
			such that $u_0(a)\geq\frac{1}{{a}^{n-1}}$.
			 
			If $u_0$ is negative at $b>a$, there exists $c\in]a,b[$
			such that $u_0(c)=0$ (we can assume that $u_0'(b)=0$).
			Using Lagrange's Theorem, there exists $d\in[a,c]$ such that 
			$u_0'(d)\leq-\frac{1}{{a}^{n-1}}$. As $d\geq a$, we have
			${d}^{n-1}u_0'(d)\leq-1$
			and therefore there exists $e\in[d,b]$ such that 
			$(r^{n-1}u_0'(r))'|_{r=e}\geq 1$, (we can take $e$ such that $e^{n-1}u_0(e)<1$).
			
						
			If $u$ is negative at $b<a$, there exists $c<a$ such that $u_0(c)=0$.
			As $u_0(a)>1$, there exists $d\in]c,a[$ such that 
			$u_0'(d)\geq1$. Considering the boundary condition $u_0'(0)\leq0$, there exists
			$e\in[0,d[$ such that $u_0'(e)=0$ and $u_0'(r)>0$ for all $r\in]e,d]$. Therefore
			there exists $f\in[e,d]$ such that $u_0''(f)\geq1$ and $u_0'(f)>0$ 
			(we can take $f$ such that $f^{n-1}u_0(f)<1$).
		 
			In both cases, we know that for some $r_0$ we have
			$(r^{n-1}u_0'(r))'|_{r=r_0}\geq1$, and $r_0^{n-1}u_0(r_0)<1$.
			Therefore we would get
				\[
					-(r^{n-1}u_0'(r))'|_{r=r_0}+\lambda r_0^{n-1}u_0(r_0)+M\leq -1+\lambda+M<0
				\]
			which is a contradiction.
				
		\end{proof}
		
		For a given function $u(r)\in C[0,1]$, consider the boundary value problem
			\[
				-v''(r)-\frac{n-1}{r}v'(r)+\lambda v(r)=
				f\left(u(r),\omega_n\int_0^1s^{n-1}g\left(v(s)\right)\,ds\right)+\lambda u(r),
			\]
		with $v'(0)=0=v(1)$. Using the operator $L$ 
		defined in the beginning of this section, 
		this equation is equivalent	to the fixed point equation in $C[0,1]$
			\begin{equation}\label{pontfix}
				v= L_0^{-1}\left(f\left(u,\omega_n\int_0^1 s^{n-1}
				g\left(v\left(s\right)\right)\,ds\right)+\lambda u\right)\equiv \Phi_u v.
			\end{equation}
		\begin{nota}
			Using a comparison method as the one in the proof of lemma 
			\ref{princmaxseml}, we get $\left\|L_0^{-1}\right\|\leq~\frac{1}{2n}$ in $C[0,1]$.
		\end{nota}
		\begin{lema}\label{contrac}
			If $f(u,v)$ is $k_1$-Lipschitz in $v$, $g$ is $k_2$-Lipschitz, and
			$k_1k_2\omega_n<2 n^2$, then $\Phi_u$ has a unique fixed point.
		\end{lema}
		\begin{proof}
			We have
				\begin{align*}
					\left|\Phi_uv_2(r)-\Phi_uv_1(r)\right|&\leq 
					\frac{1}{2n}k_1\left|\omega_n\int_0^1s^{n-1}g(v_2(s))\,ds
					-\omega_n\int_0^1s^{n-1}g(v_1(s))\,ds\right|\leq \\
					&\leq \frac{1}{2n}k_1k_2\omega_n\int_0^1s^{n-1}\left|v_2(s)-v_1(s)\right|\,ds\leq
					\frac{k_1k_2\omega_n}{2n^2}\left\|v_2-v_1\right\|_{\infty},
					\end{align*}
				so that
					\[
						\left\|\Phi_uv_2(r)-\Phi_uv_1(r)\right\|_{\infty}\leq
						\frac{k_1k_2\omega_n}{2n^2}\left\|v_2-v_1\right\|_{\infty}
					\]
				and therefore $\Phi_u$ is a contraction mapping.
		\end{proof}
			
		\begin{lema}
			Let $f$ and $g$ be functions defined as in the lemma above, 
			$\lambda>0$ such that $k_1k_2\omega_n + \lambda<1$, 
			and suppose that
				\[
					f(u_2,v)-f(u_1,v)\geq -\lambda (u_2-u_1),
				\]
			for all $r\in[0,1]$, $v\in \mathbb{R}, \text{ and } u_1\leq u_2$.
					
			Let $u_1(r)\leq u_2(r)$ be two given functions defined in $[0,1]$ and 
			$v_1(r)$, $v_2(r)$ the two respective solutions of \eqref{pontfix}.
			Then $v_1(r)\leq v_2(r)$.
		\end{lema}
		\begin{proof}
			We have
				\begin{align*}
					&-(v_2-v_1)''-\frac{n-1}{r}(v_2-v_1)'+\lambda(v_2-v_1)=\\
					&=\lambda(u_2-u_1)
					+f\left(u_2,\omega_n\int_0^1s^{n-1}g\left(v_2\right)\,ds\right)-
					f\left(u_1,\omega_n\int_0^1s^{n-1}g\left(v_2\right)\,ds\right)+\\
					&\qquad\qquad\quad\;\;\,\,+f\left(u_1,\omega_n
					\int_0^1s^{n-1}g\left(v_2\right)\,ds\right)-
					f\left(u_1,\omega_n\int_0^1s^{n-1}g\left(v_1\right)\,ds\right)\geq\\
					&\geq -k_1k_2\omega_n\int_0^1s^{n-1}\left|v_2-v_1\right|\,ds.
				\end{align*}
			The conclusion follows from Lemma \ref{princmaxseml}.
		\end{proof}
		
		\begin{teo}\label{subsobre2}
			Suppose that $f(u,v)$ is $k_1$-Lipschitz in $v$, $g$ is $k_2$-Lipschitz.
Suppose that for some $\lambda>0$  
			such that $k_1k_2\omega_n + \lambda<1$, 
				\[
					f(u_2,v)-f(u_1,v)\geq -\lambda (u_2-u_1),
				\]
			for all $v\in \mathbb{R}, \text{ and } u_1\leq u_2$.
			Let $\alpha_0$ and $\beta_0$ be
			a lower and an upper solution of \eqref{fcomr}--\eqref{fcomr2} res\-pectively, with $\alpha_0\le\beta_0$ in $[0,1]$. 
			If we take $\left(\alpha_n\right)_{n\in\mathbb{N}_0}$ and
			$\left(\beta_n\right)_{n\in\mathbb{N}_0}$ such that, according to lemma \ref{contrac},
				\[
					\alpha_{n+1}=\Phi_{\alpha_n}\alpha_{n+1}\quad \text{and} \quad
					\beta_{n+1}=\Phi_{\beta_n}\beta_{n+1}, \text{ for all } n\in\mathbb{N}_0,
				\]
			we obtain
				\[
					\alpha_0\leq\alpha_1\leq\cdots\leq\alpha_n\leq\cdots\leq\beta_n\leq\cdots\leq
					\beta_1\leq\beta_0.
				\]
			The monotone bounded sequences $\left(\alpha_n\right)_{n\in\mathbb{N}_0}$,
			$\left(\beta_n\right)_{n\in\mathbb{N}_0}$ defined above are convergent in $C[0,1]$
			to solutions of \eqref{fcomr}--\eqref{fcomr2}.
		\end{teo}
		\begin{proof}
		%	Since
			%	\[
			%		L\alpha_1=f\left(\alpha_0,\omega_n\int_0^1 s^{n-1}
			%		g\left(\alpha_1\left(s\right)\right)\,ds\right)+\lambda\alpha_0
		%		\]
		%	and
		%		\[
		%			L\alpha_0\leq f\left(\alpha_0,\omega_n\int_0^1 s^{n-1}
		%			g\left(\alpha_0\left(s\right)\right)\,ds\right)+\lambda\alpha_0, 
		%		\]
			The computation used  here is similar to another one used in \cite{dJwGaW02}. We have
				\begin{align*}
						L\left(\alpha_1-\alpha_0\right)&\geq 
						f\left(\alpha_0,\omega_n\int_0^1 s^{n-1}
					g\left(\alpha_1\left(s\right)\right)\,ds\right)-
					f\left(\alpha_0,\omega_n\int_0^1 s^{n-1}
					g\left(\alpha_0\left(s\right)\right)\,ds\right)\geq \\
					&\geq-k_1k_2\omega_n\int_0^1\left|\alpha_1(s)-\alpha_0(s)\right|\,ds
				\end{align*}
				with
				\[
					\left(\alpha_1-\alpha_0\right)'(0)\leq 0,\;\; 
					\left(\alpha_1-\alpha_0\right)(1)\geq 0,
				\]
			and, therefore, using Lemma \ref{princmaxseml}, we get $\alpha_0\leq \alpha_1$.
			Let us prove that $\alpha_1\leq \beta_0$. This comes from
				\begin{align*}
						L\left(\beta_0-\alpha_1\right)&\geq 
						f\left(\beta_0,\omega_n\int_0^1 s^{n-1}
					g\left(\beta_0\left(s\right)\right)\,ds\right)
					+\lambda\beta_0\,-f\left(\alpha_0,\omega_n\int_0^1 s^{n-1}
					g\left(\beta_0\left(s\right)\right)\,ds\right)+\\
					&+f\left(\alpha_0,\omega_n\int_0^1 s^{n-1}
					g\left(\beta_0\left(s\right)\right)\,ds\right)-
					f\left(\alpha_0,\omega_n\int_0^1 s^{n-1}
					g\left(\alpha_1\left(s\right)\right)\,ds\right)
					-\lambda\alpha_0\geq \\
					&\geq-\lambda\left(\beta_0-\alpha_0\right)+\lambda\left(\beta_0-\alpha_0\right)
					-k_1k_2\omega_n\int_0^1\left|\beta_0(s)-\alpha_1(s)\right|\,ds=\\
					&=-k_1k_2\omega_n\int_0^1\left|\beta_0(s)-\alpha_1(s)\right|\,ds.
				\end{align*}
			Applying this Lemma in the following iterations,
			we prove that
				\[
					\alpha_0\leq\alpha_1\leq\cdots\leq\alpha_n\leq\cdots\leq\beta_n\leq\cdots\leq
					\beta_1\leq\beta_0
				\]
				 as in 
			the proof of Theorem \ref{teosubsobre}.
			
			Concerning the convergence of the sequences, there is a slight difference
			from the usual method, because in each iteration we use a different operator.
			But, as
				\[
				\alpha_{n+1}(r)= L_0^{-1}\left(f\left(u\alpha_n r),\omega_n\int_0^1 s^{n-1}
				g\left(\alpha_{n+1}\left(s\right)\right)\,ds\right)+\lambda \alpha_n\right)
				\]
			and $\left\|\alpha_n\right\|_{\infty}\leq 
			\max{\left(\left\|\alpha_0\right\|_{\infty},\left\|\beta_0\right\|_{\infty}\right)}$,
			we have that $\left\|\alpha_{n+1}\right\|_{C^1}$ is bounded, and, therefore, 
			using Àrzela-Ascoli Theorem, there exists a convergent subsequence of $\alpha_n$. 
			Considering the monotonicity of $\alpha_n$, we get 
			the conclusion by the standard argument.
		\end{proof}	

\

\begin{nota} It is not difficult to prove that  the monotone sequences defined in theorem 4.10 converge in fact to {\it extremal solutions} of the boundary value problem (18)-(19).\end{nota}
		
		
		%As we were initially interested in positive solutions, let us show a result that gives us
		%a positive lower solution:
		
		\begin{expl}
			Suppose that
				\[
					\liminf_{(a,b)\to(0^+,0^+)}\frac{f(a,\frac{\omega_n}{n}g(b))}{a}>\lambda_1.
				\]
		%	where $\lambda_1$ is the first eigenvalue of $Lu=0$.
		and 	there exists $k>0$ such that $f(k,\omega_ng(k)/n)<0$. Suppose in addition that $f$ and $g$ satisfy the assumptions of Theorem \ref{subsobre2}.
			

		Then there exists a {\em positive}  solution of \eqref{fcomr}--\eqref{fcomr2}. This solution may be approximated by  monotone sequences. In fact, a simple calculation shows that for $\epsilon>0$ small enough, $\epsilon\phi$ is a positive lower solution of \eqref{fcomr}--\eqref{fcomr2}. The constant $k$ is clearly an upper solution. The statement follows.
		\end{expl}
				
		
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	\end{document}